Pozitsiya va Impuls fazosi: Versiyalar orasidagi farq

[[Kvant mexanika|Kvant mexanikasi]] pozitsiya va impuls o'rtasidagi ikkilanishning ikkita asosiy misolini keltiradi: Geyzenberg noaniqlik printsipi <math>\Delta</math>''x'' <math>\Delta</math>''p'' ≥ ''ħ'' /2 pozitsiya va impulsni bir vaqtning o'zida ixtiyoriy aniqlik bilan bilish mumkin emasligini va [[De Broyl toʻlqini|de Broyl munosabati]] '''p''' = ''ħ'''''k'''. erkin zarrachaning impulsi va to'lqin vektori o'zaro proportsionaldir.<ref>{{Kitobkitob manbasi|title=Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles|first=R.|last=Eisberg|edition=2nd|publisher=John Wiley & Sons|year=1985|isbn=978-0-471-87373-0|url=https://archive.org/details/quantumphysicsof00eisb}}</ref> Shu nuqtai nazardan, bir ma'noli bo'lsa, " [[impuls]] " va "to'lqin vektor" atamalari bir-birining o'rnida ishlatiladi. Biroq, de Broyl munosabati kristalda to'g'ri emas.

Ko'pincha Lagranj mexanikasida Lagrangian ''L'' ('''q''', ''d'' '''q''' / ''dt'', ''t'') konfiguratsiya maydonida bo'ladi, bu erda '''q''' = (''q'' ₁, ''q'' ₂ ,..., ''q _n'') umumlashtirilgan koordinatalarning <math>\eta</math> - tuplesidir.. Eyler-Lagranj harakat tenglamalari<math display="block">\frac{d}{dt}\frac{\partial L}{\partial \dot{q}_i} = \frac{\partial L}{\partial q_i} \,,\quad \dot{q}_i \equiv \frac{dq_i}{dt}\,. </math>(Bir haddan tashqari nuqta bir martalik hosilani bildiradi). Har bir umumlashtirilgan koordinata uchun kanonik impuls ta'rifini kiritish<math display="block"> p_i = \frac{\partial L}{\partial \dot{q}_i} \,, </math>Eyler-Lagranj tenglamalari shaklni oladi<math display="block">\dot{p}_i = \frac{\partial L}{\partial q_i} \,. </math>Lagrangian impuls fazosida ham ifodalanishi mumkin,<ref>{{Kitobkitob manbasi|url=https://books.google.com/books?id=1J2hzvX2Xh8C|title=Analytical Mechanics|isbn=978-0-521-57572-0|last=Hand|first=Louis N|date=1998|page=190}}</ref> ''L'' ′('''p''', ''d'' '''p''' / ''dt'', ''t''), bu erda '''p''' = (''p'' ₁, ''p'' ₂,..., ''p _n'') ''n'' -tupledir. umumlashtirilgan moment. Lagranjning umumlashtirilgan koordinata fazosining umumiy differentsialidagi o'zgaruvchilarni o'zgartirish uchun Legendre transformatsiyasi amalga oshiriladi;<math display="block">dL = \sum_{i=1}^n \left(\frac{\partial L }{\partial q_i}dq_i + \frac{\partial L }{\partial \dot{q}_i}d\dot{q}_i\right) + \frac{\partial L }{\partial t}dt = \sum_{i=1}^n (\dot{p}_i dq_i + p_i d\dot{q}_i) + \frac{\partial L }{\partial t}dt \,, </math>Bu erda umumlashtirilgan impulsning ta'rifi va Eyler-Lagrange tenglamalari ''L'' ning qisman hosilalari o'rnini egalladi. Differensiallar uchun mahsulot qoidasi<ref group="nb">For two functions {{Math|''u''}} and {{Math|''v''}}, the differential of the product is {{Math|1=''d''(''uv'') = ''udv'' + ''vdu''}}.</ref> umumlashtirilgan koordinatalar va tezliklardagi differentsiallarni umumlashtirilgan momentdagi differentsiallar va ularning vaqt hosilalari bilan almashish imkonini beradi,<math display="block">\dot{p}_i dq_i = d(q_i\dot{p}_i) - q_i d\dot{p}_i </math><math display="block"> p_i d\dot{q}_i = d(\dot{q}_i p_i) - \dot{q}_i d p_i </math>almashtirilgandan so'ng soddalashtiriladi va qayta tartibga solinadi<math display="block"> d\left[L - \sum_{i=1}^n(q_i\dot{p}_i + \dot{q}_i p_i)\right] = -\sum_{i=1}^n (\dot{q}_i d p_i + q_i d\dot{p}_i) + \frac{\partial L }{\partial t}dt \,. </math>Endi impuls fazosining to'liq differensiali Lagranjian ''L'' ′ dir

[[Kvant mexanika|Kvant mexanikasida]] zarracha kvant holati bilan tavsiflanadi. Bu kvant holatini asosiy holatlarning superpozitsiyasi (ya'ni vaznli yig'indi sifatida chiziqli birikma) sifatida ko'rsatish mumkin. Printsipial jihatdan bazaviy holatlar to'plamini tanlash erkindir, agar ular bo'sh joyni qamrab olsa. Pozitsiya operatorining xos funksiyalari bazis funksiyalar toʻplami sifatida tanlansa, holat haqida pozitsiya fazosida [[To'lqin funksiya|toʻlqin funksiyasi]] {{Math|''ψ''('''r''')}} sifatida soʻz boradi (uzunlik boʻyicha [[fazo]] haqidagi oddiy tushunchamiz). '''r''' pozitsiyasi bo'yicha tanish bo'lgan Shredinger tenglamasi pozitsiyani tasvirlashda kvant mexanikasiga misoldir.<ref name="peleg">{{Kitobkitob manbasi|title=Quantum Mechanics (Schaum's Outline Series)|first=Y.|last=Peleg|edition=2nd|publisher=McGraw Hill|year=2010|isbn=978-0-07-162358-2}}</ref>

Bazis funksiyalar toʻplami sifatida boshqa operatorning xos funksiyalarini tanlab, bir xil holatning bir qancha turli koʻrinishlariga erishish mumkin. Agar impuls operatorining xos funktsiyalari bazis funktsiyalar to'plami sifatida tanlansa, natijada to'lqin funksiyasi <math>\phi(\mathbf{k})</math> impuls fazosidagi to’lqin funksiyasi deyiladi.<ref name="peleg">{{Kitobkitob manbasi|title=Quantum Mechanics (Schaum's Outline Series)|first=Y.|last=Peleg|edition=2nd|publisher=McGraw Hill|year=2010|isbn=978-0-07-162358-2}}<cite class="citation book cs1" data-ve-ignore="true" id="CITEREFPelegPniniZaarurHecht2010">Peleg, Y.; Pnini, R.; Zaarur, E.; Hecht, E. (2010). ''Quantum Mechanics (Schaum's Outline Series)'' (2nd&nbsp;ed.). McGraw Hill. [[ISBN]]&nbsp;[[Maxsus:Kitob manbalari/978-0-07-162358-2|<bdi>978-0-07-162358-2</bdi>]].</cite></ref>

To'lqin funktsiyasining momentum ifodasi Furye transformatsiyasi va chastota sohasi tushunchasi bilan juda chambarchas bog'liq. Kvant-mexanik zarracha impulsga mutanosib chastotaga ega boʻlganligi uchun (yuqorida berilgan de-Broyl tenglamasi), zarrachani impuls komponentlari yigʻindisi sifatida tasvirlash uni chastota komponentlari yigʻindisi (yaʼni Furye konvertatsiyasi) sifatida tasvirlashga tengdir.<ref>{{Kitobkitob manbasi|title=Quantum Mechanics|first=E.|last=Abers|publisher=Addison Wesley, Prentice Hall Inc|year=2004|isbn=978-0-13-146100-0}}</ref> Bu biz o'zimizdan qanday qilib bir vakillikdan ikkinchisiga o'tishimiz mumkinligini so'raganimizda aniq bo'ladi.

Aksincha, impuls fazosida uch o'lchovli to'lqin funktsiyasi <math>\phi(\mathbf{k})</math> ortogonal bazis funktsiyalarining vaznli yig'indisi sifatida ifodalanishi mumkin <math>\phi_j(\mathbf{k})</math> ,<math display="block">\phi(\mathbf{k}) = \sum_j \psi_j \phi_j(\mathbf{k}),</math>yoki integral sifatida,<math display="block">\phi(\mathbf{k}) = \int_{\mathbf{r}\text{-space}} \psi(\mathbf{r}) \phi_{\mathbf{r}}(\mathbf{k}) \mathrm d^3\mathbf{r}.</math>Pozitsiya operatori tomonidan berilgan<math display="block">\mathbf{\hat r} = i \hbar\frac{\partial}{\partial \mathbf p} = i\frac{\partial}{\partial \mathbf{k}}</math>xos funksiyalar bilan<math display="block">\phi_{\mathbf{r}}(\mathbf{k}) = \frac{1}{\left(\sqrt{2\pi}\right)^3} e^{-i \mathbf{k}\cdot\mathbf{r}}</math>va xos qiymatlar '''r'''. Shunday qilib, shunga o'xshash parchalanish <math>\phi(\mathbf{k})</math> teskari Furye konvertatsiyasi bo'lib chiqadigan ushbu operatorning xos funktsiyalari nuqtai nazaridan amalga oshirilishi mumkin,<ref name="Penrose">{{Kitobkitob manbasi|last=R. Penrose|title=[[The Road to Reality]]|publisher=Vintage books|year=2007|isbn=978-0-679-77631-4}}</ref><math display="block">\phi(\mathbf{k})=\frac{1}{(\sqrt{2\pi})^3} \int_{\mathbf{r}\text{-space}} \psi(\mathbf{r}) e^{-i \mathbf{k}\cdot\mathbf{r}} \mathrm d^3\mathbf{r}.</math>