Quantum Osilator Harmonik

8/10/2019 Quantum Osilator Harmonik

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Quantum osilator harmonikDari Wikipedia, ensiklopedia bebas

"QHO" beralih ke halaman ini. Itu juga merupakan kode bandara IATAuntuk semua bandara di daerah

Houston.

Beberapa lintasan dariosilator harmonikmenurut hukum Newtontentangmekanika klasik(AB), dan

menurutpersamaan Schrdingerdarimekanika kuantum(!)" #ada AB, partikel (direpresentasikan sebagai bola

melekat padamusim semi) berosilasi bolak$balik" Di !, beberapa solusi untuk #ersamaan Schrdinger ditampilkan, di

mana sumbu hori%ontal adalah posisi, dan sumbu &ertikal adalah bagian n'ata (biru) atau bagian imainer (merah)

dariungsi gelombang" , D, *, +, tetapi tidak , !, adalah eigenstates energi" ! adalah negara 'ang koheren,

keadaan kuantum 'ang mendekati lintasan klasik"

-he osilator harmonik kuantumadalah kuantum mekanikanalog dariosilator harmonik klasik" .arena

sewenang$wenangpotensialbiasan'a dapat diperkirakan sebagaipotensi harmonikdi sekitar stabil titik

ekuilibrium, itu adalah salah satu sistem model 'ang paling penting dalam mekanika kuantum" Selain itu, salah

satu sistem kuantum mekanik beberapa 'ang merupakan, tepatsolusi analitis'ang diketahui" /01/21/31

Satu$dimensi osilator harmonik / sunting1

Hamiltonian dan energi eigenstates / sunting1

4epresentasi ungsi gelombang untuk pertama delapan eigenstates terikat, n5 6 sampai 7" Sumbu hori%ontal

menunukkan posisix.atatan8 raik tidak dinormalisasi, dan tanda$tanda dari beberapa ungsi berbeda dari 'ang

diberikan dalam teks"
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"

Berikut commutatorsdapat dengan mudah diperoleh dengan menggantipergantian hubungan

kanonik,

Dan operator !amilton dapat din'atakan sebagai

sehingga eigenstateNuga eigenstate energi"

!asil panen properti pergantian

dan sama,

:ni berarti bahwabekera pada < nuntuk memproduksi, hingga konstan perkalian, < n$

0, danbertindak atas < nuntuk menghasilkan < n0" @ntuk alasan ini, ini disebut>menurunkan

operator>, dana >operator membesarkan>" Dua operator bersama$sama disebut operator

tangga"Dalam teori medan kuantum, danaadalah alternati disebut >pemusnahan> dan

>penciptaan> operator karena mereka menghancurkan dan menciptakan partikel, 'ang sesuai

dengan kuanta energi kita"

=engingat setiap eigenstate energi, kita dapat bertindak di atasn'a dengan operator

menurunkan, a,untuk menghasilkan eigenstate lain dengan sedikit energi" Dengan aplikasi

berulang dari operator menurunkan, tampakn'a bahwa kita dapat menghasilkan energieigenstates keE= - .Namun, karena

eigen$nomor terkecil adalah 6, dan

"

Dalam hal ini, aplikasi berikutn'a dari operator menurunkan han'a akan menghasilkan nol kets,

bukan eigenstates energi tambahan" Selain itu, kami telah menunukkan di atas bahwa

Akhirn'a, dengan bertindak pada < 6dengan operator peningkatan dan mengalikan dengan aktor

normalisasi 'ang sesuai, kita dapat menghasilkan sebuah set tak terbatas eigenstates energi

,

sehingga
http://translate.googleusercontent.com/translate_c?depth=1&hl=id&prev=/search%3Fq%3Dthe%2Bstep%2Bpotential%2Bquantum%2Bmechanics%26biw%3D1920%26bih%3D979&rurl=translate.google.com&sl=en&u=http://en.wikipedia.org/wiki/Commutator&usg=ALkJrhh6pl1ozkNwDpSvNr0JU17BtPUGbwhttp://translate.googleusercontent.com/translate_c?depth=1&hl=id&prev=/search%3Fq%3Dthe%2Bstep%2Bpotential%2Bquantum%2Bmechanics%26biw%3D1920%26bih%3D979&rurl=translate.google.com&sl=en&u=http://en.wikipedia.org/wiki/Canonical_commutation_relation&usg=ALkJrhglevY-7o9DMKTn44ysHZMo3pjChwhttp://translate.googleusercontent.com/translate_c?depth=1&hl=id&prev=/search%3Fq%3Dthe%2Bstep%2Bpotential%2Bquantum%2Bmechanics%26biw%3D1920%26bih%3D979&rurl=translate.google.com&sl=en&u=http://en.wikipedia.org/wiki/Canonical_commutation_relation&usg=ALkJrhglevY-7o9DMKTn44ysHZMo3pjChwhttp://translate.googleusercontent.com/translate_c?depth=1&hl=id&prev=/search%3Fq%3Dthe%2Bstep%2Bpotential%2Bquantum%2Bmechanics%26biw%3D1920%26bih%3D979&rurl=translate.google.com&sl=en&u=http://en.wikipedia.org/wiki/Canonical_commutation_relation&usg=ALkJrhglevY-7o9DMKTn44ysHZMo3pjChwhttp://translate.googleusercontent.com/translate_c?depth=1&hl=id&prev=/search%3Fq%3Dthe%2Bstep%2Bpotential%2Bquantum%2Bmechanics%26biw%3D1920%26bih%3D979&rurl=translate.google.com&sl=en&u=http://en.wikipedia.org/wiki/Canonical_commutation_relation&usg=ALkJrhglevY-7o9DMKTn44ysHZMo3pjChwhttp://translate.googleusercontent.com/translate_c?depth=1&hl=id&prev=/search%3Fq%3Dthe%2Bstep%2Bpotential%2Bquantum%2Bmechanics%26biw%3D1920%26bih%3D979&rurl=translate.google.com&sl=en&u=http://en.wikipedia.org/wiki/Ladder_operator&usg=ALkJrhgdrwHRMfscJ6gHPDSzIUv-tz051whttp://translate.googleusercontent.com/translate_c?depth=1&hl=id&prev=/search%3Fq%3Dthe%2Bstep%2Bpotential%2Bquantum%2Bmechanics%26biw%3D1920%26bih%3D979&rurl=translate.google.com&sl=en&u=http://en.wikipedia.org/wiki/Ladder_operator&usg=ALkJrhgdrwHRMfscJ6gHPDSzIUv-tz051whttp://translate.googleusercontent.com/translate_c?depth=1&hl=id&prev=/search%3Fq%3Dthe%2Bstep%2Bpotential%2Bquantum%2Bmechanics%26biw%3D1920%26bih%3D979&rurl=translate.google.com&sl=en&u=http://en.wikipedia.org/wiki/Commutator&usg=ALkJrhh6pl1ozkNwDpSvNr0JU17BtPUGbwhttp://translate.googleusercontent.com/translate_c?depth=1&hl=id&prev=/search%3Fq%3Dthe%2Bstep%2Bpotential%2Bquantum%2Bmechanics%26biw%3D1920%26bih%3D979&rurl=translate.google.com&sl=en&u=http://en.wikipedia.org/wiki/Canonical_commutation_relation&usg=ALkJrhglevY-7o9DMKTn44ysHZMo3pjChwhttp://translate.googleusercontent.com/translate_c?depth=1&hl=id&prev=/search%3Fq%3Dthe%2Bstep%2Bpotential%2Bquantum%2Bmechanics%26biw%3D1920%26bih%3D979&rurl=translate.google.com&sl=en&u=http://en.wikipedia.org/wiki/Canonical_commutation_relation&usg=ALkJrhglevY-7o9DMKTn44ysHZMo3pjChwhttp://translate.googleusercontent.com/translate_c?depth=1&hl=id&prev=/search%3Fq%3Dthe%2Bstep%2Bpotential%2Bquantum%2Bmechanics%26biw%3D1920%26bih%3D979&rurl=translate.google.com&sl=en&u=http://en.wikipedia.org/wiki/Ladder_operator&usg=ALkJrhgdrwHRMfscJ6gHPDSzIUv-tz051whttp://translate.googleusercontent.com/translate_c?depth=1&hl=id&prev=/search%3Fq%3Dthe%2Bstep%2Bpotential%2Bquantum%2Bmechanics%26biw%3D1920%26bih%3D979&rurl=translate.google.com&sl=en&u=http://en.wikipedia.org/wiki/Ladder_operator&usg=ALkJrhgdrwHRMfscJ6gHPDSzIUv-tz051w


5/11

,

'ang sesuai dengan spektrum energi 'ang diberikan dalam bagian sebelumn'a"

*igenstates sewenang$wenang dapat din'atakan dalam < 6,

"

Bukti8

.eadaan dasar < 6dalam representasi posisi ditentukan oleh< 65 6,

dan karenan'a

dan sebagain'a, seperti pada bagian sebelumn'a"

Panjang alam dan energi skala / sunting1

-he osilator harmonik kuantum memiliki skala alami untuk panang dan energi, 'ang dapat

digunakan untuk men'ederhanakan masalah" :ni dapat ditemukan

oleh nondimensionali%ation" !asiln'a adalah bahwa, ika kita mengukur energi dalam

satuan dan arak dalam satuan/ (MQ),maka !amiltonian menadi

sedangkan eigenunctions energi dan eigen&alues menadi

di manaH n(x)adalah polinomial !ermite"

@ntuk menghindari kebingungan, kita tidak akan mengadopsi >unit alami> dalam artikel

ini" Namun, mereka sering berguna ketika melakukan perhitungan, dengan melewati

kekacauan" Sebagai contoh, solusi mendasar( ungsi reen) dariH-i t,'ang tergantung waktu

?perator Schroedinger untuk osilator ini, han'a bermuara pada kernel =ehler, /1
http://translate.googleusercontent.com/translate_c?depth=1&hl=id&prev=/search%3Fq%3Dthe%2Bstep%2Bpotential%2Bquantum%2Bmechanics%26biw%3D1920%26bih%3D979&rurl=translate.google.com&sl=en&u=http://en.wikipedia.org/w/index.php%3Ftitle%3DQuantum_harmonic_oscillator%26action%3Dedit%26section%3D4&usg=ALkJrhiM3TrHL98dkGdoeDCE8aRPOvXbgQhttp://translate.googleusercontent.com/translate_c?depth=1&hl=id&prev=/search%3Fq%3Dthe%2Bstep%2Bpotential%2Bquantum%2Bmechanics%26biw%3D1920%26bih%3D979&rurl=translate.google.com&sl=en&u=http://en.wikipedia.org/wiki/Nondimensionalization&usg=ALkJrhgijlnIh5Ah_Kylx4MtOBDZeBIQ1g#Quantum_harmonic_oscillatorhttp://translate.googleusercontent.com/translate_c?depth=1&hl=id&prev=/search%3Fq%3Dthe%2Bstep%2Bpotential%2Bquantum%2Bmechanics%26biw%3D1920%26bih%3D979&rurl=translate.google.com&sl=en&u=http://en.wikipedia.org/wiki/Hermite_polynomials&usg=ALkJrhintHERT-WaJy_nymZoeni-jdkvaQhttp://translate.googleusercontent.com/translate_c?depth=1&hl=id&prev=/search%3Fq%3Dthe%2Bstep%2Bpotential%2Bquantum%2Bmechanics%26biw%3D1920%26bih%3D979&rurl=translate.google.com&sl=en&u=http://en.wikipedia.org/wiki/Fundamental_solution&usg=ALkJrhgIRX0eBhjXoQyIQYeYMeGGm3BlCghttp://translate.googleusercontent.com/translate_c?depth=1&hl=id&prev=/search%3Fq%3Dthe%2Bstep%2Bpotential%2Bquantum%2Bmechanics%26biw%3D1920%26bih%3D979&rurl=translate.google.com&sl=en&u=http://en.wikipedia.org/wiki/Green%27s_function&usg=ALkJrhiy-S7GR-xm0s366AuvUpyjM57t9ghttp://translate.googleusercontent.com/translate_c?depth=1&hl=id&prev=/search%3Fq%3Dthe%2Bstep%2Bpotential%2Bquantum%2Bmechanics%26biw%3D1920%26bih%3D979&rurl=translate.google.com&sl=en&u=http://en.wikipedia.org/wiki/Mehler_kernel&usg=ALkJrhidUMjecTb6SMxwuYufhpgCMxoe3ghttp://translate.googleusercontent.com/translate_c?depth=1&hl=id&prev=/search%3Fq%3Dthe%2Bstep%2Bpotential%2Bquantum%2Bmechanics%26biw%3D1920%26bih%3D979&rurl=translate.google.com&sl=en&u=http://en.wikipedia.org/wiki/Quantum_harmonic_oscillator&usg=ALkJrhgQ3x4Vq9i-zU23Mf1N8lWR8-uKVg#cite_note-4http://translate.googleusercontent.com/translate_c?depth=1&hl=id&prev=/search%3Fq%3Dthe%2Bstep%2Bpotential%2Bquantum%2Bmechanics%26biw%3D1920%26bih%3D979&rurl=translate.google.com&sl=en&u=http://en.wikipedia.org/w/index.php%3Ftitle%3DQuantum_harmonic_oscillator%26action%3Dedit%26section%3D4&usg=ALkJrhiM3TrHL98dkGdoeDCE8aRPOvXbgQhttp://translate.googleusercontent.com/translate_c?depth=1&hl=id&prev=/search%3Fq%3Dthe%2Bstep%2Bpotential%2Bquantum%2Bmechanics%26biw%3D1920%26bih%3D979&rurl=translate.google.com&sl=en&u=http://en.wikipedia.org/wiki/Nondimensionalization&usg=ALkJrhgijlnIh5Ah_Kylx4MtOBDZeBIQ1g#Quantum_harmonic_oscillatorhttp://translate.googleusercontent.com/translate_c?depth=1&hl=id&prev=/search%3Fq%3Dthe%2Bstep%2Bpotential%2Bquantum%2Bmechanics%26biw%3D1920%26bih%3D979&rurl=translate.google.com&sl=en&u=http://en.wikipedia.org/wiki/Hermite_polynomials&usg=ALkJrhintHERT-WaJy_nymZoeni-jdkvaQhttp://translate.googleusercontent.com/translate_c?depth=1&hl=id&prev=/search%3Fq%3Dthe%2Bstep%2Bpotential%2Bquantum%2Bmechanics%26biw%3D1920%26bih%3D979&rurl=translate.google.com&sl=en&u=http://en.wikipedia.org/wiki/Fundamental_solution&usg=ALkJrhgIRX0eBhjXoQyIQYeYMeGGm3BlCghttp://translate.googleusercontent.com/translate_c?depth=1&hl=id&prev=/search%3Fq%3Dthe%2Bstep%2Bpotential%2Bquantum%2Bmechanics%26biw%3D1920%26bih%3D979&rurl=translate.google.com&sl=en&u=http://en.wikipedia.org/wiki/Green%27s_function&usg=ALkJrhiy-S7GR-xm0s366AuvUpyjM57t9ghttp://translate.googleusercontent.com/translate_c?depth=1&hl=id&prev=/search%3Fq%3Dthe%2Bstep%2Bpotential%2Bquantum%2Bmechanics%26biw%3D1920%26bih%3D979&rurl=translate.google.com&sl=en&u=http://en.wikipedia.org/wiki/Mehler_kernel&usg=ALkJrhidUMjecTb6SMxwuYufhpgCMxoe3ghttp://translate.googleusercontent.com/translate_c?depth=1&hl=id&prev=/search%3Fq%3Dthe%2Bstep%2Bpotential%2Bquantum%2Bmechanics%26biw%3D1920%26bih%3D979&rurl=translate.google.com&sl=en&u=http://en.wikipedia.org/wiki/Quantum_harmonic_oscillator&usg=ALkJrhgQ3x4Vq9i-zU23Mf1N8lWR8-uKVg#cite_note-4


6/11

dimanaK (x, y, 0)= (x-y).

Solusi ruang fase / sunting1

Dalam ormulasi ruang asemekanika kuantum, solusi untuk osilator harmonik kuantum

dalam beberapa representasi 'ang berbedadaridistribusi Cuasiprobabilit'dapat ditulis dalam

bentuk tertutup" ang paling ban'ak digunakan di antaran'a adalah untuk ungsi Wigner, 'ang

memiliki solusi

dimana

dan nadalah polinomial Eaguerre" ontoh ini menunukkan bagaimana polinomial !ermite dan

Eaguerre polinomial saling terkait melalui transormasi Wigner$We'l"

#$dimensiosilator harmonik / sunting1

?silator harmonik satu dimensi adalah mudah digeneralisasikan ke dimensi #!di mana #5 0, 2,

3, """" Dalam satu dimensi, posisi partikel ditentukan oleh satu koordinat,x.Di #dimensi, ini

digantikan oleh #posisi koordinat, 'ang kami beri labelx0,""",x #.Sesuai untuk setiap posisi

koordinat momentum, kami labelpini 0,""",p #.-he hubungan pergantian kanonikantara operator

ini

!amiltonian untuk sistem ini adalah

"

Sebagai bentuk !amiltonian ini membuat elas, osilator harmonik #$dimensipersis analog

dengan #osilator harmonik satu dimensi independen dengan massa 'ang sama dan konstanta

pegas" Dalam hal ini, kuantitasx0,""",x #akan mengacu pada posisi masing$masing dari

partikel #.:ni adalah properti n'aman dari potensial, 'ang memungkinkan energi potensial

untuk dipisahkan ke dalam istilah tergantung pada satu koordinat masing$masing"

#engamatan ini membuat solusi sederhana" @ntuk satu set tertentu dari bilangan kuantum nFG

eigenunctions energi untuk osilator #$dimensi'ang din'atakan dalam eigenunctions 0$dimensi

sebagai8

Dalam metode ?perator tangga, kita mendeinisikan #set operator tangga,
http://translate.googleusercontent.com/translate_c?depth=1&hl=id&prev=/search%3Fq%3Dthe%2Bstep%2Bpotential%2Bquantum%2Bmechanics%26biw%3D1920%26bih%3D979&rurl=translate.google.com&sl=en&u=http://en.wikipedia.org/w/index.php%3Ftitle%3DQuantum_harmonic_oscillator%26action%3Dedit%26section%3D5&usg=ALkJrhhNLtUbAJCQ2SHXUrMv8h0wdZ14tQhttp://translate.googleusercontent.com/translate_c?depth=1&hl=id&prev=/search%3Fq%3Dthe%2Bstep%2Bpotential%2Bquantum%2Bmechanics%26biw%3D1920%26bih%3D979&rurl=translate.google.com&sl=en&u=http://en.wikipedia.org/wiki/Phase_space_formulation&usg=ALkJrhjQh1Jf5HKxDkpLMqUyEFzdkfPlcghttp://translate.googleusercontent.com/translate_c?depth=1&hl=id&prev=/search%3Fq%3Dthe%2Bstep%2Bpotential%2Bquantum%2Bmechanics%26biw%3D1920%26bih%3D979&rurl=translate.google.com&sl=en&u=http://en.wikipedia.org/wiki/Phase_space_formulation&usg=ALkJrhjQh1Jf5HKxDkpLMqUyEFzdkfPlcghttp://translate.googleusercontent.com/translate_c?depth=1&hl=id&prev=/search%3Fq%3Dthe%2Bstep%2Bpotential%2Bquantum%2Bmechanics%26biw%3D1920%26bih%3D979&rurl=translate.google.com&sl=en&u=http://en.wikipedia.org/wiki/Quasiprobability_distribution&usg=ALkJrhj86HfHPJINALXNvP6uf_nbgo9xGQ#Fock_statehttp://translate.googleusercontent.com/translate_c?depth=1&hl=id&prev=/search%3Fq%3Dthe%2Bstep%2Bpotential%2Bquantum%2Bmechanics%26biw%3D1920%26bih%3D979&rurl=translate.google.com&sl=en&u=http://en.wikipedia.org/wiki/Quasiprobability_distribution&usg=ALkJrhj86HfHPJINALXNvP6uf_nbgo9xGQhttp://translate.googleusercontent.com/translate_c?depth=1&hl=id&prev=/search%3Fq%3Dthe%2Bstep%2Bpotential%2Bquantum%2Bmechanics%26biw%3D1920%26bih%3D979&rurl=translate.google.com&sl=en&u=http://en.wikipedia.org/wiki/Quasiprobability_distribution&usg=ALkJrhj86HfHPJINALXNvP6uf_nbgo9xGQhttp://translate.googleusercontent.com/translate_c?depth=1&hl=id&prev=/search%3Fq%3Dthe%2Bstep%2Bpotential%2Bquantum%2Bmechanics%26biw%3D1920%26bih%3D979&rurl=translate.google.com&sl=en&u=http://en.wikipedia.org/wiki/Wigner_function&usg=ALkJrhhA6SLfB7GCRhq2fSD0urawiZVHHwhttp://translate.googleusercontent.com/translate_c?depth=1&hl=id&prev=/search%3Fq%3Dthe%2Bstep%2Bpotential%2Bquantum%2Bmechanics%26biw%3D1920%26bih%3D979&rurl=translate.google.com&sl=en&u=http://en.wikipedia.org/wiki/Laguerre_polynomials&usg=ALkJrhg1wsYsouT5stI3XO7_CVJ9y3OrHghttp://translate.googleusercontent.com/translate_c?depth=1&hl=id&prev=/search%3Fq%3Dthe%2Bstep%2Bpotential%2Bquantum%2Bmechanics%26biw%3D1920%26bih%3D979&rurl=translate.google.com&sl=en&u=http://en.wikipedia.org/wiki/Wigner%25E2%2580%2593Weyl_transform&usg=ALkJrhhgto2oKOivzgzv_Hykir4n7iQdVghttp://translate.googleusercontent.com/translate_c?depth=1&hl=id&prev=/search%3Fq%3Dthe%2Bstep%2Bpotential%2Bquantum%2Bmechanics%26biw%3D1920%26bih%3D979&rurl=translate.google.com&sl=en&u=http://en.wikipedia.org/w/index.php%3Ftitle%3DQuantum_harmonic_oscillator%26action%3Dedit%26section%3D6&usg=ALkJrhgn8rWacXRC0PMI37hfwZF9UW-0Dwhttp://translate.googleusercontent.com/translate_c?depth=1&hl=id&prev=/search%3Fq%3Dthe%2Bstep%2Bpotential%2Bquantum%2Bmechanics%26biw%3D1920%26bih%3D979&rurl=translate.google.com&sl=en&u=http://en.wikipedia.org/wiki/Coordinate_system&usg=ALkJrhiyvIfHNSLc5cZ5DnAASOys8nX4cwhttp://translate.googleusercontent.com/translate_c?depth=1&hl=id&prev=/search%3Fq%3Dthe%2Bstep%2Bpotential%2Bquantum%2Bmechanics%26biw%3D1920%26bih%3D979&rurl=translate.google.com&sl=en&u=http://en.wikipedia.org/wiki/Coordinate_system&usg=ALkJrhiyvIfHNSLc5cZ5DnAASOys8nX4cwhttp://translate.googleusercontent.com/translate_c?depth=1&hl=id&prev=/search%3Fq%3Dthe%2Bstep%2Bpotential%2Bquantum%2Bmechanics%26biw%3D1920%26bih%3D979&rurl=translate.google.com&sl=en&u=http://en.wikipedia.org/wiki/Canonical_commutation_relations&usg=ALkJrhjdVb0wTdakqE5wwYmLuy-kCQzjighttp://translate.googleusercontent.com/translate_c?depth=1&hl=id&prev=/search%3Fq%3Dthe%2Bstep%2Bpotential%2Bquantum%2Bmechanics%26biw%3D1920%26bih%3D979&rurl=translate.google.com&sl=en&u=http://en.wikipedia.org/w/index.php%3Ftitle%3DQuantum_harmonic_oscillator%26action%3Dedit%26section%3D5&usg=ALkJrhhNLtUbAJCQ2SHXUrMv8h0wdZ14tQhttp://translate.googleusercontent.com/translate_c?depth=1&hl=id&prev=/search%3Fq%3Dthe%2Bstep%2Bpotential%2Bquantum%2Bmechanics%26biw%3D1920%26bih%3D979&rurl=translate.google.com&sl=en&u=http://en.wikipedia.org/wiki/Phase_space_formulation&usg=ALkJrhjQh1Jf5HKxDkpLMqUyEFzdkfPlcghttp://translate.googleusercontent.com/translate_c?depth=1&hl=id&prev=/search%3Fq%3Dthe%2Bstep%2Bpotential%2Bquantum%2Bmechanics%26biw%3D1920%26bih%3D979&rurl=translate.google.com&sl=en&u=http://en.wikipedia.org/wiki/Quasiprobability_distribution&usg=ALkJrhj86HfHPJINALXNvP6uf_nbgo9xGQ#Fock_statehttp://translate.googleusercontent.com/translate_c?depth=1&hl=id&prev=/search%3Fq%3Dthe%2Bstep%2Bpotential%2Bquantum%2Bmechanics%26biw%3D1920%26bih%3D979&rurl=translate.google.com&sl=en&u=http://en.wikipedia.org/wiki/Quasiprobability_distribution&usg=ALkJrhj86HfHPJINALXNvP6uf_nbgo9xGQhttp://translate.googleusercontent.com/translate_c?depth=1&hl=id&prev=/search%3Fq%3Dthe%2Bstep%2Bpotential%2Bquantum%2Bmechanics%26biw%3D1920%26bih%3D979&rurl=translate.google.com&sl=en&u=http://en.wikipedia.org/wiki/Wigner_function&usg=ALkJrhhA6SLfB7GCRhq2fSD0urawiZVHHwhttp://translate.googleusercontent.com/translate_c?depth=1&hl=id&prev=/search%3Fq%3Dthe%2Bstep%2Bpotential%2Bquantum%2Bmechanics%26biw%3D1920%26bih%3D979&rurl=translate.google.com&sl=en&u=http://en.wikipedia.org/wiki/Laguerre_polynomials&usg=ALkJrhg1wsYsouT5stI3XO7_CVJ9y3OrHghttp://translate.googleusercontent.com/translate_c?depth=1&hl=id&prev=/search%3Fq%3Dthe%2Bstep%2Bpotential%2Bquantum%2Bmechanics%26biw%3D1920%26bih%3D979&rurl=translate.google.com&sl=en&u=http://en.wikipedia.org/wiki/Wigner%25E2%2580%2593Weyl_transform&usg=ALkJrhhgto2oKOivzgzv_Hykir4n7iQdVghttp://translate.googleusercontent.com/translate_c?depth=1&hl=id&prev=/search%3Fq%3Dthe%2Bstep%2Bpotential%2Bquantum%2Bmechanics%26biw%3D1920%26bih%3D979&rurl=translate.google.com&sl=en&u=http://en.wikipedia.org/w/index.php%3Ftitle%3DQuantum_harmonic_oscillator%26action%3Dedit%26section%3D6&usg=ALkJrhgn8rWacXRC0PMI37hfwZF9UW-0Dwhttp://translate.googleusercontent.com/translate_c?depth=1&hl=id&prev=/search%3Fq%3Dthe%2Bstep%2Bpotential%2Bquantum%2Bmechanics%26biw%3D1920%26bih%3D979&rurl=translate.google.com&sl=en&u=http://en.wikipedia.org/wiki/Coordinate_system&usg=ALkJrhiyvIfHNSLc5cZ5DnAASOys8nX4cwhttp://translate.googleusercontent.com/translate_c?depth=1&hl=id&prev=/search%3Fq%3Dthe%2Bstep%2Bpotential%2Bquantum%2Bmechanics%26biw%3D1920%26bih%3D979&rurl=translate.google.com&sl=en&u=http://en.wikipedia.org/wiki/Canonical_commutation_relations&usg=ALkJrhjdVb0wTdakqE5wwYmLuy-kCQzjig


7/11

"

Dengan prosedur analog dengan kasus satu dimensi, kita kemudian dapat menunukkan bahwa

masing$masing idanH

ioperator 'ang lebih rendah dan meningkatkan energi dengan 9 masing$masing" !amiltonian ini

!amiltonian ini adalah in&arian dalam grup simetri dinamis % (#(kelompok kesatuan dalam

dimensi #!'ang dideinisikan oleh

dimana merupakan elemen dalam representasi matriks mendeinisikan % (#.

-ingkat energi dari sistem ini adalah

"

Seperti dalam kasus satu dimensi, energi 'ang terkuantisasi" *nergi keadaan dasar adalah #kali

energi satu dimensi, seperti 'ang kita harapkan menggunakan analogi #osilator satu dimensi

independen" Ada satu perbedaan lagi8 dalam kasus satu$dimensi, setiap tingkat energi sesuai

dengan keadaan kuantum 'ang unik" Di #$dimensi!kecuali untuk keadaan dasar, tingkat energi

'ang merosot!'ang berarti ada beberapa negara dengan energi 'ang sama"

Degenerasi dapat dihitung dengan relati mudah" Sebagai contoh, pertimbangkan kasus 3

dimensi8 -entukan n5 n0 n2 n3"Semua negara bagian dengan n'ang sama akan memiliki

energi 'ang sama" @ntuk ntertentu, kita

memilih ntertentu 0".emudian n2 n35 n$ n0"Ada n$ n0 0 kelompok 'ang

mungkin &n2,n3Gn2dapat mengambil nilai$nilai 6 sampai n$" #0,dan untuk

setiap n2nilai n3adalah tetap" -ingkat degenerasi karena itu8

+ormula untuk umum #dan n/ngmenadi dimensi ntereduksi simetristhrepresentasi kekuatan

kelompok kesatuan % (#'

.asus khusus # ) *!'ang diberikan di atas, berikut langsung dari persamaan umum ini" Namun

ini, han'a berlaku untuk partikel dibedakan, atau satu partikel di N dimensi (sebagai dimensi

dibedakan)" @ntuk kasus #boson dalam satu dimensi perangkap harmonik, degenerasi

timbangan sebagai seumlah cara untuk partisi integer nmenggunakan bilangan bulat kurang dari

atau sama dengan #.


8/11

!al ini muncul karena kendala menempatkan #Cuanta menadi ket negara di

mana dan , ang merupakan kendala 'ang sama seperti pada

partisi integer"

Contoh: 3D isotropik osilator harmonik / sunting1

-he Schrdinger persamaan dari bola 'ang simetris$tiga$dimensi osilator harmonik dapat

diselesaikan secara eksplisit oleh pemisahan &ariabel, lihat artikel iniuntuk kasus ini" #rosedur ini

analog dengan pemisahan dilakukan dalam atom hidrogen sepertimasalah, tetapi dengan potensi

bola simetris

dimana adalah massa dari masalah" (.arena makan digunakan di bawah ini untuk bilangan

kuantum magnetik, massa ditunukkan dengan , Bukann'a m!seperti sebelumn'a dalam artikel

ini")

Solusin'a berbun'i

dimana

adalah konstanta normalisasi"

'angumum Eaguerre polinomial" @rutan kdari polinomial adalah bilangan bulat non$negati"

adalah ungsi harmonik bola"

adalah mengurangi konstanta #lanck8 "

*nergi eigen&alue adalah

*nergi 'ang biasan'a digambarkan dengan satubilangan kuantum

.arena kadalah bilangan bulat non$negati, untuk setiap nbahkan kita

miliki dan untuk setiap nganil 'ang kita miliki

" =agnetik kuantum bilangan madalah bilangan bulat memuaskan , Sehingga

untuk setiap ndan lada +,berbeda -lkeadaan kuantum, diberi label oleh m.Dengan demikian,

degenerasi pada tingkat nadalah
http://translate.googleusercontent.com/translate_c?depth=1&hl=id&prev=/search%3Fq%3Dthe%2Bstep%2Bpotential%2Bquantum%2Bmechanics%26biw%3D1920%26bih%3D979&rurl=translate.google.com&sl=en&u=http://en.wikipedia.org/w/index.php%3Ftitle%3DQuantum_harmonic_oscillator%26action%3Dedit%26section%3D7&usg=ALkJrhh_gReBhUY2qKc5h-DvF8epgISB2whttp://translate.googleusercontent.com/translate_c?depth=1&hl=id&prev=/search%3Fq%3Dthe%2Bstep%2Bpotential%2Bquantum%2Bmechanics%26biw%3D1920%26bih%3D979&rurl=translate.google.com&sl=en&u=http://en.wikipedia.org/wiki/Particle_in_a_spherically_symmetric_potential&usg=ALkJrhgR3NBYSXeFJncBNB8wF3g9txQR0w#3D_isotropic_harmonic_oscillatorhttp://translate.googleusercontent.com/translate_c?depth=1&hl=id&prev=/search%3Fq%3Dthe%2Bstep%2Bpotential%2Bquantum%2Bmechanics%26biw%3D1920%26bih%3D979&rurl=translate.google.com&sl=en&u=http://en.wikipedia.org/wiki/Particle_in_a_spherically_symmetric_potential&usg=ALkJrhgR3NBYSXeFJncBNB8wF3g9txQR0w#3D_isotropic_harmonic_oscillatorhttp://translate.googleusercontent.com/translate_c?depth=1&hl=id&prev=/search%3Fq%3Dthe%2Bstep%2Bpotential%2Bquantum%2Bmechanics%26biw%3D1920%26bih%3D979&rurl=translate.google.com&sl=en&u=http://en.wikipedia.org/wiki/Hydrogen-like_atom&usg=ALkJrhhX4gFasi-B5OZKHbsLrF3dHdPBYQ#Schr.C3.B6dinger_equation_in_a_spherically_symmetric_potentialhttp://translate.googleusercontent.com/translate_c?depth=1&hl=id&prev=/search%3Fq%3Dthe%2Bstep%2Bpotential%2Bquantum%2Bmechanics%26biw%3D1920%26bih%3D979&rurl=translate.google.com&sl=en&u=http://en.wikipedia.org/wiki/Hydrogen-like_atom&usg=ALkJrhhX4gFasi-B5OZKHbsLrF3dHdPBYQ#Schr.C3.B6dinger_equation_in_a_spherically_symmetric_potentialhttp://translate.googleusercontent.com/translate_c?depth=1&hl=id&prev=/search%3Fq%3Dthe%2Bstep%2Bpotential%2Bquantum%2Bmechanics%26biw%3D1920%26bih%3D979&rurl=translate.google.com&sl=en&u=http://en.wikipedia.org/wiki/Particle_in_a_spherically_symmetric_potential&usg=ALkJrhgR3NBYSXeFJncBNB8wF3g9txQR0whttp://translate.googleusercontent.com/translate_c?depth=1&hl=id&prev=/search%3Fq%3Dthe%2Bstep%2Bpotential%2Bquantum%2Bmechanics%26biw%3D1920%26bih%3D979&rurl=translate.google.com&sl=en&u=http://en.wikipedia.org/wiki/Particle_in_a_spherically_symmetric_potential&usg=ALkJrhgR3NBYSXeFJncBNB8wF3g9txQR0whttp://translate.googleusercontent.com/translate_c?depth=1&hl=id&prev=/search%3Fq%3Dthe%2Bstep%2Bpotential%2Bquantum%2Bmechanics%26biw%3D1920%26bih%3D979&rurl=translate.google.com&sl=en&u=http://en.wikipedia.org/wiki/Laguerre_polynomials&usg=ALkJrhg1wsYsouT5stI3XO7_CVJ9y3OrHg#Generalized_Laguerre_polynomialshttp://translate.googleusercontent.com/translate_c?depth=1&hl=id&prev=/search%3Fq%3Dthe%2Bstep%2Bpotential%2Bquantum%2Bmechanics%26biw%3D1920%26bih%3D979&rurl=translate.google.com&sl=en&u=http://en.wikipedia.org/wiki/Laguerre_polynomials&usg=ALkJrhg1wsYsouT5stI3XO7_CVJ9y3OrHg#Generalized_Laguerre_polynomialshttp://translate.googleusercontent.com/translate_c?depth=1&hl=id&prev=/search%3Fq%3Dthe%2Bstep%2Bpotential%2Bquantum%2Bmechanics%26biw%3D1920%26bih%3D979&rurl=translate.google.com&sl=en&u=http://en.wikipedia.org/wiki/Spherical_harmonics&usg=ALkJrhiLgn7D0hOf0FykSpVqwUyyRhKbVghttp://translate.googleusercontent.com/translate_c?depth=1&hl=id&prev=/search%3Fq%3Dthe%2Bstep%2Bpotential%2Bquantum%2Bmechanics%26biw%3D1920%26bih%3D979&rurl=translate.google.com&sl=en&u=http://en.wikipedia.org/wiki/Spherical_harmonics&usg=ALkJrhiLgn7D0hOf0FykSpVqwUyyRhKbVghttp://translate.googleusercontent.com/translate_c?depth=1&hl=id&prev=/search%3Fq%3Dthe%2Bstep%2Bpotential%2Bquantum%2Bmechanics%26biw%3D1920%26bih%3D979&rurl=translate.google.com&sl=en&u=http://en.wikipedia.org/wiki/Planck_constant&usg=ALkJrhgivk_Y8yhbRtcz9fuZXWu5cf5elQhttp://translate.googleusercontent.com/translate_c?depth=1&hl=id&prev=/search%3Fq%3Dthe%2Bstep%2Bpotential%2Bquantum%2Bmechanics%26biw%3D1920%26bih%3D979&rurl=translate.google.com&sl=en&u=http://en.wikipedia.org/wiki/Quantum_number&usg=ALkJrhhS6L0LJHux6-jonhGciZKxN-Udbghttp://translate.googleusercontent.com/translate_c?depth=1&hl=id&prev=/search%3Fq%3Dthe%2Bstep%2Bpotential%2Bquantum%2Bmechanics%26biw%3D1920%26bih%3D979&rurl=translate.google.com&sl=en&u=http://en.wikipedia.org/wiki/Quantum_number&usg=ALkJrhhS6L0LJHux6-jonhGciZKxN-Udbghttp://translate.googleusercontent.com/translate_c?depth=1&hl=id&prev=/search%3Fq%3Dthe%2Bstep%2Bpotential%2Bquantum%2Bmechanics%26biw%3D1920%26bih%3D979&rurl=translate.google.com&sl=en&u=http://en.wikipedia.org/wiki/Quantum_state&usg=ALkJrhjZsYeaw9yv5EFqYu1bxapJwM1bswhttp://translate.googleusercontent.com/translate_c?depth=1&hl=id&prev=/search%3Fq%3Dthe%2Bstep%2Bpotential%2Bquantum%2Bmechanics%26biw%3D1920%26bih%3D979&rurl=translate.google.com&sl=en&u=http://en.wikipedia.org/wiki/Quantum_state&usg=ALkJrhjZsYeaw9yv5EFqYu1bxapJwM1bswhttp://translate.googleusercontent.com/translate_c?depth=1&hl=id&prev=/search%3Fq%3Dthe%2Bstep%2Bpotential%2Bquantum%2Bmechanics%26biw%3D1920%26bih%3D979&rurl=translate.google.com&sl=en&u=http://en.wikipedia.org/w/index.php%3Ftitle%3DQuantum_harmonic_oscillator%26action%3Dedit%26section%3D7&usg=ALkJrhh_gReBhUY2qKc5h-DvF8epgISB2whttp://translate.googleusercontent.com/translate_c?depth=1&hl=id&prev=/search%3Fq%3Dthe%2Bstep%2Bpotential%2Bquantum%2Bmechanics%26biw%3D1920%26bih%3D979&rurl=translate.google.com&sl=en&u=http://en.wikipedia.org/wiki/Particle_in_a_spherically_symmetric_potential&usg=ALkJrhgR3NBYSXeFJncBNB8wF3g9txQR0w#3D_isotropic_harmonic_oscillatorhttp://translate.googleusercontent.com/translate_c?depth=1&hl=id&prev=/search%3Fq%3Dthe%2Bstep%2Bpotential%2Bquantum%2Bmechanics%26biw%3D1920%26bih%3D979&rurl=translate.google.com&sl=en&u=http://en.wikipedia.org/wiki/Hydrogen-like_atom&usg=ALkJrhhX4gFasi-B5OZKHbsLrF3dHdPBYQ#Schr.C3.B6dinger_equation_in_a_spherically_symmetric_potentialhttp://translate.googleusercontent.com/translate_c?depth=1&hl=id&prev=/search%3Fq%3Dthe%2Bstep%2Bpotential%2Bquantum%2Bmechanics%26biw%3D1920%26bih%3D979&rurl=translate.google.com&sl=en&u=http://en.wikipedia.org/wiki/Particle_in_a_spherically_symmetric_potential&usg=ALkJrhgR3NBYSXeFJncBNB8wF3g9txQR0whttp://translate.googleusercontent.com/translate_c?depth=1&hl=id&prev=/search%3Fq%3Dthe%2Bstep%2Bpotential%2Bquantum%2Bmechanics%26biw%3D1920%26bih%3D979&rurl=translate.google.com&sl=en&u=http://en.wikipedia.org/wiki/Particle_in_a_spherically_symmetric_potential&usg=ALkJrhgR3NBYSXeFJncBNB8wF3g9txQR0whttp://translate.googleusercontent.com/translate_c?depth=1&hl=id&prev=/search%3Fq%3Dthe%2Bstep%2Bpotential%2Bquantum%2Bmechanics%26biw%3D1920%26bih%3D979&rurl=translate.google.com&sl=en&u=http://en.wikipedia.org/wiki/Laguerre_polynomials&usg=ALkJrhg1wsYsouT5stI3XO7_CVJ9y3OrHg#Generalized_Laguerre_polynomialshttp://translate.googleusercontent.com/translate_c?depth=1&hl=id&prev=/search%3Fq%3Dthe%2Bstep%2Bpotential%2Bquantum%2Bmechanics%26biw%3D1920%26bih%3D979&rurl=translate.google.com&sl=en&u=http://en.wikipedia.org/wiki/Spherical_harmonics&usg=ALkJrhiLgn7D0hOf0FykSpVqwUyyRhKbVghttp://translate.googleusercontent.com/translate_c?depth=1&hl=id&prev=/search%3Fq%3Dthe%2Bstep%2Bpotential%2Bquantum%2Bmechanics%26biw%3D1920%26bih%3D979&rurl=translate.google.com&sl=en&u=http://en.wikipedia.org/wiki/Planck_constant&usg=ALkJrhgivk_Y8yhbRtcz9fuZXWu5cf5elQhttp://translate.googleusercontent.com/translate_c?depth=1&hl=id&prev=/search%3Fq%3Dthe%2Bstep%2Bpotential%2Bquantum%2Bmechanics%26biw%3D1920%26bih%3D979&rurl=translate.google.com&sl=en&u=http://en.wikipedia.org/wiki/Quantum_number&usg=ALkJrhhS6L0LJHux6-jonhGciZKxN-Udbghttp://translate.googleusercontent.com/translate_c?depth=1&hl=id&prev=/search%3Fq%3Dthe%2Bstep%2Bpotential%2Bquantum%2Bmechanics%26biw%3D1920%26bih%3D979&rurl=translate.google.com&sl=en&u=http://en.wikipedia.org/wiki/Quantum_state&usg=ALkJrhjZsYeaw9yv5EFqYu1bxapJwM1bsw


9/11

di mana umlahn'a dimulai dari 6 atau 0, tergantung pada apakah nadalah genap atau

ganil" !asil ini sesuai dengan rumus dimensi di atas, dan umlah dimensi dari representasi

simetris % (*!kelompok degenerasi 'ang rele&an"

?silator harmonik kisi8 onon / sunting1

.ita dapat memperluas gagasan osilator harmonik dengan satu kisi ban'ak

partikel" #ertimbangkan kuantum mekanik satu dimensi rantai harmonikdari #atom identik" :ni

adalah model kuantum mekanik sederhana kisi, dan kita akan melihat bagaimanaonontimbul

dari itu" +ormalisme bahwa kami akan mengembangkan untuk model ini adalah mudah

digeneralisasikan ke dua dan tiga dimensi"

Seperti pada bagian sebelumn'a, kita menunukkan posisi massa oleh , ang diukur

dari posisi kesetimbangan mereka ('aitu ika partikel iberada pada posisi

kesetimbangan") Dalam dua atau lebih dimensi, adalah umlah &ektor" -he!amiltonianuntuk

sistem ini adalah

di mana madalah massa setiap atom (dengan asumsi sama untuk semua), dan dan adalah

posisi dan momentumoperator untuk atom th idan umlah tersebut dibuat selama tetangga

terdekat (nn)" Namun, sebagai salah satu harapkan, dalam kisi bisa uga muncul gelombang 'ang

dapat berperilaku sebagai partikel" @ntuk pengobatan gelombang, adalah kebiasaan untuk

memperlakukan dengan ruang +ourier'ang menggunakanmode 'angnormaldari wa&e&ectorsebagai &ariabel bukan koordinat partikel" Iumlah mode normal 'ang

sama untuk partikel namun ruang +ouriersangat berguna mengingatperiodisitasdari sistem"

.ami memperkenalkan, kemudian, satu setN>koordinat normal> , Dideinisikan

sebagai transormasi +ourier diskritdarixs, danN>konugat momentum> dideinisikan sebagai

transormasi +ourier daris,

Iumlah k nakan berubah menadi bilangan gelombangdari phonon, 'aitu 2J dibagi

dengan panang gelombang" Dibutuhkan pada nilai$nilai terkuantisasi, karena umlah atom

terbatas"

#ilihan ini mempertahankan hubungan pergantian diinginkan baik ruang n'ata atau gelombang

ruang &ektor

Dari hasil umum
http://translate.googleusercontent.com/translate_c?depth=1&hl=id&prev=/search%3Fq%3Dthe%2Bstep%2Bpotential%2Bquantum%2Bmechanics%26biw%3D1920%26bih%3D979&rurl=translate.google.com&sl=en&u=http://en.wikipedia.org/w/index.php%3Ftitle%3DQuantum_harmonic_oscillator%26action%3Dedit%26section%3D8&usg=ALkJrhhPtJyCrtVgKG2Em1LehtVlPujmtwhttp://translate.googleusercontent.com/translate_c?depth=1&hl=id&prev=/search%3Fq%3Dthe%2Bstep%2Bpotential%2Bquantum%2Bmechanics%26biw%3D1920%26bih%3D979&rurl=translate.google.com&sl=en&u=http://en.wikipedia.org/w/index.php%3Ftitle%3DHarmonic_chain%26action%3Dedit%26redlink%3D1&usg=ALkJrhjXIq-ib5qEEtea7TNlqkDBr_iqkghttp://translate.googleusercontent.com/translate_c?depth=1&hl=id&prev=/search%3Fq%3Dthe%2Bstep%2Bpotential%2Bquantum%2Bmechanics%26biw%3D1920%26bih%3D979&rurl=translate.google.com&sl=en&u=http://en.wikipedia.org/wiki/Phonon&usg=ALkJrhhQyds9wMowV5qzR3MuyeJ8co7HlQhttp://translate.googleusercontent.com/translate_c?depth=1&hl=id&prev=/search%3Fq%3Dthe%2Bstep%2Bpotential%2Bquantum%2Bmechanics%26biw%3D1920%26bih%3D979&rurl=translate.google.com&sl=en&u=http://en.wikipedia.org/wiki/Phonon&usg=ALkJrhhQyds9wMowV5qzR3MuyeJ8co7HlQhttp://translate.googleusercontent.com/translate_c?depth=1&hl=id&prev=/search%3Fq%3Dthe%2Bstep%2Bpotential%2Bquantum%2Bmechanics%26biw%3D1920%26bih%3D979&rurl=translate.google.com&sl=en&u=http://en.wikipedia.org/wiki/Hamiltonian_(quantum_mechanics)&usg=ALkJrhhy18Kj64-yQhKEtLUyY9ka2uCxKghttp://translate.googleusercontent.com/translate_c?depth=1&hl=id&prev=/search%3Fq%3Dthe%2Bstep%2Bpotential%2Bquantum%2Bmechanics%26biw%3D1920%26bih%3D979&rurl=translate.google.com&sl=en&u=http://en.wikipedia.org/wiki/Hamiltonian_(quantum_mechanics)&usg=ALkJrhhy18Kj64-yQhKEtLUyY9ka2uCxKghttp://translate.googleusercontent.com/translate_c?depth=1&hl=id&prev=/search%3Fq%3Dthe%2Bstep%2Bpotential%2Bquantum%2Bmechanics%26biw%3D1920%26bih%3D979&rurl=translate.google.com&sl=en&u=http://en.wikipedia.org/wiki/Hamiltonian_(quantum_mechanics)&usg=ALkJrhhy18Kj64-yQhKEtLUyY9ka2uCxKghttp://translate.googleusercontent.com/translate_c?depth=1&hl=id&prev=/search%3Fq%3Dthe%2Bstep%2Bpotential%2Bquantum%2Bmechanics%26biw%3D1920%26bih%3D979&rurl=translate.google.com&sl=en&u=http://en.wikipedia.org/wiki/Momentum&usg=ALkJrhjKOjKCSpGEci941JdrsSCN6v0_oAhttp://translate.googleusercontent.com/translate_c?depth=1&hl=id&prev=/search%3Fq%3Dthe%2Bstep%2Bpotential%2Bquantum%2Bmechanics%26biw%3D1920%26bih%3D979&rurl=translate.google.com&sl=en&u=http://en.wikipedia.org/wiki/Waves&usg=ALkJrhi9r-MPyGBClccWT3FjyaxtkhdMfwhttp://translate.googleusercontent.com/translate_c?depth=1&hl=id&prev=/search%3Fq%3Dthe%2Bstep%2Bpotential%2Bquantum%2Bmechanics%26biw%3D1920%26bih%3D979&rurl=translate.google.com&sl=en&u=http://en.wikipedia.org/wiki/Fourier_space&usg=ALkJrhhpgPGg4yjcbTItGcJL2UNDkuNa-Qhttp://translate.googleusercontent.com/translate_c?depth=1&hl=id&prev=/search%3Fq%3Dthe%2Bstep%2Bpotential%2Bquantum%2Bmechanics%26biw%3D1920%26bih%3D979&rurl=translate.google.com&sl=en&u=http://en.wikipedia.org/wiki/Fourier_space&usg=ALkJrhhpgPGg4yjcbTItGcJL2UNDkuNa-Qhttp://translate.googleusercontent.com/translate_c?depth=1&hl=id&prev=/search%3Fq%3Dthe%2Bstep%2Bpotential%2Bquantum%2Bmechanics%26biw%3D1920%26bih%3D979&rurl=translate.google.com&sl=en&u=http://en.wikipedia.org/wiki/Normal_modes&usg=ALkJrhhhABUEeJRllIX9Icjr8PtEvUqnSghttp://translate.googleusercontent.com/translate_c?depth=1&hl=id&prev=/search%3Fq%3Dthe%2Bstep%2Bpotential%2Bquantum%2Bmechanics%26biw%3D1920%26bih%3D979&rurl=translate.google.com&sl=en&u=http://en.wikipedia.org/wiki/Normal_modes&usg=ALkJrhhhABUEeJRllIX9Icjr8PtEvUqnSghttp://translate.googleusercontent.com/translate_c?depth=1&hl=id&prev=/search%3Fq%3Dthe%2Bstep%2Bpotential%2Bquantum%2Bmechanics%26biw%3D1920%26bih%3D979&rurl=translate.google.com&sl=en&u=http://en.wikipedia.org/wiki/Normal_modes&usg=ALkJrhhhABUEeJRllIX9Icjr8PtEvUqnSghttp://translate.googleusercontent.com/translate_c?depth=1&hl=id&prev=/search%3Fq%3Dthe%2Bstep%2Bpotential%2Bquantum%2Bmechanics%26biw%3D1920%26bih%3D979&rurl=translate.google.com&sl=en&u=http://en.wikipedia.org/wiki/Wavevector&usg=ALkJrhgoSSRMDDGreZOVnf6WXNLdbEzwiwhttp://translate.googleusercontent.com/translate_c?depth=1&hl=id&prev=/search%3Fq%3Dthe%2Bstep%2Bpotential%2Bquantum%2Bmechanics%26biw%3D1920%26bih%3D979&rurl=translate.google.com&sl=en&u=http://en.wikipedia.org/wiki/Wavevector&usg=ALkJrhgoSSRMDDGreZOVnf6WXNLdbEzwiwhttp://translate.googleusercontent.com/translate_c?depth=1&hl=id&prev=/search%3Fq%3Dthe%2Bstep%2Bpotential%2Bquantum%2Bmechanics%26biw%3D1920%26bih%3D979&rurl=translate.google.com&sl=en&u=http://en.wikipedia.org/wiki/Fourier_space&usg=ALkJrhhpgPGg4yjcbTItGcJL2UNDkuNa-Qhttp://translate.googleusercontent.com/translate_c?depth=1&hl=id&prev=/search%3Fq%3Dthe%2Bstep%2Bpotential%2Bquantum%2Bmechanics%26biw%3D1920%26bih%3D979&rurl=translate.google.com&sl=en&u=http://en.wikipedia.org/wiki/Fourier_series&usg=ALkJrhj2aMzRp0iGmp_YtvsexvT8lSkcqAhttp://translate.googleusercon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10/11

mudah untuk menunukkan, melalui trigonometri dasar, bahwa istilah energi potensial adalah

dimana

-he !amiltonian dapat ditulis dalam ruang &ektor gelombang sebagai

#erhatikan bahwa kopling antara &ariabel posisi telah berubah auh, ika Qdan s

'anghermitian('ang mereka tidak), 'ang berubah !amiltonian akan menggambarkanNosilator

harmonik un/oupled.

Bentuk kuantisasi tergantung pada pilihan kondisi batas, untuk kesederhanaan, kamimemaksakan kondisi batasperiodik!mendeinisikan (N+1)th atom sebagai setara dengan atom

pertama" Secara isik, hal ini sesuai dengan bergabung rantai pada uung$uungn'a" .uantisasi

'ang dihasilkan adalah

Batas atas untuk nberasal dari panang gelombang minimum, 'ang dua kali kisi arak,

sepertiyan!dibahas di atas"

Nilai$nilai eigen osilator harmonik atau tingkat energi untuk modus k

Iika kita mengabaikan titik nolenergi maka tingkat merata spasi di

,2 ,3 ,...

Iadi umlah 'ang tepatdari energi,harus diberikan ke osilator harmonik kisi untuk

mendorong ke tingkat energi berikutn'a" Dibandingkan dengan otonkasus ketika medan

elektromagnetik'ang Cuantised, kuantum energi getaran disebut phonon"

Semua sistem kuantum menunukkan seperti gelombang dan partikel$seperti properti" #artikel$

seperti properti phonon adalah dipahami dengan menggunakan metode kuantisasi

kedua#engoperasian dan teknik dielaskan kemudian" /K1

Aplikasi / sunting1

etaran darimolekul diatomikadalah contoh dari &ersi dua$tubuh osilator harmonik

kuantum" Dalam hal ini, rekuensi sudut diberikan oleh
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11/11

dimana adalah mengurangi massadan ditentukan oleh

massa dari dua atom"/L1

Atom !ookeadalah model sederhana dari heliumatom menggunakan osilator harmonik kuantum

#emodelan onon, seperti dibahas di atas

harge, C, dengan massa, m, dalam medan magnet seragam, B, adalah contoh dari kuantum

osilator harmonik satu dimensi" /71

4eerensi / sunting1

0" ^riiths, Da&id I"(266)" 0engantar Quantum 1e/hani/s(2nd ed")" #rentice !all" :SBN6$03$

M6K32L$"

2" ^Eibo, 4ichard E"(2662)" 0engantar 1ekanika 2uantum.Addison$Wesle'":SBN6$M6K3$M70$K"

3" ^4ashid, =uneer A"(266L)">-ransisi amplitudo untuk bergantung waktu osilator harmonik linier

dengan Einear tergantung waktu istilah ditambahkan ke !amiltonian>(#D+$#ower#oint =icrosot) 1A

3ashid $. 4enter 5or Ad6an/ed 1atematika dan 7isika"#usat Nasional untuk +isika" Diakses 2606"

" ^#auli, W" (2666), 1ekanika 8elombang. 9olume : dari 0auli e/tures on 0h;si/s(Do&er Buku$

buku tentang +isika) :SBN O7M$6ML0L20"

K" ^=ahan, D (0OM0)" ban;ak 5isika partikel.New ork8" Springer :SBN636LL33MK"

L" ^>Quantum !armonic ?scillator>" H;per0h;si/s.Diakses pada 2 September 266O"

7" ^1ekanika kuantum teori non$relati6istik" Amsterdam8 Butterworth !einemann" 0O77" p" "

KK :SBNO7M66M6K63ML 66M6K63MO#eriksa

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Quantum Osilator Harmonik