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UTPA STEM/CBI Courses/Numerical Methods and Statistics/Taylor Series

Course Title: MECE 2450 Numerical Methods and Statistics

Lecture Topic: Using the Taylor Series

Instructor: Javier A. Kypuros, Ph.D.

Institution: UTPA

Backwards Design
Course Objectives

 * Primary Objectives- By the next class period students will be able to:
 * use the Taylor series to approximate 1st- and 2nd-order derivatives.

 * Sub Objectives- The objectives will require that students be able to:
 * flow chart or pseudocode,
 * program "For" and "Do" loops,
 * program "if-then-else" statements,
 * numerically approximate derivatives.

 * Difficulties- Students may have difficulty:
 * translating a problem into logic and numerical algorithms,
 * understanding flowchart/pseudocode syntax, and
 * learning MATLAB.

The Taylor-Series is used through-out engineering to approximate complex functions that model physical phenomena. Non-linear functions like $$\sin {t}$$ and $$e^t$$ which appear in many areas of engineering can be approximated using the Taylor Series.
 * Real-World Contexts- There are many ways that students can use this material in the real-world, such as:

Model of Knowledge

 * Concept Map
 * text
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 * Content Priorities
 * Enduring Understanding
 * text
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 * Important to Do and Know
 * text
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 * Worth Being Familiar with
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Assessment of Learning

 * Formative Assessment
 * In Class (groups)
 * text
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 * Homework (individual)
 * text
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 * Summative Assessment
 * text
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Legacy Cycle
OBJECTIVE

By the next class period, students will be able to: The objectives will require that students be able to:
 * text
 * text
 * text
 * text
 * text
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THE CHALLENGE

How can you estimate the instantaneous velocity and acceleration of a vehicle during the course of a drag race if the instrument panel does not function?

GENERATE IDEAS

 * You can use a stop watch and the odometer (if it functions) to measure distance traveled at various intervals.

 * Use GPS to track position at various time intervals.

 * Alternatively you can use markers on the side of the track that are at known distances and a stop watch.

MULTIPLE PERSPECTIVES

The professor will lead a discussion asking students probing questions. The idea is to lead them to a first-order approximation of instantaneous velocity. $$ v_{instantaneous} \approx \Delta v / \Delta t $$
 * If you travel to San Antonio from Edinburg and it takes you 4 hours, what is your average speed?
 * On interstates there are markers on the side of the road. If you have a watch, how might you estimate your current velocity?
 * How can you improve your velocity estimates?

RESEARCH & REVISE

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TEST YOUR METTLE

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GO PUBLIC

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Pre-Lesson Quiz

 * 1) question
 * 2) question
 * 3) question
 * 4) question
 * 5) question
 * 6) question
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 * 20) question

Test Your Mettle Quiz
Saldivar.jess 04:50, 6 February 2010 (UTC)Don't forget to include the concept map and objective of this course
 * 1) question
 * 2) question
 * 3) question
 * 4) question
 * 5) question
 * 6) question
 * 7) question
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 * 20) question


text:
Sign language

 * Sign language.

Possible specific studies
Topics have been classified according to connections shown on the Wikipedia article, these classifications are not to be taken as authoritative.

BANZSL
 * Signed English
 * Manual alphabet
 * Manually Coded Language
 * Home sign
 * BSL
 * Auslan
 * NZSL
 * Maritime Sign Language
 * South African Sign Language

 * French Sign Language
 * Quebec Sign Language
 * /ASL (American Sign Language)
 * Bolivian Sign Language
 * Thai Sign Language
 * Ugandan Sign Language
 * Kenyan Sign Language
 * Philippine Sign Language
 * Malaysian Sign Language.

 * Irish Sign Language
 * [[w:Austro-Hungarian Sign Language
 * Russian Sign Language (Moldovan, Ukrainian)
 * Lithuanian Sign Language
 * Georgian Sign Language
 * Mongolian Sign Language
 * Czech Sign Language
 * Hungarian Sign Language
 * Slovak Sign Language
 * Romanian Sign Language
 * Bulgarian Sign Language
 * Polish Sign Language
 * Dutch Sign Language
 * Flemish Sign Language
 * Belgian-French Sign Language
 * Spanish Sign Language,
 * Mexican Sign Language
 * Brazilian Sign Language (LIBRAS)

 * Scandinavian Sign Language family
 * Finnish Sign Language
 * Swedish Sign Language
 * [[w:Norwegian Sign Language]
 * Danish Sign Language
 * Icelandic Sign Language

 * German Sign Language
 * Israeli Sign Language

 * Nicaraguan Sign Language
 * Al-Sayyid Bedouin Sign Language
 * Providence Island Sign Language.
 * Eastern Arabic Sign Language.

 * Martha's Vineyard Sign Language

 * Kata Kolok in a village in Bali
 * Adamorobe Sign Language in Ghana
 * Yucatec Maya sign language in Mexico
 * Plains Indian Sign Language

 * Australian Aboriginal sign languages
 * Warlpiri Sign Language
 * Warumungu language
 * Diyari language,
 * Kaytetye language and Akitiri Sign Language
 * Arrernte language
 * Warlmanpa language

 * Japanese Sign Language family.
 * Japanese Sign Language
 * Taiwanese Sign Language
 * Korean Sign Language

Written forms of sign languages

 * SignWriting
 * Hamburg Notation System
 * Stokoe notation

Primate use of sign language

 * Main article: Great ape language –Primate use of sign language

There have been several notable examples of scientists teaching non-human primates basic signs in order to communicate with humans. Notable examples are:-
 * Chimpanzees: Washoe and Loulis
 * Gorillas: Michael and Koko.

Gestural theory of human language origins

 * Main article: Origin of language –Gestural theory

The gestural theory states that vocal human language developed from a gestural sign language. An important question for gestural theory is what caused the shift to vocalization.

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

title: Sign language


text:
User:BP83

Read more at:  See more at:

title: User:BP83


text:
User talk:Usman22

what is the difference between electrical & electronics? --Usman22 03:49, 21 November 2009 (UTC)

title: User talk:Usman22


text:
New Zealand Law/Criminal/Conspiracy

Statutory Provisions
Really? This is missing ? Congratulations: you are a giant flapping twat atop the Millard with a southerly howling in, smattering you with the penquin shit it picked up from the Ross Shelf. You are as obvious and as helpful as a penis instead of a nose on the face of a priest in an Italian preschool at Christmas

Definition
Conspiracy involves:
 * (i) an intention to agree; and
 * (ii) a common design to put the agreement into effect. (R v Gemmell)

There is no need to take any steps but there must be agreement that such steps will be taken. Conspiracy is complete on the making of an agreement.

Three types of conspiracy
There are 3 types of conspiracy:
 * (a) simple;
 * (b) circle or wheel; and
 * (c) chain conspiracy.

Impossibility
Impossibility is available as a 'defence' to conspiracy.

R v Gemmell

 * Facts

 * Gemmell and principal discussed a robbery.
 * G refused to take part in an aggravated robbery.
 * But on the day of robbery, G reluctantly agreed to drive principal and others to post office.

 * Held
 * 1) There is difference in agreeing to commit one crime and aiding & abetting another.
 * 2) Aiding & abetting aggravated robbery cannot transform an agreement to commit (simple) robbery into agreement to commit aggravated robbery.
 * 3) Conspiracy requires:
 * (a) intention common to the conspirators; and
 * (b) a manifestation of that intention by mutual consultation or agreement

R v Sanders

 * An agreement made outside NZ to commit a crime in NZ is actionable in NZ if any act in furtherance of the conspiracy is performed in NZ.

R v Richards

 * Facts
 * Thompson regularly bought heroin from Richards.
 * Myers, Thompson's friend, asked Thompson to help him score some drugs.
 * Thomspon got the drugs for Myers from Richards.
 * Thompson was charged with conspiracy to supply and manufacture.

 * Held
 * 1) A conspiracy is committed when the agreement is complete.
 * 2) The agreement must occur before the “object crime”. An agreement to supply and then later deliver, is sufficient.
 * 3) There must be a causal link btwn the agreement and crime taking place. If a party will have no influence over intended conduct (eg. acquiescence to a course of conduct) – there is no conspiracy.
 * 4) An exchange of promises is implied in some circumstances.
 * 5) The defendant encouraged supply, and knew that manufacture was necessary, therefore he agreed to it.
 * 6) There was no intention to actively assist in manufacture but support of whole operation means defendant is a conspirator to manufacture.
 * 7) A person who organises or encourages at the outset can be a conspirator if, at the time of the agreement, he expressly or impliedly commits to be a supporter in principle of the project, available to assist in ways that may not yet be specified.
 * 8) A drug user not only agrees that others should make and supply drugs but that he will continue to actively encourage and receive drugs.
 * 9) For the Crown to establish conspiracy, it must prove 4 things:
 * (a) that the accused formed a common intention w/ at least one conspirator
 * (b) that intention was expressed by agreement
 * (c) the course of conduct, if carried out, would be a crime; and
 * (d) that the agreement included intentional promise to actively assist AND made commission more likely (e.g influential, indicated commitment)

title: New Zealand Law/Criminal/Conspiracy


text:
File:BCs.svg

Boundary Conditions

title: File:BCs.svg


text:
User talk:Dry.eats

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title: User talk:Dry.eats


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User talk:Lwitiko mwakatuma

 Hello Lwitiko mwakatuma, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

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title: User talk:Lwitiko mwakatuma


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User talk:Tradervic

 Hello Tradervic, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

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title: User talk:Tradervic


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User talk:London Bridge

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title: User talk:London Bridge


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User talk:NickM~enwikiversity

 Hello NickM, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

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Your account will be renamed
Hello,

The developer team at Wikimedia is making some changes to how accounts work, as part of our on-going efforts to provide new and better tools for our users like cross-wiki notifications. These changes will mean you have the same account name everywhere. This will let us give you new features that will help you edit and discuss better, and allow more flexible user permissions for tools. One of the side-effects of this is that user accounts will now have to be unique across all 900 Wikimedia wikis. See the announcement for more information.

Unfortunately, your account clashes with another account also called NickM. To make sure that both of you can use all Wikimedia projects in future, we have reserved the name NickM~enwikiversity that only you will have. If you like it, you don't have to do anything. If you do not like it, you can pick out a different name.

Your account will still work as before, and you will be credited for all your edits made so far, but you will have to use the new account name when you log in.

Sorry for the inconvenience.

Yours, Keegan Peterzell Community Liaison, Wikimedia Foundation 23:45, 17 March 2015 (UTC)

Renamed
 This account has been renamed as part of single-user login finalisation. If you own this account you can |log in using your previous username and password for more information. If you do not like this account's new name, you can choose your own using this form after logging in: . -- Keegan (WMF) (talk) 06:17, 19 April 2015 (UTC)

title: User talk:NickM~enwikiversity


text:
User:Wissutak/Homework No.3 Which Operating System is best

According to http://www.w3schools.com/browsers/browsers_os.asp

The lastest statistic of people using different operating system was done in October,2009.

 * Win7 4.4%
 * Vista 18.6%
 * Win2003 1.5%
 * WinXP 63.3%
 * W2000 0.7%
 * Linux 4.2%
 * Mac 6.8%

What operating systems should install on your netbook?

I think Window7 is the best operating system to install in the netbook. The resons are shown below.
 * Window7 gives the netbook a faster system.
 * Window7 helps saving power, which will result in longer battery life.
 * Window7 can run the better graphic than WindowXP.
 * Windows7 will come built in with a sensor technology.
 * Window7 has better RAM.


title: File:4.A.DFT.pdf


title: File:5.A.Fourier.pdf


text:
User:Tarunuday

Hey, This is tarun, from Kochi, India. I'm studying in Grade 11, pursuing my dream of getting an admission into an IIT.

title: User:Tarunuday


text:
User:Graeme E. Smith/Interwiki Sandbox

Watchlist Menu

 * Wikibooks Watchlist
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User_talk Menu

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title: User:Graeme E. Smith/Interwiki Sandbox


text:
User:OpenScientist/Social networks for scientists

Purpose of this page
This page is intended to provide an overview of the usage of social networking platforms for and by scientists. It can be considered as an update to this blog post from mid-2008. For discussion, please use this FriendFeed thread.


text:
User:Cheerios~enwikiversity

Welcome to my homepage! I don't know what I wan't on here yet, so for now, I'll call this under construction.

title: User:Cheerios~enwikiversity


text:
User talk:Justlogic

 Hello Justlogic, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

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 Woof! My name is Jack Russell. I am a dog and a Wikiversity mascot. I am pretty new around Wikiversity. Perhaps we can learn together! Tail wag...

title: User talk:Justlogic


text:
User:Jasonghuna

हमे तो अपने ने लूटा गैरों में कहाँ दम था. मेरी हड्डी वहाँ टूटी, जहाँ हॉस्पिटल बन्द था.

मुझे जिस एम्बुलेन्स में डाला, उसका पेट्रोल ख़त्म था. मुझे रिक्शे में इसलिए बैठाया, क्योंकि उसका किराया कम था.

मुझे डॉक्टरों ने उठाया, नर्सों में कहाँ दम था. मुझे जिस बेड पर लेटाया, उसके नीचे बम था.

title: User:Jasonghuna


text:
User:Katknave/Toronto Silver Plating Company

TORONTO SILVER PLATE COMPANY - Toronto - Canada

Founded in 1882 by James A. Watts and others. A building was erected at the corner of 570 King Street West, and is still standing today. The Toronto Preservation Board has designated the 'Toronto Silver Plate Building' an Ontario Heritage site. In 1885 Toronto S.P. Company began the production of Holmes & Edwards flatware. The firm was purchased in 1914 by Wm. A. Rogers & Co Ltd (Canada). The company stopped using the company silverplate mark in 1929.


title: User:Dariant3


text:
User talk:Aomauma

On Sun, Oct 4, 2009 at 10:13 PM, ormuma kanratana  wrote:

2.5GHz Quad-core PowerPC G5

16GB 533 DDR2 ECC SDRAM- 8x2GB

2x500GB Serial ATA - 7200rpm

QUADRO FX 4500 512MB SDRAM

Apple Cinema HD Display (30" flat panel)

Apple Cinema HD Display (30" flat panel)

AirPort Extreme + Bluetooth built-in

Apple USB Modem

16x SuperDrive DL (DVD+R DL/DVD±RW/CD-RW)

Fibre Channel PCI Express Card (w/ SFP-SFP cable)

Apple Wireless Keyboard & Mouse - U.S. English

Mac OS X - U.S. English

iWork '05 preinstalled

Accessory kit

AppleCare Protection Plan for Power Mac (w/or w/o Display) - Auto-enroll

total 800000+

which os is best for you
in my opinion, i prefer Linux over Windows due to this following reasons Linux has less risk for viruses and malware. It is also free for applications and is also far more flexible than Windows. Moreover, Bugs and security holes are patched much quicker and its improving at a faster rate than Windows. Linux has multiple routes for support.

title: User talk:Aomauma


text:
User talk:Irwja75p

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title: User talk:Irwja75p


text:
User talk:AdamCollet

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title: User talk:AdamCollet


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User talk:Myidbe

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Feel free to leave a message on my talk page if you have any questions or need help. --mikeu talk 08:18, 22 November 2009 (UTC)

title: User talk:Myidbe


text:
User:CJMcGill

CJMcgill will contribute to this forum as practical, and benefit from it as possible.

title: User:CJMcGill


text:
User talk:Youthinkyouknowme

Please do not add nonsense to Wikiversity; it is considered vandalism. If you would like to experiment, use the sandbox. Thank you. Adambro 11:42, 22 November 2009 (UTC)

title: User talk:Youthinkyouknowme


text:
User:Nboruett

Norbert Boruett works for an organization that is has been introducing innovative medical education in kenya for the last ten years and most recently we have spread to other countries.

title: User:Nboruett


text:
User:Jeffswin

Ladies and Gentlemen, I am here to experiment with Wikiversity. Do you have any recommendations where I should start? What courses have you found enlightening? Gracias, Jeff

title: User:Jeffswin


text:
User:Phantomsteve

My home wiki is the English Wikipedia where my user page is here.

title: User:Phantomsteve


text:
Anishinabe/Culture/A brief history/Extra credit

Extra credit essays
When you have completed the extra credit assignment for this lesson, please link to it here. Please read Wikiversity's essay guidelines before posting.


text:
User:Chiefreyes

ΑΘΣ

title: User:Chiefreyes


title: File:Benedic2HW7graphs.jpg


text:
User:Egm6321.f09.team7/HW7

Problem Statement
Pg. 37-1 []

Show that the following relation is true:

$$x\frac12ln(x\frac{1+x}{1-x})-1 = xtanh^{-1}(x)-1$$

Problem Solution
We can recall from HW 6 Problem 13 the proof of $$\frac12ln(\frac{1+x}{1-x}) = tanh^{-1}(x)$$

From here a few easy steps give us the proof we are looking for.

First recognize that the relation from the last homework above is part of our original relation if we isolate it we have:

$$x(\frac12ln(\frac{1+x}{1-x}))-1 = x(tanh^{-1}(x))-1$$

we can cross out the relation from the previous homework and are left with:

$$x\cancel{(\frac12ln(\frac{1+x}{1-x}))}-1 = x\cancel{(tanh^{-1}(x))}-1$$ = $$x(1)-1=x(1)-1$$

or

x-1 = x-1

proving the relation is true

Problem Statement
Pg. 37-1 []

From the transparencies equation (2) is defined as:

$$Q_n(x) = \underline{P}_n(x)tanh^{-1}(x)-2 \sum\nolimits_{j=1,3,5}^{J} \frac{2n-2j+1}{(2n-j+1)j}\underline{P}_{n-j}(x)$$

Using (2)show the relationship of the odd or evenness of Q_n to "n"

Problem Solution
First we need to make some definitions within the equation number (2)

The function $$tanh^{-1}(x)$$ is an odd function both from the definition and also as explained on the transparency

Next the previous homework proved that the polynomial $$\underline{P}_n(x)$$ is an even function when n is even

It follows that the function $$\underline{P}_{n-j}(x)$$ is odd for even n since n-j would be odd (j always being odd)

we recognize that the order of the functions is $$tanh^{-1}(x)$$ multiplied by $$\underline{P}_n(x)$$ then subtracted by $$\underline{P}_{n-j}(x)$$ defining an odd function as "O" and an even function as "E" we will have the following two possibilities:

for an even n:

E * O - O leads to O - O which finally gives us O

for odd n:

O * O - E leads to E - E which finally gives us E

(The relationships for odd and even functions were explained in the previous homework if further information is needed)

So the final result for Qn is:

n=odd Qn=even, n=even Qn=odd

Problem Statement
Pg. 37-1 []

Plot the polynmial $$/underline{P}_{n}$$ for n 0 through 4 and $$Q_{n}$$ for n 0 through 4

Problem Statement
'''See attached file contains the Problem Statement! '''

Problem Solution
'''See attached file contains the solution! '''

Problem Statement
'''See attached file contains the Problem Statement! '''

Problem Solution
'''See attached file contains the solution! '''

Problem Statement/Problem Solution
Please see attached!

Signatures
Egm6321.f09.team7.hua 01:03, 9 December 2009 (UTC) --Egm6321.f09.team7.benedict 01:47, 23 November 2009 (UTC) --Egm6321.f09.team7.mm 19:03, 29 November 2009 (UTC))

--User:Emg6321.f09.blanco 01:17, 2 December 2009 (UTC))

title: User:Egm6321.f09.team7/HW7


text:
User talk:Sallyll1000

 Hello Sallyll1000, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

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title: User talk:Sallyll1000


text:
User:Sallyll1000

CSF Bill: Schedule 1

title: User:Sallyll1000


text:
Educational technology/Educational Technology Lesson Plans

 * {| cellpadding="4" cellspacing="0" width="350px"

 * bgcolor="#e0ddd3" style="border: 0px #cccccc solid;" |Subject: Physical science
 * }

 * {| cellpadding="4" cellspacing="0" width="350px"

 * bgcolor="#e0ddd3" style="border: 0px #cccccc solid;" |Sub-Subject: Electric circuit and the Scientific Method
 * }

 * {| cellpadding="4" cellspacing="0" width="350px"

 * bgcolor="#e0ddd3" style="border: 0px #cccccc solid;" |Length/Duration: 1 90-100 min. period
 * }

 * {| cellpadding="4" cellspacing="0" width="350px"

 * bgcolor="#e0ddd3" style="border: 0px #cccccc solid;" |Technologies Used: Electric Circuit Simulator, Create a Graph, Moodle
 * }

 * {| cellpadding="4" cellspacing="0" width="350px"

 * bgcolor="#e0ddd3" style="border: 0px #cccccc solid;" |Link: http://en.wikiversity.org/wiki/Currents,_and_Circuits,_and_Variables,_Oh_My!
 * }
 * }

 * {| cellpadding="4" cellspacing="0" width="350px"

 * bgcolor="#e0ddd3" style="border: 0px #cccccc solid;" |Subject: Advisory course
 * }

 * {| cellpadding="4" cellspacing="0" width="350px"

 * bgcolor="#e0ddd3" style="border: 0px #cccccc solid;" |Sub-Subject: Focus on self, culture, community and college
 * }

 * {| cellpadding="4" cellspacing="0" width="350px"

 * bgcolor="#e0ddd3" style="border: 0px #cccccc solid;" |Length/Duration: 3-4 days
 * }

 * {| cellpadding="4" cellspacing="0" width="350px"

 * bgcolor="#e0ddd3" style="border: 0px #cccccc solid;" |Technologies Used: iMovie, PikiWiki, Memory Miner
 * }

 * {| cellpadding="4" cellspacing="0" width="350px"

 * bgcolor="#e0ddd3" style="border: 0px #cccccc solid;" |Link: http://en.wikiversity.org/wiki/Who_Am_I%3F
 * }
 * }

 * {| cellpadding="4" cellspacing="0" width="350px"

 * bgcolor="#e0ddd3" style="border: 0px #cccccc solid;" |Subject: Climate change awareness
 * }

 * {| cellpadding="4" cellspacing="0" width="350px"

 * bgcolor="#e0ddd3" style="border: 0px #cccccc solid;" |Length/Duration: 2 hours
 * }

 * {| cellpadding="4" cellspacing="0" width="350px"

 * bgcolor="#e0ddd3" style="border: 0px #cccccc solid;" |Technologies Used: YouTube, Blogger, Facebook, Make Beliefs Comix
 * }

 * {| cellpadding="4" cellspacing="0" width="350px"

 * bgcolor="#e0ddd3" style="border: 0px #cccccc solid;" |Link: http://en.wikiversity.org/wiki/Visual_thinking_strategies
 * }
 * }

 * {| cellpadding="4" cellspacing="0" width="350px"

 * bgcolor="#e0ddd3" style="border: 0px #cccccc solid;" |Subject: Core subjects and electives
 * }

 * {| cellpadding="4" cellspacing="0" width="350px"

 * bgcolor="#e0ddd3" style="border: 0px #cccccc solid;" |Sub-Subject: Character education/Earth charter principles
 * }

 * {| cellpadding="4" cellspacing="0" width="350px"

 * bgcolor="#e0ddd3" style="border: 0px #cccccc solid;" |Length/Duration: 2 class periods (1 hour each)/on-going
 * }

 * {| cellpadding="4" cellspacing="0" width="350px"

 * bgcolor="#e0ddd3" style="border: 0px #cccccc solid;" |Technologies Used: Wikispaces
 * }

 * {| cellpadding="4" cellspacing="0" width="350px"

 * bgcolor="#e0ddd3" style="border: 0px #cccccc solid;" |Link: http://en.wikiversity.org/wiki/Fostering_Classroom_Norms:_Using_Wikispaces
 * }
 * }

 * {| cellpadding="4" cellspacing="0" width="350px"

 * bgcolor="#e0ddd3" style="border: 0px #cccccc solid;" |Subject: Stem cells
 * }

 * {| cellpadding="4" cellspacing="0" width="350px"

 * bgcolor="#e0ddd3" style="border: 0px #cccccc solid;" |Sub-Subject: Developing and sharing opinions on a controversial topic
 * }

 * {| cellpadding="4" cellspacing="0" width="350px"

 * bgcolor="#e0ddd3" style="border: 0px #cccccc solid;" |Length/Duration: 2 workshops, 3 hours in length
 * }

 * {| cellpadding="4" cellspacing="0" width="350px"

 * bgcolor="#e0ddd3" style="border: 0px #cccccc solid;" |Technologies Used: Go!Animate, Blogger
 * }

 * {| cellpadding="4" cellspacing="0" width="350px"

 * bgcolor="#e0ddd3" style="border: 0px #cccccc solid;" |Link: http://en.wikiversity.org/wiki/Stem_cell_workshop
 * }
 * }

 * {| cellpadding="4" cellspacing="0" width="350px"

 * bgcolor="#e0ddd3" style="border: 0px #cccccc solid;" |Subject: Humanities course
 * }

 * {| cellpadding="4" cellspacing="0" width="350px"

 * bgcolor="#e0ddd3" style="border: 0px #cccccc solid;" |Sub-Subject: Normal vs not normal
 * }

 * {| cellpadding="4" cellspacing="0" width="350px"

 * bgcolor="#e0ddd3" style="border: 0px #cccccc solid;" |Length/Duration: 1-2 days
 * }

 * {| cellpadding="4" cellspacing="0" width="350px"

 * bgcolor="#e0ddd3" style="border: 0px #cccccc solid;" |Technologies Used: Glogster
 * }

 * {| cellpadding="4" cellspacing="0" width="350px"

 * bgcolor="#e0ddd3" style="border: 0px #cccccc solid;" |Link: http://en.wikiversity.org/wiki/Normal_vs_not_normal
 * }
 * }

 * {| cellpadding="4" cellspacing="0" width="350px"

 * bgcolor="#e0ddd3" style="border: 0px #cccccc solid;" |Subject: Biology
 * }

 * {| cellpadding="4" cellspacing="0" width="350px"

 * bgcolor="#e0ddd3" style="border: 0px #cccccc solid;" |Sub-Subject: Assessments
 * }

 * {| cellpadding="4" cellspacing="0" width="350px"

 * bgcolor="#e0ddd3" style="border: 0px #cccccc solid;" |Length/Duration: Varies
 * }

 * {| cellpadding="4" cellspacing="0" width="350px"

 * bgcolor="#e0ddd3" style="border: 0px #cccccc solid;" |Technologies Used: Google Docs, aniBOOM, ToonDoo, Windows Movie Maker
 * }

 * {| cellpadding="4" cellspacing="0" width="350px"

 * bgcolor="#e0ddd3" style="border: 0px #cccccc solid;" |Link: http://en.wikiversity.org/wiki/Learning_is_Active
 * }
 * }

 * {| cellpadding="4" cellspacing="0" width="350px"

 * bgcolor="#e0ddd3" style="border: 0px #cccccc solid;" |Subject: Biology
 * }

 * {| cellpadding="4" cellspacing="0" width="350px"

 * bgcolor="#e0ddd3" style="border: 0px #cccccc solid;" |Sub-Subject: Cardiovascular system
 * }

 * {| cellpadding="4" cellspacing="0" width="350px"

 * bgcolor="#e0ddd3" style="border: 0px #cccccc solid;" |Length/Duration: 3-4 80-minute class periods
 * }

 * {| cellpadding="4" cellspacing="0" width="350px"

 * bgcolor="#e0ddd3" style="border: 0px #cccccc solid;" |Technologies Used: Google Docs, Google Groups, Vernier Software and Probeware
 * }

 * {| cellpadding="4" cellspacing="0" width="350px"

 * bgcolor="#e0ddd3" style="border: 0px #cccccc solid;" |Link: http://en.wikiversity.org/wiki/Action_-_Reflection
 * }
 * }

 * {| cellpadding="4" cellspacing="0" width="350px"

 * bgcolor="#e0ddd3" style="border: 0px #cccccc solid;" |Subject: Online resources
 * }

 * {| cellpadding="4" cellspacing="0" width="350px"

 * bgcolor="#e0ddd3" style="border: 0px #cccccc solid;" |Sub-Subject: Autism
 * }

 * {| cellpadding="4" cellspacing="0" width="350px"

 * bgcolor="#e0ddd3" style="border: 0px #cccccc solid;" |Length/Duration: 30 minutes
 * }

 * {| cellpadding="4" cellspacing="0" width="350px"

 * bgcolor="#e0ddd3" style="border: 0px #cccccc solid;" |Technologies Used: Links to various websites useful in the Pre-school Special Day Class
 * }

 * {| cellpadding="4" cellspacing="0" width="350px"

 * bgcolor="#e0ddd3" style="border: 0px #cccccc solid;" |Link: http://en.wikiversity.org/wiki/Online_Autism_Resources_for_Early_Childhood_Special_Education_Parents_and_Teachers
 * }
 * }

 * {| cellpadding="4" cellspacing="0" width="350px"

 * bgcolor="#e0ddd3" style="border: 0px #cccccc solid;" |Subject: Parent Education
 * }

 * {| cellpadding="4" cellspacing="0" width="350px"

 * bgcolor="#e0ddd3" style="border: 0px #cccccc solid;" |Sub-Subject: Communication devices
 * }

 * {| cellpadding="4" cellspacing="0" width="350px"

 * bgcolor="#e0ddd3" style="border: 0px #cccccc solid;" |Length/Duration: 45 minutes
 * }

 * {| cellpadding="4" cellspacing="0" width="350px"

 * bgcolor="#e0ddd3" style="border: 0px #cccccc solid;" |Technologies Used: PECS, GoTalk, Tech/Speak, Dynavox/Mayer-Johnson, Proloquo2Go
 * }

 * {| cellpadding="4" cellspacing="0" width="350px"

 * bgcolor="#e0ddd3" style="border: 0px #cccccc solid;" |Link: http://en.wikiversity.org/wiki/Action_-_Reflection
 * }
 * }

 * {| cellpadding="4" cellspacing="0" width="350px"

 * bgcolor="#e0ddd3" style="border: 0px #cccccc solid;" |Subject: AVID
 * }

 * {| cellpadding="4" cellspacing="0" width="350px"

 * bgcolor="#e0ddd3" style="border: 0px #cccccc solid;" |Sub-Subject: Self-esteem/Paradigm recognition
 * }

 * {| cellpadding="4" cellspacing="0" width="350px"

 * bgcolor="#e0ddd3" style="border: 0px #cccccc solid;" |Length/Duration: 2 60 min. periods
 * }

 * {| cellpadding="4" cellspacing="0" width="350px"

 * bgcolor="#e0ddd3" style="border: 0px #cccccc solid;" |Technologies Used: Jing, ComicLife, digital camera
 * }

 * {| cellpadding="4" cellspacing="0" width="350px"

 * bgcolor="#e0ddd3" style="border: 0px #cccccc solid;" |Link: http://en.wikiversity.org/wiki/Staying_true_to_my_Paradigm%3F#Elaborate.2F_Extensions_and_Links
 * }
 * }

 * {| cellpadding="4" cellspacing="0" width="350px"

 * bgcolor="#e0ddd3" style="border: 0px #cccccc solid;" |Subject: Online communication efficacy
 * }

 * {| cellpadding="4" cellspacing="0" width="350px"

 * bgcolor="#e0ddd3" style="border: 0px #cccccc solid;" |Sub-Subject: Improving international online dialogues
 * }

 * {| cellpadding="4" cellspacing="0" width="350px"

 * bgcolor="#e0ddd3" style="border: 0px #cccccc solid;" |Length/Duration: 60 min.
 * }

 * {| cellpadding="4" cellspacing="0" width="350px"

 * bgcolor="#e0ddd3" style="border: 0px #cccccc solid;" |Technologies Used: Secondary Translation to Other Common Languages
 * }

 * {| cellpadding="4" cellspacing="0" width="350px"

 * bgcolor="#e0ddd3" style="border: 0px #cccccc solid;" |Link: http://en.wikiversity.org/wiki/Online_Communication_Efficacy
 * }
 * }

 * {| cellpadding="4" cellspacing="0" width="350px"

 * bgcolor="#e0ddd3" style="border: 0px #cccccc solid;" |Subject: The Truth is Out There
 * }

 * {| cellpadding="4" cellspacing="0" width="350px"

 * bgcolor="#e0ddd3" style="border: 0px #cccccc solid;" |Sub-Subject: Seeking and spreading the truth online
 * }

 * {| cellpadding="4" cellspacing="0" width="350px"

 * bgcolor="#e0ddd3" style="border: 0px #cccccc solid;" |Length/Duration: 60 min.
 * }

 * {| cellpadding="4" cellspacing="0" width="350px"

 * bgcolor="#e0ddd3" style="border: 0px #cccccc solid;" |Technologies Used: Computer Accessed - Internet Research and Information Web Sites
 * }

 * {| cellpadding="4" cellspacing="0" width="350px"

 * bgcolor="#e0ddd3" style="border: 0px #cccccc solid;" |Link: http://en.wikiversity.org/wiki/The_Truth_is_Out_There
 * }
 * }

 * {| cellpadding="4" cellspacing="0" width="350px"

 * bgcolor="#e0ddd3" style="border: 0px #cccccc solid;" |Subject: Advanced Earth Science
 * }

 * {| cellpadding="4" cellspacing="0" width="350px"

 * bgcolor="#e0ddd3" style="border: 0px #cccccc solid;" |Sub-Subject: Topographic maps and remote sensing
 * }

 * {| cellpadding="4" cellspacing="0" width="350px"

 * bgcolor="#e0ddd3" style="border: 0px #cccccc solid;" |Length/Duration: 2 class periods/1 block period: 1 hr 40 min.
 * }

 * {| cellpadding="4" cellspacing="0" width="350px"

 * bgcolor="#e0ddd3" style="border: 0px #cccccc solid;" |Technologies Used: Google Earth, Jing, Facebook
 * }

 * {| cellpadding="4" cellspacing="0" width="350px"

 * bgcolor="#e0ddd3" style="border: 0px #cccccc solid;" |Link: http://en.wikiversity.org/wiki/Mapping_my_world
 * }
 * }

 * {| cellpadding="4" cellspacing="0" width="350px"

 * bgcolor="#e0ddd3" style="border: 0px #cccccc solid;" |Subject: Earth Science
 * }

 * {| cellpadding="4" cellspacing="0" width="350px"

 * bgcolor="#e0ddd3" style="border: 0px #cccccc solid;" |Sub-Subject: Variables & communicating in science
 * }

 * {| cellpadding="4" cellspacing="0" width="350px"

 * bgcolor="#e0ddd3" style="border: 0px #cccccc solid;" |Length/Duration: 50 min.
 * }

 * {| cellpadding="4" cellspacing="0" width="350px"

 * bgcolor="#e0ddd3" style="border: 0px #cccccc solid;" |Technologies Used: Masses and Spring Lab Simulator, Create a Graph
 * }

 * {| cellpadding="4" cellspacing="0" width="350px"

 * bgcolor="#e0ddd3" style="border: 0px #cccccc solid;" |Link: http://en.wikiversity.org/wiki/Secondary_Science_Lesson
 * }
 * }

Resources:
 * Educational Technologies
 * Lesson Plan Template 1
 * Lesson Plan Template 2


text:
User:Pupei

Operating system which is the most popular for net book is windows -Windows is creditable. There are about 90% of people prefers Windows Operating system (mostly is Window 7 and Window XP) rather than other operating system for example, Linux (less than 10% of users), Mac OS X, and Android. -windows is extremely well supported by every software company in the world except for Apple

-Windows is cheaper -Windows make a good marketing -Windows has long been used and it is the most used, people find difficult to switch to a new operating system.

title: User:Pupei


text:
Physics/Essays/Fedosin/Quantum Gravitational Resonator

Quantum Gravitational Resonator (QGR) – closed topological object of the three dimensional space, in the general case – ‘’cavity’’ of arbitrary form, which has definite ‘’surface’’ and ‘’thickness’’. The QGR can have “infinite” phase shifted oscillations of gravitational field strength and gravitational torsion field, due to the quantum properties of QGR.

History
Considering that the theory of the gravitational resonator is based on the Maxwell-like gravitational equations and Quantum Electromagnetic Resonator (QER), therefore the QGR history is close connected with the QER history.

Gravitational LC circuit
The gravitational LC circuit can be composed by analogy with the electromagnetic LC circuit, and gravitational field strength and gravitational torsion field oscillate in the circuit as a result of oscillating mass current.

Gravitational voltage on gravitational inductance is:
 * $$V_{gL} = -L_g\cdot \frac{d I_{gL}}{d t}. \ $$

Gravitational mass current through gravitational capacitance is:
 * $$I_{gC} = C_g\cdot \frac{d V_{gC}}{d t}. \ $$

Differentiating these equations with respect to the time variable, we obtain:
 * $$\frac{d V_{gL}}{d t} = -L_g \frac{d^2I_{gL}}{dt^2}, \qquad \frac{d I_{gC}}{d t} = C_g \frac{d^2V_{gC}}{dt^2}.$$

Considering the following relationships for voltages and currents:
 * $$V_{gL} = V_{gC} =V_g, \qquad I_{gL} = I_{gC}=I_g ,\ $$

we obtain the following differential equations for gravitational oscillations:
 * $$~\frac{d^2 I_g}{dt^2} + \frac{1}{L_g C_g}I_g = 0, \qquad \frac{d^2 V_g}{dt^2} + \frac{1}{L_g C_g}V_g = 0. \quad \quad \quad \quad \quad (1) \  $$

Furthermore, considering the following relationships between voltage and mass, current and flux of gravitational torsion field:
 * $$ m = C_g V_g, \qquad \Phi = L_g I_g $$

the above oscillation equation can be rewritten in the form:
 * $$\frac{d^2 m}{dt^2} + \frac{1}{L_g C_g} m = 0. \quad \quad \quad \quad \quad (2) \ $$

This equation has the partial solution:
 * $$ m(t) = m_0 \sin (\omega_g t) ,\ $$

where
 * $$\omega_g = \frac{1}{\sqrt{L_g C_g}} \ $$

is the resonance frequency, and
 * $$\rho_{LC} = \sqrt{\frac{L_g}{C_g}} ,\ $$

is the gravitational characteristic impedance.

For the sake of completeness we can present the differential equation for the flux of gravitational torsion field in the form:
 * $$\frac{d^2 \Phi}{dt^2} + \frac{1}{L_g C_g}\Phi = 0. \quad \quad \quad \quad \quad (3) \ $$

The realization of gravitational LC circuit is described in a section of maxwell-like gravitational equations.

Quantum gravitational LC circuit oscillator
Inductance momentum quantum operator in the electric-like gravitational mass space can be presented in the following form:
 * $$\hat p_{gm} = -i\hbar \frac{d}{dm}, \quad \quad \quad \quad \quad \hat p_{gm}^* = i\hbar \frac{d}{dm},\quad \quad \quad \quad \quad (4a) \ $$

where $$\hbar \ $$ is reduced Plank constant, $$\hat p_{gm}^* \ $$ is the complex-conjugate momentum operator, $$ m \ $$ is the induced mass.

Capacitance momentum quantum operator in the magnetic-like gravitational mass space can be presented in the following form:
 * $$\hat p_{g\Phi} = -i\hbar \frac{d}{d\Phi}, \quad \quad \quad \quad \quad \hat p_{g\Phi}^* = i\hbar \frac{d}{d\Phi},\quad \quad \quad \quad \quad (4b) \ $$

where $$\Phi \ $$ is the induced torsion field flux, which is imitated by electric-like gravitational mass current ($$i_g \ $$):
 * $$\Phi = L_g \cdot i_g. \ $$

We can introduce the third momentum quantum operator in the current form:
 * $$\hat p_{gi} = -\frac{i\hbar}{L_g} \frac{d}{di_g}, \quad \quad \quad \quad \quad  \hat p_{gi}^* = \frac{i\hbar}{L_g} \frac{d}{di_g},\quad \quad \quad \quad \quad (4c) \ $$

These quantum momentum operators defines three Hamilton operators:
 * $$\hat H_{gLm} = -\frac{\hbar^2}{2L_g}\cdot \frac{d^2}{dm^2} + \frac{L_g \omega_0^2}{2}\hat m^2 \quad \quad \quad \quad \quad (5a) \ $$
 * $$\hat H_{gC\Phi} = -\frac{\hbar^2}{2C_g}\cdot \frac{d^2}{d\Phi^2} + \frac{C_g \omega_0^2}{2}\hat \Phi^2 \quad \quad \quad \quad \quad (5b) \ $$
 * $$\hat H_{gLi} = -\frac{\hbar^2\omega_0^2}{2L_g}\cdot \frac{d^2}{di_g^2} + \frac{L_g \omega_0}{2}\hat i_g^2, \quad \quad \quad \quad \quad (5c) \ $$

where $$\omega_0 = \frac{1}{\sqrt{L_g C_g}} \ $$ is the resonance frequency. We consider the case without dissipation ($$R_g = 0 \ $$). The only difference of the gravitational charge spaces and gravitational current spaces from the traditional 3D- coordinate space is that it is one dimensional (1D). Schrodinger equation for the gravitational quantum LC circuit could be defined in three form:
 * $$-\frac{\hbar^2}{2L_g}\frac{d^2 \Psi}{dm^2} + \frac{L_g\omega_0^2}{2}m^2\Psi = W\Psi \quad \quad \quad \quad \quad (6a) \ $$
 * $$-\frac{\hbar^2}{2C_g}\frac{d^2 \Psi}{d\Phi^2} + \frac{C_g\omega_0^2}{2}\Phi^2\Psi = W\Psi \quad \quad \quad \quad \quad (6b) \ $$
 * $$-\frac{\hbar^2\omega_0^2}{2L_g}\frac{d^2 \Psi}{di_g^2} + \frac{L_g\omega_0}{2}i_g^2\Psi = W\Psi. \quad \quad \quad \quad \quad (6c) \ $$

To solve these equations we should to introduce the following dimensionless variables:
 * $$\xi_m = \frac{m}{m_0}; \quad \quad m_0 = \sqrt{\frac{\hbar}{L_g\omega_0}}; \quad \quad \lambda_m = \frac{2W}{\hbar\omega_0} \quad \quad (7a) \ $$
 * $$\xi_{\Phi} = \frac{\Phi}{\Phi_0}; \quad \quad \Phi_0 = \sqrt{\frac{\hbar}{C_g\omega_0}}; \quad \quad \lambda_{\Phi} = \frac{2W}{\hbar\omega_0} \quad \quad (7b) \ $$
 * $$\xi_i = \frac{i_g}{i_{g0}}; \quad \quad i_{g0} = \sqrt{\frac{\hbar \omega_0}{L_g}}; \quad \quad \lambda_i = \frac{2W}{\hbar\omega_0}. \quad \quad (7c) \ $$

where $$m_0 \ $$ is scaling induced electric-like gravitational mass; $$\Phi_0 \ $$ is scaling induced gravitational torsion field flux and $$i_{g0} \ $$ is scaling induced mass current.

Then the Schrodinger equation will take the form of the differential equation of Chebyshev-Ermidt:
 * $$\left(\frac{d^2}{d\xi^2} + \lambda - \xi^2\right)\Psi = 0. \ $$

The eigen values of the Hamiltonian will be:
 * $$W_n = \hbar \omega_0(n + 1/2), \quad \quad n = 0,1,2,\dots \ $$

where at $$n = 0 \ $$ we shall have zero oscillation:
 * $$W_0 = \hbar \omega_0/2. \ $$

In the general case the scaling mass and torsion flux can be rewritten in the form:
 * $$m_0 = \sqrt{\frac{\hbar}{L_g\omega_0}} = \frac{m_P}{\sqrt{4\pi }}= \sqrt {\frac {\hbar c}{4 \pi G}} ,\ $$
 * $$\Phi_0 = \sqrt{\frac{\hbar}{C_g\omega_0}} = \frac{h}{m_P \sqrt{\pi} } = \sqrt {\frac {4 \pi G \hbar }{ c } }, \ $$

where $$m_P \ $$ is the Planck mass, $$c  \ $$ is the speed of light, $$G  \ $$ is the gravitational constant.

These three equations (4) form the base of the nonrelativistic quantum gravidynamics, which considers elementary particles from the intrinsic point of view. Note that, the standard quantum electrodynamics considers elementary particles from the external point of view.

Gravitational resonator as quantum LC circuit
Due to Luryi density of states (DOS) approach we can define gravitational quantum capacitance as:
 * $$C_g = m_g^2\cdot D_{2D}\cdot S_g, \ $$

and quantum inductance as:
 * $$L_g = \Phi_g^2\cdot D_{2D}\cdot S_g, \ $$

where $$S_g \ $$ is the resonator surface area, $$D_{2D} = \frac{m_0}{\pi \hbar^2} \ $$ is two dimensional (2D) DOS, $$m_0  \ $$ is the carrier mass, $$m_g \ $$ is the induced gravitational mass, and $$\Phi_g \ $$ is the gravitational torsion field flux.

Energy stored on quantum capacitance is:
 * $$W_{Cg} = \frac{m_g^2}{2 C_g } = \frac{1}{2D_{2D}S_g}. \ $$

Energy stored on quantum inductance is:
 * $$W_{Lg} = \frac{\Phi_g^2}{2 L_g }= \frac{1}{2D_{2D}S_g} = W_{Cg}. \ $$

Resonator angular frequency is:
 * $$\omega_{gR} = \frac{1}{\sqrt{L_g C_g}} = \frac{1}{m_g \Phi_g D_{2D}S_g}. \ $$

Energy conservation law for zero oscillation is:
 * $$W_{gR} = \frac {1}{2} \hbar \omega_{gR} = \frac{\hbar }{2 m_g \Phi_g D_{2D}S_g} = W_{Cg} = W_{Lg}. \ $$

This equation can be rewritten as:
 * $$m_g \Phi_g = \hbar . \ $$

Characteristic gravitational resonator impedance is:
 * $$ \rho_g = \sqrt{\frac{L_g}{C_g}} = \frac{\Phi_g}{m_g} = 2 \alpha \frac{\Phi_{g0}}{m_S}= \rho_{g0},  \ $$

where $$ \alpha \ $$ is the fine structure constant, $$\Phi_{g0} = h/m_S  \ $$ is the gravitational torsion flux quantum, $$ h  \ $$ is the Planck constant,  $$m_S  \ $$ is the Stoney mass, $$ \rho_{g0}\ $$ is the gravitational characteristic impedance of free space.

Considering above equations, we can find out the following induced mass and induced gravitational torsion flux:
 * $$m_g = \frac{m_S}{\sqrt{4\pi \alpha}} ,\ $$
 * $$\Phi_g = \sqrt{\frac{\alpha}{\pi}}\frac{h}{m_S} . \ $$

Note, that these induced quantities maintain the energy balance between resonator oscillation energy and total energy on capacitance and inductance
 * $$\hbar \omega_{gR} = W_{gL}(t) + W_{gC}(t). \ $$

Since capacitance oscillations are phase shifted ($$\psi = \pi /2 \ $$) with respect to inductance oscillations, therefore we get:
 * $$W_{gL} = \begin{cases} 0, & \mbox{at }t=0; \psi=0\mbox{ and}\, t=\frac{T_R}{2};\psi=\pi \\ W_L, & \mbox{at }t=\frac{T_R}{4};\psi=\frac{\pi}{4} \mbox{ and} \, t=\frac{3T_R}{4};\psi=\frac{3\pi}{4} \end{cases} \ $$

 * $$W_{gC} = \begin{cases} W_C, & \mbox{at }t=0; \psi=0\mbox{ and} \, t=\frac{T_R}{2};\psi=\pi \\ 0, & \mbox{at }t=\frac{T_R}{4};\psi=\frac{\pi}{4} \mbox{ and} \, t=\frac{3T_R}{4};\psi=\frac{3\pi}{4} \end{cases} \ $$

where $$T_R = \frac{2\pi}{\omega_{gR}} \ $$ is the oscillation period.

Planckion resonator
Planckion radius is:
 * $$r_P = \frac{\lambda_P}{2\pi }, \ $$

where $$\lambda_P = \frac {h}{m_P c} \ $$ is the Compton wavelength of planckion, $$c  \ $$ is the speed of light, $$ m_P  \ $$ is the Planck mass.

Planckion surface scaling parameter is:
 * $$S_P = 2\pi r_P^2 = \frac{\lambda_P^2}{2\pi}. \ $$

Planckion angular frequency is:
 * $$ \omega_P = \frac{m_Pc^2}{ \hbar} = \frac{2\pi c}{\lambda_P}. \ $$

Planckion density of states is:
 * $$D_P = \frac{1}{S_P W_P} = \frac {1}{ S_P \hbar \omega_P } = \frac{m_P}{2\pi \hbar^2}. \ $$

Standard DOS quantum resonator approach yields the following values for the gravitational reactive quantum parameters:
 * $$C_P = m_g^2 D_P S_P = \frac {m_S^2 }{4\pi \alpha} \frac{m_P}{2\pi \hbar^2} \frac{\lambda_P^2}{2\pi} = \frac{ \varepsilon_g \lambda_P }{2\pi } = \frac {m_P}{4\pi c^2}, \ $$

where $$ \varepsilon_g = \frac {1}{4\pi G} \ $$ is the gravitoelectric gravitational constant in the set of selfconsistent gravitational constants, and
 * $$L_P = \Phi_g^2 D_P S_P = \frac {\Phi_0^2}{4 \pi \beta} D_P S_P = \frac {\alpha h^2}{\pi m_S^2 } D_P S_P = \frac {\alpha h^2}{\pi m_S^2 }\frac{m_P}{2\pi \hbar^2} \frac{\lambda_P^2}{2\pi} = \frac{\mu_g \lambda_P }{ 2\pi }, \ $$

where $$ \mu_g = \frac {4 \pi G }{ c^2} \ $$ is the gravitomagnetic gravitational constant of selfconsistent gravitational constants, $$ \beta = \frac {1}{4\alpha} \ $$ is the gravitational torsion coupling constant, which is equal to magnetic coupling constant.

Thus, s.c. free planckion can be considered as discoid quantum resonator which has radius $$ r_P \ $$.

Bohr atom as a gravitational quantum resonator
The gravitational quantum capacitance for Bohr atom is:
 * $$C_\Gamma = m^2_R  D_B S_B = \varepsilon _\Gamma a_B, \  $$

where $$ a_B \ $$ is the Bohr radius, $$S_B = \pi a_B^2 \ $$ is the flat surface area, $$ ~ m_R = \frac { \sqrt {m_p m_e}}{2 \sqrt \pi } $$ is the induced mass, $$ D_B= \frac {m_e}{\pi \hbar^2} \ $$ is the density of states, $$ \varepsilon _\Gamma = \frac {1 }{4 \pi \Gamma } $$ is the gravitoelectric gravitational constant of selfconsistent gravitational constants in the field of strong gravitation, $$\Gamma $$ is the strong gravitational constant, $$ m_p \ $$ and $$ m_e \ $$ are masses of proton and electron.

The gravitational quantum inductance is:
 * $$ L_\Gamma = \phi_\Gamma ^2 D_B S_B = \mu _\Gamma a_B, \  $$

where $$ \mu_\Gamma = \frac {4 \pi \Gamma }{ c^2 } $$ is the gravitomagnetic gravitational constant of selfconsistent gravitational constants in the field of strong gravitation, and the induced gravitational torsion flux is:
 * $$ \phi_\Gamma = \frac {\alpha h }{\sqrt {\pi m_p m_e } }= \frac {2 \alpha \sqrt{m_e} }{ \sqrt {\pi m_p } } \sigma_e = \frac {2 \alpha \sqrt{m_p} }{ \sqrt {\pi m_e } } \Phi_\Gamma = \frac {2 \sqrt{m_p} }{ \alpha \sqrt {\pi m_e } } \Phi_\Omega ,$$

where $$ \sigma_e \ $$ is the velocity circulation quantum, $$ \Phi_\Gamma = \frac{h}{2 m_p} $$ is the strong gravitational torsion flux quantum, which is related to proton with its mass $$ m_p $$.

Here the strong gravitational electron torsion flux for the first energy level is:
 * $$ \Phi_{\Omega } = \Omega S_B = \frac { \mu_\Gamma m_e }{4 \pi a_B } \sigma_e = \frac { \Gamma m_e }{c^2 a_B }\sigma_e  = \frac { \Gamma h }{2c^2 a_B }= \frac { \pi  \alpha \Gamma m_e }{c }= \alpha^2 \Phi_\Gamma, \ $$

where $$ \Omega \ $$ is the gravitational torsion field of strong gravitation in electron disc.

The gravitational wave impedance is:
 * $$ \rho_\Gamma = \sqrt {\frac{ L_\Gamma }{ C_\Gamma }}= \sqrt{\frac {\mu _\Gamma }{\varepsilon _\Gamma }} = \frac{4\pi \Gamma }{c} = 6.346\cdot 10^{21}\, \mathrm {m^2/(s\cdot kg)}. \ $$

The resonance frequency of gravitational oscillation is:
 * $$\omega_\Gamma = \frac{1}{\sqrt{L_\Gamma C_\Gamma }} = \frac{ c}{a_B} = \frac {\omega_B}{\alpha }, \ $$

where $$\omega_B = \frac{c \alpha }{ a_B } \ $$ is the angular frequency of electron rotation in atom.

For the quantum gravitational resonator approach we can derive the following maximal values for the energies stored on capacitance and inductance:
 * $$W_C = \frac{m_R^2}{2 C_\Gamma } = \frac{\hbar \omega_B }{2} = \frac{\alpha \hbar \omega_\Gamma }{2}= W_B, \ $$
 * $$W_L = \frac{ \phi_\Gamma ^2}{2 L_\Gamma } = W_B .\ $$

The energy $$ W_B \ $$ of the wave of strong gravitation in the electron matter has the same value as in case of rotating electromagnetic wave, and can be associated with the mass:
 * $$m_{Bmin} = \frac{W_B}{c^2} = \frac{\hbar \omega_B}{2 c^2} = \frac{\alpha^2}{2}m_e << m_e, \ $$

which could be named as the minimal mass-energy of the quantum resonator.

One way to explain the minimal mass-energy $$m_{Bmin}$$ is the supposition that Planck constant can be used at all matter levels including the level of star. As a result of the approach one should introduce different scales such as Planck scale, Stoney scale, Natural scale, with the proper masses and lengths. But such proper masses do not relate with the real particles.

Another way recognizes the similarity of matter levels and SPФ symmetry as the principles of matter structure where the action constants depend on the matter levels. For example there is the stellar Planck constant at the star level that describes star systems without any auxiliary mass and scales.

Reference Books

 * Johnstone Stoney, Phil. Trans. Roy. Soc. 11, (1881)
 * Stratton J.A.(1941). Electromagnetic Theory. New York, London: McGraw-Hill.p.615. djvu
 * Детлаф А.А., Яворский Б.М., Милковская Л.Б.(1977). Курс физики. Том 2. Электричество и магнетизм (4-е издание). М.: Высшая школа, "Reference Book on Electricity"  djvu
 * Гольдштейн Л.Д., Зернов Н.В. (1971). Электромагнитные поля. 2- издание. Москва: Советское Радио. 664с. "Electromagnetic Fields" djvu


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Physics/Essays/Fedosin/Quantum Electromagnetic Resonator

A Quantum Electromagnetic Resonator (QER) – a closed topological object of the three dimensional space, in the general case – is a ‘’cavity’’ of arbitrary form, which has definite ‘’surface’’ and ‘’thickness’’. The QER has “infinite” phase shifted oscillations of electromagnetic field, due to its quantum properties.

History
It has happened, that such physical values as capacitance and inductance have no interest in the modern quantum electrodynamics. Furthermore, they are neglected even in classical electrodynamics, where electric and magnetic fields are dominated. The point is that, the last are not included in evident form in the Maxwell's equations, and so the resulting solutions includes fields only. Yes, sometimes these coefficients were obtained from the solutions of Maxwell equations, but it was very rarely.

It is known too, that s.c. “field approach” in electrodynamics that considered point charges leads to the “fool infinities”, as the interaction radius trends to zero. Furthermore, these “fool infinities” are presented in quantum electrodynamics too, where power methods are developed to compensate them.

Contrary to theoretical physics, applied physics has widely used reactive parameters, such as capacitance and inductance, firstly in electrotechnics and then in applied radiotechnics. Now reactive parameters are widely used in information technologies, which are based on the generation, transmission and radiation of electromagnetic waves of different frequencies.

The present day situation (without proper development of the theory of reactive parameters such as inductance, capacitance and electromagnetic resonator) prevents or slows developments in information technology and quantum computing. Note that, mechanical harmonic oscillator was considered in quantum mechanics in the early fourth decade of the twentieth century, when quantum theory was developed. However, quantum consideration of the $$LC - \ $$ circuit was started only by Louisell (1973). Since then, there have been no practical examples of quantum capacitance and inductance, so this approach did not obtain proper consideration.

A theoretically correct introduction of quantum capacitance, based on the density of states, first was presented by Luryi (1988) for QHE. However, Luryi did not introduce quantum inductance, and this approach was not considered in quantum LC circuits and resonators. A year later, Yakymakha (1989) considered an example of series and parallel quantum $$LC - \ $$ circuits (characteristic impedances) during QHE explanation (integer and fractional). However this work did not consider the Schrodinger equation for the quantum LC circuit.

For the first time, both quantum values, capacitance and inductance, were considered by Yakymakha (1994), during spectroscopic investigations of MOSFETs at the very low frequencies (sound range). The flat quantum capacitances and inductances here had thicknesses about the Compton wave length of an electron, and its characteristic impedance – the wave impedance of free space.

And three year later, Devoret (1997) presented a complete theory of the Quantum LC circuit (applied to the Josephson junction). Possible application of the quantum LC circuits and resonators in the quantum computation are considered by Devoret (2004).

Classical electromagnetic resonator
In the general case, a classical electromagnetic resonator (CER) is a cavity resonator. It is a hollow closed conductor such as a metal box or a cavity within a metal block, containing electromagnetic waves reflecting back and forth between the cavity's walls. Therefore, a CER has infinite resonance frequencies, due to the three dimensions. For example, the rectangular CER has the following resonance frequencies:
 * $$\omega_{mnp} = \frac{1}{\sqrt{\varepsilon_0 \varepsilon_r \mu_0 \mu_r}}\sqrt{(\frac{m\pi}{a})^2 + (\frac{n\pi}{b})^2 + (\frac{p\pi}{l})^2}, \ $$

where $$m,n,p = integer. \ $$; $$a, b, l \ $$ are the width, thickness and length correspondingly, $$\varepsilon_0 \ $$ is the electric constant, $$\varepsilon_r \ $$ is the relative permittivity, $$\mu_0 \ $$ is the  vacuum permeability, $$\mu _r \ $$ is the relative permeability.

Contrary to the classical LC circuit, both electric and magnetic fields are displaced in the same volume of CER.

These oscillating electromagnetic fields, in the classical case, are like standing waves, that form electromagnetic waves, that could be radiated to the external world.

Now CERs are widely used in the radio frequency range (centimetre and decimetre diapason). Furthermore, they are also used in quantum electronics, which deal with monochromic light waves.

Quantum LC circuit oscillator
In classical physics we have the following correspondence between mechanical and electrodynamical physical parameters:

Magnetic inductance and mechanical mass:
 * $$L \leftrightarrow m \ $$;

Electric capacitance and reciprocal elasticity:
 * $$C \leftrightarrow 1/k \ $$;

Electric charge and coordinate displacement:
 * $$q \leftrightarrow x \ $$.

The inductance momentum quantum operator in the electric charge space can be presented in the following form:
 * $$\hat p_{Lq} = -i\hbar \frac{d}{dq}, \quad \quad \quad \quad \quad \hat p_{Lq}^* = i\hbar \frac{d}{dq},\quad \quad \quad \quad \quad (1a) \ $$

where $$\hbar \ $$ is the reduced Planck constant, $$\hat p_L^* \ $$ is the complex-conjugate momentum operator.

The capacitance momentum quantum operator in magnetic charge space can be presented in the following form:

 * $$\hat p_{C\phi} = -i\hbar \frac{d}{d\phi}, \quad \quad \quad \quad \quad \hat p_{C\phi}^* = i\hbar \frac{d}{d\phi},\quad \quad \quad \quad \quad (1b) \ $$

where $$\phi \ $$ is the induced magnetic flux.

Considering that there is no free magnetic flux, but it could be imitated by electric current ($$i \ $$):
 * $$\phi = L\cdot i, \ $$

we may introduce the third momentum quantum operator in the current form:
 * $$\hat p_{Ci} = -\frac{i\hbar}{L} \frac{d}{di}, \quad \quad \quad \quad \quad  \hat p_{Ci}^* = \frac{i\hbar}{L} \frac{d}{di},\quad \quad \quad \quad \quad (1c) \ $$

These quantum momentum operators define three Hamilton operators:
 * $$\hat H_{Lq} = -\frac{\hbar^2}{2L}\cdot \frac{d^2}{dq^2} + \frac{L\omega_0^2}{2}\hat q^2 \quad \quad \quad \quad \quad (2a) \ $$
 * $$\hat H_{C\phi} = -\frac{\hbar^2}{2C}\cdot \frac{d^2}{d\phi^2} + \frac{C\omega_0^2}{2}\hat \phi^2 \quad \quad \quad \quad \quad (2b) \ $$
 * $$\hat H_{Ci} = -\frac{\hbar^2\omega_0^2}{2L}\cdot \frac{d^2}{di^2} + \frac{L\omega_0}{2}\hat i^2, \quad \quad \quad \quad \quad (2c) \ $$

where $$\omega_0 = \frac{1}{\sqrt{LC}} \ $$ is the resonance frequency.

We consider the case without dissipation ($$R = 0 \ $$).

The only difference of the charge spaces and current spaces here from the traditional 3D- coordinate space is that they are one dimensional (1D).

The Schrodinger equation for the quantum LC circuit may be defined in three forms:
 * $$-\frac{\hbar^2}{2L}\frac{d^2 \Psi}{dq^2} + \frac{L\omega_0^2}{2}q^2\Psi = W\Psi \quad \quad \quad \quad \quad (3a) \ $$
 * $$-\frac{\hbar^2}{2C}\frac{d^2 \Psi}{d\phi^2} + \frac{C\omega_0^2}{2}\phi^2\Psi = W\Psi \quad \quad \quad \quad \quad (3b) \ $$
 * $$-\frac{\hbar^2\omega_0^2}{2L}\frac{d^2 \Psi}{di^2} + \frac{L\omega_0}{2}i^2\Psi = W\Psi. \quad \quad \quad \quad \quad (3c) \ $$

To solve these equations we introduce the following dimensionless variables:
 * $$\xi_q = \frac{q}{q_0}; \quad \quad q_0 = \sqrt{\frac{\hbar}{L\omega_0}}; \quad \quad \lambda_q = \frac{2W}{\hbar\omega_0} \quad \quad (4a) \ $$
 * $$\xi_{\phi} = \frac{\phi}{\phi_0}; \quad \quad \phi_0 = \sqrt{\frac{\hbar}{C\omega_0}}; \quad \quad \lambda_{\phi} = \frac{2W}{\hbar\omega_0} \quad \quad (4a) \ $$
 * $$\xi_i = \frac{i}{i_0}; \quad \quad i_0 = \sqrt{\frac{\hbar \omega_0}{L}}; \quad \quad \lambda_i = \frac{2W}{\hbar\omega_0}. \quad \quad (4a) \ $$

where $$q_0 \ $$ is scaling induced electric charge; $$\phi_0 \ $$ is scaling induced magnetic flux and $$i_0 \ $$ is scaling induced electric current.

Then the Schrodinger equation will take the form of the differential Chebyshev-Ermidt equation:
 * $$(\frac{d^2}{d\xi^2} + \lambda - \xi^2)\Psi = 0. \ $$

The eigenvalues of the Hamiltonian will be:
 * $$W_n = \hbar \omega_0(n + 1/2), \ $$

where at $$n = 0 \ $$ we shall have zero oscillation:
 * $$W_0 = \hbar \omega_0/2. \ $$

In the general case the scaling charge and flux can be rewritten in the form:
 * $$q_0 = \sqrt{\frac{\hbar}{L\omega_0}} = \frac{e}{\sqrt{4\pi \alpha }} \ $$
 * $$\phi_0 = \sqrt{\frac{\hbar}{C\omega_0}} = \sqrt{\frac{\alpha}{\pi}}\cdot \frac{h}{e}, \ $$

where $$\alpha \ $$ is the fine structure constant.

Furthermore, the scaling current and voltage will be here:
 * $$I_0 = \frac{\phi_0}{L} = \sqrt{\frac{\pi}{\alpha}}\cdot \frac{ec}{\lambda_0} \ $$
 * $$V_0 = \frac{q_0}{C} = \sqrt{4\pi \alpha}\cdot \frac{ch}{e\lambda_0}, \ $$

where $$\lambda_0 = \frac{h}{m_0c} $$ is the particle wavelength.

These scaling parameters were obtained by using the following quantum resonator parameters:
 * $$ Z_0 = 2\alpha R_K \quad \quad (5a) \ $$

for characteristic impedance, and
 * $$\omega_0 = \frac{2\pi c}{\lambda_0} \quad \quad (5b) \ $$

for resonance frequency, and $$ R_K=\frac {h}{e^2} \ $$ is the von Klitzing constant. With this we have: $$ L=\frac {\mu_0 \lambda_0}{2 \pi}, \ $$ $$ C=\frac {\varepsilon_0 \lambda_0}{2 \pi}. \ $$

These three equations (3) form the base of a nonrelativistic quantum electrodynamics, which considers elementary particles from an intrinsic point of view. Standard quantum electrodynamics considers elementary particles from an external point of view.

Resonator as quantum LC circuit
The Luryi density of states (DOS) approach defines quantum capacitance as:
 * $$C_{QR} = q_R^2\cdot D_{2D}\cdot S_R, \ $$

and quantum inductance as:
 * $$L_{QR} = \phi_R^2\cdot D_{2D}\cdot S_R, \ $$

where $$S_R \ $$ is the resonator surface area, $$D_{2D} = \frac{m}{\pi \hbar^2} \ $$ is two dimensional (2D) DOS, $$q_R \ $$ is the induced electric charge, and $$\phi_R \ $$ is the induced magnetic flux. Note that these charges should be defined afterward.

The energy stored on quantum capacitance:
 * $$W_{CR} = \frac{q_R^2}{2 C_{QR}}= \frac{1}{2D_{2D}S_R}. \ $$

The energy stored on quantum inductance:
 * $$W_{LR} = \frac{\phi_R^2}{ 2L_{QR}} = \frac{1}{2D_{2D}S_R} = W_{CR}. \ $$

The resonator angular frequency is:
 * $$\omega_{QR} = \frac{1}{\sqrt{L_{QR}C_{QR}}} = \frac{1}{q_R\phi_RD_{2D}S_R}. \ $$

The energy conservation law for zero oscillation:
 * $$W_{QR} = \frac {1}{2} \hbar \omega_{QR} = \frac{\hbar }{2 q_R\phi_RD_{2D}S_R} = W_{CR} = W_{LR}. \ $$

This equation can be rewritten as:
 * $$q_R\phi_R = \hbar, \ $$

from which it is evident that induced charge $$q_R \ $$ and induced magnetic flux $$\phi_R \ $$ are connected to electric and magnetic fluxes in the resonator.

The characteristic resonator impedance is:
 * $$ Z_{QR} = \sqrt{\frac{L_{QR}}{C_{QR}}} = \frac{\phi_R}{q_R} = \begin{cases} \frac{\phi_0}{e}=R_K, & \mbox{for Quantum Hall Effect }  \\ 2\alpha \frac{\phi_0}{e}=2\alpha R_K = Z_0, & \mbox{for other } \end{cases} \ $$

where $$\phi_0 = h/e \ $$ is the magnetic flux quantum, $$ Z_0 = 2\alpha\cdot h/e^2 = \mu_0 c = \frac{1}{\varepsilon_0 c } = \sqrt {\frac{\mu_0 }{\varepsilon_0 }}$$ is the impedance of free space.

Considering the above equations, we have the following electric and magnetic sets of fluxes:
 * $$q_R = \begin{cases} q_{R1} = \frac{e}{\sqrt{2\pi}}, & \mbox{for Quantum Hall Effect } \\ q_{R2} = \frac{e}{\sqrt{4\pi \alpha}}, & \mbox{for other } \end{cases} \ $$

 * $$\phi_R = \begin{cases} \phi_{R1} = \frac{\phi_0}{\sqrt{2{\pi}}}, & \mbox{for Quantum Hall Effect } \\ \phi_{R2} = \frac {\phi_0 }{\sqrt{4 \pi \beta }}, & \mbox{for other } \end{cases} \ $$

where $$\beta = \frac{\phi_0^2}{2\mu_0hc} = \frac{1}{4\alpha} \ $$ is the magnetic coupling constant.

These values are not the real charge and magnetic flux, but maximal quantities that maintain the energy balance between resonator oscillation energy and total energy on capacitance and inductance:
 * $$\hbar \omega_{QR} = W_{QL}(t) + W_{QC}(t). \ $$

Since the capacitance oscillations are phase shifted ($$\psi = \pi /2 \ $$) with respect to inductance oscillations, we get:
 * $$W_{QL} = \begin{cases} 0, & \mbox{at }t=0; \psi=0\mbox{ and} \quad t=\frac{T_{QR}}{2};\psi=\pi \\ W_{QL}, & \mbox{at }t=\frac{T_{QR}}{4};\psi=\frac{\pi}{4} \mbox{ and}\quad t=\frac{3T_{QR}}{4};\psi=\frac{3\pi}{4} \end{cases} \ $$

 * $$W_{QC} = \begin{cases} W_{QC}, & \mbox{at }t=0; \psi=0\mbox{ and}\quad t=\frac{T_{QR}}{2};\psi=\pi \\ 0, & \mbox{at }t=\frac{T_{QR}}{4};\psi=\frac{\pi}{4} \mbox{ and}\quad t=\frac{3T_{QR}}{4};\psi=\frac{3\pi}{4} \end{cases} \ $$

where $$T_{QR} = \frac{2\pi}{\omega_{QR}} \ $$ is the oscillation period.

De Broglie electromagnetic resonator
De Broglie matter wave can be considered for the electric charge and magnetic flux. Actually, using the de Broglie approach in the form of Blohintzev we can derive the following properties of charge quantum resonator (de Broglie charge bubble).

De Broglie electric charge
The electric charge in the de Broglie wave approach can be presented by the following wave function:
 * $$\Psi(q_E,t) = A_E exp [i(\frac{\phi_Mq_E}{\hbar} - \frac{W_Lt}{\hbar})] = A_E exp [i(\gamma_E)] \ $$

with
 * $$W_L = \hbar \omega_L \ $$
 * $$\phi_M = \hbar k_E, \ $$

where $$k_E = \frac{2\pi}{q_{\lambda}}$$ is electric charge "wave vector" and $$q_{\lambda} $$ is electric charge "wave length". The phase function
 * $$\gamma_E = k_Eq_E - \omega_Lt \ $$

has differential
 * $$d\gamma_E = k_Edq_E - \omega_Ldt = 0. \ $$

So, the phase current is:
 * $$i_{ph} = \frac{dq_E}{dt} = \frac{\omega_L}{k_E} \ $$

Considering wave energy in the form
 * $$W_L = \frac{\phi_M^2}{2L} = \frac{\hbar^2k_E^2}{2L} = \hbar \omega_L, \ $$

where $$L \ $$ is de Broglie quantum inductance, and frequency
 * $$\omega_L = \frac{\hbar k_E^2}{2L} \ $$

de Broglie "group current" will be:
 * $$i_g = \frac{d\omega_L}{dk_E} = \frac{\hbar k_E}{L}. \ $$

On the other hand, we have a limit for the group current:
 * $$i_g = \frac{ec}{\lambda}. \ $$

Equating group currents, we may derive the de Broglie quantum inductance:
 * $$L = \frac{h}{eq_{\lambda}}\cdot \frac{\lambda}{c}. \ $$

De Broglie magnetic flux
Magnetic flux in the de Broglie wave approach can be presented by the following wave function:
 * $$\Psi(q_M,t) = A_M exp [i(\frac{\phi_E q_M}{\hbar} - \frac{W_Ct}{\hbar})] = A_M exp [i(\gamma_M)] \ $$

with
 * $$W_C = \hbar \omega_C \ $$
 * $$\phi_E = \hbar k_M, \ $$

where $$ q_M $$ is the space coordinate, $$k_M = \frac{2\pi}{q_{M\lambda}}$$ is magnetic flux "wave vector" and $$q_{M\lambda} $$ is magnetic flux "wave length". The phase function
 * $$\gamma_M = k_Mq_M - \omega_Ct \ $$

has differential
 * $$d\gamma_M = k_Mdq_M - \omega_Cdt = 0. \ $$

The phase voltage will be:
 * $$v_{ph} = \frac{dq_M}{dt} = \frac{\omega_C}{k_M} \ $$

Considering the wave energy in the form
 * $$W_C = \frac{\phi_E^2}{2C} = \frac{\hbar^2k_M^2}{2C} = \hbar \omega_C, \ $$

where $$C \ $$ is de Broglie quantum cdpacitance, and frequency
 * $$\omega_C = \frac{\hbar k_M^2}{2C} \ $$

the de Broglie "group voltage" will be:
 * $$v_g = \frac{d\omega_C}{dk_M} = \frac{\hbar k_M}{C}. \ $$

On the other hand we have the following limit for the group voltage:
 * $$v_g = \frac{hc}{e}\cdot \frac{1}{\lambda}. \ $$

Equating group voltages, we may derive de Broglie capacitance:
 * $$C = \frac{e}{q_{M\lambda}}\cdot \frac{\lambda}{c}. \ $$

De Broglie LC circuit
Using separated electric and magnetic de Broglie charge approaches, we should consider the pilot wave approach to maintain electric and magnetic charges in a stable form. However, when we consider the composite de Broglie wave function in the following form:
 * $$\Psi(q_M,t) = A_MA_E exp [i(\frac{\phi_Mq_E}{\hbar} - \frac{W_Lt}{\hbar})] exp [i(\frac{\phi_Eq_M}{\hbar} - \frac{W_Ct}{\hbar})] \ $$

with
 * $$W_C = \hbar \omega_C = W_L = \hbar \omega_L \ $$
 * $$\phi_E = \hbar k_M, \ $$
 * $$\phi_M = \hbar k_E, \ $$

then separated electric charge and magnetic flux produce a composite quantum resonator.

Considering that the de Broglie particle should be consistent with the quantum resonator, we determine the electric charge and magnetic flux induced in the LC circuit:
 * $$q_{\lambda} = \gamma_q e \ $$
 * $$q_{M\lambda} = \gamma_{\phi} \frac{h}{e}. \ $$

The reactive parameters in that case will be:
 * $$L(\lambda) = \frac{R_K}{\gamma_q}\cdot \frac{\lambda}{c} \ $$
 * $$C(\lambda) = \frac{1}{\gamma_{\phi}R_K}\cdot \frac{\lambda}{c}. \ $$

Characteristic impedance for the de Broglie charged particle resonator:
 * $$ Z_{\lambda} = \sqrt{\frac{L(\lambda)}{C(\lambda)}} = R_K\sqrt{\frac{\gamma_{\phi}}{\gamma_q}} = 2\alpha R_K. \ $$

Resonance frequency for the de Broglie charged particle resonator:
 * $$\omega(\lambda) = \frac{1}{\sqrt{L(\lambda)C(\lambda)}} = \frac{c}{\lambda}\sqrt{\gamma_{\phi}\gamma_q} = \frac{2\pi c}{\lambda}. \ $$

Thus, we have the following algebraic system for unknown parameters:
 * $$\sqrt{\gamma_{\phi}\gamma_q} = 2\pi \ $$
 * $$\sqrt{\frac{\gamma_{\phi}}{\gamma_q}} = 2\alpha. \ $$

The solution of this system is as follows:
 * $$\gamma_q = \frac{\pi}{\alpha} \quad \quad \gamma_{\phi} = 4\pi \alpha \ $$

Then, induced electric charge and magnetic flux may be rewritten as:
 * $$q_{\lambda} = \gamma_q e \ = \frac{\pi}{\alpha}\cdot e \ $$
 * $$q_{M\lambda} = \gamma_{\phi} \frac{h}{e} = 4\pi \alpha \cdot \frac{h}{e}, \ $$

and reactive parameters are:
 * $$L(\lambda) = \frac{\alpha R_K}{\pi}\cdot \frac{\lambda}{c} \ $$
 * $$C(\lambda) = \frac{1}{4\pi \alpha R_K}\cdot \frac{\lambda}{c}. \ $$

So, quantum oscillations of electric charge and magnetic flux make the charged de Broglie bubble stable in time.

Bohr atomic resonator
General information on Bohr atom. Bohr radius is:
 * $$a_B = \frac{\lambda_0}{2\pi \alpha}, \ $$

where $$\alpha \ $$ is the electric  fine structure constant, $$\lambda_0 \ $$ is the Compton wavelength of electron.

Bohr surface scaling parameter for electron disc is:
 * $$S_B = 2\pi a_B^2. \ $$

Bohr angular frequency is:
 * $$\omega_B = \frac{\hbar}{ma_B^2}, \ $$

where $$\hbar \ $$ is reduced Plank constant and $$m \ $$ is electron mass.

Bohr density of states is:
 * $$D_B = \frac{1}{\hbar \omega_B S_B} = \frac{m}{2\pi \hbar^2}. \ $$

Standard DOS quantum resonator approach yields the following values for the Bohr atom reactive quantum parameters:
 * $$C_{QRB} = q_{R2}^2D_BS_B = \frac{e^2}{4\pi \alpha} \frac{m}{2\pi \hbar^2} S_B = \frac{\varepsilon_0}{\lambda_0}S_B \ $$

 * $$L_{QRB} = \phi_{R2}^2D_BS_B =  \frac{\phi_0^2}{4\pi \beta} \frac{m}{2\pi \hbar^2} S_B = \frac{\mu_0}{\lambda_0}S_B, \ $$

where $$\phi_0 = h/e \ $$ is the magnetic flux quantum, $$\beta = \frac{\phi_0^2}{2\mu_0hc} = \frac{1}{4\alpha}  \ $$ is the magnetic coupling constant.

Thus, s.c. "Bohr atom" could be considered as discoid quantum resonator which has radius $$a_B \ $$ and thickness $$\lambda_0 \ $$.

Electron resonator
Let suppose that the electron has its mass defined by the quantum oscillations, more precisely by the oscillations of the quantum LC circuit:
 * $$ \hbar \omega_e = m_e c^2. \ $$

The oscillation length will be:
 * $$x_e = \frac{\lambda_0}{2\pi}, \ $$

where $$\lambda_0 = \frac{h}{m_e c} \ $$ is the Compton wavelength of the electron.

Further, let us consider that the mass $$ m_e \ $$ is uniformly distributed on the electron surface:
 * $$S_e = 2\pi x_e^2 = \frac{\lambda_0^2}{2\pi}. \ $$

Electron angular frequency is:
 * $$\omega_e = \frac{m_e c^2}{\hbar} = \frac{2\pi c}{\lambda_0}, \ $$

where $$c \ $$ is the speed of light.

Then we can find out the density of states for this mass:
 * $$D_e = \frac{1}{S_e}\frac{1}{m_e c^2} = \frac{m_e}{2\pi \hbar^2}. \ $$

Standard DOS quantum resonator approach yields the following values for the electron reactive quantum parameters:
 * $$C_{QRe} = q_{R2}^2D_e S_e = \frac{e^2}{4\pi \alpha}\frac{m_e}{2\pi \hbar^2}S_e = \frac{\varepsilon_0}{\lambda_0}S_e = \frac {\varepsilon_0 \lambda_0}{2 \pi}. \ $$
 * $$L_{QRe} = \phi_{R2}^2D_e S_e = \frac{\phi_0^2}{4\pi \beta}\frac{m_e}{2\pi \hbar^2}S_e = \frac{\mu_0}{\lambda_0}S_e =\frac {\mu_0 \lambda_0}{2 \pi}. \ $$

Thus, s.c. "free electron" could be considered as discoid quantum resonator which has radius $$x_e \ $$.

Photon resonator
As it is known, the photon momentum is defined as:
 * $$p_p = \frac{ W_p }{c}, \ $$

where $$c \ $$ is the speed of light, $$ W_p= \hbar \omega_p \ $$ is the energy of a photon.

So, the "effective (energy) photon mass" could be defined as:
 * $$m_p = \frac{\hbar \omega_p}{c^2}. \ $$

Then the length scaling parameter of harmonic oscillator will be:
 * $$x_p = \sqrt{\frac{\hbar}{m_p\omega_p}} = \frac{\lambda_p}{2\pi}, \ $$

where $$\lambda_p = \frac{2\pi c}{\omega_p} \ $$ is the photon wavelength.

Photon surface scaling parameter is:
 * $$S_p = 4\pi x_p^2 = \frac{\lambda_p^2}{\pi}. \ $$

Photon angular frequency is:
 * $$\omega_p = \frac{2\pi c}{\lambda_p}. \ $$

Photon density of states is:
 * $$D_p = \frac{1}{S_p W_p} = \frac{\pi}{\lambda_p hc}. \ $$

Standard DOS quantum resonator approach yields the following values for the photon reactive quantum parameters:
 * $$C_{QRp} = q_{p}^2 D_p S_p = \frac{e^2}{4\pi \alpha}\frac{\pi}{\lambda_p hc}S_p = \frac{\varepsilon_0}{2\lambda_p}S_p = \frac {\varepsilon_0 \lambda_p}{2 \pi},\ $$
 * $$L_{QRp} = \phi_{p}^2D_pS_p = \frac{\phi_0^2}{4\pi \beta}\frac{\pi}{\lambda_p hc }S_p = \frac{\mu_0}{2\lambda_p}S_p =\frac {\mu_0 \lambda_p}{2 \pi}. \ $$

Thus, photon can be considered as quantum resonator which has radius $$x_p \ $$.

Quantum Hall resonator
General information on the Hall resonator. Cyclotron frequency is:
 * $$\omega_c = \frac{eB}{m}, \ $$

where $$e \ $$ is the elementary charge, $$B \ $$ is the magnetic field induction and $$m \ $$ is the electron effective mass in solid.

Magnetic length is:
 * $$l_B = \sqrt{\frac{\hbar}{e B}}, \ $$

Scaling area parameter is:
 * $$S_H = 2\pi l_B^2 = \frac{h}{eB} = \frac{1}{n_H}, \ $$

where $$n_H \ $$ is the electron surface density.

Density of states is:
 * $$D_H = \frac{n_H}{\hbar \omega_c} = \frac{m}{2\pi \hbar^2}. \ $$

Density of states quantum capacitance is:
 * $$C_H = q_H^2 \cdot D_{H}\cdot S_H = \frac{e^2}{4\pi } \frac{m}{2 \pi \hbar^2} S_H = \alpha \frac{\varepsilon_0}{\lambda_0}S_H \ $$

or in another form:
 * $$C_H = \frac{\varepsilon_0}{d_C}S_H, \ $$

where $$d_C = \frac{\lambda_0}{\alpha} \ $$ is the capacitance thickness.

Density of states quantum inductance is:
 * $$L_H = \phi_H^2 \cdot D_{H}\cdot S_H = \frac{\phi_0^2}{4\pi } \frac{m}{2 \pi \hbar^2} S_H= \beta \frac{\mu_0}{\lambda_0}S_H \ $$

or in another form:
 * $$L_H = \frac{\mu_0}{d_L}S_H, \ $$

where $$d_L = \frac{\lambda_0}{\beta} = 4 \alpha \lambda_0 \ $$ is the inductance thickness.

Thus, quantum Hall resonator has the cylindrical form with radius $$l_B \ $$ and two different thickness: $$d_C \ $$ for capacitance and $$d_L \ $$ for inductance.

Flat atom resonator
A very low frequency resonance was discovered by Yakymakha (1994) with mesoscopic parameters:

the angular frequency: $$\omega_0 = \frac{1}{\sqrt{L_0C_0} } = 5088 \ $$  rad/s,

the surface scaling parameter: $$S_0 = \frac{h}{m\omega_0} = 1.4196\cdot 10^{-7} \ $$ m2 ,

the quantum capacitance: $$C_0 = \frac{1}{\omega_0 Z_0} = 5.1805\cdot 10^{-7} \ $$ F, where $$Z_0 \ $$ is the  impedance of free space,

the quantum inductance: $$L_0 = \frac{Z_0}{\omega_0} = 7.3524 \cdot 10^{-2} \ $$ H.

In the case of quantum resonator DOS will be:
 * $$D_0 = \frac{1}{\hbar \omega_0 S_0} = \frac{m}{2\pi \hbar^2}. \ $$

Quantum resonator capacitance is:
 * $$C_{QR0} = q_{R2}^2D_0S_0 = \frac{\varepsilon_0}{\lambda_0}S_0, \ $$

where $$ \lambda_0 = \frac {h}{m_0 c} \ $$.

Quantum resonator inductance is:
 * $$L_{QR0} = \phi_{R2}^2D_0S_0 = \frac{\mu_0}{\lambda_0}S_0. \ $$

Note that, even in the mesoscopic case we have the reactive parameter thickness about Compton wave length of electron.

Reference books

 * Stratton J.A.(1941). Electromagnetic Theory. New York, London: McGraw-Hill.p.615. djvu
 * Детлаф А.А., Яворский Б.М., Милковская Л.Б.(1977). Курс физики. Том 2. Электричество и магнетизм (4-е издание). М.: Высшая школа, "Reference Book on Electricity"  djvu
 * Гольдштейн Л.Д., Зернов Н.В. (1971). Электромагнитные поля. 2- издание. Москва: Советское Радио. 664с. "Electromagnetic Fields" djvu

Related papers

 * Boris Ya. Zel’dovich. Impedance and parametric excitation of oscillators. UFN, 2008, v. 178, No 5 PDF