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Materials Science and Engineering/List of Topics

Introduction to Solid State Chemistry
Lecture 1

 * Taxonomy of Chemical Species
 * Origins of Modern Chemistry

Lecture 2

 * Classification Schemes of the Elements
 * Mendeleyev and the Periodic Table
 * Atomic Structure

Lecture 3

 * Rutherford Model of the Atom
 * Bohr Model of the Atom

Lecture 4

 * Atomic Spectra of Hydrogen
 * Matter-Energy Interactions Involving Atomic Hydrogen

Lecture 5

 * The Shell Model (Bohr-Sommerfeld Model) and Multi-electron Atoms
 * Quantum Numbers: n, l, m, s

Lecture 6

 * De Broglie, Heisenberg, and Schrodinger
 * The Aufbau Principle
 * Pauli Exlusion Principle
 * Hund's Rules
 * Photoelectron Spectroscopy
 * Average Valence Electron Energy

Lecture 7

 * Octet Stability by Electron Transfer: Ionic Bonding
 * Properties of Ionic Compounds: Crystal Lattice Energy

Lecture 8

 * Born-Haber Cycle
 * Octet Stability by Electron Sharing
 * Covalent Bonding
 * Lewis Structures
 * Hybridization

Lecture 9

 * Electronegativity
 * Partial Charge
 * Polar Bonds and Polar Molecules
 * Ionic Character and Covalent Bonds
 * Pauling's Calculation of Heteronuclear Bond Energies

Lecture 10

 * LCAO MO
 * Energy Level Diagrams of H2, He2, Li2
 * Hybridization
 * Double Bonds and Triple Bonds
 * Paramagnetism and Diamagnetism

Lecture 11

 * The Shapes of Molecules
 * Electron Domain Theory
 * Secondary Bonding

Lecture 12

 * Metallic Bonding
 * Band Theory of Solids (Heitler and London)
 * Band Gaps in Metals
 * Semiconductors
 * Insulators
 * Absorption Edge of a Semiconductor

Lecture 13

 * Intrinsic and Extrinsic Semiconductors
 * Doping
 * Compound Semiconductors
 * Molten Semiconductors

Lecture 14

 * Introduction to the Solid State
 * The Seven Crystal Systems
 * The Fourteen Bravais Lattices

Lecture 15

 * Simple Cubic
 * Face-Centered Cubic
 * Body-Centered Cubic
 * Diamond Cubic
 * Crystal Coordinate Systems
 * Miller Indices

Lecture 16

 * Characterization of Atomic Structure
 * The Generation of X-Rays and Moseley's Law

Lecture 17

 * X-Ray Spectra
 * Bragg's Law

Lecture 18

 * X-Ray Diffraction of Crystals
 * Diffractometry
 * Debye-Scherrer
 * Laue
 * Crystal Symmetry

Lecture 19

 * Defects in Crystals
 * Point Defects
 * Line Defects
 * Interfacial Defects
 * Voids

Lecture 20

 * Amorphous Solids
 * Glass Formation
 * Inorganic Glasses
 * Silicates

Lecture 21

 * Engineered Glasses
 * Network Formers
 * Network Modifiers
 * Intermediates
 * Properties of Silicate Glasses
 * Metallic Glass

Lecture 22

 * Chemical Kinetics
 * The Rate Equation
 * Order of Reaction
 * Rate Laws of Zeroth, First, and Second Order Reactions
 * Temperature Dependence of Rate of Reaction

Lecture 23

 * Fick's First Law and Steady-state Diffusion
 * Dependence of the Diffusion Coefficient on Temperature and on Atomic *Arrangement

Lecture 24

 * Fick's Second Law (FSL)
 * Transient-State Diffusion
 * Error Function Solutions to FSL

Lecture 25

 * Solutions
 * Solute
 * Solvent
 * Solubility Rules
 * Solubility Producet

Lecture 26

 * Acids and Bases
 * Arrhenius
 * Bronsted-Lowry
 * Lewis Definition
 * Acid Strength and pH

Lecture 27

 * Basic Concepts of Organic Chemistry
 * Alkanes
 * Alkenes
 * Alkynes
 * Aromatics
 * Functional Groups
 * Alcohols and Ethers Aldehydes and Ketones
 * Esters
 * Amines

Lecture 28

 * Organic Glasses - Polymers
 * Synthesis by Addition Polymerization and by Condensation *Polymerization

Lecture 29

 * Structure-Property Relationship in Polymers
 * Crystalline Polymers

Lecture 30

 * Biochemisty
 * The Amino Acids
 * Peptides
 * Proteins

Lecture 31

 * Protein Structure
 * Primary
 * Secondary
 * Tertiary
 * Denaturing of Proteins

Lecture 32

 * Ipids: Self Assembly into Bilayers
 * Nucleic Acids
 * DNA
 * Encoding Information of Protein Synthesis
 * Electrochemistry of Batteries and Fuel Cells

Lecture 34

Two-Component Phase Diagrams: Limited Solid Solubility Lever Rule

Quantum Mechanics
Quantum Mechanics
 * Origins of Quantum Physics
 * Postulates and Mathematical Formalism of Quantum Mechanics
 * Mathematical Tools of Quantum Mechanics
 * The Schrodinger Equation
 * One-Dimensional Arrangements
 * Angular Momentum
 * Three-Dimensional Arrangements
 * Rotations and Addition of Angular Momentum
 * Identical Particles
 * Approximation Methods of Stationary States
 * Time-Dependent Pertubation Theory

Materials
Materials

 * Structure of Crystalline Solids
 * Imperfections in Solids
 * Phase Diagrams
 * Phase Transformations
 * Metal Alloys
 * Structures and Properties of Ceramics
 * Structures and Properties of Ceramics
 * Applications and Processing of Ceramics
 * Polymer Structures
 * Characteristics, Applications, and Processing of Polymers
 * Composites
 * Corrosion and Degradation of Materials

Thermodynamics
Thermodynamics

 * Zeroth Law of Thermodynamics
 * Equations of State
 * First Law of Thermodynamics
 * Second Law of Thermodynamics
 * Third Law of Thermodynamics
 * Chemical Equilibrium
 * Phase Equilibrium
 * Electrochemical Equilibrium
 * Ionic Equilibria

Statistical Mechanics
Statistical Mechanics

Probability

 * General Defintions

Classical Statistical Mechanics

 * Clasical Mechanics
 * Equations of Motion
 * Langrangian Formulation
 * Microscopic State of a System
 * Equilibrium Ensembles

Quantum Statistical Mechanics

 * Vibrations of a Solid

Kinetics
Kinetics

 * Introduction
 * Irreversible Thermodynamics
 * Driving Forces and Fluxes for Diffusion
 * Self-Diffusion and Interdiffusion
 * Diffusion Potential
 * The Diffusion Equation
 * Diffusion in Crystals
 * Diffusion in Noncrystalline Materials
 * Surface Evolution Due to Capillary Forces
 * Surface Evolution Due to Capillary Forces
 * Particle Coarsening
 * Grain Growth
 * Anisotropic Surface, Diffusional Creep, and Sintering
 * General Features of Phase Transformations
 * Spinodal Decomposition and Continuous Ordering
 * Nucleation
 * Diffusional Growth
 * Morphological Stability of Moving Interfaces
 * Kinetics of Nucleation and Growth Transformations

Mechanics of Materials
Mechanics of Materials

 * Introduction to Elasticity
 * Tensors
 * Stress and Strain
 * Generalized Hooke's Law
 * Physical Origin of Elastic Moduli
 * Composites
 * Cellular Solids
 * Viscoelasticity
 * Continuum Plasticity
 * Dislocation Mechanics
 * Mechanism of Low Temperature Plasticity
 * Continuum Creep Response
 * Mechanisms of Creep Deformation

Conductors and Resistors
[[Image:Hall effect.png|thumb|300px|Hall effect diagram, showing electron flow (rather than [[conventional current]]). Legend: 1. Electrons (not conventional current!) 2. Hall element, or Hall sensor 3. Magnets 4. Magnetic field 5. Power source Description:

In drawing "A", the Hall element takes on a negative charge at the top edge (symbolised by the blue color) and positive at the lower edge (red color). In "B" and "C", either the electric current or the magnetic field is reversed, causing the polarization to reverse. Reversing both current and magnetic field (drawing "D") causes the Hall element to again assume a negative charge at the upper edge.]]

 * Ohm's Law
 * Ohm's law states that, in an electrical circuit, the current passing through a conductor between two points is directly proportional to the potential difference (i.e. voltage drop or voltage) across the two points, and inversely proportional to the resistance between them.
 * Average kinetic energy: $$\frac{3 k_B T}{2}$$
 * Average thermal velocity: $$v_{th} = \left ( \frac{3 k_B T}{m} \right )^{1/2}$$
 * Charge carrier density: $$\mathbf J = - eN_e v_D$$
 * $$v_D = - \mu_e \mathbf E$$
 * Conductivity: $$\sigma = e N_e \mu_e\;$$
 * Acceleration: $$\mathbf a = \frac{-e \mathbf E}{m}$$
 * Average drift velocity of the free-electron gas is the acceleration times the average time between collisions: $$\mu_e = \frac{e \tau}{m}$$
 * Conductivity: $$\sigma = \frac{N_e e^2 \tau}{m}$$

 * Drude Model
 * The Drude model of electrical conduction was developed in the 1900s by Paul Drude to explain the transport properties of electrons in materials (especially metals). The Drude model is the application of kinetic theory to electrons in a solid. It assumes that the material contains immobile positive ions and an "electron gas" of classical, non-interacting electrons of density n, each of whose motion is damped by a frictional force due to collisions of the electrons with the ions, characterized by a relaxation time τ.
 * Resistance Modeled as Viscosity
 * Hall Effect
 * The Hall effect refers to the potential difference (Hall voltage) on the opposite sides of an electrical conductor through which an electric current is flowing, created by a magnetic field applied perpendicular to the current.
 * A particle of charge $$q$$ moving with a velocity $$\mathbf v$$ in a magnetic field $$\mathbf B$$ experiences force: $$\mathbf F = q \mathbf v \times \mathbf B$$n
 * AC/DC
 * An alternating current (AC) is an electrical current whose magnitude and direction vary cyclically, as opposed to direct current, whose direction remains constant. The usual waveform of an AC power circuit is a sine wave, as this results in the most efficient transmission of energy. However in certain applications different waveforms are used, such as triangular or square waves. Used generically, AC refers to the form in which electricity is delivered to businesses and residences. However, audio and radio signals carried on electrical wire are also examples of alternating current. In these applications, an important goal is often the recovery of information encoded (or modulated) onto the AC signal.
 * Heat Conductivity
 * In physics, thermal conductivity, k, is the property of a material that indicates its ability to conduct heat. It is used primarily in Fourier's Law for heat conduction.
 * Heat Capacity
 * Heat capacity (symbol: Cp) — as distinct from specific heat capacity — is the measure of the heat energy required to increase the temperature of an object by a certain temperature interval. Heat capacity is an extensive property because its value is proportional to the amount of material in the object; for example, a bathtub of water has a greater heat capacity than a cup of water.

Optical Properties of Conductors

$$ \int_{\Sigma} \nabla \times \mathbf{F} \cdot d\mathbf{\Sigma} = \oint_{\partial\Sigma} \mathbf{F} \cdot d \mathbf{r}, $$
 * Del operator
 * In vector calculus, del is a vector differential operator represented by the nabla symbol: $$\nabla$$. Del is a mathematical tool serving primarily as a convention for mathematical notation; it makes many equations easier to comprehend, write, and remember. Depending on the way del is applied, it can describe the gradient (slope), divergence (degree to which something converges or diverges) or curl (rotational motion at points in a fluid). More intuitive descriptions of each of the many operations del performs can be found below.
 * Gradient
 * In vector calculus, the gradient of a scalar field is a vector field which points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change.
 * Divergence
 * In vector calculus, the divergence is an operator that measures the magnitude of a vector field's source or sink at a given point; the divergence of a vector field is a (signed) scalar. For a vector field that denotes the velocity of air expanding as it is heated, the divergence of the velocity field would have a positive value because the air expands. If the air cools and contracts, the divergence is negative. The divergence could be thought of as a measure of the change in density.
 * Curl
 * In vector calculus, curl is a vector operator that shows a vector field's "rate of rotation", that is the direction of the axis of rotation and the magnitude of the rotation. It can also be described as the circulation density. In many European countries the operator is called rot (short for rotor) instead of curl.
 * Permittivity
 * Permittivity is a physical quantity that describes how an electric field affects and is affected by a dielectric medium, and is determined by the ability of a material to polarize in response to the field, and thereby reduce the total electric field inside the material. Thus, permittivity relates to a material's ability to transmit (or "permit") an electric field. It is directly related to electric susceptibility. For example, in a capacitor, an increased permittivity allows the same charge to be stored with a smaller electric field (and thus a smaller voltage), leading to an increased capacitance.
 * Stoke's Theorem
 * Stokes' theorem (or Stokes's theorem) in differential geometry is a statement about the integration of differential forms which generalizes several theorems from vector calculus. It is named after Sir George Gabriel Stokes (1819–1903), although the first known statement of the theorem is by William Thomson (Lord Kelvin) and appears in a letter of his to Stokes.
 * The classical Kelvin-Stokes theorem:
 * Derivation of the Wave Equation with Maxwell's Equations
 * The wave equation is an important second-order linear partial differential equation that describes the propagation of a variety of waves, such as sound waves, light waves and water waves. It arises in fields such as acoustics, electromagnetics, and fluid dynamics. Historically, the problem of a vibrating string such as that of a musical instrument was studied by Jean le Rond d'Alembert, Leonhard Euler, Daniel Bernoulli, and Joseph-Louis Lagrange.
 * Skin Depth
 * When an electromagnetic wave interacts with a conductive material, mobile charges within the material are made to oscillate back and forth with the same frequency as the impinging fields. The movement of these charges, usually electrons, constitutes an alternating electric current, the magnitude of which is greatest at the conductor's surface. The decline in current density versus depth is known as the skin effect and the skin depth is a measure of the distance over which the current falls to 1/e of its original value. A gradual change in phase accompanies the change in magnitude, so that, at a given time and at appropriate depths, the current can be flowing in the opposite direction to that at the surface.
 * Ionosphere
 * The ionosphere is the uppermost part of the atmosphere, distinguished because it is ionized by solar radiation. It plays an important part in atmospheric electricity and forms the inner edge of the magnetosphere. It has practical importance because, among other functions, it influences radio propagation to distant places on the Earth. It is located in the Thermosphere.
 * Effective Mass
 * In solid state physics, a particle's effective mass is the mass it seems to carry in the semiclassical model of transport in a crystal. It can be shown that, under most conditions, electrons and holes in a crystal respond to electric and magnetic fields almost as if they were free particles in a vacuum, but with a different mass. This mass is usually stated in units of the ordinary mass of an electron me (9.11×10-31 kg).

Insulators and Capacitors

 * Polarization:
 * In classical electromagnetism, the polarization density (or electric polarization, or simply polarization) is the vector field that expresses the density of permanent or induced electric dipole moments in a dielectric material. The polarization vector P is defined as the dipole moment per unit volume. The SI unit of measure is coulombs per square metre.
 * Capacitor:
 * A capacitor is an electrical/electronic device that can store energy in the electric field between a pair of conductors (called "plates"). The process of storing energy in the capacitor is known as "charging", and involves electric charges of equal magnitude, but opposite polarity, building up on each plate. The capacitor's capacitance (C) is a measure of the amount of charge (Q) stored on each plate for a given potential difference or voltage (V) which appears between the plates:

$$C = {Q \over V}$$ $$C \approx \frac{\varepsilon A}{d}; A \gg d^2$$ $$ E_\mathrm{stored} = {1 \over 2}  C V^2 = {1 \over 2} {Q^2 \over C} = {1 \over 2} {V Q} $$ $$\frac{C'}{C}=\frac{Q'}{Q}=\epsilon_r$$
 * Dielectric constant:
 * The relative static permittivity (or static relative permittivity) of a material under given conditions is a measure of the extent to which it concentrates electrostatic lines of flux. It is the ratio of the amount of stored electrical energy when a potential is applied, relative to the permittivity of a vacuum. The relative static permittivity is the same as the relative permittivity evaluated for a frequency of zero.
 * Electric displacement:
 * In physics, the electric displacement field or electric induction is a vector field \mathbf{D} that appears in Maxwell's equations. It accounts for the effects of unbound charges within materials. "D" stands for "displacement," as in the related concept of displacement current in dielectrics.
 * Electronic polarizability:
 * The theoretical dielectric strength of a material is an intrinsic property of the bulk material and is dependent on the configuration of the material or the electrodes with which the field is applied. At breakdown, the electric field frees bound electrons. If the applied electric field is sufficiently high, free electrons may become accelerated to velocities that can liberate additional electrons during collisions with neutral atoms or molecules in a process called avalanche breakdown. Breakdown occurs quite abruptly (typically in nanoseconds), resulting in the formation of an electrically conductive path and a disruptive discharge through the material. For solid materials, a breakdown event severely degrades, or even destroys, its insulating capability.
 * Dielectric strength:
 * The theoretical dielectric strength of a material is an intrinsic property of the bulk material and is dependent on the configuration of the material or the electrodes with which the field is applied. At breakdown, the electric field frees bound electrons. If the applied electric field is sufficiently high, free electrons may become accelerated to velocities that can liberate additional electrons during collisions with neutral atoms or molecules in a process called avalanche breakdown. Breakdown occurs quite abruptly (typically in nanoseconds)., resulting in the formation of an electrically conductive path and a disruptive discharge through the material. For solid materials, a breakdown event severely degrades, or even destroys, its insulating capability. Dielectric strength (MV/m) of various common materials:

Optical Properties of Insulators
$$\nabla^2 \mathbf E = \mu \epsilon \frac{\partial^2 \mathbf E}{\partial t^2}$$ $$\nabla^2 \mathbf B = \mu \epsilon \frac{\partial^2 \mathbf B}{\partial t^2}$$ $$n = \epsilon_r^{1/2}$$ $$\epsilon_r = n^2\;$$ $$n_1 \sin \theta_1 = n_2 \sin \theta_2\;$$
 * Wave equations in a dielectric
 * Relationship between dielectric constant and index of refraction
 * Snell's law
 * Total internal reflection
 * Total internal reflection is an optical phenomenon that occurs when a ray of light strikes a medium boundary at an angle larger than the critical angle with respect to the normal to the surface. If the refractive index is lower on the other side of the boundary no light can pass through, so effectively all of the light is reflected. The critical angle is the angle of incidence above which the total internal reflection occurs.
 * Dispersion
 * In optics, dispersion is the phenomenon that the phase velocity of a wave depends on its frequency. In a prism, dispersion causes the spatial separation of a white light into spectral components of different wavelengths. Dispersion is most often described in light waves, but it may happen to any kind of wave that interacts with a medium or can be confined to a waveguide, such as sound waves. Dispersion is sometimes called chromatic dispersion to emphasize its wavelength-dependent nature.
 * Attenuation
 * Attenuation is the reduction in amplitude and intensity of a signal. Signals may be attenuated exponentially by transmission through a medium, in which case attenuation is usually reported in dB with respect to distance traveled through the medium. Attenuation can also be understood to be the opposite of amplification. Attenuation is an important property in telecommunications and ultrasound applications because of its importance in determining signal strength as a function of distance. Attenuation is usually measured in units of decibels per unit length of medium (dB/cm, dB/km, etc) and is represented by the attenuation coefficient of the medium in question. Attenuation decreases the intensity of electromagnetic radiation due to absorption or scattering of photons. Attenuation does not include the decrease in intensity due to inverse-square law geometric spreading. Therefore, calculation of the total change in intensity involves both the inverse-square law and an estimation of attenuation over the path.
 * Diffraction
 * Diffraction refers to various phenomena associated with the bending of waves when they interact with obstacles in their path. It occurs with any type of wave, including sound waves, water waves, and electromagnetic waves such as visible light, x-rays and radio waves. As physical objects have wave-like properties, diffraction also occurs with matter and can be studied according to the principles of quantum mechanics. While diffraction always occurs when propagating waves encounter obstacles in their paths, its effects are generally most pronounced for waves where the wavelength is on the order of the size of the diffracting objects. The complex patterns resulting from the intensity of a diffracted wave are a result of interference between different parts of a wave that traveled to the observer by different paths.


text:
Materials Science and Engineering/List of Topics/Quantum Mechanics

Origins of Quantum Physics

Postulates and Mathematical Formalism of Quantum Mechanics

Mathematical Tools of Quantum Mechanics

The Schrodinger Equation

One-Dimensional Arrangements

Angular Momentum

Three-Dimensional Arrangements

Rotations and Addition of Angular Momentum

Identical Particles

Approximation Methods of Stationary States

Time-Dependent Pertubation Theory

Reference
Nouredine Zettili, "Quantum Mechanics: Concepts and Application". John Wiley & Sons, LTD. New York, 2001

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/


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text:
Materials Science and Engineering/List of Topics/Quantum Mechanics/Origins of Quantum Physics

Blackbody Radiation
In physics, a black body is an object that absorbs all electromagnetic radiation that falls onto it. No radiation passes through it and none is reflected. It is this lack of both transmission and reflection to which the name refers. These properties make black bodies ideal sources of thermal radiation. That is, the amount and spectrum of electromagnetic radiation they emit is directly related to their temperature. Black bodies below around 700 K (430 °C) produce very little radiation at visible wavelengths and appear black. Black bodies above this temperature however, produce radiation at visible wavelengths starting at red, going through orange, yellow, and white before ending up at blue as the temperature increases.

The term "black body" was introduced by Gustav Kirchhoff in 1860. The light emitted by a black body is called black-body radiation (or cavity radiation), and has a special place in the history of quantum mechanics.

Photoelectric Effect
The photoelectric effect is a quantum electronic phenomenon in which electrons are emitted from matter after the absorption of energy from electromagnetic radiation such as x-rays or visible light.[1] The emitted electrons can be referred to as photoelectrons in this context. The effect is also termed the Hertz Effect, due to its discovery by Heinrich Rudolf Hertz, although the term has generally fallen out of use.

Study of the photoelectric effect led to important steps in understanding the quantum nature of light and electrons and influenced the formation of the concept of wave–particle duality.

Compton Effect
In physics, Compton scattering or the Compton effect is the decrease in energy (increase in wavelength) of an X-ray or gamma ray photon, when it interacts with matter. Inverse Compton scattering also exists, where the photon gains energy (decreasing in wavelength) upon interaction with matter. The amount the wavelength increases by is called the Compton shift. Although nuclear compton scattering exists, Compton scattering usually refers to the interaction involving only the electrons of an atom. The Compton effect was observed by Arthur Holly Compton in 1923 and further verified by his graduate student Y. H. Woo in the years following. Arthur Compton earned the 1927 Nobel Prize in Physics for the discovery.

The effect is important because it demonstrates that light cannot be explained purely as a wave phenomenon. Thomson scattering, the classical theory of an electromagnetic wave scattered by charged particles, cannot explain any shift in wavelength. Light must behave as if it consists of particles in order to explain the Compton scattering. Compton's experiment convinced physicists that light can behave as a stream of particles whose energy is proportional to the frequency.

The interaction between electrons and high energy photons results in the electron being given part of the energy (making it recoil), and a photon containing the remaining energy being emitted in a different direction from the original, so that the overall momentum of the system is conserved. If the photon still has enough energy left, the process may be repeated. If the photon has sufficient energy (in general a few eV, right around the energy of visible light), it can even eject an electron from its host atom entirely (a process known as the Photoelectric effect).

Pair Production
Pair production refers to the creation of an elementary particle and its antiparticle, usually from a photon (or another neutral boson). This is allowed, provided there is enough energy available to create the pair – at least the total rest mass energy of the two particles – and that the situation allows both energy and momentum to be conserved (though not necessarily on shell). All other conserved quantum numbers (angular momentum, electric charge) of the produced particles must sum to zero — thus the created particles shall have opposite values of each (for instance, if one particle has strangeness +1 then another one must have strangeness −1).

de Broglie's Hypothesis: Matter Waves
In physics, the de Broglie hypothesis (pronounced /brœj/, as French breuil, close to "broy") is the statement that all matter (any object) has a wave-like nature (wave-particle duality). The de Broglie relations show that the wavelength is inversely proportional to the momentum of a particle and that the frequency is directly proportional to the particle's kinetic energy. The hypothesis was advanced by Louis de Broglie in 1924 in his PhD thesis; he was awarded the Nobel Prize for Physics in 1929 for this work, which made him the first person to receive a Nobel Prize on a PhD thesis.

Experimental Confirmation of de Broglie's Hypothesis
In 1927 at Bell Labs, Clinton Davisson and Lester Germer fired slow-moving electrons at a crystalline nickel target. The angular dependence of the reflected electron intensity was measured, and was determined to have the same diffraction pattern as those predicted by Bragg for X-Rays. Before the acceptance of the de Broglie hypothesis, diffraction was a property that was thought to be only exhibited by waves. Therefore, the presence of any diffraction effects by matter demonstrated the wave-like nature of matter. When the de Broglie wavelength was inserted into the Bragg condition, the observed diffraction pattern was predicted, thereby experimentally confirming the de Broglie hypothesis for electrons.

This was a pivotal result in the development of quantum mechanics. Just as Arthur Compton demonstrated the particle nature of light, the Davisson-Germer experiment showed the wave-nature of matter, and completed the theory of wave-particle duality. For physicists this idea was important because it means that not only can any particle exhibit wave characteristics, but that one can use wave equations to describe phenomena in matter if one uses the de Broglie wavelength.

Since the original Davisson-Germer experiment for electrons, the de Broglie hypothesis has been confirmed for other elementary particles.

Experimental
Since the demonstrations of wave-like properties in photons and electrons, similar experiments have been conducted with neutrons and protons. Among the most famous experiments are those of Estermann and Otto Stern in 1929. Authors of similar recent experiments with atoms and molecules, described below, claim that these larger particles also act like waves.

A dramatic series of experiments emphasizing the action of gravity in relation to wave–particle duality were conducted in the 1970s using the neutron interferometer. Neutrons, one of the components of the atomic nucleus, provide much of the mass of a nucleus and thus of ordinary matter. In the neutron interferometer, they act as quantum-mechanical waves directly subject to the force of gravity. While the results were not surprising since gravity was known to act on everything, including light (see tests of general relativity and the Pound-Rebka falling photon experiment), the self-interference of the quantum mechanical wave of a massive fermion in a gravitational field had never been experimentally confirmed before.

In 1999, the diffraction of C60 fullerenes by researchers from the University of Vienna was reported.[7] Fullerenes are comparatively large and massive objects, having an atomic mass of about 720 u. The de Broglie wavelength is 2.5 pm, whereas the diameter of the molecule is about 1 nm, i.e. about 400 times larger. As of 2005, this is the largest object for which quantum-mechanical wave-like properties have been directly observed in far-field diffraction.

In 2003 the Vienna group also demonstrated the wave nature of tetraphenylporphyrin[8] – a flat biodye with an extension of about 2 nm and a mass of 614 u. For this demonstration they employed a near-field Talbot Lau interferometer.[9][10] In the same interferometer they also found interference fringes for C60F48., a fluorinated buckyball with a mass of about 1600 u, composed of 108 atoms[8] Large molecules are already so complex that they give experimental access to some aspects of the quantum-classical interface, i.e. to certain decoherence mechanisms.[11][12]

Whether objects heavier than the Planck mass (about the weight of a large bacterium) have a de Broglie wavelength is theoretically unclear and experimentally unreachable; above the Planck mass a particle's Compton wavelength would be smaller than the Planck length and its own Schwarzschild radius, a scale at which current theories of physics may break down or need to be replaced by more general ones.

de Broglie Equation
The first de Broglie equation relates the wavelength $$\lambda$$ to the particle momentum $$~p~$$ as

 * $$\lambda = \frac{h}{p} = \frac {h}{\gamma mv} = \frac {h}{mv} \sqrt{1 - \frac{v^2}{c^2}}$$

where $$~h~$$ is Planck's constant,  $$~m~$$ is the particle's rest mass, $$~v~$$ is the particle's velocity, $$~\gamma~$$ is the Lorentz factor, and $$~c~$$ is the speed of light in a vacuum.

The greater the energy, the larger the frequency and the shorter (smaller) the wavelength. Given the relationship between wavelength and frequency, it follows that short wavelengths are more energetic than long wavelengths. The second de Broglie equation relates the frequency of the wave associated to a particle to the total energy of the particle such that

 * $$f = \frac{E}{h} = \frac{\gamma\,mc^2}{h} = \frac {1}{\sqrt{1 - \frac{v^2}{c^2}}} \cdot \frac{mc^2}{h}$$

where $$~f~$$ is the frequency and $$~E~$$ is the total energy. The two equations are often written as

 * $$p = \hbar k$$
 * $$E = \hbar \omega$$

where $$~\hbar=h/(2\pi)~$$ is the reduced Planck's constant (also known as Dirac's constant, pronounced "h-bar"), $$~k~$$ is the wavenumber, and $$~\omega~$$ is the angular frequency.

Quantum View of Particles and Waves
In physics and chemistry, wave–particle duality is the concept that all matter exhibits both wave-like and particle-like properties. A central concept of quantum mechanics, duality addresses the inadequacy of classical concepts like "particle" and "wave" in fully describing the behaviour of objects. Various interpretations of quantum mechanics attempt to explain this ostensible paradox.

The idea of duality is rooted in a debate over the nature of light and matter dating back to the 1600s, when competing theories of light were proposed by Christiaan Huygens and Isaac Newton. Through the work of Albert Einstein, Louis de Broglie and many others, current scientific theory holds that all particles also have a wave nature. This phenomenon has been verified not only for elementary particles, but also for compound particles like atoms and even molecules. In fact, according to traditional formulations of non-relativistic quantum mechanics, wave–particle duality applies to all objects, even macroscopic ones; we can't detect wave properties of macroscopic objects due to their small wavelengths.

Heisenberg Uncertainty Principle
In quantum physics, the outcome of even an ideal measurement of a system is not deterministic, but instead is characterized by a probability distribution, and the larger the associated standard deviation is, the more "uncertain" we might say that that characteristic is for the system. The Heisenberg uncertainty principle, or HUP, gives a lower bound on the product of the standard deviations of position and momentum for a system, implying that it is impossible to have a particle that has an arbitrarily well-defined position and momentum simultaneously. More precisely, the products of the standard deviations in each of the three spatial dimensions are bounded by

 * $$(\Delta x)(\Delta p_x) \geq \hbar/2$$

 * $$(\Delta y)(\Delta p_y) \geq \hbar/2$$

 * $$(\Delta z)(\Delta p_z) \geq \hbar/2$$

where $$\hbar$$ is the reduced Planck constant; Δx, Δy, and Δz are the standard deviations of the three coordinates of position; and Δpx, Δpy, and Δpz are the standard deviations of the three components of momentum. The principle generalizes to many other pairs of quantities besides position and momentum (for example, angular momentum about two different axes), and can be derived directly from the axioms of quantum mechanics (particularly the de Broglie relations).

Note that the uncertainties in question are characteristic of the mathematical quantities themselves. In any real-world measurement, there will be additional uncertainties created by the non-ideal and imperfect measurement process. The uncertainty principle holds true regardless of whether the measurements are ideal (sometimes called von Neumann measurements) or non-ideal (Landau measurements). Note also that the product of the uncertainties, of order 10−35 joule-seconds, is so small that the uncertainty principle has negligible effect on objects of macroscopic scale, despite its importance for atoms and subatomic particles.

The uncertainty principle was an important step in the development of quantum mechanics when it was discovered by Werner Heisenberg in 1927. It is often confused with the observer effect.

Probabilistic Interpretation

 * 1) 	A system is completely described by a wave function $$\psi$$, which represents an observer's knowledge of the system. (Heisenberg)
 * 2) 	The description of nature is essentially probabilistic. The probability of an event is related to the square of the amplitude of the wave function related to it. (Max Born)
 * 3)       Heisenberg's uncertainty principle states the observed fact  that it is not possible to know the values of all of the properties of the system at the same time; those properties that are not known with precision must be described by probabilities.
 * 4) 	(Complementary Principle) Matter exhibits a wave-particle duality. An experiment can show the particle-like properties of matter, or wave-like properties, but not both at the same time.(Niels Bohr)
 * 5) 	Measuring devices are essentially classical devices, and measure classical properties such as position and momentum.
 * 6)       The Correspondence Principle of Bohr and Heisenberg: the quantum mechanical description of large systems should closely approximate to the classical description.

Atomic Transitions
In physics, atomic spectral lines are of two types:

 * An emission line is formed when an electron makes a transition from a particular discrete energy level of an atom, to a lower energy state, emitting a photon of a particular energy and wavelength. A spectrum of many such photons will show an emission spike at the wavelength associated with these photons.
 * An absorption line is formed when an electron makes a transition from a lower to a higher discrete energy state, with a photon being absorbed in the process. These absorbed photons generally come from background continuum radiation and a spectrum will show a drop in the continuum radiation at the wavelength associated with the absorbed photons.

The two states must be bound states in which the electron is bound to the atom, so the transition is sometimes referred to as a "bound–bound" transition, as opposed to a transition in which the electron is ejected out of the atom completely ("bound–free" transition) into a continuum state, leaving an ionized atom, and generating continuum radiation.

A photon with an energy equal to the energy difference between the levels is released or absorbed in the process. The frequency ν at which the spectral line occurs is related to the photon energy E by Planck's law E = hν where h is Planck's constant.

Spectroscopy
Spectroscopy is the study of the interaction between radiation (electromagnetic radiation, or light, as well as particle radiation) and matter. Spectrometry is the measurement of these interactions and an instrument which performs such measurements is a spectrometer or spectrograph. A plot of the interaction is referred to as a spectrum.

Historically, spectroscopy referred to a branch of science in which visible light was used for the theoretical study of the structure of matter and for qualitative and quantitative analyses. Recently, however, the definition has broadened as new techniques have been developed that utilise not only visible light, but many other forms of radiation.

Spectroscopy is often used in physical and analytical chemistry for the identification of substances through the spectrum emitted from or absorbed by them. Spectroscopy is also heavily used in astronomy and remote sensing. Most large telescopes have spectrometers, which are used either to measure the chemical composition and physical properties of astronomical objects or to measure their velocities from the Doppler shift of their spectral lines.

Rutherford Planetary Model of the Atom
The Rutherford model or planetary model was a model of the atom devised by Ernest Rutherford. Rutherford directed the famous Geiger-Marsden experiment in (1909), which suggested to Rutherford's analysis (1911) that the Plum pudding model (of J. J. Thomson) of the atom was incorrect. Rutherford's new model for the atom, based on the experimental results, had a number of essential modern features, including a relatively high central charge concentrated into a very small volume in comparison to the rest of the atom.

Bohr Model of the Hydrogen Atom
In atomic physics, the Bohr model depicts the atom as a small, positively charged nucleus surrounded by electrons that travel in circular orbits around the nucleus — similar in structure to the solar system, but with electrostatic forces providing attraction, rather than gravity. This was an improvement on the earlier cubic model (1902), the plum-pudding model (1904), the Saturnian model (1904), and the Rutherford model (1911). Since the Bohr model is a quantum-physics based modification of the Rutherford model, many sources combine the two, referring to the Rutherford-Bohr model.

Introduced by Niels Bohr in 1913, the model's key success lay in explaining the Rydberg formula for the spectral emission lines of atomic hydrogen; while the Rydberg formula had been known experimentally, it did not gain a theoretical underpinning until the Bohr model was introduced. Not only did the Bohr model explain the reason for the structure of the Rydberg formula, but it provided a justification for its empirical results in terms of fundamental physical constants.

The Bohr model is a primitive model of the hydrogen atom. As a theory, it can be derived as a first-order approximation of the hydrogen atom using the broader and much more accurate quantum mechanics, and thus may be considered to be an obsolete scientific theory. However, because of its simplicity, and its correct results for selected systems (see below for application), the Bohr model is still commonly taught to introduce students to quantum mechanics, before moving on to the more accurate but more complex valence shell atom. A related model was originally proposed by Arthur Erich Haas in 1910, but was rejected.

Reference
Nouredine Zettili, "Quantum Mechanics: Concepts and Application". John Wiley & Sons, LTD. New York, 2001


text:
Materials Science and Engineering/List of Topics/Quantum Mechanics/Mathematical Tools of Quantum Mechanics

The Linear Space and Wave Functions
In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. More formally, a vector space is a set on which two operations, called (vector) addition and (scalar) multiplication, are defined and satisfy certain natural axioms which are listed below. Vector spaces are the basic objects of study in linear algebra, and are used throughout mathematics, science, and engineering.

The most familiar vector spaces are two- and three-dimensional Euclidean spaces. Vectors in these spaces are ordered pairs or triples of real numbers, and are often represented as geometric vectors which are quantities with a magnitude and a direction, usually depicted as arrows. These vectors may be added together using the parallelogram rule (vector addition) or multiplied by real numbers (scalar multiplication). The behavior of geometric vectors under these operations provides a good intuitive model for the behavior of vectors in more abstract vector spaces, which need not have a geometric interpretation. For example, the set of (real) polynomials forms a vector space.

The Hilbert Space
The mathematical concept of a Hilbert space, named after the German mathematician David Hilbert, generalizes the notion of Euclidean space in a way that extends methods of vector algebra from the two-dimensional plane and three-dimensional space to infinite-dimensional spaces. In more formal terms, a Hilbert space is an inner product space — an abstract vector space in which distances and angles can be measured — which is "complete", meaning that if a sequence of vectors approaches a limit, then that limit is guaranteed to be in the space as well.

Hilbert spaces arise naturally and frequently in mathematics, physics, and engineering, typically as infinite-dimensional function spaces. They are indispensable tools in the theories of partial differential equations, quantum mechanics, and signal processing. The recognition of a common algebraic structure within these diverse fields generated a greater conceptual understanding, and the success of Hilbert space methods ushered in a very fruitful era for functional analysis.

Geometric intuition plays an important role in many aspects of Hilbert space theory. An element of a Hilbert space can be uniquely specified by its coordinates with respect to an orthonormal basis, in analogy with Cartesian coordinates in the plane. This means that Hilbert space can also usefully be thought of in terms of infinite sequences that are square-summable. Linear operators on a Hilbert space are likewise fairly concrete objects: in good cases, they are simply transformations that stretch the space by different factors in mutually perpendicular directions.

Dimension and Basis of a Vector Space
In linear algebra, a basis is a set of vectors that, in a linear combination, can represent every vector in a given vector space, and such that no element of the set can be represented as a linear combination of the others. In other words, a basis is a linearly independent spanning set.

Square-Integrable: Wave Functions
A real- or complex-valued function of a real or complex variable is square-integrable on an interval if the integral of the square of its absolute value, over that interval, is finite. The set of all measurable functions that are square-integrable forms a Hilbert space, the so-called L2 space.

This is especially useful in quantum mechanics as wave functions must be square integrable over all space if a physically possible solution is to be obtained from the theory.

Dirac Notation
Bra-ket notation is the standard notation for describing quantum states in the theory of quantum mechanics. It can also be used to denote abstract vectors and linear functionals in pure mathematics. It is so called because the inner product (or dot product) of two states is denoted by a bracket, $$\langle\phi|\psi\rangle$$, consisting of a left part, $$\langle\phi|$$, called the bra, and a right part, $$|\psi\rangle$$, called the ket. The notation was invented by Paul Dirac, and is also known as Dirac notation.

General Definitions
In mathematics, an operator is a function, that operates on (or modifies) another function. Often, an "operator" is a function that acts on functions to produce other functions (the sense in which Oliver Heaviside used the term); or it may be a generalization of such a function, as in linear algebra, where some of the terminology reflects the origin of the subject in operations on the functions that are solutions of differential equations. An operator can perform a function on any number of operands (inputs) though most often there is only one operand.

An operator might also be called an operation, but the point of view is different. For instance, one can say "the operation of addition" (but not the "operator of addition") when focusing on the operands and result. One says "addition operator" when focusing on the process of addition, or from the more abstract viewpoint, the function +: S×S → S.

Hermitian Adjoint
Suppose H is a Hilbert space, with inner product $$\langle\cdot,\cdot\rangle$$. Consider a continuous linear operator A : H → H (this is the same as a bounded operator).

Using the Riesz representation theorem, one can show that there exists a unique continuous linear operator A* : H → H with the following property:

 * $$ \lang Ax, y \rang = \lang x , A^* y \rang \quad \mbox{for all } x,y\in H$$

This operator A* is the adjoint of A.

Projection Operators
In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself such that P2 = P. Projections map the whole vector space to a subspace and leave the points in that subspace unchanged.[1]

Though abstract, this definition of "projection" formalizes and generalizes the idea of graphical projection. One can also consider the effect of a projection on a geometrical object by examining the effect of the projection on points in the object.

Uncertainty Relation between Two

 * $$(\Delta x)(\Delta p_x) \geq \hbar/2$$

 * $$(\Delta y)(\Delta p_y) \geq \hbar/2$$

 * $$(\Delta z)(\Delta p_z) \geq \hbar/2$$

Inverse Operator
In mathematics, if ƒ is a function from A to B, then an inverse function for ƒ is a function in the opposite direction, from B to A, with the property that a round trip (a composition) returns each element to itself. Not every function has an inverse; those that do are called invertible.

For example, let ƒ be the function that converts a temperature in degrees Celsius to a temperature in degrees Fahrenheit:
 * $$ f(C) = \tfrac95 C + 32 ; \,\!$$

then its inverse function converts degrees Fahrenheit to degrees Celsius:
 * $$ f^{-1}(F) = \tfrac59 (F - 32) . \,\!$$

Or, suppose ƒ assigns each child in a family of three the year of its birth. An inverse function would tell us which child was born in a given year. However, if the family has twins (or triplets) then we cannot know which to name for their common birth year. As well, if we are given a year in which no child was born then we cannot name a child. But if each child was born in a separate year, and if we restrict attention to the three years in which a child was born, then we do have an inverse function. For example,
 * $$\begin{align}

f(\text{Alan})&=2005, \quad & f(\text{Bree})&=2007 , \quad & f(\text{Cary})&=2001 \\ f^{-1}(2001)&=\text{Cary}, \quad & f^{-1}(2005)&=\text{Alan} , \quad & f^{-1}(2007)&=\text{Bree} \end{align} $$

Unitary Operator
In functional analysis, a branch of mathematics, a unitary operator is a bounded linear operator U : H → H on a Hilbert space H satisfying

 * $$U^*U=UU^*=I$$

where U∗ is the adjoint of U, and I : H → H is the identity operator. This property is equivalent to the following:

 * 1) The range of U is dense, and
 * 2) U preserves the inner product 〈,  〉 on the Hilbert space, i.e. for all vectors x and y in the Hilbert space,
 * $$\langle Ux, Uy \rangle = \langle x, y \rangle.$$

To see this, notice that U preserves the inner product implies U is an isometry (thus, a bounded linear operator). The fact that U has dense range ensures it has a bounded inverse U−1. It is clear that U−1 = U∗.

Thus, unitary operators are just automorphisms of Hilbert spaces, i.e., they preserve the structure (in this case, the linear space structure, the inner product, and hence the topology) of the space on which they act. The group of all unitary operators from a given Hilbert space H to itself is sometimes referred to as the Hilbert group of H, denoted Hilb(H).

Eigenvalues and Eigenvectors
In mathematics, a vector may be thought of as an arrow. It has a length, called its magnitude, and it points in some particular direction. A linear transformation may be considered to operate on a vector to change it, usually changing both its magnitude and its direction. An of a given linear transformation is a vector which is multiplied by a constant called the  during that transformation. The direction of the eigenvector is either unchanged by that transformation (for positive eigenvalues) or reversed (for negative eigenvalues).

For example, an eigenvalue of +2 means that the eigenvector is doubled in length and points in the same direction. An eigenvalue of +1 means that the eigenvector stays the same, while an eigenvalue of −1 means that the eigenvector is reversed in direction. An eigenspace of a given transformation is the set of all eigenvectors of that transformation that have the same eigenvalue, together with the zero vector (which has no direction). An eigenspace is an example of a subspace of a vector space.

In linear algebra, every linear transformation between finite-dimensional vector spaces can be given by a matrix, which is a rectangular array of numbers arranged in rows and columns. Standard methods for finding eigenvalues, eigenvectors, and eigenspaces of a given matrix are discussed below.

These concepts play a major role in several branches of both pure and applied mathematics — appearing prominently in linear algebra, functional analysis, and to a lesser extent in nonlinear mathematics.

Many kinds of mathematical objects can be treated as vectors: functions, harmonic modes, quantum states, and frequencies, for example. In these cases, the concept of direction loses its ordinary meaning, and is given an abstract definition. Even so, if this abstract direction is unchanged by a given linear transformation, the prefix "eigen" is used, as in eigenfunction, eigenmode, eigenstate, and eigenfrequency.

Reference
Nouredine Zettili, "Quantum Mechanics: Concepts and Application". John Wiley & Sons, LTD. New York, 2001


text:
Materials Science and Engineering/List of Topics/Quantum Mechanics/Postulates of Quantum Mechanics

Mathematical Formalism and Postulates of Quantum Mechanics
The mathematical formulation of quantum mechanics is the body of mathematical formalisms which permits a rigorous description of quantum mechanics. It is distinguished from mathematical formalisms for theories developed prior to the early 1900s by the use of abstract mathematical structures, such as infinite-dimensional Hilbert spaces and operators on these spaces. Many of these structures were drawn from functional analysis, a research area within pure mathematics that developed in parallel with, and was influenced by, the needs of quantum mechanics. In brief, values of physical observables such as energy and momentum were no longer considered as values of functions on phase space, but as eigenvalues of linear operators.

This formulation of quantum mechanics continues to be used today. At the heart of the description are ideas of quantum state and quantum observable which, for systems of atomic scale, are radically different from those used in previous models of physical reality. While the mathematics permits calculation of many quantities that can be measured experimentally, there is a definite theoretical limit to values that can be simultaneously measured. This limitation was first elucidated by Heisenberg through a thought experiment, and is represented mathematically in the new formalism by the non-commutativity of quantum observables.

Prior to the emergence of quantum mechanics as a separate theory, the mathematics used in physics consisted mainly of differential geometry and partial differential equations; probability theory was used in statistical mechanics. Geometric intuition clearly played a strong role in the first two and, accordingly, theories of relativity were formulated entirely in terms of geometric concepts. The phenomenology of quantum physics arose roughly between 1895 and 1915, and for the 10 to 15 years before the emergence of quantum theory (around 1925) physicists continued to think of quantum theory within the confines of what is now called classical physics, and in particular within the same mathematical structures. The most sophisticated example of this is the Sommerfeld-Wilson-Ishiwara quantization rule, which was formulated entirely on the classical phase space.

The Wavefunction
A wave function is a mathematical tool used in quantum mechanics to describe any physical system. It is a function from a space that consists of the possible states of the system into the complex numbers. The laws of quantum mechanics (i.e. the Schrödinger equation) describe how the wave function evolves over time. The values of the wave function are probability amplitudes — complex numbers — the squares of the absolute values of which, give the probability distribution that the system will be in any of the possible states.

For example, in an atom with a single electron, such as hydrogen or ionized helium, the wave function of the electron provides a complete description of how the electron behaves. It can be decomposed into a series of atomic orbitals which form a basis for the possible wave functions. For atoms with more than one electron (or any system with multiple particles), the underlying space is the possible configurations of all the electrons and the wave function describes the probabilities of those configurations.

Observables and Operators
In mathematics, an operator is a function, that operates on (or modifies) another function. Often, an "operator" is a function that acts on functions to produce other functions (the sense in which Oliver Heaviside used the term).

In physics, particularly in quantum physics, a system observable is a property of the system state that can be determined by some sequence of physical operations. For example, these operations might involve submitting the system to various electromagnetic fields and eventually reading a value off some gauge. In systems governed by classical mechanics, any experimentally observable value can be shown to be given by a real-valued function on the set of all possible system states. In quantum physics, on the other hand, the relation between system state and the value of an observable is more subtle, requiring some basic linear algebra to explain. In the mathematical formulation of quantum mechanics, states are given by non-zero vectors in a Hilbert space V (where two vectors are considered to specify the same state if, and only if, they are scalar multiples of each other) and observables are given by self-adjoint operators on V. However, as indicated below, not every self-adjoint operator corresponds to a physically meaningful observable. For the case of a system of particles, the space V consists of functions called wave functions.

In quantum mechanics, measurement of observables exhibits some seemingly mysterious phenomena. This often leads to many misconceptions about the nature of quantum mechanics itself. The facts of the matter, however, are far more ordinary. Specifically, if a system is in a state described by a wave function, the measurement process affects the state in a non-deterministic, but statistically predictable way. In particular, after a measurement is applied, the state description by a single wave function may be destroyed, being replaced by a statistical ensemble of wave functions. The irreversible nature of measurement operations in quantum physics is sometimes referred to as the measurement problem and is described mathematically by quantum operations. By the structure of quantum operations, this description is mathematically equivalent to that offered by relative state interpretation where the original system is regarded as a subsystem of a larger system and the state of the original system is given by the partial trace of the state of the larger system.

Physically meaningful observables must also satisfy transformation laws which relate observations performed by different observers in different frames of reference. These transformation laws are automorphisms of the state space, that is bijective transformations which preserve some mathematical property. In the case of quantum mechanics, the requisite automorphisms are unitary (or antiunitary) linear transformations of the Hilbert space V. Under Galilean relativity or special relativity, the mathematics of frames of reference is particularly simple, and in fact restricts considerably the set of physically meaningful observables.

Postulates of Quantum Mechanics
The following summary of the mathematical framework of quantum mechanics can be partly traced back to von Neumann's postulates.
 * Each physical system is associated with a (topologically) separable complex Hilbert space H with inner product $$\langle\phi\mid\psi\rangle$$. Rays (one-dimensional subspaces) in H are associated with states of the system. In other words, physical states can be identified with equivalence classes of vectors of length 1 in H, where two vectors represent the same state if they differ only by a phase factor. Separability is a mathematically convenient hypothesis, with the physical interpretation that countably many observations are enough to uniquely determine the state.
 * The Hilbert space of a composite system is the Hilbert space tensor product of the state spaces associated with the component systems. For a non-relativistic system consisting of a finite number of distinguishable particles, the component systems are the individual particles.
 * Physical symmetries act on the Hilbert space of quantum states unitarily or antiunitarily (supersymmetry is another matter entirely).
 * Physical observables are represented by densely-defined self-adjoint operators on H.
 * The expected value (in the sense of probability theory) of the observable A for the system in state represented by the unit vector $$\left|\psi\right\rangle\in H$$ is
 * $$\langle\psi\mid A\mid\psi\rangle$$
 * By spectral theory, we can associate a probability measure to the values of A in any state ψ. We can also show that the possible values of the observable A in any state must belong to the spectrum of A.  In the special case A has only discrete spectrum, the possible outcomes of measuring A are its eigenvalues.

 * More generally, a state can be represented by a so-called density operator, which is a trace class, nonnegative self-adjoint operator $$\rho$$ normalized to be of trace 1. The expected value of A in the state $$\rho$$ is
 * $$ \operatorname{tr}(A\rho)$$
 * If $$\rho_\psi$$ is the orthogonal projector onto the one-dimensional subspace of H spanned by $$\left|\psi\right\rangle$$, then
 * $$ \operatorname{tr}(A\rho_\psi)=\left\langle\psi\mid A\mid\psi\right\rangle$$
 * Density operators are those that are in the closure of the convex hull of the one-dimensional orthogonal projectors. Conversely, one-dimensional orthogonal projectors are extreme points of the set of density operators. Physicists also call one-dimensional orthogonal projectors pure states and other density operators mixed states.

The Uncertainty Principle
The Uncertainty Principle is a mathematical principle which follows from the definition of operators of momentum and position (namely, lack of commutativity between them) and which explains the behavior of the universe at atomic and subatomic scales.

The Uncertainty Principle was developed as an answer to the question: How does one measure the location of an electron around a nucleus if an electron is a wave? When quantum mechanics was developed, it was seen to be a relation between the classical and quantum descriptions of a system using wave mechanics.

In March 1926, working in Niels Bohr's institute, Werner Heisenberg formulated the principle of uncertainty thereby laying the foundation of what became known as the Copenhagen interpretation of quantum mechanics. Heisenberg had been studying the papers of Paul Dirac and Jordan. Heisenberg discovered a problem with measurement of basic variables in the equations. His analysis showed that uncertainties, or imprecisions, always turned up if one tried to measure the position and the momentum of a particle at the same time. Heisenberg concluded that these uncertainties or imprecisions in the measurements were not the fault of the experimenter, but fundamental in nature and are inherent mathematical properties of operators in quantum mechanics arising from definitions of these operators.

The term Copenhagen interpretation of quantum mechanics was often used interchangeably with and as a synonym for Heisenberg's Uncertainty Principle by detractors who believed in fate and determinism and saw the common features of the Bohr-Heisenberg theories as a threat. Within the widely but not universally accepted Copenhagen interpretation of quantum mechanics (i.e. it was not accepted by Einstein or other physicists such as Alfred Lande), the uncertainty principle is taken to mean that on an elementary level, the physical universe does not exist in a deterministic form, but rather as a collection of probabilities, or potentials. For example, the pattern (probability distribution) produced by millions of photons passing through a diffraction slit can be calculated using quantum mechanics, but the exact path of each photon cannot be predicted by any known method. The Copenhagen interpretation holds that it cannot be predicted by any method, not even with theoretically infinitely precise measurements.

If one goes even further to the direct interpretation that classical physics and ordinary language are only approximations to a completely quantum reality, then the probabilities are assigned to these approximations and are no longer fundamental. The equations of quantum mechanics themselves specify the progression of the quantum state of any isolated system uniquely.

Uncertainty Principle Versus Observer Effect
The uncertainty principle in quantum mechanics is sometimes erroneously explained by claiming that the measurement of position necessarily disturbs a particle's momentum, and vice versa—i.e., that the uncertainty principle is a manifestation of the observer effect. Indeed, Heisenberg himself may have initially offered explanations which suggested this view. Prior to the more modern understanding, a measurement was often visualized as a physical disturbance inflicted directly on the measured system, being sometimes illustrated as a thought experiment called Heisenberg's microscope. For instance, when measuring the position of an electron, one imagines shining a light on it, thus disturbing the electron and producing the quantum mechanical uncertainties in its position.

Equating the uncertainty principle and the observer effect mischaracterizes the way measurement in quantum mechanics is understood. The uncertainty principle is an inequality related to the statistical spreading of the wave function whereas the observer effect creates a systematic error. Because the uncertainty principle is a statistical property of the wave function, it appears most readily in a sequence of several measurements.

Consider a hypothetical experiment in which a physicist prepares an ensemble of 2N particles in the same way (so that each is in precisely the same initial quantum state). Suppose further that the physicist is using perfect measuring equipment and that N is sufficiently large so that the net result is statistically significant. For the first N particles of this ensemble, the position would be measured and recorded, giving a probability distribution for position. For the remaining N particles, momentum would be measured, giving a probability distribution for momentum. Finally, the product of the standard deviations would be computed, giving a value of at least $$\hbar/2$$. If the position and momentum had been measured subsequently for the same particle, then the results of the second measurement would not reflect the original state, due to a correct application of the observer effect. But in this experiment, no such claim is made. The physicist never attempts to measure the position and momentum of a single particle but measures them for a different set of N particles from the same initial state. One measurement cannot affect the other. Moreover, although each measurement collapses the quantum state of the particle, the probability distribution resulting from these measurements will correctly reflect the quantum state as it existed before the measurement. Consequently, the uncertainty principle should be considered an intrinsic smearing of statistical information instead of a limitation on measuring equipment.

In any case, it is now understood that the uncertainties in the system exist prior to and independent of the measurement, and the uncertainty principle is therefore independent of the observer effect.

Consequences of quantum symmetry
Remarkably, there exists a realm of physics for which mathematical assertions of simple symmetries in real objects cease to be approximations. That is the domain of quantum physics, which for the most part is the physics of very small, very simple objects such as electrons, protons, light, and atoms.

Unlike everyday objects, objects such as electrons have very limited numbers of configurations, called states, in which they can exist. This means that when symmetry operations such as exchanging the positions of components are applied to them, the resulting new configurations often cannot be distinguished from the originals no matter how diligent an observer is. Consequently, for sufficiently small and simple objects the generic mathematical symmetry assertion F(x) = x ceases to be approximate, and instead becomes an experimentally precise and accurate description of the situation in the real world.

While it makes sense that symmetries could become exact when applied to very simple objects, the immediate intuition is that such a detail should not affect the physics of such objects in any significant way. This is in part because it is very difficult to view the concept of exact similarity as physically meaningful. Our mental picture of such situations is invariably the same one we use for large objects: We picture objects or configurations that are very, very similar, but for which if we could "look closer" we would still be able to tell the difference.

However, the assumption that exact symmetries in very small objects should not make any difference in their physics was discovered in the early 1900s to be spectacularly incorrect. The situation was succinctly summarized by Richard Feynman in the direct transcripts of his Feynman Lectures on Physics, Volume III, Section 3.4, Identical particles. (Unfortunately, the quote was edited out of the printed version of the same lecture.)

 * "... if there is a physical situation in which it is impossible to tell which way it happened, it always interferes; it never fails."

The word "interferes" in this context is a quick way of saying that such objects fall under the rules of quantum mechanics, in which they behave more like waves that interfere than like everyday large objects.

In short, when an object becomes so simple that a symmetry assertion of the form F(x) = x becomes an exact statement of experimentally verifiable sameness, x ceases to follow the rules of classical physics and must instead be modeled using the more complex—and often far less intuitive—rules of quantum physics.

This transition also provides an important insight into why the mathematics of symmetry are so deeply intertwined with those of quantum mechanics. When physical systems make the transition from symmetries that are approximate to ones that are exact, the mathematical expressions of those symmetries cease to be approximations and are transformed into precise definitions of the underlying nature of the objects. From that point on, the correlation of such objects to their mathematical descriptions becomes so close that it is difficult to separate the two.

The Correspondence Principle
In physics, the correspondence principle is a quantitative tool in the old quantum theory, explicitly formulated by Niels Bohr in 1923. It says that the behavior of quantum mechanical systems reproduce classical physics in the limit of large quantum numbers.

Expectation Values
Quantum physics shows an inherent statistical behaviour: The measured outcome of an experiment will generally not be the same if the experiment is repeated several times. Only the statistical mean of the measured values, averaged over a large number of runs of the experiment, is a repeatable quantity. Quantum theory does not, in fact, predict the result of individual measurements, but only their statistical mean. This predicted mean value is called the expectation value.

While the computation of the mean value of experimental results is very much the same as in classical statistics, its mathematical representation in the formalism of quantum theory differs significantly from classical measure theory.

Formalism in quantum mechanics
In quantum theory, an experimental setup is described by the observable $$A$$ to be measured, and the state $$\sigma$$ of the system. The expectation value of $$A$$ in the state $$\sigma$$ is denoted as $$\langle A \rangle_\sigma$$.

Mathematically, $$A$$ is a selfadjoint operator on a Hilbert space. In the most commonly used case in quantum mechanics, $$\sigma$$ is a pure state, described by a normalized vector $$\psi$$ in the Hilbert space. The expectation value of $$A$$ in the state $$\psi$$ is defined as

(1)     $$ \langle A \rangle_\psi = \langle \psi | A \psi \rangle $$.

If dynamics is considered, either the vector $$\psi$$ or the operator $$A$$ is taken to be time-dependent, depending on whether the Schrödinger picture or Heisenberg picture is used. The time-dependence of the expectation value does not depend on this choice, however.

If $$A$$ has a complete set of eigenvectors $$\phi_j$$, with eigenvalues $$a_j$$, then (1) can be expressed as

(2)     $$ \langle A \rangle_\psi = \sum_j a_j |\langle \psi | \phi_j \rangle|^2 $$.

This expression is similar to the arithmetic mean, and illustrates the physical meaning of the mathematical formalism: The eigenvalues $$a_j$$ are the possible outcomes of the experiment, and their corresponding coefficient $$|\langle \psi | \phi_j \rangle|^2$$ is the probability that this outcome will occur; it is often called the transition probability.

A particularly simple case arises when $$A$$ is a projection, and thus has only the eigenvalues 0 and 1. This physically corresponds to a "yes-no" type of experiment. In this case, the expectation value is the probability that the experiment results in "1", and it can be computed as

(3)     $$ \langle A \rangle_\psi = \| A \psi \|^2$$.

In quantum theory, also operators with non-discrete spectrum are in use, such as the position operator $$Q$$ in quantum mechanics. This operator does not have eigenvalues, but has a completely continuous spectrum. In this case, the vector $$\psi$$ can be written as a complex-valued function $$\psi(x)$$ on the spectrum of $$Q$$ (usually the real line). For the expectation value of the position operator, one then has the formula

(4)     $$ \langle Q \rangle_\psi = \int \, x \, |\psi(x)|^2 \, dx$$.

A similar formula holds for the momentum operator $$P$$, in systems where it has continuous spectrum.

All the above formulae are valid for pure states $$\sigma$$ only. Prominently in thermodynamics, also mixed states are of importance; these are described by a positive trace-class operator $$\rho = \sum_i \rho_i | \psi_i \rangle \langle \psi_i |$$, the statistical operator or density matrix. The expectation value then can be obtained as

(5)     $$ \langle A \rangle_\rho = \mathrm{Trace} (\rho A) =  \sum_i \rho_i \langle \psi_i | A \psi_i \rangle = \sum_i \rho_i \langle A \rangle_{\psi_i} $$.

The Ehrenfest Theorem
The Ehrenfest theorem, named after Paul Ehrenfest, relates the time derivative of the expectation value for a quantum mechanical operator to the commutator of that operator with the Hamiltonian of the system. It is

 * $$\frac{d}{dt}\langle A\rangle = \frac{1}{i\hbar}\langle [A,H] \rangle + \left\langle \frac{\partial A}{\partial t}\right\rangle $$

where A is some QM operator and $$\langle A\rangle$$ is its expectation value. Ehrenfest's theorem fits neatly into the Heisenberg picture of quantum mechanics.

Ehrenfest's theorem is closely related to Liouville's theorem from Hamiltonian mechanics, which involves the Poisson bracket instead of a commutator. In fact, it is a general rule of thumb that a theorem in quantum mechanics which contains a commutator can be turned into a theorem in classical mechanics by changing the commutator into a Poisson bracket and multiplying by $$i\hbar$$.

The theorem can be shown to follow from the Lindblad equation, a master equation for the time evolution of a mixed state.

Derivation
Suppose some system is presently in a quantum state $$\Phi$$. If we want to know the instantaneous time derivative of the expectation value of A, that is, by definition

 * $$ \frac{d}{dt}\langle A\rangle = \frac{d}{dt}\int \Phi^* A \Phi~dx^3 = \int \left( \frac{\partial \Phi^*}{\partial t} \right) A\Phi~dx^3 + \int \Phi^* \left( \frac{\partial A}{\partial t}\right) \Phi~dx^3 +\int \Phi^* A \left( \frac{\partial \Phi}{\partial t} \right) ~dx^3 $$

 * $$ = \int \left( \frac{\partial \Phi^*}{\partial t} \right) A\Phi~dx^3 + \left\langle \frac{\partial A}{\partial t}\right\rangle + \int \Phi^* A \left( \frac{\partial \Phi}{\partial t} \right) ~dx^3, $$

where we are integrating over all space. Often (but not always) the operator A is time independent, so that its derivative is zero and we can ignore the middle term. If we apply the Schrödinger equation, we find that

 * $$\frac{\partial \Phi}{\partial t} = \frac{1}{i\hbar}H\Phi$$

and

 * $$\frac{\partial \Phi^*}{\partial t} = \frac{-1}{i\hbar}\Phi^*H^* = \frac{-1}{i\hbar}\Phi^*H.$$

Notice $$H=H^*$$ because the Hamiltonian is hermitian. Placing this into the above equation we have

 * $$\frac{d}{dt}\langle A\rangle = \frac{1}{i\hbar}\int \Phi^* (AH-HA) \Phi~dx^3 + \left\langle \frac{\partial A}{\partial t}\right\rangle = \frac{1}{i\hbar}\langle [A,H]\rangle + \left\langle \frac{\partial A}{\partial t}\right\rangle.$$

Wave Packets
In physics, a wave packet is an envelope or packet containing an arbitrary number of wave forms. In quantum mechanics the wave packet is ascribed a special significance: it is interpreted to be a "probability wave" describing the probability that a particle or particles in a particular state will be measured to have a given position and momentum.

By applying the Schrödinger equation in quantum mechanics it is possible to deduce the time evolution of a system, similar to the process of the Hamiltonian formalism in classical mechanics. The wave packet is a mathematical solution to the Schrödinger equation. The square of the area under the wave packet solution is interpreted to be the probability density of finding the particle in a region.

In the coordinate representation of the wave (such as the Cartesian coordinate system) the position of the wave is given by the position of the packet. Moreover, the narrower the wave packet, and therefore the better defined the position of the wave packet, the larger the uncertainty in the momentum of the wave. This trade-off is known as the Heisenberg uncertainty principle.

Reference
Nouredine Zettili, "Quantum Mechanics: Concepts and Application". John Wiley & Sons, LTD. New York, 2001


text:
Materials Science and Engineering/List of Topics/Quantum Mechanics/One Dimensional Problems

Reference
Nouredine Zettili, "Quantum Mechanics: Concepts and Application". John Wiley & Sons, LTD. New York, 2001


text:
Materials Science and Engineering/List of Topics/Quantum Mechanics/Angular Momentum

Reference
Nouredine Zettili, "Quantum Mechanics: Concepts and Application". John Wiley & Sons, LTD. New York, 2001


text:
Materials Science and Engineering/List of Topics/Quantum Mechanics/Three-Dimensional Problems

Reference
Nouredine Zettili, "Quantum Mechanics: Concepts and Application". John Wiley & Sons, LTD. New York, 2001


text:
Materials Science and Engineering/List of Topics/Quantum Mechanics/Rotations and Addition of Angular Momentum

Reference
Nouredine Zettili, "Quantum Mechanics: Concepts and Application". John Wiley & Sons, LTD. New York, 2001


text:
Materials Science and Engineering/List of Topics/Quantum Mechanics/Identical Particles

Reference
Nouredine Zettili, "Quantum Mechanics: Concepts and Application". John Wiley & Sons, LTD. New York, 2001


text:
Materials Science and Engineering/List of Topics/Quantum Mechanics/Approximation Methods of Stationary States

Reference
Nouredine Zettili, "Quantum Mechanics: Concepts and Application". John Wiley & Sons, LTD. New York, 2001


text:
Materials Science and Engineering/List of Topics/Quantum Mechanics/Time-Dependent Pertubation Theory

Reference
Nouredine Zettili, "Quantum Mechanics: Concepts and Application". John Wiley & Sons, LTD. New York, 2001


text:
Materials Science and Engineering/List of Topics/Quantum Mechanics/Scattering Theory

Reference
Nouredine Zettili, "Quantum Mechanics: Concepts and Application". John Wiley & Sons, LTD. New York, 2001


text:
User talk:Scforth

Welcome
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User talk:Sertalpbilal

Welcome
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And don't forget to explore Wikiversity with the links to your left. Be bold, and see you around Wikiversity! Erkan Yilmaz ( Wikiversity:Chat, wiki blog ) 00:05, 21 December 2007 (UTC)

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Materials Science and Engineering/List of Topics/Quantum Mechanics/Schrodinger Equation

In physics, the Schrödinger equation, proposed by the Austrian physicist Erwin Schrödinger in 1926, describes the space- and time-dependence of quantum mechanical systems. It is of central importance in non-relativistic quantum mechanics, playing a role for microscopic particles analogous to Newton's second law in classical mechanics for macroscopic particles. Microscopic particles include elementary particles, such as electrons, as well as systems of particles, such as atomic nuclei. Macroscopic particles vary in mass from cells to the galactic superclusters.

Historical background and development
Schrödinger's equation follows very naturally from earlier developments:

In 1905, by considering the photoelectric effect, Albert Einstein had published his
 * $$E = h f\;$$

formula for the relation between the energy E and frequency f of the quanta of radiation (photons), where h is Planck's constant.

In 1924 Louis de Broglie presented his de Broglie hypothesis which states that all particles (not just photons) have an associated wavefunction $$\Psi\;$$ with properties:
 * $$p=h / \lambda\;$$, where $$\lambda\,$$ is the wavelength of the wave and p the momentum of the particle.

De Broglie showed that this was consistent with Einstein's formula and special relativity so that
 * $$E = h f\;$$

still holds, but now this is hypothesized to hold for all particles, not just photons anymore.

Expressed in terms of angular frequency $$\omega = 2\pi f\;$$ and wavenumber $$k = 2\pi / \lambda\;$$, with $$\hbar = h / 2 \pi\;$$ we get:
 * $$E=\hbar \omega$$

and
 * $$\mathbf{p}=\hbar \mathbf{k}\;$$

where we have expressed p and k as vectors.

Schrödinger's great insight, late in 1925, was to express the phase of a plane wave as a complex phase factor:
 * $$\psi \approx e^{i(\mathbf{k}\cdot\mathbf{x}- \omega t)}$$

and to realize that since
 * $$ \frac{\partial}{\partial t} \psi = -i\omega \psi $$

then
 * $$ E \psi = \hbar \omega \psi = i\hbar\frac{\partial}{\partial t} \psi $$

and similarly since:
 * $$ \frac{\partial}{\partial x} \psi = i k_x \psi $$

then
 * $$ p_x \psi = \hbar k_x \psi = -i\hbar\frac{\partial}{\partial x} \psi $$

and hence:
 * $$ p_x^2 \psi = -\hbar^2\frac{\partial^2}{\partial x^2} \psi $$

so that, again for a plane wave, he got:
 * $$ p^2 \psi = (p_x^2 + p_y^2 + p_z^2) \psi = -\hbar^2\left(\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}\right) \psi = -\hbar^2\nabla^2 \psi $$

And by inserting these expressions into the Newtonian formula for a particle with total energy E, mass m, moving in a potential V:
 * $$E=\frac{p^2}{2m}+V$$ (simply the sum of the kinetic energy and potential energy; the plane wave model assumed V = 0)

he got his famed equation for a single particle in the 3-dimensional case in the presence of a potential:
 * $$i\hbar\frac{\partial}{\partial t}\Psi=-\frac{\hbar^2}{2m}\nabla^2\Psi + V\Psi$$

Using this equation, Schrödinger computed the spectral lines for hydrogen by treating a hydrogen atom's single negatively charged electron as a wave, $$\psi\;$$, moving in a potential well, V, created by the positively charged proton. This computation tallied with experiment, the Bohr model and also the results of Werner Heisenberg's matrix mechanics - but without having to introduce Heisenberg's concept of non-commuting observables. Schrödinger published his wave equation and the spectral analysis of hydrogen in a series of four papers in 1926.

The Schrödinger equation defines the behaviour of $$\psi\;$$, but does not interpret what $$\psi\;$$ is. Schrödinger tried unsuccessfully to interpret it as a charge density. In 1926 Max Born, just a few days after Schrödinger's fourth and final paper was published, successfully interpreted $$\psi\;$$ as a probability amplitude, although Schrödinger was never reconciled to this statistical or probabilistic approach.

Mathematical formulation
In the mathematical formulation of quantum mechanics, a physical system is associated with a complex Hilbert space such that each instantaneous state of the system is described by a ray in that space. The nonzero elements of a Hilbert space are by definition normalizable and it is convenient, although not necessary, to represent a state by an element of the ray which is normalized to unity. This vector is often somewhat loosely referred to as wave function, although in a more rigorous formulation of quantum mechanics a wave function is a special case of a state vector. (In fact, a wave function is a state in the position representation, see below). A state vector encodes the probabilities for the outcomes of all possible measurements applied to the system. It contains all information of the system that is knowable in a quantum mechanical sense. As the state of a system generally changes over time, the state vector is a function of time. The Schrödinger equation provides a quantitative description of the rate of change of the state vector.

In Dirac's bra-ket notation at time $$t$$ the state is given by the ket $$|\psi(t)\rangle$$. The time-dependent Schrödinger equation, giving the time evolution of the ket, is:

 * $$H(t)\left|\psi\left(t\right)\right\rangle = \mathrm{i}\hbar \frac{d}{d t} \left| \psi \left(t\right) \right\rangle$$

where $$\mathrm{i}$$ is the imaginary unit, $$t$$ is time, $$d/d t$$ is the derivative with respect to $$t$$, $$\hbar$$ is the reduced Planck's constant (Planck's constant divided by $$2\pi\,$$), $$\psi(t)\,$$ is the time dependent state vector, and $$H(t)$$ is the Hamiltonian (a self-adjoint operator acting on the state space). If one assumes a certain representation for $$\psi\,$$, for instance position or momentum representation, the state vector is assumed to depend on more variables than time alone, and the time derivative must be replaced by the partial derivative $$\partial / \partial t. $$

The Hamiltonian describes the total energy of the system. As with the force occurring in Newton's second law, its form is not provided by the Schrödinger equation, but must be independently determined from the physical properties of the system.

Time-independent Schrödinger equation
For many real-world problems the energy operator ($$H$$), does not depend on time. In such problems, there may exist a static, or time independent solution. For such a solution the time-dependent Schrödinger equation simplifies to the time-independent Schrödinger equation, which has the well-known appearance $$H\Psi = E\Psi\,$$.

An example of a simple one-dimensional time-independent Schrödinger equation for a particle of mass m, moving in a potential U(x) is:
 * $$ -\frac{\hbar^2}{2 m} \frac{d^2 \psi (x)}{dx^2} + U(x) \psi (x) = E \psi (x). $$

The analogous 3-dimensional time-independent equation is, :
 * $$ \left[-\frac{\hbar^2}{2 m} \nabla^2 + U(\mathbf{r}) \right] \psi (\mathbf{r}) = E \psi (\mathbf{r}), $$

or
 * $$ -\frac{\hbar^2}{2 m} \nabla^2 \psi + (U - E) \psi = 0, $$

where $$ \nabla $$ is the del operator.

For every time-independent Hamiltonian, $$H$$, there exists a set of quantum states, $$\left|\psi_n\right\rang$$, known as energy eigenstates, and corresponding real numbers $$E_n$$ satisfying the eigenvalue equation,

 * $$ H \left|\psi_n\right\rang = E_n \left|\psi_n \right\rang. $$

Such a state possesses a definite total energy, whose value $$E_n$$ is the eigenvalue of the Hamiltonian. The corresponding eigenvector $$\psi_n\,$$ is normalizable to unity. This eigenvalue equation is referred to as the time-independent Schrödinger equation. We purposely left out the variable(s) on which the wavefunction $$\psi_n\,$$ depends. In the first example above it depends on the single variable x and in the second on x, y, and z—the components of the vector r. In both cases the Schrödinger equation has the same appearance, but its Hamilton operator is defined on different function (state, Hilbert) spaces. In the first example the function space consists of functions of one variable and in the second example the function space consists of functions of three variables.

Self-adjoint operators, such as the Hamiltonian, have the property that their eigenvalues are always real numbers, as we would expect, since the energy is a physically observable quantity. Sometimes more than one linearly independent state vector correspond to the same energy $$E_n$$. If the maximum number of linearly independent eigenvectors corresponding to $$E_n$$ equals k, we say that the energy level $$E_n$$ is k-fold degenerate. When k=1 the energy level is called non-degenerate.

On inserting a solution of the time-independent Schrödinger equation into the full Schrödinger equation, we get

 * $$\mathrm{i} \hbar \frac{\partial}{\partial t} \left| \psi_n \left(t\right) \right\rangle = E_n \left|\psi_n\left(t\right)\right\rang. $$

It is relatively easy to solve this equation. One finds that the energy eigenstates (i.e., solutions of the time-independent Schrödinger equation) change as a function of time only trivially, namely, only by a complex phase:

 * $$ \left| \psi \left(t\right) \right\rangle = \mathrm{e}^{-\mathrm{i} Et / \hbar} \left|\psi\left(0\right)\right\rang. $$

It immediately follows that the probability amplitude,
 * $$\psi(t)^*\psi(t) = \mathrm{e}^{\mathrm{i} Et / \hbar}\mathrm{e}^{-\mathrm{i} Et / \hbar}

\psi(0)^*\psi(0) = |\psi(0)|^2, $$ is time-independent. Because of a similar cancellation of phase factors in bra and ket, all average (expectation) values of time-independent observables (physical quantities) computed from $$\psi(t)\,$$ are  time-independent.

Energy eigenstates are convenient to work with because they form a complete set of states. That is, the eigenvectors $$ \left\{\left|n\right\rang\right\} $$ form a basis for the state space. We introduced here the short-hand notation $$|\,n\,\rang = \psi_n$$. Then any state vector that is a solution of the time-dependent Schrödinger equation (with a time-independent $$H$$) $$ \left|\psi\left(t\right)\right\rang $$ can be written as a linear superposition of energy eigenstates:

 * $$\left|\psi\left(t\right)\right\rang = \sum_n c_n(t) \left|n\right\rang \quad,\quad H \left|n\right\rang = E_n \left|n\right\rang \quad,\quad \sum_n \left|c_n\left(t\right)\right|^2 = 1.$$

(The last equation enforces the requirement that $$ \left|\psi\left(t\right)\right\rang $$, like all state vectors, may be normalized to a unit vector.) Applying the Hamiltonian operator to each side of the first equation, the time-dependent Schrödinger equation in the left-hand side and using the fact that the energy basis vectors are by definition linearly independent, we readily obtain

 * $$\mathrm{i}\hbar \frac{\partial c_n}{\partial t} = E_n c_n\left(t\right).$$

Therefore, if we know the decomposition of $$ \left|\psi\left(t\right)\right\rang $$ into the energy basis at time $$t = 0$$, its value at any subsequent time is given simply by

 * $$\left|\psi\left(t\right)\right\rang = \sum_n \mathrm{e}^{-\mathrm{i}E_nt/\hbar} c_n\left(0\right) \left|n\right\rang. $$

Note that when some values $$c_n(0)\,$$ are not equal to zero for differing energy values $$E_n\,$$, the left-hand side is not an eigenvector of the energy operator $$H$$. The left-hand is an eigenvector when the only $$c_n(0)\,$$-values not equal to zero belong the same energy, so that $$\mathrm{e}^{-\mathrm{i}E_nt/\hbar}$$ can be factored out. In many real-world application this is the case and the state vector $$\psi(t)\,$$ (containing time only in its phase factor) is then a solution of the time-independent Schrödinger equation.

Example
Let $$|\,1\,\rangle$$ and $$|\,2\,\rangle$$ be degenerate eigenstates of the time-independent Hamiltonian $$H\,$$:

H\,|\,1\,\rangle = E |\,1\,\rangle \quad \hbox{and} \quad H\,|\,2\,\rangle = E |\,2\,\rangle. $$ Suppose a solution $$\psi(t)\,$$ of the full (time-dependent) Schrödinger equation of $$H\,$$ has the form at t = 0:

c_1 |\,1\,\rangle + c_2 |\,2\,\rangle. $$ Hence, because of the discussion above, at t > 0 :
 * \,\psi(0)\,\rangle =

\mathrm{e}^{-\mathrm{i}Et/\hbar} c_1 |\,1\,\rangle + \mathrm{e}^{-\mathrm{i}Et/\hbar} c_2 |\,2\,\rangle = \mathrm{e}^{-\mathrm{i}Et/\hbar} \left( c_1 |\,1\,\rangle + c_2 |\,2\,\rangle\right) = \mathrm{e}^{-\mathrm{i}Et/\hbar}|\,\psi(0)\,\rangle, $$ which shows that $$\psi(t)\,$$ only depends on time in a trivial way (through its phase), also in the case of degeneracy.
 * \,\psi(t)\,\rangle =

Apply now $$H\,$$:

H\,|\,\psi(t)\,\rangle = \mathrm{e}^{-\mathrm{i}Et/\hbar} c_1 E\,|\,1\,\rangle + \mathrm{e}^{-\mathrm{i}Et/\hbar} c_2 E\, |\,2\,\rangle = E\mathrm{e}^{-\mathrm{i}Et/\hbar} \left( c_1 |\,1\,\rangle + c_2 |\,2\,\rangle\right) $$

= E \mathrm{e}^{-\mathrm{i}Et/\hbar}|\,\psi(0)\,\rangle = E\,|\,\psi(t)\,\rangle. $$ Conclusion: The wavefunction $$\psi(t)\,$$ with the given initial condition (its form at t = 0), remains a solution of the time-independent Schrödinger equation $$H\psi(t) = E\psi(t)$$ for all times t > 0.


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