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The Sorrows of Young Werther

The Sorrows of Young Werther (Die Leiden des jungen Werthers), or simply Werther, is a 1774 epistolary novel by Johann Wolfgang Goethe, which appeared as a revised edition in 1787. It was one of the main novels in the Sturm und Drang period in German literature, and influenced the later Romantic movement. Goethe, aged 24 at the time, finished Werther in five and a half weeks of intensive writing in January to March 1774. It instantly placed him among the foremost international literary celebrities and was among the best known of his works. The novel is made up of biographical and auto-biographical facts in relation to two triangular relationships and one individual: Goethe, Christian Kestner, and Charlotte Buff (who married Kestner); Goethe, Peter Anton Brentano, Maximiliane von La Roche (who married Brentano), and Karl Wilhelm Jerusalem, who died by suicide on the night of Oct 29 or 30, 1772. He shot himself in the head with a pistol borrowed from Kestner. The novel was adapted as the opera Werther by Jules Massenet in 1892.

Plot summary
Most of The Sorrows of Young Werther, a story about a young man's extreme response to unrequited love, is presented as a collection of letters written by Werther, a young artist of a sensitive and passionate temperament, to his friend Wilhelm. These give an intimate account of his stay in the fictional village of Wahlheim (based on Garbenheim, near Wetzlar), whose peasants have enchanted him with their simple ways. There he meets Charlotte, a beautiful young girl who takes care of her siblings after the death of their mother. Werther falls in love with Charlotte despite knowing beforehand that she is engaged to a man named Albert, eleven years her senior.

Despite the pain it causes him, Werther spends the next few months cultivating a close friendship with them both. His sorrow eventually becomes so unbearable that he is forced to leave Wahlheim for Weimar, where he makes the acquaintance of Fräulein von B. He suffers great embarrassment when he forgetfully visits a friend and unexpectedly has to face there the weekly gathering of the entire aristocratic set. He is not tolerated and asked to leave since he is not a nobleman. He then returns to Wahlheim, where he suffers still more than before, partly because Charlotte and Albert are now married. Every day becomes a torturing reminder that Charlotte will never be able to requite his love. She, out of pity for her friend and respect for her husband, decides that Werther must not visit her so frequently. He visits her one final time, and they are both overcome with emotion after he recites to her a passage of his own translation of Ossian.

Even before that incident, Werther had hinted at the idea that one member of the love triangle – Charlotte, Albert or Werther himself – had to die to resolve the situation. Unable to hurt anyone else or seriously consider murder, Werther sees no other choice but to take his own life. After composing a farewell letter to be found after his death, he writes to Albert asking for his two pistols, on the pretext that he is going "on a journey". Charlotte receives the request with great emotion and sends the pistols. Werther then shoots himself in the head, but does not die until twelve hours later. He is buried between two linden trees that he had mentioned frequently in his letters. The funeral is not attended by any clergy, or by Albert or Charlotte. The book ends with an intimation that Charlotte may die of a broken heart: "I shall say nothing of...Charlotte's grief. ... Charlotte's life was despaired of."

Effect on Goethe
Werther was one of Goethe's few works aligned with the aesthetic, social and philosophical ideals that pervaded the German proto-Romantic movement known as Sturm und Drang, before he and Friedrich von Schiller moved into Weimar Classicism. The novel was published anonymously, and Goethe distanced himself from it in his later years, regretting the fame it had brought him and the consequent attention to his own youthful love of Charlotte Buff, then already engaged to Johann Christian Kestner. Although he wrote Werther at the age of 24, it was all for which some of his visitors in his old age knew him. Goethe had changed his views of literature radically by then, even denouncing the Romantic movement as "everything that is sick."

Goethe described the powerful impact the book had on him, writing that even if Werther had been a brother of his whom he had killed, he could not have been more haunted by his vengeful ghost. Yet, Goethe substantially reworked the book for the 1787 edition and acknowledged the great personal and emotional influence that The Sorrows of Young Werther could exert on forlorn young lovers who discovered it. He later commented to his secretary Johann Peter Eckermann on January 2, 1824 (as it was recorded by Eckermann and published in his book, Gespräche mit Goethe):

"'(Sup.) Fri., Jan. 2. (...) The conversation now turned on “Werther.” “That,” said Goethe, “is a creation which I, like the pelican, fed with the blood of my own heart. It contains so much from the innermost recesses of my breast—so much feeling and thought, that it might easily be spread into a novel of ten such volumes. [...] (...) “On considering more closely the much-talked-of ‘Werther’ period, we discover that it does not belong to the course of universal culture, but to the career of life in every individual, who, with an innate free natural instinct, must accommodate himself to the narrow limits of an antiquated world. Obstructed fortune, restrained activity, unfulfilled wishes, are not the calamities of any particular time, but those of every individual man; and it would be bad, indeed, if every one had not, once in his life, known a time when ‘Werther’ seemed as if it had been written for him alone.”'"

Cultural impact
The Sorrows of Young Werther turned Goethe, previously an unknown author, into a literary celebrity almost overnight. Napoleon Bonaparte considered it one of the great works of European literature, having written a Goethe-inspired soliloquy in his youth and carried Werther with him on his campaigning to Egypt. It also started the phenomenon known as "Werther Fever," which caused young men throughout Europe to dress in the clothing style described for Werther in the novel. Items of merchandising such as prints, decorated Meissen porcelain and even a perfume were produced. Thomas Carlyle coined an epithet, "Wertherism", to describe the self-indulgency of the age that the phenomenon represented. When Goethe completed Werther, he likened his mood to one experienced “after a general confession, joyous and free and entitled to a new life”. For Goethe the Werther effect was a cathartic one, freeing himself from the despair in his life.

The book reputedly also led to some of the first known examples of copycat suicide. The men were often dressed in the same clothing "as Goethe's description of Werther and using similar pistols." Often the book was found at the scene of the suicide. Rüdiger Safranski, a modern biographer of Goethe, dismisses the Werther Effect "as only a persistent rumor." Nonetheless, this aspect of "Werther Fever" was watched with concern by the authorities – both the novel and the Werther clothing style were banned in Leipzig in 1775; the novel was also banned in Denmark and Italy. It was also watched with fascination by fellow authors. One of these, Friedrich Nicolai, decided to create a satirical piece with a happy ending, entitled Die Freuden des jungen Werthers ("The Joys of Young Werther"), in which Albert, having realized what Werther is up to, loaded chicken's blood into the pistol, thereby foiling Werther's suicide, and happily concedes Charlotte to him. After some initial difficulties, Werther sheds his passionate youthful side and reintegrates himself into society as a respectable citizen.

The Hebrew translation was popular among youths in the Zionist communities in British Mandate of Palestine in the 1930s and 1940s and was blamed for the suicide of several young men considered to have emulated Werther.

Goethe, however, was not pleased with the "Freuden" and started a literary war with Nicolai that lasted all his life, writing a poem titled "Nicolai auf Werthers Grabe" ("Nicolai on Werther's grave"), in which Nicolai (here a passing nameless pedestrian) defecates on Werther's grave, so desecrating the memory of a Werther from which Goethe had distanced himself in the meantime, as he had from the Sturm und Drang. This argument was continued in his collection of short and critical poems the Xenien  and his play Faust.

Alternative versions and appearances

 * Goethe's work was the basis for the 1892 opera Werther by Jules Massenet.
 * In Mary Shelley's Frankenstein, Frankenstein's monster finds the book in a leather portmanteau, along with two others – Plutarch's Lives of the Noble Greeks and Romans, and Milton's Paradise Lost. He sees Werther's case as similar to his own, of one rejected by those he loved.
 * The book influenced Ugo Foscolo's The Last Letters of Jacopo Ortis, which tells of a young man who commits suicide, out of desperation caused not only by love, but by the political situation of Italy before Italian unification. This is taken to be the first Italian epistolary novel.
 * Thomas Carlyle, who incidentally translated Goethe's novel Wilhelm Meister into English, frequently refers to and parodies Werther's relationship in his 1836 novel Sartor Resartus.
 * The statistician Karl Pearson's first book was The New Werther.
 * William Makepeace Thackeray wrote a poem satirizing Goethe's story entitled "Sorrows of Werther".
 * Henri Pouctal made a film adaptation in 1910, cosidered to be lost.

 * Thomas Mann's 1939 novel Lotte in Weimar recounts a fictional reunion between Goethe and his youthful passion, Charlotte Buff, as elderlies.
 * In 1968 Jean-Pierre Lajournade made a metacinematic critical reading of the novel in Werther (aka Les Souffrances du jeune Werther), a film made for TV infused with the aesthetics and the politics of the time.
 * Spanish filmmaker Pilar Miró adapted the novel in 1986 in Werther.
 * A 2002 episode of the Canadian television series History Bites titled "Love & Death" is about the cultural impact of Werther, with Bob Bainborough satirically portraying Goethe in 1780 as a guest on a talk show spoofing The Rosie O'Donnell Show. Goethe wants to discuss his newest work, an adaptation of Iphigenia in Tauris, but is annoyed by having to deal with obsessive fans of Werther.
 * Ulrich Plenzdorf, a GDR poet, wrote a satirical novel (and play) called Die neuen Leiden des jungen W. ("The New Sorrows of Young W."), transposing the events into an East German setting, with the protagonist as an ineffectual teenager rebelling against the system.
 * In William Hill Brown's The Power of Sympathy, the novel appears next to Harrington's unsealed suicide note.
 * The 2010 German film Goethe! is a fictional account of the relations between the young Goethe, Charlotte Buff and her fiancé Kestner, which at times draws on that of Werther, Charlotte and Albert.
 * The 2014 novel The Sorrows of Young Mike by John Zelazny is a loosely autobiographical parody of Goethe's novel.
 * In the 2015 game, The Witcher 3: Wild Hunt's Blood and Wine expansion pack, there is a treasure hunt called "The Suffering of Young Francois", where a man named François seeks help from a witch to make a woman named Charlotte, who is engaged with Albert, fall in love with him. The witch tricked François, making a Spriggan appear in the state and murder everyone. When François learns of this, he hangs himself.
 * The story is read in the first episode of the 2019 series Rookie Historian Goo Hae-ryung.
 * The story is read to the dragon Temeraire by Captain William Laurence in Naomi Novik’s novel Black Powder War, the third book in the Temeraire series.

English translations

 * The Sorrows of Werter, trans. Daniel Malthus (1779)
 * Werter and Charlotte, trans. unknown (1786)
 * The Sorrows of Werter, trans. M. Aubry (1789)
 * The Letters of Werter, trans. unknown (1799)
 * The Sorrows of Werter, trans. William Render (1801)
 * The Sorrows of Werter, trans. Frederick Gotzberg (1802)
 * The Sorrows of Werter, trans. Dr. Pratt (1809)
 * The Sorrows of Werter, trans. R. Dillon Boylan (1854)

 * originally publ. by Random House.
 * The Sufferings of Young Werther, trans. Stanley Corngold (2011)
 * The Sufferings of Young Werther, trans. Stanley Corngold (2011)
 * The Sufferings of Young Werther, trans. Stanley Corngold (2011)
 * The Sufferings of Young Werther, trans. Stanley Corngold (2011)

title: The Sorrows of Young Werther


text:
Talk:I

Should there be a note on uppercase vs lowercase of the pronoun?

 * Russian Я (ya) shouldn't be linked to this page. This was a mix-up from the fact that Я translates to the pronoun "I"

In computing
I think that "i" stands for (current) iteration, which is why "i" is used as a loop counter. --Joshtek 15:54, 19 August 2005 (UTC)
 * I think thats part of the reason. also i believe that "i" was the first integer variable in fortran. Plugwash
 * You seem to be correct: http://www.kirupa.com/forum/showthread.php?t=58108&page=1&pp=15 Joshtek

A Question
Why on earth does the "i" has a dot? I can understand the dot in Turkish, for distinguishing reasons, but how come there is the period on top also in latin and other languages, and for so many centuries? Ciacchi 16:48, 17 March 2006 (UTC)
 * Oh, nevermind, I found it out. Ciacchi 16:06, 13 April 2006 (UTC)

Phil and Jim?
Um, are Phil and Jim people we should know? If so, please expand their significance in relation to using the word "I". Otherwise, please remove this bullet point from the Meanings of I section. Thanks Apatterno 05:00, 28 March 2006 (UTC)

I don't get it either.......... -) Dance3600#*-* (talk) 08:34, 21 April 2014 (UTC)

Apple's use of the 'i'
Should there be a reference to the use of i in the market world? Such as iPod, iPhone, iRiver, iLife, iTunes etc. --Reverieuk 01:02, 13 January 2007 (UTC) icandy-pram, i agree there should be something along the lines of "....one of the few single letters which is copyrighted/trademarked...." or something, cos there seems to be a lot of "i" products around - not just apple.

When did the short i as in "bill" evolve?
When did the short i as in "bill" evolve? Does anybody know. Imhotep is the earliest name I know of with the short I sound, so maybe it is as old as written language? —The preceding unsigned comment was added by TedHuntington (talk • contribs) 22:23, 29 March 2007 (UTC).
 * The premise of this question seems to be wrong. We can speak about how the English came to have a short close front unrounded vowel or rather a near-close near-front unrounded vowel. I think this happened as Proto-Germanic evolved. The question, when did this sound appear in any language is un-answerable, but probably humans have been making this sound and using it in their language for as long as the language has existed and the shape of the human mouth has been in the form it is. It would thus be a lot older than written language. Stefán 23:52, 29 March 2007 (UTC)

Yes, I guess that is my question: when did the short-i as in "dill" evolve in human language. We can certainly trace the short-a as in "cat", and the a in "ape" to English since no other language has them. So you are basically stating that, to your knowledge, native people still speak the short-i in "dill" when saying the most ancient words of written history, such as the name "Imhotep"? Perhaps that is about the best any human can do today for the origins of the short-i in "dill". We can say too that the letters that sound the same probably evolved together. For example, b,d,g,k,p,t are all related and l,m,n,r are all related and may have been created around the same time during evolution. TedHuntington 00:01, 6 April 2007 (UTC)
 * This question still seems bizarre. Firstly, I can't make out if you're asking about eg. the existence of specifically in "dill" etc, or in any word in any language whatsoever. Also, we cannot tell what the first spoken languages sounded like, or when did some specific speech sound first appear, but there is no reason to suspect any to be particularly new. The physical features required for speech have existed for tens of thousands of years, but historical linguistics is able to probe the past only to the time-depth of about ten thousand years with any reliability, and even that ten thousand years is a lot; it is sufficient for virtually any sound to change into any other sound.
 * As an aside, it doesn't hold that "no other language has the a sounds of 'ape' or 'cat'". Finnish, for instance, has the latter and it seems that it has preserved it all the way from Proto-Uralic, some 6,000 years ago at least (while its latest English incarnation is only a few centuries old). Similarly, all the vowels in "Imhotep" are English adaptations and do not reflect actual Egyptian pronunciation (which is largely unclear, but do see Egyptian language). -- Trɔpʏliʊm • blah 22:03, 18 August 2008 (UTC)

 * We also don't know exactly how any given word was pronounced thousands of years ago (though for some features there is useful indirect evidence). English-speakers pronounce Imhotep with  because it comes naturally to us, not because he did. —Tamfang (talk) 06:41, 11 July 2011 (UTC)

Its use in Mathematics
Why is there no mention of the letter "i" as a mathematical symbol for the square root of minus 1? I put this fact on the page entitled "I" but someone has deleted it. Would the person who did that please explain why on this discussion page A.S.A.P. Paggle

 * Each letter has numerous uses as a symbol; they are listed at I (disambiguation) because it would be clutter to list them all in I. —Tamfang (talk) 06:44, 11 July 2011 (UTC)

Hieroglyphs
Why the hell do people always put up bs "related" hierogrlyphs. The arm hieroglyph is an ayin - not related to the yodh. In fact, hieroglyphs has a related glyph - the yodh! Why isn't that there?211.30.58.23 (talk) 00:06, 10 April 2008 (UTC)

Capitalisation of the letter I in the English language
Should there not be a part emphasising the strange way we capitalise the letter I in the English language when referring to ourselves? The article mentions the Turkish capital and dotted i versions, so i don't see why there isn't a section about the English differences. There's also a lot of reasoning behind it (which i still don't understand) which could be interesting to add to the article. For example, some sources ( http://www.alt-usage-english.org/excerpts/fxwhyisi.html ) say it's used to avoid misreading, which seems strange given the fact that a capital I commonly looks like an uncapped L(i = I - l = L). Other sources say it's akin to capitalising the first letter of a name when referring to another, such as "Tom", though this doesn't make much sense since we don't capitalise the M in me, nor the W in we. -Evaristé93 (talk) 04:28, 23 July 2011 (UTC)

 * This article is about the letter, not the word. Take it to I (pronoun). —Tamfang (talk) 19:25, 1 August 2011 (UTC)

 * What is I 78.146.102.144 (talk) 02:04, 22 December 2011 (UTC)
 * I 84.13.46.167 (talk) 10:59, 23 December 2011 (UTC)

wiki is a lie — Preceding unsigned comment added by 199.43.174.141 (talk) 15:08, 30 October 2015 (UTC)

Inverted question and exclamation marks
The symbol is not like either of these. I suggest remove this statement. P0mbal (talk) 23:06, 16 November 2016 (UTC)

 * A lowercase letter "i" looks pretty similar to an inverted exclamation point "¡", especially in some fonts. —Granger (talk · contribs) 23:09, 16 November 2016 (UTC)

 * Agree, but the inverted question mark as used in Spanish? Don't think so.P0mbal (talk) 23:18, 16 November 2016 (UTC)

Palochka
"For the similar letter in the Cyrillic alphabet, see palochka." So a link to Palochka should be in the Disambiguation section, surely? P0mbal (talk) 23:16, 16 November 2016 (UTC)

Name of the letter
Wouldn't the name of this letter be written out as "eye", rather than just itself? --NovaBrunswick 11:09, 16 April 2017 (UTC)

Semi-protected edit request on 5 May 2018
Add right after  209.52.88.173 (talk) 18:47, 5 May 2018 (UTC)
 * Yes check.svg Done L293D (☎ • ✎) 19:01, 5 May 2018 (UTC)

Semi-protected edit request on 10 February 2019
130.193.233.251 (talk) 11:13, 10 February 2019 (UTC)
 * Red question icon with gradient background.svg Not done: it's not clear what changes you want to be made. Please mention the specific changes in a "change X to Y" format and provide a reliable source if appropriate. DannyS712 (talk) 18:28, 10 February 2019 (UTC)

Semi-protected edit request on 8 November 2019
I need help on wiki?Christian boyz (talk) 20:17, 8 November 2019 (UTC) Christian boyz (talk) 20:17, 8 November 2019 (UTC)
 * Red question icon with gradient background.svg Not done: it's not clear what changes you want to be made. Please mention the specific changes in a "change X to Y" format and provide a reliable source if appropriate. Dorsetonian (talk) 20:27, 8 November 2019 (UTC)

Dotted capital I
Neither this article nor the one about Dotted I (as found in Turkish as a distinct letter) mentions the fact that capital I in France is sometimes dotted. Nor does the article on tittle mention this. Does anyone know anything about this (is it only in France where capital I is optionally dotted?)? shall I add the info? For example, see https://asterix.com/en/the-collection/albums/asterix-and-the-chieftains-daughter/ or https://asterix.com/le-nouvel-album-dasterix-sortira-le-21-octobre-2021/ or https://www.ac-rennes.fr/ouverture-d-un-enseignement-a-distance-du-breton-des-la-rentree-2021-122312 (the last being a Breton example, see pictures of the classroom walls - the other two examples being in French). — Preceding unsigned comment added by 31.208.199.100 (talk) 11:11, 10 February 2023 (UTC)
 * Do you have any examples that are not comics? Perhaps Hergé was doing this for effect, to indicate that the heroes are speaking a Breton dialect of Gaulish? Either way, we would need a reliable source (like Grand Larousse encyclopédique or the Academie Française) that recognises it formally before we could use it (examples of use are not valid citations). The fact that it is not mentioned at French orthography makes me doubt strongly that it is standard French. --𝕁𝕄𝔽 (talk) 19:43, 10 February 2023 (UTC)

Remove /ai̯/ from Portuguese
I gives /ai̯/ as a possible value for "i" in Portuguese just because of English words like "site", which isn't seen as a native word but as an English word. Following this logic, you'd have say that /ej/ is also a value for "a" in Portuguese and all other languages that use the Latin alphabet just because of "Facebook", /h~x/ is a possible value for J just because of the Spanish name Juan, and so on and on. Please remove this from the article and leave out only the actual single native pronunciation for the letter i: /i/. -- 𠔻 (talk) 23:00, 28 April 2023 (UTC)

But are they serifs?
Back when I was in school I always saw the uppercase letter "I" drawn with full length bars, of the same length as the bar at the top of the letter "T". And we were taught it that way despite not learning to draw anything with serifs. The idea that the bars on the letter are actually serifs seems to me to be arguable. 2600:1002:B035:C350:7B18:A5E9:C097:3B56 (talk) 21:19, 9 October 2023 (UTC)
 * (For convenience, I have attached an explanatory graphic copied from typeface anatomy.)
 * I was taught the same, because at school it is important to be able to distinguish between the handwritten forms of capital $⟨i⟩$, little $⟨L⟩$ figure $⟨1⟩$ and figure $⟨7⟩$. In print, it is usually obvious from context so is less critical for adult readers.
 * But to come back to your original question, I'm afraid we are left with "if it walks like a duck and quacks like a duck, then it's a duck". The bars top and bottom of the $⟨I⟩$ have the same characteristics as serifs on other letters, so they are serifs (even if the typeface is classified as sans-serif). Or at least they are until to you can find a reliable source that says otherwise. --𝕁𝕄𝔽 (talk) 23:20, 9 October 2023 (UTC)
 * Have either of you ever seen a typeface with serifs on those crossbars, as on T? —Tamfang (talk) 04:32, 23 October 2023 (UTC)
 * Yes, good point that I think really answers the question. The little bars at the top and bottom of capital I are not serifs, they are part of the letter. If it had serifs, they would look like the one on the end of the "arm" of the E in the image above (pointer 17). (I don't recall ever noticing a 'normal' (not 'display') typeface with serifs on capital I. And many sans typefaces have bars top and bottom. See for example Times New Roman: I, Courier New: I , Verdana: I .) Consequently, I have stricken my earlier reply. . --𝕁𝕄𝔽 (talk) 10:30, 23 October 2023 (UTC)
 * Weird, I thought my point argued the other way. —Tamfang (talk) 00:24, 30 October 2023 (UTC)

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

title: Talk:I


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Mount Monadnock

Mount Monadnock, or Grand Monadnock, is a mountain in the towns of Jaffrey and Dublin, New Hampshire. It is the most prominent mountain peak in southern New Hampshire and is the highest point in Cheshire County. It lies 38 mi southwest of Concord and 62 mi northwest of Boston. At 3165 ft, Mount Monadnock is nearly 1000 ft higher than any other mountain peak within 30 mi and rises 2000 ft above the surrounding landscape. Monadnock's bare, isolated, and rocky summit provides expansive views. It is known for being featured in the writings of Ralph Waldo Emerson and Henry David Thoreau.

Mt. Monadnock has long been cited as one of the most frequently climbed mountains in the world. It bears a number of hiking trails, including the 110 mi Metacomet-Monadnock Trail and the 50 mi Monadnock-Sunapee Greenway. The summit is barren largely because of fires set by early settlers. The first major fire, set in 1800 to clear the lower slopes for pasture, swept through the stands of virgin red spruce on the summit and flanks of the mountain. Between 1810 and 1820, local farmers, who believed that wolves were denning in the blowdowns, set fire to the mountain again. The conflagration raged for weeks, destroying the topsoil and denuding the mountain above 2000 ft.

The term "monadnock" is used by American geologists to describe any isolated mountain formed from the exposure of a harder rock as a result of the erosion of a softer one once surrounding it (a landform termed "inselberg ("island-peak") elsewhere in the world).

Name
The word "monadnock" is derived from an Abenaki word used to describe a mountain. Loosely translated, it means "mountain that stands alone", although the exact meaning of the word is uncertain. The term was adopted by early settlers of southern New Hampshire and later by American geologists as an alternative term for an inselberg or isolated mountain. Mount Monadnock is often called Grand Monadnock, to differentiate it from other Vermont and New Hampshire peaks with "Monadnock" in their names.

Its official name on federal maps is "Monadnock Mountain".

The transcendentalists
Ralph Waldo Emerson, Henry David Thoreau, and Margaret Fuller visited the mountain and wrote fondly of it. Emerson was a frequent visitor, and made the mountain the subject of "Monadnoc", one of his most famous poems. Thoreau visited the mountain four times between 1844 and 1860, and spent a great deal of time observing and cataloging natural phenomena. He is regarded as having written one of the first serious naturalist inventories of the mountain. A bog near the summit of Mount Monadnock and a rocky lookout off the Cliff Walk trail are named after him; another lookout is named after Emerson.

The Halfway House and other structures
In 1858, Moses Cudworth of Rindge opened the "Halfway House" hotel on the south side of the mountain, roughly halfway from the base to the summit. The "Toll Road" was built to service it. By that time the popularity of the mountain was booming, and it was not long before Cudworth enlarged the hotel to accommodate 100 guests. On busy summer days, the stables at the Halfway House held as many as 75 horses. The Halfway House became public property when hundreds of residents of the nearby towns formed a coalition to buy the Toll Road and hotel, and worked to prevent a radio tower from being constructed on the summit. After the hotel burned down in 1954, a concession stand operated at the site until 1969. It and the toll road were both closed to public vehicles. Moses Spring, with its source in a hole drilled through a rock behind the site, is one of the few remaining artifacts of the hotel years. The foundations of two water tanks, and the nearby reservoir that fed them, are extant on the hillside above the Old Halfway House.

A small firewarden's hut was located on the summit of Mount Monadnock and operated from 1911 to 1948, when it was decommissioned with the advent of modern means of detecting forest fires. The hut was used as a snack bar concession and hikers' shelter until 1969; it was removed in 1972. A small cabin, located farther down the mountain, served as the fire lookout's residence. It, too, has been removed.

A private dwelling, 400 ft south of the site of the former Halfway House, is the last remaining inholding on the mountain above 1000 ft.

Hiking history
The earliest recorded ascent of Mount Monadnock took place in 1725 by Captain Samuel Willard and fourteen rangers under his command who camped at the top and used the summit as a lookout while patrolling for Native Americans. Before the practice came to be frowned upon, many early hikers carved their names in the summit; the earliest such engraving reads "S. Dakin, 1801" and is attributed to a local town clerk.

Notable "power hiking" records associated with the mountain include that of Garry Harrington, who hiked to the summit 16 times in a 24-hour period, and Larry Davis, who claimed to have hiked to the summit daily for 2,850 consecutive days (7.8 years). Davis has hiked the mountain over 7,250 times in the past 35 years.

Monadnock is often claimed to be the second-most frequently climbed mountain in the world, after Mount Fuji in Japan. Monadnock is climbed by 125,000 hikers yearly, while Mount Fuji sees 200,000-300,000 hikers yearly. However, according to UNESCO, Tai Shan in China receives more than 2 million visitors a year, far surpassing the other two peaks in popularity.

Bio-geography, ecology, and geology

Due to fires in the early 19th century which burned a majority of the top 300 ft around the summit, there is little soil cover on the peak and surrounding uplands. Soils are mapped as Rock Outcrop over a large area around the summit, while low-lying areas have classic podzol stony fine sandy loam profiles usually mapped as Lyman or Tunbridge series. Although the mountain has recovered to the degree that its landscape appears natural, Mount Monadnock is an example of a mountain which has an artificial tree line, occurring well below the climatic tree line of similar mountains in the region.

From the base to the summit, floral species diminish with increasingly shallow soil levels. The summit is home to many different specialized sub-alpine species which can retain moisture for long periods of time. Alpine and sub-alpine species include mountain ash, cotton grass, sheep laurel, mountain sandwort, and the mountain cranberry. Krummholz, trees stunted by harsh weather, are found on Mount Monadnock, as are several alpine bogs. Lower elevations on Mount Monadnock are clad in northern hardwood forest species; middle elevations support stands of red spruce. Before the fires, Mount Monadnock was totally covered in a red spruce forest. Since the summit has been barren of soil cover, red spruce have been slowly ascending back towards the top in a process known as ecological succession.

The mountain is largely composed of 400-million-year-old schist and quartzite rock primarily associated with the Devonian Littleton Formation, which extends south into Massachusetts and north into the White Mountains. At lower elevations on the mountain, and stratigraphically below the Littleton Formation, the bedrock is from the Silurian period, consisting of Rangeley Formation schists, Perry Mountain Formation quartzites, the Francestown Formation granulite, and the Warner Formation granulite. Structurally, the mountain is part of an overturned syncline – called a fold nappe – that was caused by the compressional forces of the Acadian orogeny. Dramatic small- to medium-scale metamorphic folds are visible on many of the rock faces of the mountain, including the famous Billings Fold (a recumbent syncline found about 450 ft west of the summit), shown in the 1942 edition of Marland P. Billings' Structural Geology. In addition to impressive folds, the Devonian Littleton Formation shows large pseudomorphs of sillimanite after andalusite that occur as 4 in "turkey tracks".

Climate
Mount Monadnock has a humid continental climate (Köppen Dfb) with subarctic climate (Köppen Dfc) characteristics.

Hydrology
Mount Monadnock lies on the divide between the Connecticut River and Merrimack River watersheds. The northern slopes of the mountain drain to Howe Reservoir, then via Minnewawa Brook to the Ashuelot River, the Connecticut River, and ultimately Long Island Sound. Gleason Brook and Mountain Brook flow off the western slopes, then via Shaker Brook to the South Branch of the Ashuelot River. The southwestern slopes of the mountain drain to Fassett Brook, then Quarry Brook, before reaching the South Branch of the Ashuelot.

To the southeast of the mountain, Mead Brook and Stony Brook flow to Mountain Brook, a tributary of the Contoocook River, which flows north to the Merrimack River and ultimately the Gulf of Maine. The eastern slopes of the mountain drain to Thorndike Pond, the outlet of which (Stanley Brook) flows northeast to Nubanusit Brook, then into the Contoocook and the Merrimack.

Recreation

Mount Monadnock is open to hiking, backpacking, picnicking, and snowshoeing. Backcountry skiing is possible on some of the lower trails. A seasonal campground east of the mountain is maintained by the state of New Hampshire, but camping is not allowed anywhere else on the mountain. A per-vehicle fee is charged to park at the Old Toll Road and State Park Headquarters trailheads. There are no roads to the summit, and the Old Toll Road, which leads to the Halfway House site, is closed to vehicles. ATVs are not allowed on the mountain. Good views of the mountain can be had from a number of regional roads and highways, especially from New Hampshire Route 124.

Trails

Mount Monadnock is criss-crossed by many miles of well-maintained hiking trails. The most popular trails are the White Dot Trail, 2.2 mi, the shortest but steepest ascent to summit, accessible from the Monadnock State Park headquarters and campground on the southeast side of the mountain; and the White Arrow Trail, 2.3 mi to summit, considered the easiest ascent, via the Old Toll Road (Halfway House) trailhead on the south side of the mountain. Other major trails include:

 * The Marlboro Trail, on the west side of the mountain, 2.2 mi to summit
 * The Dublin Path, also part of the Monadnock-Sunapee Greenway, on the north side, 2.4 mi to summit
 * The Pumpelly Trail, on the northeast side, the longest ascent, 4.4 mi to summit. This trail boasts panoramic views for longer than the other trails, as it hugs the ridge.
 * The Birchtoft Trail (Gilson Pond trailhead) and several connecting trails, providing access to the summit from the east side of the mountain

Other important trails include the White Cross Trail, an alternative parallel to the White Dot Trail, and the Cliff Walk, a scenic detour that shadows a ledge to the east of the White Arrow Trail.

Mount Monadnock is the northern terminus of the Metacomet-Monadnock Trail (also known as the "M&M Trail"), a 110 mi hiking trail stretching south through Massachusetts to the Connecticut border. The trail ascends the mountain from the southwest; however, there is no trailhead where that trail crosses Route 124 at the foot of the mountain, and the highway there is marked with "no parking" signs. The closest trailhead southbound on the M&M Trail is on Bullard Road, which leads south from Route 124 half a mile west of the M&M Trail's crossing of the state highway.

The mountain is also the southern terminus of the Monadnock-Sunapee Greenway, a 50 mi highland trail connecting Mount Monadnock and Mount Sunapee. A third long-distance trail project, the Wantastiquet-Monadnock Greenway, aims at linking Mount Monadnock with Pisgah State Park and Wantastiquet Mountain, a prominent peak in Hinsdale, New Hampshire, overlooking the town of Brattleboro, Vermont, and the Connecticut River.

Hiking the mountain

The trip to the summit and back via one of the shorter trails, which usually takes less than six hours, is considered moderately challenging for the average hiker but requires no technical skills during the late spring to early fall season. Throughout the year wind and weather on the exposed, rocky summit cause harsher conditions than below the tree line. Winter storms and summer thunderstorms can be life-threatening.

Winter snow and ice can remain on the trails well into May, making hiking hazardous or impossible without special equipment. The compacting of snow due to winter recreation on the more frequently used trails tends to lengthen the time it takes for the snow and ice to melt off. Stabilizers, crampons, ice axes, ski poles, and snowshoes may be helpful or necessary for winter and early spring ascents, depending on ice and snow cover.

Fall hiking generally requires warmer clothing, an awareness of weather conditions, and earlier ascent times. As there are no reliable water sources on the mountain and little shelter from the sun above 2000 ft, dehydration and heat stroke are potential hazards.

The mountain and surrounding area are black bear habitat, although problem encounters with bears are rare.

Advice on hiking can be solicited from Memorial Day through Columbus Day and select off-season weekends at the park headquarters on the southeast side of the mountain. Trail descriptions and maps of Mount Monadnock are published by a variety of sources. The Society for the Protection of New Hampshire Forests prints a guidebook with maps, trail descriptions, history, geology, ecology, and other trivia on the mountain.

Conservation
Most of the mountain has been protected and is not subject to development. The Society for the Protection of New Hampshire Forests is the major landowner, holding over 4000 acre in its Monadnock Reservation. Lands on Monadnock are leased to the state for management purposes. An additional 1000 acre on the mountain are owned directly by the state of New Hampshire as Monadnock State Park. The town of Jaffrey also owns portions of the mountain.

In 2000, the Metacomet-Monadnock Trail was included in a study by the National Park Service for possible inclusion in a new National Scenic Trail, now tentatively called the New England National Scenic Trail, which would also include the Mattabesett and Metacomet trails in Connecticut, giving it some of the status accorded to the Appalachian Trail. However, there seems to be some uncertainty whether or not the New Hampshire section of the Metacomet-Monadnock Trail will be included in the final NST designation.

Tributes
Beside Emerson and Thoreau, other artists and writers have been inspired by Mount Monadnock, including:

 * William Preston Phelps (1848–1917) was known as "the Painter of Mount Monadnock".
 * Many other American artists have painted Mount Monadnock, among them Abbott Handerson Thayer, Rockwell Kent and Richard Whitney.
 * The American composer Alan Hovhaness (1911–2000), who climbed Mount Monadnock several times during his youth, composed the symphonic fantasy entitled Monadnock, Op. 2, No. 1, circa 1935.
 * H. P. Lovecraft, the early 20th century writer of horror, science fiction and fantasy, authored the poem "To Templeton and Mount Monadnock".
 * Galway Kinnell, a Pulitzer Prize–winning poet, wrote the poem "Flower Herding on Mount Monadnock".
 * Poet Edwin Arlington Robinson, who summered in Monadnock's shadow from 1911 to 1934, wrote the poem "Monadnock Through the Trees".
 * Part of the yearly Mount Monadnock Celebration of Dance takes place on the summit of Mount Monadnock at the autumn equinox.

Several United States Navy ships were named USS Monadnock in the nineteenth and twentieth centuries.

In 1961, F. Nelson Blount established the Monadnock, Steamtown & Northern Railroad, a local steam-powered tourist railroad named after the mountain and Steamtown USA, Blount's accompanying steam museum. The MS&N operated in the region until 1967, when it was folded in with the Green Mountain Railroad. An image of the peak made up the railroad's logo.

Other nearby peaks named Monadnock

 * Pack Monadnock and North Pack Monadnock are two peaks in the Wapack Range, 11 mi to the east across the Contoocook River valley from Mount Monadnock, in Greenfield and Peterborough, New Hampshire. Pack is a Native American word for "little".
 * Little Monadnock Mountain is a peak located 7 mi southwest of Mount Monadnock in Troy and Fitzwilliam, New Hampshire. Like Mount Monadnock, it is crossed by the Metacomet-Monadnock Trail.
 * Monadnock Mountain is the name of a mountain in the Northeast Kingdom of Vermont.

Trails

 * Metacomet-Monadnock Trail
 * Monadnock-Sunapee Greenway Trail Club
 * Maps, Trails and More Information

title: Mount Monadnock


text:
Zeeman effect

The Zeeman effect is the effect of splitting of a spectral line into several components in the presence of a static magnetic field. It is named after the Dutch physicist Pieter Zeeman, who discovered it in 1896 and received a Nobel prize for this discovery. It is analogous to the Stark effect, the splitting of a spectral line into several components in the presence of an electric field. Also similar to the Stark effect, transitions between different components have, in general, different intensities, with some being entirely forbidden (in the dipole approximation), as governed by the selection rules.

Since the distance between the Zeeman sub-levels is a function of magnetic field strength, this effect can be used to measure magnetic field strength, e.g. that of the Sun and other stars or in laboratory plasmas.

Discovery
In 1896 Zeeman learned that his laboratory had one of Henry Augustus Rowland's highest resolving Rowland grating, an imaging spectrographic mirror. Zeeman had read James Clerk Maxwell's article in Encyclopædia Britannica describing Michael Faraday's failed attempts to influence light with magnetism. Zeeman wondered if the new spectrographic techniques could succeed where early efforts had not.

When illuminated by a slit shaped source, the grating produces a long array of slit images corresponding to different wavelengths. Zeeman placed a piece of asbestos soaked in salt water into a Bunsen burner flame at the source of the grating: he could easily see two lines for sodium light emission. Energizing a 10 kilogauss magnet around the flame he observed a slight broadening of the sodium images.

When Zeeman switched to cadmium at the source he observed the images split when the magnet was energized. These splitting could be analyzed with Hendrik Lorentz's then new electron theory. In retrospect we now know that the magnetic effects on sodium require quantum mechanical treatment. Zeeman and Lorentz were awarded the 1902 Nobel prize; in his acceptance speech Zeeman explained his apparatus and showed slides of the spectrographic images.

Nomenclature
Historically, one distinguishes between the normal and an anomalous Zeeman effect (discovered by Thomas Preston in Dublin, Ireland ). The anomalous effect appears on transitions where the net spin of the electrons is non-zero. It was called "anomalous" because the electron spin had not yet been discovered, and so there was no good explanation for it at the time that Zeeman observed the effect. Wolfgang Pauli recalled that when asked by a colleague as to why he looked unhappy, he replied, "How can one look happy when he is thinking about the anomalous Zeeman effect?"

At higher magnetic field strength the effect ceases to be linear. At even higher field strengths, comparable to the strength of the atom's internal field, the electron coupling is disturbed and the spectral lines rearrange. This is called the Paschen–Back effect.

In the modern scientific literature, these terms are rarely used, with a tendency to use just the "Zeeman effect". Another rarely used obscure term is inverse Zeeman effect, referring to the Zeeman effect in an absorption spectral line.

A similar effect, splitting of the nuclear energy levels in the presence of a magnetic field, is referred to as the nuclear Zeeman effect.

Theoretical presentation
The total Hamiltonian of an atom in a magnetic field is

 * $$H = H_0 + V_{\rm M},\ $$

where $$H_0$$ is the unperturbed Hamiltonian of the atom, and $$V_{\rm M}$$ is the perturbation due to the magnetic field:

 * $$V_{\rm M} = -\vec{\mu} \cdot \vec{B},$$

where $$\vec{\mu}$$ is the magnetic moment of the atom. The magnetic moment consists of the electronic and nuclear parts; however, the latter is many orders of magnitude smaller and will be neglected here. Therefore,

 * $$\vec{\mu} \approx -\frac{\mu_{\rm B} g \vec{J}}{\hbar},$$

where $$\mu_{\rm B}$$ is the Bohr magneton, $$\vec{J}$$ is the total electronic angular momentum, and $$g$$ is the Landé g-factor. A more accurate approach is to take into account that the operator of the magnetic moment of an electron is a sum of the contributions of the orbital angular momentum $$\vec L$$ and the spin angular momentum $$\vec S$$, with each multiplied by the appropriate gyromagnetic ratio:
 * $$\vec{\mu} = -\frac{\mu_{\rm B} (g_l \vec{L} + g_s \vec{S})}{\hbar},$$

where $$g_l = 1$$ and $$g_s \approx 2.0023193$$ (the latter is called the anomalous gyromagnetic ratio; the deviation of the value from 2 is due to the effects of quantum electrodynamics). In the case of the LS coupling, one can sum over all electrons in the atom:

 * $$g \vec{J} = \left\langle\sum_i (g_l \vec{l_i} + g_s \vec{s_i})\right\rangle = \left\langle (g_l\vec{L} + g_s \vec{S})\right\rangle,$$

where $$\vec{L}$$ and $$\vec{S}$$ are the total spin momentum and spin of the atom, and averaging is done over a state with a given value of the total angular momentum.

If the interaction term $$V_M$$ is small (less than the fine structure), it can be treated as a perturbation; this is the Zeeman effect proper. In the Paschen–Back effect, described below, $$V_M$$ exceeds the LS coupling significantly (but is still small compared to $$H_{0}$$). In ultra-strong magnetic fields, the magnetic-field interaction may exceed $$H_0$$, in which case the atom can no longer exist in its normal meaning, and one talks about Landau levels instead. There are intermediate cases which are more complex than these limit cases.

Weak field (Zeeman effect)
If the spin–orbit interaction dominates over the effect of the external magnetic field, $$ \vec L$$ and $$ \vec S$$ are not separately conserved, only the total angular momentum $$ \vec J = \vec L + \vec S$$ is. The spin and orbital angular momentum vectors can be thought of as precessing about the (fixed) total angular momentum vector $$ \vec J$$. The (time-)"averaged" spin vector is then the projection of the spin onto the direction of $$ \vec J$$:

 * $$\vec S_{\rm avg} = \frac{(\vec S \cdot \vec J)}{J^2} \vec J$$

and for the (time-)"averaged" orbital vector:

 * $$\vec L_{\rm avg} = \frac{(\vec L \cdot \vec J)}{J^2} \vec J.$$

Thus,

 * $$\langle V_{\rm M} \rangle = \frac{\mu_{\rm B}}{\hbar} \vec J\left(g_L\frac{\vec L \cdot \vec J}{J^2} + g_S\frac{\vec S \cdot \vec J}{J^2}\right) \cdot \vec B.$$

Using $$ \vec L = \vec J - \vec S$$ and squaring both sides, we get

 * $$\vec S \cdot \vec J = \frac{1}{2}(J^2 + S^2 - L^2) = \frac{\hbar^2}{2}[j(j+1) - l(l+1) + s(s+1)],$$

and: using $$ \vec S = \vec J - \vec L$$ and squaring both sides, we get

 * $$\vec L \cdot \vec J = \frac{1}{2}(J^2 - S^2 + L^2) = \frac{\hbar^2}{2}[j(j+1) + l(l+1) - s(s+1)].$$

Combining everything and taking $$ J_z = \hbar m_j$$, we obtain the magnetic potential energy of the atom in the applied external magnetic field,

\begin{align} V_{\rm M} &= \mu_{\rm B} B m_j \left[ g_L\frac{j(j+1) + l(l+1) - s(s+1)}{2j(j+1)} + g_S\frac{j(j+1) - l(l+1) + s(s+1)}{2j(j+1)} \right]\\ &= \mu_{\rm B} B m_j \left[1 + (g_S-1)\frac{j(j+1) - l(l+1) + s(s+1)}{2j(j+1)} \right], \\ &= \mu_{\rm B} B m_j g_j \end{align} $$

where the quantity in square brackets is the Landé g-factor gJ of the atom ($$g_L = 1$$ and $$g_S \approx 2$$) and $$m_j$$ is the z-component of the total angular momentum. For a single electron above filled shells $$s = 1/2$$ and $$ j = l \pm s $$, the Landé g-factor can be simplified into:

 * $$ g_j = 1 \pm \frac{g_S-1}{2l+1} $$

Taking $$V_m$$ to be the perturbation, the Zeeman correction to the energy is

\begin{align} E_{\rm Z}^{(1)} = \langle n l j m_j | H_{\rm Z}^' | n l j m_j \rangle = \langle V_M \rangle_\Psi = \mu_{\rm B} g_J B_{\rm ext} m_j \end{align} $$

Example: Lyman-alpha transition in hydrogen
The Lyman-alpha transition in hydrogen in the presence of the spin–orbit interaction involves the transitions

 * $$2P_{1/2} \to 1S_{1/2}$$ and $$2P_{3/2} \to 1S_{1/2}.$$

In the presence of an external magnetic field, the weak-field Zeeman effect splits the 1S1/2 and 2P1/2 levels into 2 states each ($$m_j = 1/2, -1/2$$) and the 2P3/2 level into 4 states ($$m_j = 3/2, 1/2, -1/2, -3/2$$). The Landé g-factors for the three levels are:

 * $$g_J = 2$$ for $$1S_{1/2}$$ (j=1/2, l=0)

 * $$g_J = 2/3$$ for $$2P_{1/2}$$ (j=1/2, l=1)

 * $$g_J = 4/3$$ for $$2P_{3/2}$$ (j=3/2, l=1).

Note in particular that the size of the energy splitting is different for the different orbitals, because the gJ values are different. On the left, fine structure splitting is depicted. This splitting occurs even in the absence of a magnetic field, as it is due to spin–orbit coupling. Depicted on the right is the additional Zeeman splitting, which occurs in the presence of magnetic fields.

Strong field (Paschen–Back effect)
The Paschen–Back effect is the splitting of atomic energy levels in the presence of a strong magnetic field. This occurs when an external magnetic field is sufficiently strong to disrupt the coupling between orbital ($$\vec{L}$$) and spin ($$\vec{S}$$) angular momenta. This effect is the strong-field limit of the Zeeman effect. When $$s = 0$$, the two effects are equivalent. The effect was named after the German physicists Friedrich Paschen and Ernst E. A. Back.

When the magnetic-field perturbation significantly exceeds the spin–orbit interaction, one can safely assume $$[H_{0}, S] = 0$$. This allows the expectation values of $$L_{z}$$ and $$S_{z}$$ to be easily evaluated for a state $$|\psi\rangle $$. The energies are simply

 * $$ E_{z} = \left\langle \psi \left| H_{0} + \frac{B_{z}\mu_{\rm B}}{\hbar}(L_{z}+g_{s}S_z) \right|\psi\right\rangle = E_{0} + B_z\mu_{\rm B} (m_l + g_{s}m_s). $$

The above may be read as implying that the LS-coupling is completely broken by the external field. However $$m_l$$ and $$m_s$$ are still "good" quantum numbers. Together with the selection rules for an electric dipole transition, i.e., $$\Delta s = 0, \Delta m_s = 0, \Delta l = \pm 1, \Delta m_l = 0, \pm 1$$ this allows to ignore the spin degree of freedom altogether. As a result, only three spectral lines will be visible, corresponding to the $$\Delta m_l = 0, \pm 1$$ selection rule. The splitting $$\Delta E = B \mu_{\rm B} \Delta m_l$$ is independent of the unperturbed energies and electronic configurations of the levels being considered.

More precisely, if $$s \ne 0$$, each of these three components is actually a group of several transitions due to the residual spin–orbit coupling and relativistic corrections (which are of the same order, known as 'fine structure'). The first-order perturbation theory with these corrections yields the following formula for the hydrogen atom in the Paschen–Back limit:
 * $$ E_{z+fs} = E_{z} + \frac{m_e c^2 \alpha^4}{2 n^3} \left\{ \frac{3}{4n} - \left[ \frac{l(l+1) - m_l m_s}{l(l+1/2)(l+1) } \right]\right\}.$$

Example: Lyman-alpha transition in hydrogen
In this example, the fine-structure corrections are ignored.

Intermediate field for j = 1/2
In the magnetic dipole approximation, the Hamiltonian which includes both the hyperfine and Zeeman interactions is
 * $$	H = h A \vec I \cdot \vec J - \vec \mu \cdot \vec B $$
 * $$	H = h A \vec I \cdot\vec J + ( \mu_{\rm B} g_J\vec J + \mu_{\rm N} g_I\vec I ) \cdot \vec {\rm B} $$

where $$A$$ is the hyperfine splitting (in Hz) at zero applied magnetic field, $$\mu_{\rm B}$$ and $$\mu_{\rm N}$$ are the Bohr magneton and nuclear magneton respectively, $$\vec J$$ and $$\vec I$$ are the electron and nuclear angular momentum operators and $$g_J$$ is the Landé g-factor: $$ g_J = g_L\frac{J(J+1) + L(L+1) - S(S+1)}{2J(J+1)} + g_S\frac{J(J+1) - L(L+1) + S(S+1)}{2J(J+1)}.$$

In the case of weak magnetic fields, the Zeeman interaction can be treated as a perturbation to the $$|F,m_f \rangle$$ basis. In the high field regime, the magnetic field becomes so strong that the Zeeman effect will dominate, and one must use a more complete basis of $$|I,J,m_I,m_J\rangle$$ or just $$|m_I,m_J \rangle$$ since $$I$$ and $$J$$ will be constant within a given level. To get the complete picture, including intermediate field strengths, we must consider eigenstates which are superpositions of the $$|F,m_F \rangle $$ and $$|m_I,m_J \rangle $$ basis states. For $$J = 1/2$$, the Hamiltonian can be solved analytically, resulting in the Breit–Rabi formula. Notably, the electric quadrupole interaction is zero for $$L = 0$$ ($$J = 1/2$$), so this formula is fairly accurate.

We now utilize quantum mechanical ladder operators, which are defined for a general angular momentum operator $$L$$ as

 * $$ L_{\pm} \equiv L_x \pm iL_y $$

These ladder operators have the property
 * $$ L_{\pm}|L_,m_L \rangle = \sqrt{(L \mp m_L)(L \pm m_L +1)} |L,m_L \pm 1 \rangle$$

as long as $$m_L$$ lies in the range $${-L, \dots ... ,L}$$ (otherwise, they return zero). Using ladder operators $$J_{\pm}$$ and $$I_{\pm}$$ We can rewrite the Hamiltonian as

 * $$ H = h A I_z J_z + \frac{hA}{2}(J_+ I_- + J_- I_+) + \mu_{\rm B} B g_J J_z + \mu_{\rm N} B g_I I_z$$

We can now see that at all times, the total angular momentum projection $$m_F = m_J + m_I$$ will be conserved. This is because both $$J_z$$ and $$I_z$$ leave states with definite $$ m_J $$ and $$ m_I $$ unchanged, while $$ J_+ I_- $$ and $$ J_- I_+ $$ either increase $$ m_J $$ and decrease $$ m_I $$ or vice versa, so the sum is always unaffected. Furthermore, since $$J = 1/2$$ there are only two possible values of $$m_J$$ which are $$\pm 1/2$$. Therefore, for every value of $$ m_F $$ there are only two possible states, and we can define them as the basis:

 * $$|\pm\rangle \equiv |m_J = \pm 1/2, m_I = m_F \mp 1/2 \rangle $$

This pair of states is a two-level quantum mechanical system. Now we can determine the matrix elements of the Hamiltonian:

 * $$ \langle \pm |H|\pm \rangle = -\frac{1}{4} hA + \mu_{\rm N} B g_I m_F \pm \frac{1}{2} (hAm_F + \mu_{\rm B} B g_J- \mu_{\rm N} B g_I))$$
 * $$ \langle \pm |H| \mp \rangle = \frac{1}{2} hA \sqrt{(I + 1/2)^2 - m_F^2}$$

Solving for the eigenvalues of this matrix – as can be done by hand (see two-level quantum mechanical system), or more easily, with a computer algebra system – we arrive at the energy shifts:

 * $$ \Delta E_{F=I\pm1/2} = -\frac{h \Delta W }{2(2I+1)} + \mu_{\rm N} g_I m_F B \pm \frac{h \Delta W}{2}\sqrt{1 + \frac{2m_F x }{I+1/2}+ x^2 }$$
 * $$x \equiv \frac{B(\mu_{\rm B} g_J - \mu_{\rm N} g_I)}{h \Delta W} \quad \quad \Delta W= A \left(I+\frac{1}{2}\right)$$

where $$\Delta W$$ is the splitting (in units of Hz) between two hyperfine sublevels in the absence of magnetic field $$B$$, $$x$$ is referred to as the 'field strength parameter' (Note: for $$m_F = \pm(I+1/2)$$ the expression under the square root is an exact square, and so the last term should be replaced by $$+\frac{h\Delta W}{2}(1\pm x)$$). This equation is known as the Breit–Rabi formula and is useful for systems with one valence electron in an $$s$$ ($$J = 1/2$$) level.

Note that index $$F$$ in $$\Delta E_{F=I\pm1/2}$$ should be considered not as total angular momentum of the atom but as asymptotic total angular momentum. It is equal to total angular momentum only if $$B=0$$ otherwise eigenvectors corresponding different eigenvalues of the Hamiltonian are the superpositions of states with different $$F$$ but equal $$m_F$$ (the only exceptions are $$|F=I+1/2,m_F=\pm F \rangle$$).

Astrophysics
George Ellery Hale was the first to notice the Zeeman effect in the solar spectra, indicating the existence of strong magnetic fields in sunspots. Such fields can be quite high, on the order of 0.1 tesla or higher. Today, the Zeeman effect is used to produce magnetograms showing the variation of magnetic field on the Sun.

Laser cooling
The Zeeman effect is utilized in many laser cooling applications such as a magneto-optical trap and the Zeeman slower.

Spintronics
Zeeman-energy mediated coupling of spin and orbital motions is used in spintronics for controlling electron spins in quantum dots through electric dipole spin resonance.

Metrology
Old high-precision frequency standards, i.e. hyperfine structure transition-based atomic clocks, may require periodic fine-tuning due to exposure to magnetic fields. This is carried out by measuring the Zeeman effect on specific hyperfine structure transition levels of the source element (cesium) and applying a uniformly precise, low-strength magnetic field to said source, in a process known as degaussing.

The Zeeman effect may also be utilized to improve accuracy in atomic absorption spectroscopy.

Biology
A theory about the magnetic sense of birds assumes that a protein in the retina is changed due to the Zeeman effect.

Nuclear spectroscopy
The nuclear Zeeman effect is important in such applications as nuclear magnetic resonance spectroscopy, magnetic resonance imaging (MRI), and Mössbauer spectroscopy.

Other
The electron spin resonance spectroscopy is based on the Zeeman effect.

Demonstrations
The Zeeman effect can be demonstrated by placing a sodium vapor source in a powerful electromagnet and viewing a sodium vapor lamp through the magnet opening (see diagram). With magnet off, the sodium vapor source will block the lamp light; when the magnet is turned on the lamp light will be visible through the vapor.

The sodium vapor can be created by sealing sodium metal in an evacuated glass tube and heating it while the tube is in the magnet.

Alternatively, salt (sodium chloride) on a ceramic stick can be placed in the flame of Bunsen burner as the sodium vapor source. When the magnetic field is energized, the lamp image will be brighter. However, the magnetic field also affects the flame, making the observation depend upon more than just the Zeeman effect. These issues also plagued Zeeman's original work; he devoted considerable effort to ensure his observations were truly an effect of magnetism on light emission.

When salt is added to the Bunsen burner, it dissociates to give sodium and chloride. The sodium atoms get excited due to photons from the sodium vapour lamp, with electrons excited from 3s to 3p states, absorbing light in the process. The sodium vapour lamp emits light at 589nm, which has precisely the energy to excite an electron of a sodium atom. If it was an atom of another element, like chlorine, shadow will not be formed. When a magnetic field is applied, due to the Zeeman effect the spectral line of sodium gets split into several components. This means the energy difference between the 3s and 3p atomic orbitals will change. As the sodium vapour lamp don't precisely deliver the right frequency any more, light doesn't get absorbed and passes through, resulting in the shadow dimming. As the magnetic field strength is increased, the shift in the spectral lines increases and lamp light is transmitted.

Historical

 * (Chapter 16 provides a comprehensive treatment, as of 1935.)

title: Zeeman effect