清华大学精品课程 量子力学
Upcoming SlideShare
Loading in...5
×
 

Like this? Share it with your network

Share

清华大学精品课程 量子力学

on

  • 5,103 views

 

Statistics

Views

Total Views
5,103
Views on SlideShare
5,060
Embed Views
43

Actions

Likes
1
Downloads
42
Comments
0

3 Embeds 43

http://littlesujin.yo2.cn 21
http://littlesujin.wordpress.com.cn 19
http://www.slideshare.net 3

Accessibility

Categories

Upload Details

Uploaded via as Adobe PDF

Usage Rights

© All Rights Reserved

Report content

Flagged as inappropriate Flag as inappropriate
Flag as inappropriate

Select your reason for flagging this presentation as inappropriate.

Cancel
  • Full Name Full Name Comment goes here.
    Are you sure you want to
    Your message goes here
    Processing…
Post Comment
Edit your comment

清华大学精品课程 量子力学 Presentation Transcript

  • 1. Quantum Mechanics I L + D&E + S Lecture + Discussion&Exercise + Seminar Professor: 庄鹏飞 (High Energy Nuclear Physics) Doctor students: 何联毅 (High Energy Nuclear Physics) 屈真 (High Energy Nuclear Physics) 梅佳伟 (High Temperature Superconductivity) 清华大学精品课程, 北京市精品课程
  • 2. 1.充分认识量子力学在科学研究中的重要性 量子力学(高等量子力学),量子场论: 原子分子物理,光学,凝聚态,核物理,粒子物理,…… 结论: 没有量子力学,几乎不能做任何物质科学研究! 2.充分认识学习量子力学的困难 1)经典物理在日常生活中有对应现象, 量子力学很难找到 日常生活对应 2)量子力学与经典物理的思想方法有本质不同 3)既难于理解,也难于处理,需要更多数学 3.有哪些要求 1) 分析力学,高等数学 2) 勤于思考,多做习题
  • 3. 4.我们的教学模式 讲授(Lecture)+讨论与习题(Discussion & Exercise)+专题研究(Seminar) 世界一流大学理论物理教学的通用模式 4.1: L 大班上课: 强调基本概念,基本思想,例如 Hilbert空间表述, Dirac符号,测量理论, 对称性, 等等. 参考教材: Griffiths, Sakurai,苏汝铿,曾谨言,张永德,等 4.2: D&E 小班讨论 内容: 1)联系授课(L)内容,TA提示问题或学生提示问题,讨论; 2)难题解答 特点: 1)师生共同正确,深刻理解QM; 2)有机会使学生对问题提出自己的看法(L被动,D&E和S主动) 3)理论联系实际,解答困难习题; 4)规范,开放的讨论氛围.
  • 4. 4.3: 量子力学网站: http://qm.phys.tsinghua.edu.cn 课程介绍,教师与TA联系方式,讲义,作业,答案,通知,其他 3个讨论区: 量子力学一般问题,量子力学高级论坛,量子力学教学建议 4.4: S 内容: 与科学研究相关的小课题 目的: 深入理解,应用知识,专深发展,学习科研方法,进行科研训练,培养科 学精神 方式: 教授出题,学生选题(也可以学生自己找题),教授指导,学生调研,解 决问题(?),最后报告 4.5: 时间分配 1) L,D&E在本学期,必修,4学分,共64学时,其中L为48学时,D&E为16学 时,L/D&E=3/1 2) S在下学期,选修,2学分
  • 5. 4.6: 考试方式 60-70%期末考试 + 20-30%讨论课成绩 + 10%习题 4.7: TA 共4个TA, 3个讨论课TA, 1个on-line TA 讨论课TA: 何联毅 负责基科51,52,53 共24人 屈 真 负责基科54,55,56,物理41,42 共24人 梅佳伟 负责其它22人 1)每两周主持1次小班讨论课, 2)作业全改(每周一按小班交作业至物理系,同时取回上次作业,每人准备2 个作业本), 3)on-line答疑 4)经常性的联系 on-line TA: 郝学文 1)量子力学网站运行与维护 2)on-line答疑 3)协助改作业
  • 6. 游戏规则 教师与TA: 必须认真负责 学生: L: 可来可不来,可早退,但不可影响别人。 S:下学期,可参加,可不参加,姜太公钓鱼,愿者上钩。 D&E: 必须参加。 作业: 必须交。 多看量子力学网站:http://qm.phys.tsinghua.edu.cn
  • 7. 第一章 波函数 1.1 波粒二象性 什么是波粒二象性?是指几何形状,还是指运动形态? 1)光的波粒二象性 h ε = hν , p= λ 其中, ε 、 p 是粒子的物理量,ν 、 λ 是波动物理量。 波粒二象性是指物理量的取值既具有粒子性,也具有波动性。不是指几何形状,也不是指运动形 态。 2)原子的量子论描述 a. 电子具有确定的分离轨道。“确定”是经典的,“分离”是量子的。(经典轨道是连续的) b. 跃迁 hν = Em − En ,体系的性质与两条轨道的关联相关 → 矩阵力学。 “轨道”是经典的, “两条” 是量子的。矩阵 → 不对易。 c. 跃迁几率:量子论不能给出结果。 → 量子力学 问题:如何自洽地描述微观粒子的运动? 1.2 电子双缝衍射实验 1
  • 8. 实验结果: 只开缝 1,强度分布为 I1 ( x ) = ψ 1 ( x ) ; 2 I 2 ( x ) = ψ 2 ( x) ; 2 只开缝 2, I = ψ 1 ( x) + ψ 2 ( x ) ≠ I1 + I 2 ,电子具有衍射特性,波动性。 2 同时开缝 1 和 2, 实验分析: 一次只发射一个电子,屏上开始出现随机的光斑分布,长时间后出现衍射条纹。 光斑说明粒子性,但随机说明统计性,故不是经典粒子,而是统计意义上的粒子; 衍射条纹说明波动性,但只有长时间才有统计性,故不是经典波动,而是统计意义上的波动; 合起来说明粒子的位置力学量具有统计意义上的波粒二象性。 一个电子说明波粒二象性是微观粒子的固有特性,不是多个粒子相互作用的结果。 总结: 1)观察物理量 (x ) 的取值时既观察到粒子性质,又观察到波动性。 粒子性:物理量的取值具有颗粒性,一份一份的; 波动性:物理量的取值不确定; 2)粒子性与波动性都是从力学量取值的统计意义来理解,不是指运动的空间位形。 注意: 此处的统计根源与经典统计不同。每次发射一个电子,即使初态完全相同,也仍具有统计意义上 的波粒二象性,而每一次丢一枚硬币,若初始条件完全相同,则每一次结果同。 1.3 Born 统计解释(将力学量 x 取值的粒子性与波动性统一起来) r 引入几率波函数ψ r,t), ( ⎧ 波幅的平方 ψ r,t) 2 ⎪ (r 波动性 衍射条纹强度 ∝ ⎨ , r ρ ⎪粒子出现的几率( r,t) 粒子性 ⎩ r 那么微观粒子在 t 时刻位于 r 的几率密度为 r r r r ρ r,t) ψ r,t) = ψ r,t)( r,t) ψ ∝( 2 ( ( * 注意波函数一般为复函数。 r 基本量是波函数ψ ,虽然本身不是可观察物理量,但它描述物理量 r 取值的几率。 2
  • 9. 1.4 几率波的一般性质 1)几率归一化 粒子在全空间出现的几率为 1。 r r ∫ d r ψ (r,t ) 2 = A < ∞ ,波函数平方可积 a) 若 3 2 r1 r r r r2 1 1 ψ ( r , t ) = 1 ,称 ψ ( r , t ) 为归一化波函数, ρ (r , t ) = ψ ( r , t ) 。 则 ∫d r3 A A A b) 对于某些理想(非物理)情况,波函数不能归一,例如: r rr ψ (r ,t ) = e r i ( k ⋅r −ωt ) ,波矢 k ,频率ω 。 r r2 此时 ∫ d 3 r ψ ( r , t ) = ∞ ,波函数平方不可积。 但是不能归一并不影响相对几率 r ψ ( r1 , t ) 2 与归一化无关 r ψ ( r2 , t ) 2 以后要讨论它们的归一化问题,可以用箱归一化。 c) 注意: *) 在统计解释中,ψ 的意义是通过 ψ 来定义的,ψ 本身无意义。归一化后,ψ 仍有相位不确定 2 性 r r ψ ( r , t ) = eiαψ ( r , t ) 2 2 , 统计解释是否包含了波函数全部信息? *)经典波无归一化问题 ψ 和 Cψ 是完全不同的,后者能量密度是前者 C2 倍。 2)经典粒子:确定的力学量 q, p 。 r r2 量子粒子:力学量(例如位置)不确定,只有几率确定 ρ (r , t ) ∝ ψ ( r , t ) ,导致平均值确定 3r r r2 ∫ d r r ψ (r ,t ) 。 r r (t ) = 3r r2 ∫ d r ψ (r ,t ) ⇒ 经典力学中力学量 F 的规律应该对应于量子力学中<F>的规律 1
  • 10. 例如 E = T + V → E =T +V r 3)力学量的几率分布确定 → ψ ( r , t ) 单值; r 力学量的几率分布有限 → ψ ( r , t ) 有限; r 几率分布连续 → ψ ( r , t ) 连续。 一般情况下, 但不排除存在个别孤立奇点,几率分布不连续(以 后详细讨论)。 总结: r 归一、单值、连续、有限是一般条件下几率解释对ψ ( r , t ) 的物理约束条件。 1.5 Schrödinger 方程 r 1)几率波ψ ( r , t ) 的时空演化 Schrödinger,1926: h2 r 2 ∂ r r r ih ψ ( r , t ) = ( − ∇ + V ( r, t ))ψ ( r , t ) ∂t 2m 对于自由粒子, h2 r 2 r ∂ r ψ (r,t ) = − ∇ ψ (r ,t ) ih ∂t 2m 可以证明平面波 i rr r rr ( p⋅r − Et ) r ψ ( r , t ) = Ae r i ( k ⋅r −ωt ) = Ae (由 De Brogile 关系 ε = hω , p = hk ) h 是自由 Schrödinger 方程的解。 注意: r a) 虽然一般情形时力学量取值不确定,但平面波具有确定动量 p 和能量 E。 b) S-方程是基本运动方程,地位如同经典力学中的牛顿方程,不可能推出,是量子力学基本假定 之一; r c) 方程包含因子 i,要求ψ ( r , t ) 为复函数,否则方程两边一边为虚函数,一边为实函数。所以平 rr rr i ( k ⋅r −ωt ) ,不能是 ACos(k ⋅ r − ωt ) 。 Ae 面几率波只能是 2)几率守恒 2
  • 11. h2 r 2 ∂ ψ = (− ∇ + V )ψ ih ∂t 2m h2 r 2 ∂ ψ = (− ∇ + V )ψ ∗ , ∗ (V = V ∗ ) -i h ∂t 2m r r r r2 可以证明,几率密度 ρ r,t)= ψ(r , t ) = ψ r,t)( r,t) ψ 满足连续性方程 ( ( * rrr ∂ r ρ r,t)+∇ ⋅ j ( r ,t ) = 0 , ( ∂t rr −i h ∗ r r j ( r ,t ) = (ψ ∇ψ −ψ∇ψ ∗ ) 2m r j 的物理意义时什么? 对有限空间积分: rr r r r∂ r d 3 r ρ r,t)+ ∫ d 3 r∇ ⋅ j ( r ,t ) = 0 , ∫V ∂t ( V rrr rr d d 3 r ρ r,t)=- ∫r dS ⋅ j ( r ,t ) dt ∫V ( S r r 定域几率守恒:区域 V 内几率的变化=流出面积S 的几率,故称 j 为几率流密度。 r r rr r ∂ r ρ( r,t)+∇ ⋅ jm ( r ,t ) = 0 ρ m = mρ , jm = mj 定域质量守恒: ∂t m r r rrr ∂ r 定域电荷守恒: ρ e r,t)+∇ ⋅ je ( r ,t ) = 0 ρe = eρ , je = ej ( ∂t 位置的不确定,导致质量、电荷分布的不确定,按几率分布。 若对整个空间积分: rrr rr d d 3 r ρ r,t)=- ∫ dS ⋅ j ( r ,t ) , dt ∫∞ ( ∞ 由于 rrr dS ⋅ j ( r ,t ) = 0 ∫ ∞ 故 rr ∫ d 3 r ρ r,t) ( 与时间无关,是一常数。 ∞ 3
  • 12. 意味着 a)若几率波是可以归一的,则归一化与时间无关。S-方程保证了归一性不随时间而变。 b)总几率守恒,无粒子的产生与消灭,S-方程描述的是非相对论量子力学。 rrr r r c)由 j 的形式, ∫ dS ⋅ j ( r ,t ) = 0 意味ψ ( r → ∞, t ) → 0 。 ∞ r 3)若 V 中不含与波函数相关的量,S-方程是关于ψ ( r , t ) 的线性方程。若ψ 1 ,...ψ m 是方程的解,则 它们的任意线性迭加仍是方程的解。 1.6 态函数、测量与态叠加原理 1)态函数 r2 r 粒子的位置几率分布 ψ ( r , t ) ,其他力学量的取值几率?例如动量。如果几率波只能给出 r 的 几率分布,而不能给出其他力学量的几率分布,则几率波不能完全确定体系的状态。如果几率波 r r 能给出所有物理量的几率分布,则可称ψ ( r , t ) 为体系的态函数。知道了ψ ( r , t ) ,则知道了体系力 学量的所有性质。 对于平面几率波,动量有确定取值。对于任意的几率波,频率、波矢不确定,故动量、能量 不确定,但可以由平面波展开(付里叶展开): r i rr d 3p r r ∞ ( p ⋅r − E t ) ψ (r , t ) = ∫ (2π h ) ϕ p , t) e ( h 3 /2 −∞ r i rr d 3r r r ∞ − ( p ⋅r − E t ) ψ (r , t ) e ∫ (2π h ) ϕ p , t)= ( h 3/2 −∞ 此处引入因子 1/ ( 2π h ) 是考虑到平面波的 δ 函数归一化 3/ 2 irr r r p ⋅r 1 = δ ( p) ∫d r 3 h e ( 2π h ) 3/ 2 r r 问题:ψ ( r , t ) 是位置几率幅, ϕ p, t)的物理意义是什么? ( 由 r rr r2 ∞ r = ∫ d 3r r ψ ( r , t ) −∞ rr∂ dr r2 ∞ r = ∫ d 3r r ψ ( r , t ) ∂t −∞ dt 由 S-方程 4
  • 13. r ih ∞ 3 r r r r r ( ) dr 2m ∫−∞ d r r ∇ ⋅ ψ ∗∇ψ −ψ∇ψ * = dt r 由分部积分,并考虑ψ ( r → ∞, t ) → 0 , r ih ∞ 3 r ∗ r r dr 2m ∫−∞ d r (ψ ∇ψ −ψ∇ψ * ) =− dt 再对括号中第二部分进行分部积分, r ih ∞ r r dr = − ∫ d 3 rψ ∗∇ψ m −∞ dt 由于平均值满足经典力学规律, r r r r r r dr ∞ dr = ∫ d 3 rψ ∗ (−ih∇)ψ p=m ⇒ p ≡m −∞ dt dt r 代入ψ ( r , t ) 的付里叶展开式 rr r r irr irr d 3rd 3 p1d 3 p2 ∗ r r r − (p1 ⋅r − Et ) (p 2 ⋅r − Et ) ϕ ( p1 , t ) e h (−ih∇)ϕ ( p 2 , t ) e h p =∫ (2π h) 3 r 3r 3r r i ( p2 − p1 )⋅rr rr d 3 rd p1d p2 ∗ r r ϕ ( p1 , t ) ϕ ( p 2 , t ) p2 e h =∫ (2π h)3 r r rr rrr = ∫ d 3 p1d 3 p2ϕ ∗ ( p1 , t ) ϕ ( p 2 , t ) p2δ (p2 − p1 ) r2 rr = ∫ d 3 p p ϕ ( p, t ) 与 r rr r2 ∞ r = ∫ d 3r r ψ ( r , t ) −∞ r r r2 r 进行比较,知 ϕ ( p, t ) 是动量取值为 p 的几率。由于ψ ( r , t ) 确定时, ϕ ( p, t ) 确定,并且以后可以 r r 即给定ψ ( r , t ) , 故称几率波ψ ( r , t ) 证明,其他力学量的取值几率也是确定的, 态的性质就确定了。 r r r 为态函数。由于给定 ϕ ( p, t ) 时,ψ ( r , t ) 亦确定,故 ϕ ( p, t ) 也可以称之为系统的态函数。 5
  • 14. 2 ψ (r,t ) 2 ψ (r ) ψ ( r ) ∝ δ ( r − r0 ) r0 ψ (r ) δ ( r − r0 ) ψ (r ) open 3 ψ (r ) ψ (r ) ψ1 (r ) ψ 2 (r ) ψ1 (r ) ψ 2 (r ) … … ψ1 ψ 2 …ψ n ψ1 ψ 2 …ψ n Schrodinger 1
  • 15. 1 S- 2 ψ (r,t ) ϕ ( p, t ) 3 2.1 Hilbert 1 3 en , n = 1, 2,3 3 A = ∑ an en n =1 A ⋅ B = ∑ anbm en ⋅ em n,m A⋅ A ≥ 0 en ⋅ em = δ nm ⎛ a1 ⎞ ⎜⎟ A ⋅ B = ∑ an bn = AB A = ⎜ a2 ⎟ , A = (a1 , a2 , a3 ) ⎜a ⎟ n ⎝ 3⎠ an = en ⋅ A 2 Hilbert 3 → 3 → 2
  • 16. Dirac a a a a n f ⎧ ∑ an n ⎧ ∑ an n* ⎪ ⎪ a =⎨ n a =⎨ n ⎪ ∫ df a f f ⎪ ∫ df a* f ⎩ ⎩ f ⎧ ∑ anbm n m * ⎪ a b = ⎨ n,m ⎪ ∫ dfdf ' a* b f ' f f ' ⎩ f aa ≥ 0 n m = δ nm f f ' = δ ( f − f ') ⎛ a1 ⎞ ⎧ ∑ anbn = a +b * ⎜⎟ ⎪ a ↔ a + = ( a1 an ) a a ↔a=⎜ 2⎟ a b =⎨ n * * * a2 ⎜⎟ ⎪ ∫ dfa* b f = a +b ⎩ ⎜⎟ f ⎝ an ⎠ am = m a af = f a a = ∑ an n = ∑ n a n = ∑ n n a n n n ∑n n =1 a n ∑n n = n n+ 1 n 3D ⎛1⎞ ⎛0⎞ ⎛ 0⎞ ⎜⎟ ⎜⎟ ⎜⎟ 1 = ⎜0⎟, 2 = ⎜1⎟, 3 = ⎜ 0⎟, ⎜0⎟ ⎜0⎟ ⎜1⎟ ⎝⎠ ⎝⎠ ⎝⎠ 1 = (1 0 0 ) , 2 = ( 0 1 0) , 3 = ( 0 0 1) 3
  • 17. ⎛1⎞ ⎛0⎞ ⎛0⎞ ⎜⎟ ⎜⎟ ⎜⎟ ∑ n n = ⎜ 0 ⎟ (1 0 0 ) + ⎜ 1 ⎟ ( 0 1 0 ) + ⎜ 0 ⎟ ( 0 0 1) ⎜0⎟ ⎜0⎟ ⎜1⎟ n ⎝⎠ ⎝⎠ ⎝⎠ ⎛1 0 0⎞ ⎛ 0 0 0⎞ ⎛ 0 0 0⎞ ⎜ ⎟⎜ ⎟⎜ ⎟ = ⎜0 0 0⎟ + ⎜ 0 1 0⎟ + ⎜0 0 0⎟ ⎜0 0 0⎟ ⎜0 0 0⎟ ⎜0 0 1⎟ ⎝ ⎠⎝ ⎠⎝ ⎠ ⎛1 0 0⎞ ⎜ ⎟ = ⎜0 1 0⎟ = I ⎜0 0 1⎟ ⎝ ⎠ ∫ df f =1 f 4
  • 18. 2. Hilbert T (α a + β b ) = αT a + βT b ˆ ˆ ˆ ˆ T ˆ T a → T a = a' ˆ α ˆ n T n → n' = T n = ∑ m m T n ˆ ˆ m Tmn = m T n ˆ T n = ∑ Tmn m ˆ m Tmn = m T n ˆ ˆ Hilbert T a ⎛ ⎞ a → a ' = T a = ∑ m m T n n a = ∑ ⎜ ∑ Tmn an ⎟ m ˆ ˆ m⎝n ⎠ m,n a n an a = ∑ n n a = ∑ an n n n a' = ∑ a m m ' m a m = ∑ Tmn a n ' n ⎛ a1' ⎞ ⎛ a1 ⎞ ⎛ T11 . . T1N ⎞ ⎜⎟ ⎜⎟ ⎜ ⎟ ⎜.⎟ ⎜.⎟ ⎜. .. .⎟ a' = Ta a=⎜ ⎟ a' = ⎜ ⎟ T =⎜ .. .⎟ . . ⎜.⎟ ⎜⎟ ⎜ ⎟ ⎜a ⎟ ⎜T . . TNN ⎟ ⎜ a' ⎟ ⎝ N⎠ ⎝ N1 ⎠ ⎝ N⎠ 1
  • 19. 3. T a =λ a ˆ λ a Ta = λa (T − λI )a = 0 a≠0 det(T − λ I ) = 0 T11 − λ T12 T1 T22 − λ T21 T2 =0 TN 1 TN 2 TNN λ N (T − λI )a = 0 a ⎛1 0 ⎞ T=⎜ ⎟ ⎝ 0 -1⎠ 1λ 0 λ 2 −1 = 0 =0 -1-λ 0 λ ±1 ⎛a ⎞ a=⎜ 1⎟ ⎜a ⎟ ⎝ 2⎠ ⎛ 0 0 ⎞ ⎛ α1 ⎞ ⎛1⎞ λ =1 ⎟⎜ ⎟ ⎜ ⎜⎟ 0 ⎝ 0 −2 ⎠⎝ α 2 ⎠ ⎝0⎠ ⎛0⎞ λ1 ⎜⎟ ⎝1⎠ 4. ~ Tij+ = T ji , T + = T *, * T 2
  • 20. (T ) T + =T , + =Tij ij T ∵ ( AB) + = B + A+ 1 () ( ) ( ) ( )b () + ∴ a T b ≡ a T b = a +Tb = T + a b = T + a b ≡ a T + b = a T ˆ ˆ ˆ ˆ ˆ T =T+ ˆ ˆ ˆˆ T T ( ) ( )b a T b = aT ˆ ˆ ˆ aT b T i =λ i ˆ ˆ 2 T i T = i λ* ˆ i T i = i λ* i ˆ λ i i = λ* i i λ = λ* 3 T i =λi i T j =λ j j λi ≠ λ j ˆ ˆ i T j =λ j i j ˆ ∵ λi* i j =λ j i j (λ − λ ) i j =0 i j λi ≠ λ j i j =0 i j =δ ij f f ' δ ( f − f ') g T i, j =λi i, j j = 1,...g ˆ 3
  • 21. a g g i, n = ∑ Cnj i, j n = 1, 2,...g j =1 g g ∵ T i, n = ∑ CnjT i, j =λi ∑ Cnj i, j =λi i, n ˆ ˆ j =1 j =1 λi ∴ i, n ˆ T C nj g i, m i, n = δ mn g ( g + 1) g2 − g g2 + = g Cnj 2 2 i, n Cnj i, m j , n δ ijδ mn δ mn δ ij b ˆ ˆ ˆ T' T T' T i, j =λi i, j ˆ T ' i, j =λij i, j ˆ {Tˆ , Tˆ '} {λ , λ } i, j i ij i, j i ', j ' δ ii 'δ jj ' , 4 4
  • 22. ∑i i = 1, i ∫ df f =1 f ∑i i + ∫ df f f =1 i Hilbert 5
  • 23. 5. 3 Hilbert I i M m 1 i =∑ m m i m S Smi = m i I M 2 a =∑ i i a I i I ia a i a a =∑ m m a M m M ma a m a m a = ∑ m i i a = ∑ Smi i a i i aM = S a I 3 ˆ I Tij = i T j Tmn = m T n = ∑ m i i T j j n = ∑ m i i T j n j = ∑ SmiTij Snj = ∑ SmiTij S + * ˆ ˆ ˆ * M jn i, j i, j i, j i, j TM = STI S + S 1
  • 24. 4 (S S ) = ∑ S Smj = ∑ Smi Smj = ∑ m i m j =∑ i m m j = i j = δ ij + + * * im ij m m m m S +S = 1 SS + = 1 S + = S −1 S+ ≠ S S 5 = λI a ˆ TI a I I = S λI a = λI S a = λI a = STI S + S a = STI a ˆ ˆ ˆ TM a M I I I I M λI M 6. 1 Hilbert ψ ψ Hilbert 2 r = ∫ d 3 r r ψ ( r , t ) = ∫ d 3 rψ * ( r , t ) rψ ( r ,t ) 2 p = ∫ d 3 rψ ∗ ( r , t ) (−i ∇)ψ ( r , t ) r =r ˆ p = −i ∇ ˆ r = ∫ d 3 rψ * ( r , t ) r ψ ( r , t ) ˆ p = ∫ d 3 rψ ∗ ( r , t ) p ψ ( r , t ) ˆ O r p O(r , p ) 2
  • 25. O = O(r , p) ˆˆ ˆ O = ∫ d 3rψ ∗ ( r , t ) O(r , p)ψ ( r , t ) ˆˆ ψ ˆ Hilbert F F =ψ Fψ ˆ 3 ψ Fψ =ψ Fψ , * ˆ ˆ ( ) = ⎡ ψ ( F ψ )⎤ = ( ψ ) * ψ Fψ F+ ψ ˆ ˆ ˆ ⎣ ⎦ ψ F + =F ˆˆ 4 ψ F ≡ψ Fψ ˆ F ( ) ( ∆F ) 2 ≡ ψ F− F ψ ≠0 2 ˆ δ φ ˆ F F ( ) 2 φ F− F φ =0 ˆ 3
  • 26. ( )( F - F ) φ φ F- F =0 ˆ ˆ ˆ ∵F- F ) = (( F − F ) φ ) ( + ∴ φ F- F ˆ ˆ (F- F ) φ Fφ = F φ =0 ˆ ˆ ˆ ˆ F F F ψ ψ ˆ ˆ F F F ψ ψ F F n Fn ψ F 5 F n = Fn n ˆ F Hilbert F n m n = δ mn ψ F =ψ Fψ ˆ F = ∑ ψ n n F m m ψ = ∑ ψ n Fm n m m ψ ∑ n ψ Fn = ∑ Fn n ψ 2 nψ * ˆ m,n m ,n n n nψ ψ ψ F n ψ ψ 2 nψ F n Fn 2 nψ 4
  • 27. 7. 1) x x =x x ˆ x x x ' = δ ( x − x ') ∫ dx x x =1 ψ = ∫ dx x x ψ ψ x ψ ≡ ψ ( x) ψ x = ψ x ψ = ∫ dxdx ' ψ x x x x ' x ' ψ ˆ ˆ ˆ x xxx ' ≡ x x x ' = x ' x x ' = δ ( x − x ') x ' = δ ( x − x ') x ˆ x = ∫ dxdx ' x ψ δ ( x − x ') x x ' ψ * = ∫ dx x ψ x xψ * = ∫ dxψ * ( x) xψ ( x) p = ψ p ψ = ∫ dxdx ' ψ x x p x ' x ' ψ ˆ ˆ ˆ p p xx ' ≡ x p x ' = ∫ dpdp ' x p p p p ' p ' x ' ˆ ˆ p p p ' = δ ( p − p ') p ˆ i 1 px xp= e xp 2π i i − p'x' 1 px pxx ' = ∫ dpdp ' δ (p − p ')pe e 2π i p ( x − x ') 1 = ∫ dp pe 2π ∂ ⎞ i p ( x − x ') 1⎛ = ∫ dp ⎜ −i ⎟e 2π ⎝ ∂x ⎠ ∂⎞ ⎛ 1 i p ( x − x ') ∂x ⎠ ∫ 2π = ⎜ −i ⎟ dp e ⎝ ∂ δ (x − x ') = −i ∂x 1
  • 28. ∂⎞ ⎛ p = ∫ dxdx 'ψ * ( x) ⎜ −i ⎟ δ ( x − x ')ψ ( x ') ∂x ⎠ ⎝ lim ψ ( x ) = 0 x →±∞ ∂* p = ∫ dxdx ' δ ( x − x ')ψ ( x ') i ψ ( x) ∂x ∂* = ∫ dxψ ( x) i ψ ( x) ∂x ∂⎞ ⎛ p = ∫ dxψ * ( x) ⎜ i ψ ⎟ ( x) ∂x ⎠ ⎝ ∂⎞ ⎛ pxx ' = δ ( x − x ') ⎜ −i ⎟ ∂x ⎠ ⎝ xxx ' = δ ( x − x ') x ∂⎞ ⎛ pxx ' = δ ( x − x ') ⎜ −i ⎟ ∂x ⎠ ⎝ O ( x, p ) ˆˆ ∂⎞ ⎛ Oxx ' = x O x ' =δ (x − x ')O ⎜ x, −i ˆ ⎟ ∂x ⎠ ⎝ 2) ⎛ ∂⎞ x pp ' = p x p ' = δ ( p − p ') ⎜ i ⎟, ˆ ⎝ ∂p ⎠ p pp ' = p p p ' = δ ( p − p ') p, ˆ ⎛∂ ⎞ O pp ' = p O p ' = δ ( p − p ')O ⎜ i ˆ , p⎟ ⎝ ∂p ⎠ 8. ˆ 1 ˆ x p xxx ' = δ ( x − x ') x xxx ' = ( xx ' x ) = (δ ( x '− x) x ') = δ ( x '− x) x ' = δ ( x − x ') x = xxx ' ∗ ∗ + ⎛ ⎞ ∂⎞ ∂ ⎛ pxx ' = δ ( x − x ') ⎜ −i ⎟ = δ ( x − x ') ⎜ −i ⎟ ⎜ ⎟ ∂ ( x − x ') ⎠ ∂x ⎠ ⎝ ⎝ 2
  • 29. ∗ ⎛ ∂ ⎞⎞ ⎛ ∂⎞ ⎛ ∂⎞ ⎛ = ( px ' x ) ∗ = ⎜ δ ( x '− x) ⎜ −i ⎟ ⎟ = δ ( x '− x) ⎜ i ⎟ = δ ( x − x ') ⎜ i + ⎟ p ∂x ' ⎠ ⎠ ⎝ ∂x ' ⎠ ⎝ ∂x ' ⎠ xx ' ⎝ ⎝ ⎛ ⎞ ∂ = δ ( x − x ') ⎜ −i ⎟ = pxx ' ∂ ( x − x ') ⎠ ⎝ ˆ ˆ x p 2 F n = Fn n ˆ F Fmn ≡ m F n = Fn m n = Fnδ mn ˆ F F 3 Schrödinger Schrödinger ∂ ˆ2 ˆ p + V ( x) ψ =H ψ , H= ˆ ˆ i ∂t 2m ∂ x ψ = x H ψ = ∫ dx ' x H x ' x ' ψ ˆ ˆ i ∂t ∂⎞ ∂⎞ ˆ⎛ ˆ⎛ = ∫ dx ' δ ( x − x ' ) H ⎜ x, −i ⎟ x ' ψ = H ⎜ x , −i ⎟ xψ ∂x ⎠ ∂x ⎠ ⎝ ⎝ ⎛ ⎞ ∂ ∂2 2 ψ ( x, t ) = ⎜ − + V(x) ⎟ψ ( x,t ) i ∂t ⎝ 2m ∂x 2 ⎠ ⎛ p2 ⎛ ∂ ⎞⎞ ∂ ϕ ( p, t ) = ⎜ ⎟ ⎟ ϕ ( p,t ) + V⎜i i ∂t ⎝ ∂p ⎠ ⎠ ⎝ 2m 1 rψ = e − r / a0 4 πa 3 0 rψ 2 pψ p p ψ = ∫ d 3r p r r ψ = ∫ d 3r r p rψ * ( 2a0 ) 3/ 2 i − pir 1 1 = ∫d r − r / a0 = 3 e e ( 2π ) π (a0 p 2 + π a0 3/ 2 22 ) 3 ∫ d pp 2 pψ 3 = p ∫d p 2 pψ 3 3
  • 30. 5 p2 ˆ Hψ Eψ ˆ ˆ H 2m 2 d2 ψx Eψ x 2m dx 2 i 1 px ψx e p 2mE 2π p2 (p 2mE ) ϕ ( p ) 0 ϕ ( p ) Eϕ ( p ) 2 2m ϕ ( p ) δ ( p − 2mE ) p 2mE ⎧∞ x0 V ( x) = ⎨ 6 F>0 ⎩ Fx x >0 ˆ2 ˆ p + V ( x) H ψ =Eψ , H= ˆ ˆ 2m ⎧⎛ ⎞ 2 d2 + Fx ⎟ψ ( x ) = Eψ ( x ) ⎪⎜ x >0 2 ⎨⎝ 2m dx ⎠ ⎪ ψ ( x) x >0 ⎩ ⎛ p2 d⎞ ⎟ ϕ ( p ) = Eϕ ( p ) +i F ⎜ ⎝ 2m dp ⎠ i ⎛ p3 ⎞ − Ep ⎟ ⎜ F ⎜ 6m ⎟ ϕ ( p ) = Ae ⎝ ⎠ 4
  • 31. i 1 ψ ( x ) = x ψ = ∫ dp x p p ψ = ∫ dp e ϕ ( p) px 2π i ⎛ p3 ⎛ E ⎞ ⎞ + x− ⎜ ⎜ 6mF ⎜ F ⎟ ⎟ ⎛ 1 ⎛ p3 ⎛ E ⎞ ⎞⎞ p ∞ ∞ ⎠⎟ ⎝ A A ⎝ ⎠ ∫ ∫ = = + ⎜ x − ⎟ p⎟⎟ dp cos ⎜ ⎜ dpe 2π 2π ⎝ ⎝ 6mF ⎝ F ⎠ ⎠⎠ −∞ −∞ 0 ψ ( 0) = 0 ∞ ⎛ p3 E⎞ ∫ − p⎟ = 0 → dp cos ⎜ En ⎝ 6mF ⎠ F −∞ 5
  • 32. 9. ψ A (A− A ) ( ) ( ∆A) 2 2 ≡ψ ψ=ff f = A− A ψ ˆ ˆ ˆ ˆ ˆ B ( ) ( )ψ () 2 2 ≡ ψ B− B ψ =gg ∆B g = B− B ˆ ˆ ˆ ˆ ˆ ( ∆A) ( ∆B ) 2 2 ˆ ˆ AB ( ∆A) ( ∆B ) 2 2 =ff ˆ ˆ gg α β Schwarz 2 αα ββ ≥ αβ ( ∆A) ( ∆B ) 2 2 2 ≥ fg ˆ ˆ ∵ Z 2 ⎛1 ⎞ Z = ( Re Z ) + ( Im Z ) ≥ ( Im Z ) = ⎜ ( Z − Z * ) ⎟ 2 2 2 2 ⎝ 2i ⎠ 2 ()() ⎛1 )⎞ ≥⎜ ( f g − g f 2 2 ∆A ∆B ∴ ˆ ˆ ⎟ ⎝ 2i ⎠ ( A − A )( B − B ) ψ f g =ψ = ψ AB − A B − A B + A B ψ ˆ ˆ ˆˆ ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ∵ = AB − A B − A B + A B = AB − A B ˆˆ ˆˆ ˆˆ ˆˆ ˆˆ ˆˆ g f = BA − B ˆˆ ˆ ˆ A ( ) 2 2 ()() ⎛1 ˆ ˆ ⎞ ⎛ 1 ⎡ A, B ⎤ ⎞ 2 2 ∆A ∆B ≥⎜ AB − BA ⎟ = ⎜ ∴ ˆ ˆˆ ˆˆ ˆ ⎦⎟ ⎠ ⎝ 2i ⎣ ⎝ 2i ⎠ ⎡ A, B ⎤ = AB − BA ˆˆ ˆ ˆ ˆˆ ⎣ ⎦ 1
  • 33. ψ ( ∆A) ( ∆B ) 1 ⎡ ˆ ˆ⎤ 2 ( ≥ 0) 2 2 ≥− ˆ ˆ A, B ⎦ 4⎣ ⎡ ˆ ˆ⎤ ⎣ A, B ⎦ ≠ 0 ( ∆A) ( ∆B ) 2 2 = =0 ˆ ˆ [ x, p ] ˆˆ 1 xp = ∫ dxdx ' x x x p x ' x ' ˆˆ ˆ ˆ ∂⎞ ⎛ = ∫ dxdx ' x x δ ( x − x ') ⎜ -i ⎟ x' ⎝ ∂x ⎠ ∂⎞ ⎛ = ∫ dxx x ⎜ -i ⎟x ⎝ ∂x ⎠ ∂ ⎡ ⎤ ∫ dx x ⎢ −i x −i x ˆˆ px =i x⎥ ∂x ⎦ ⎣ 2 [ x, p ] = i ( ∆x ) ( ∆p ) ≥ 2 2 ˆˆ , 4 i 1 px xψ = e 2π x 1 2 xψ = = 2π −∞ < x < ∞ =∞ 2 ( ∆x ) ( ∆p ) ≥ 2 2 4 2 F ⎡ AF , BF ⎤ = CF ˆˆ ˆ ⎣ ⎦ 2
  • 34. G ⎡ AG , BG ⎤ = AG BG − BG AG ˆˆ ˆˆ ˆˆ ⎣ ⎦ = SAF S −1SBF S −1 − SBF S −1SAF S −1 ˆ ˆ ˆ ˆ ( ) = S AF BF − BF AF S −1 ˆˆ ˆˆ = SCF S −1 ˆ =Cˆ G 3 0<x < a ⎧0 V ( x)= ⎨ ⎩∞ 0< x<a Shrödinger (x − ) ( ∆x ) 2 = ≤ a2 2 x p2 ˆ E E =T +V 2m (p− ) ( ∆p ) 2 = = p2 − p 2 2 ∵ ˆ ˆ ˆ ˆ p ( ∆p ) p2 ≥ 2 ∴ ˆ ( ∆p ) ≤ 2mE 2 ( ∆x ) ( ∆p ) ≤ 2mEa 2 2 2 2 ( ∆x ) ( ∆p ) ≥ 2 2 4 3
  • 35. 2 ( ∆x ) ( ∆p ) 2mEa ≥ ≥ 2 2 2 4 2 2mEa 2 ≥ 4 2 E≥ 8ma 2 2 Emin = 8ma 2 4 2 ⎛ 1 ⎡ ˆ ˆ⎤ ⎞ ( ∆A ) ( ∆B ) ≥⎜ 2 2 A, B ⎦ ⎟ ⎝ 2i ⎣ ⎠ 2 gg= f g 1 Schwarz ff Re f g = 0 2 g =c f 1 c Re ( c f f )=0 c = ia 2 a ff g = ia f ( ) (B − B ) ψ = ia A − A ψ ˆ ˆ ψ A= x B= p ˆˆ ˆˆ ⎛ ⎞ − p ⎟ψ ( x ) = ia ( x − x )ψ ( x ) d ⎜i ⎝ ⎠ dx ψ ( x ) = Ae ( ) 2 −a x− x /2 ei p x/ Gaussian 4
  • 36. 10. 2 ( ∆x ) ( ∆p ) ≥ 2 2 4 ( ∆A ) ∆A ≡ 2 ∆ x ⋅ ∆p ≥ 2 (t, x ) ( E, p ) xµ pµ ∆x ⋅ ∆p ≥ → ∆t ⋅ ∆E ≥ 2 2 Schroedinger ⎛ ⎞ ∂ ∂2 2 ψ ( x, t ) = ⎜ − + V ( x ) ⎟ψ ( x,t ) i ∂t ⎝ 2m ∂x 2 ⎠ x x t t t x, p, E (H − E ) ( ∆t ) ( ∆E ) 2 ∆t =? 2 = 2 ˆ t ˆ O O (t ) = ψ (t ) O ψ (t ) ˆ ⎛∂ ∂O ˆ⎛∂ ˆ ⎞ˆ ⎞ d O = ⎜ ψ (t ) O ψ (t ) + ψ (t ) ψ (t ) + ψ (t ) O ⎜ ψ (t ) ⎟ ⎟ ⎝ ∂t ∂t ⎝ ∂t ⎠ ⎠ dt ∂ ψ (t ) = H ψ (t ) ˆ i ∂t ∂ ψ (t ) = ψ (t ) H −i ˆ ∂t ∂O ˆ d 1 1 O = − ψ ( t ) HO ψ ( t ) + ψ ( t ) ψ ( t ) + ψ ( t ) OH ψ ( t ) ˆˆ ˆˆ ∂t dt i i ∂O ˆ 1 ⎡ˆ ˆ⎤ O, H ⎦ + i⎣ ∂t ˆ O d 1 ⎡ˆ ˆ⎤ O O, H ⎦ i⎣ dt ˆˆ O, H 1
  • 37. 2 2 ⎛ 1 ⎡ˆ ˆ⎤ ⎞ ⎛d ⎞ 2 ( ∆O ) ( ∆E ) ≥⎜ O, H ⎦ ⎟ = ⎜ 2 2 O⎟ ⎣ ⎝ 2i ⎠ 4 ⎝ dt ⎠ d ∆O ⋅ ∆ E ≥ O 2 dt ∆t ⋅ ∆ E ≥ 2 d ∆t =∆O/ O dt ∆t ∆O ∆t ˆ O O ψ (t ) H ψ ( t ) = E ψ (t ) ∆E = 0 ˆ ∂ ψ (t ) = H ψ (t ) = E ψ (t ) ˆ i ∂t i ψ (t ) = e ψ ( 0) - Et O (t ) = ψ (t ) O ψ (t ) ˆ ˆ O O ( t ) = ψ ( 0 ) O ψ ( 0 ) = O (0) ˆ d d ∆t =∆O/ O →∞ O =0 dt dt 11. ˆ ˆ ˆ ˆ A B A B n A n = an n B n = bn n ˆ ˆ 2
  • 38. ψ =∑ n nψ Hilbert n n ( AB − BA) ψ ∑ ( AB − BA) n n ψ ˆ ˆ ˆˆ ˆ ˆ ˆˆ n = ∑ (b A − a B ) n n ψ ˆ ˆ n n n = ∑ ( bn an − an bn ) n n ψ n =0 ⎢ A, B ⎥ = 0 ψ ˆˆ ⎣ ⎦ ˆ ˆ ˆ ˆ A B A B ˆ A n an n ⎡ A, B ⎤ = 0 ˆˆ ⎣ ⎦ AB n = BA n = an B n ˆˆ ˆˆ ˆ ˆ ˆ ˆ ˆ n Bn an Bn A A ˆ B n =bn n ˆ ˆ ˆ n B A B ˆ A ˆ2 ˆp ⎡ p, H ⎤ = 0 H= ˆˆ ˆ ˆ p p H p ⎣ ⎦ 2m p2 p, E = ˆ ˆ H p 2m ⎢ A, B ⎥ = 0 ⎢ B, C ⎥ = 0 ˆˆ ˆˆ AB BC ⎣ ⎦ ⎣ ⎦ ⎡ L2 ,L ⎤ =0 ⎡ L2 ,L ⎤ =0 ˆˆ ˆˆ L=r×p A C ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ x y ˆ ˆ ⎡ Lx , Ly ⎤ ≠ 0 ˆ ˆ ˆˆ ˆ ˆ L2 L2 Lx Lx Ly Ly ⎣ ⎦ ˆ Lz =0 3
  • 39. Schrödinger Schrödinger ∂2 ∂ 2 ψ ( x, t ) = Hψ ( x, t ), H =− + V ( x, t ) ˆ ˆ i ∂t 2m ∂x 2 Schrödinger ∂ 2 ψ (r , t ) = Hψ (r , t ), H =− ∇ 2 + V (r , t ) ˆ ˆ i ∂t 2m 1. Schrödinger ˆ t H V (r ) ψ ( r , t ) = ϕ ( r ) f (t ) ∂ ϕ ( r ) f (t ) = Hϕ ( r ) f (t ) ˆ i ∂t ϕ ( r ) f (t ) 1 df 1 ˆ = Hϕ i dt ϕ f E t r df = Ef , H ϕ = Eϕ ˆ i dt Schrödinger i − Et ψ (r , t ) = ϕ (r ) f (t ), H ϕ ( r ) = Eϕ ( r ) , f ( t ) = Ce ˆ H ϕ ( r ) = Eϕ ( r ) ˆ E Schroedinger ˆ Schrödinger t H i − En t ψ ( r , t ) = ∑ Cnψ n ( r , t ) = ∑ Cnϕn ( r ) e n n 2. ρ ( r , t ) = ψ ( r , t ) = ϕ ( r ) = ρ ( r , 0) 2 2 1 4
  • 40. j ( r , t ) = j ( r , 0) O t ∂⎞ ˆ⎛ O = ψ ( t ) O ψ ( t ) = ∫ dxψ * ( x, t ) O ⎜ x, −i ⎟ψ ( x,t ) ˆ ∂x ⎠ ⎝ ∂⎞ ˆ⎛ = ∫ dxϕ * ( x ) O ⎜ x, −i ⎟ϕ ( x ) ∂x ⎠ ⎝ V (r ) V ( −r ) = V ( r ) 2 ⎛ ⎞ 2 ∇ 2 + V ( r ) ⎟ ϕ ( r ) = Eϕ ( r ) , − ⎜ ⎝ 2m ⎠ ⎛ ⎞ 2 ∇ 2 + V ( r ) ⎟ ϕ ( − r ) = Eϕ ( − r ) ⎜− ⎝ 2m ⎠ ϕ (r ) ϕ ( −r ) E ϕ ( −r ) = Cϕ ( r ) r → −r ϕ ( r ) = Cϕ ( −r ) = C 2ϕ (r ) C = ±1 C2 =1 ϕ ( r ) = ϕ ( −r ) ϕ ( r ) = −ϕ ( −r ) V ( −r ) = V ( r ) 5
  • 41. 3. limψ ( r , t ) → 0 ψ (r,t ) r →∞ ψ 1 ( x ) ,ψ 2 ( x ) 1 E ( E − V ( x ) )ψ 2m ψ 1′′ + = 0, 1 2 ( E − V ( x ) )ψ 2m ′′ ψ2 + =0 2 2 ψ 1ψ 2 −ψ 2ψ 1′′ = 0 ′′ (ψ 1ψ 2′ −ψ 2ψ 1′ ) = 0, ' ψ 1ψ 2 −ψ 2ψ 1′ = const , ′ ψ ( x → ∞) → 0 ψ 1ψ 2 −ψ 2ψ 1' = 0 ' ψ 1′ ψ 2 ′ ψ ( x) ≠ 0 = ψ1 ψ 2 ⎛ ψ 1 ⎞′ ⎟ =0 ⎜ ln ⎝ ψ2 ⎠ ψ1 = ln C ln ψ2 ψ 1 ( x ) = Cψ 2 ( x ) 1 V (−x) = V ( x) 2 2 ( E − V ( x ))ϕ ( x ) 2m ϕ ′′ ( x ) = − 2 V ( x) ϕ ′′, ϕ ′, ϕ a) V ( x) ϕ ′′ b) a a +ε ∫ε ( E − V ( x ) )ϕ ( x ) dx 2m ϕ′(a + ε ) − ϕ′(a − ε ) = − 2 a− ϕ′ ϕ ∆V →0 1
  • 42. ϕ' ϕ ⎧ ∞, a +ε ⎪ ∫ε ( E − V ( x ) )ϕ ( x ) dx = ⎨ ϕ' ϕ ∆V → ∞ , ⎪ ϕ' ϕ a− ⎩ δ 3 V ( x ) = −γδ ( x ) = V ( − x ) ϕ x = 0 , V → −∞, ∆V → ∞, 2mE ϕ ′′ + ϕ =0 x≠0 Schroedinger 2 2mE E<0 k2 = − 2 ⎧ Aekx + A′e− kx x<0 ϕ ( x) = ⎨ . ⎩Ce + C ′e − kx x>0 kx ⎧ Ae kx x<0 ϕ ( x → ±∞ ) ϕ ( x ) = ⎨ − kx A ' = C = 0, ⎩C ′e x>0 ϕ ( x → ±∞ ) → 0 E<0 E>0 V (−x) = V ( x) ϕs ( − x ) = ϕs ( x ) , C ' = A, ⎧ Aekx x<0 ϕs ( x ) = ⎨ − kx x>0 ⎩ Ae ϕa ( − x ) = −ϕa ( x ) , C ' = − A, ⎧ Aekx x<0 ϕa ( x ) = ⎨ . − kx ⎩ − Ae x>0 A ϕa ( x ) x=0 2
  • 43. ϕs ( x ) x=0 ( E + γδ ( x ) ) ϕ 2m ϕ s′′ = − s 2 0+ ϕ s′ ( 0 ) − ϕ s′ ( 0− ) = − ∫ ( E + γδ ( x ) )ϕ ( x ) dx = − γϕ ( 0 ) 2m 2m + s s 2 2 0− 2m γA −2kA = − 2 mγ 2 E=− 22 4 ⎧0 x >a ⎪ V ( x) = ⎨ ⎪ −V0 x <a ⎩ ⎧ 2mE ⎪ ϕ ′′ + 2 ϕ = 0 x >a ⎪ ⎨ Schrödinger ⎪ϕ ′′ + 2m ( E + V0 ) ϕ = 0 x <a ⎪ ⎩ 2 ⎧ Aekx + A′e− kx x < −a ⎪ ϕ ( x ) = ⎨ Beik ′x + B′e− ik ′x −V0 < E < 0 x <a ⎪ kx ⎩ Ce + C ′e − kx x>a 2m ( E + V0 ) 2mE k ′2 = k2 = − , 2 2 ⎧ Ae kx x < −a ⎪ ϕ ( x ) = ⎨ Beik ′x + B′e − ik ′x ϕ x <a x → ±∞ ⎪ ⎩ C ′e − kx x>a lim ϕ ( x ) = 0, x →±∞ 3
  • 44. ϕ ( x) V (−x) = V ( x) : ⎧ Ae kx x < −a ⎪ ⎪ ϕ s ( x ) = ⎨ B ( eik ′x + e − ik ′x ) = 2 BCosk ′x C ' = A, B ' = B, x <a ⎪ − kx x>a ⎪ Ae ⎩ ⎧ Ae kx x < −a ⎪ ⎪ ϕa ( x ) = ⎨ B ( eik ′x − e − ik ′x ) = 2iBSink ′x C ' = − A, B ' = − B, x <a. ⎪ − kx ⎪ − Ae x>a ⎩ ϕ ,ϕ ′ x = − a, a Ae− ka = 2 BCosk ′a ϕ s , ϕ s′ x = −a , Ake − ka = 2 Bk ' Sink ′a x=a ⎧ k ′tgk ′a = k → Ens ⎪ ⎨ 1 B = Ae − ka Seck ′a ⎪ ⎩ 2 ⎧ x < −a Ae kx ⎪ − ka ϕ s ( x ) = ⎨ Ae Seck ′aCosk ′x x < a ⎪ Ae − kx x>a ⎩ ∞ ∫ ϕ ( x) 2 dx = 1 → A . s −∞ ϕ a , ϕ a′ x = −a k ′ctgk ′a = − k → a En ⎧ x < −a Ae kx ⎪ ϕa ( x ) = ⎨− Ae Csck ′aSink ′x − ka x < a, ⎪ − kx − Ae x>a ⎩ ∞ ϕa ( x ) dx = 1 → A ∫ 2 −∞ 4. ϕ ( r → ∞) = 0 → 4
  • 45. d Φ ( r → ∞, θ , ϕ ) θ = 0, π R TR T1 δ 1 V ( x ) = −γδ ( x ) 2mE ϕ ′′ + ϕ =0 x≠0 2 ⎧ Aeikx + A′e − ikx x<0 ϕ ( x) = ⎨ E >0 ⎩ Ce + C ′e − ikx x>0 ikx 2mE k2 = 2 C′ = 0 3 A A’ C ϕ x=0 A + A′ = C 2 ϕ x=0 ϕ ′ ( 0+ ) − ϕ ′ ( 0− ) = − 2m γϕ ( 0 ) 2 2m ik ( C − A + A′ ) = − γ ( A + A′ ) 1 A 2 5
  • 46. iβ mγ 1 A′ = A β= 2 A C= 1 − iβ 1 − iβ k R= T= i ⎛ * ∂ψ ∂ψ * ⎞ ⎜ψ −ψ j =− ⎟ ∂x ∂x ⎠ 2m ⎝ A′ 2 β2 R= 2 = 1+ β 2 A 2 C 1 T= = 1+ β 2 2 A R +T =1 6
  • 47. 2 ⎧0 x >a ⎪ V ( x) = ⎨ ⎪ −V0 x <a ⎩ ⎧ 2mE ⎪ ϕ quot;+ 2 ϕ = 0, x >a ⎪ ⎨ Schroedinger ⎪ϕ quot;+ 2m( E + V0 ) ϕ = 0, x <a ⎪ ⎩ 2 ⎧ Aeikx + A′e − ikx x < −a ⎪ ϕ ( x ) = ⎨ Beik ′x + B′e −ik ′x E >0 x < a, ⎪ ⎩ Ce + C ′e − ikx x>a ikx 2m ( E + V0 ) 2mE k ′2 = k2 = , 2 2 C′ = 0 5 A A’ B B’ C ϕ ,ϕ ′ ∆V − a, a x = − a, a ⎧ A′eika − Be − ik ′a − B′eik ′a = − Ae− ika ⎪ ⎪ A′ike + Bik ′e − B′ik ′e = Aike − ik ′a ik ′a − ika ika ⎨ Beik ′a + B′e− ik ′a − Ceika = 0 ⎪ ⎪ ⎩ Bik ′e − B′ik ′e ik ′a − ik ′a − Cikeika = 0 A′, B, B′, C 1 A A A’ C η Sin2k ′ae−2ika i e−2ika A′ = 2 C= A A ε ε Cos 2k ′a − i Sin2k ′a Cos 2k ′a − i Sin2k ′a 2 2 k′ k k′ k ε= η= +, − k k′ k k′ 1
  • 48. η2 Sin 2 2k ′a A′ 2 4 R= = , ε2 2 A Cos 2k ′a + Sin 2k ′a 2 2 4 2 C 1 T= = ε2 2 A Cos 2 2k ′a + Sin 2 2k ′a 4 R +T =1. k′ = k, η = 0 V0 = 0, R = 0, T = 1 a sin2k ′a = 0 V0 ≠ 0, R = 0, T = 1, b n 2π 2 2 2k ′a = nπ , n = 1, 2 En = −V0 + 8ma 2 5. E = T +V ≥ V E <V 1 V ≠∞ Schrödinger E 2
  • 49. ˆˆˆ 2 H , T ,V E ≠ T +V E = T + V ≥ V ≥ Vmin E =E E ≥ Vmin E > −V0 3 E =0 E= ω 6. 1 ˆˆˆ L=r×p→L=r×p L = ε x p , i, j, k = 1, 2,3 ˆ ˆˆ i ijk j k ⎡ xi , x j ⎤ = 0, ⎡ pi , p j ⎤ = 0, ⎡ xi , p j ⎤ = i δ ij ⎣ˆ ˆ ⎦ ⎣ˆ ˆ ⎦ ⎣ˆ ˆ ⎦ ⎡ Li , L j ⎤ = i ε ijk Lk ˆˆ ˆ ⎣ ⎦ ⎡ Lx , Ly ⎤ = i Lz ⎡ Ly , Lz ⎤ = i Lx ⎡ Lz , Lx ⎤ = i Ly ˆˆ ˆ ˆˆ ˆ ˆˆ ˆ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ˆ L2 = L2x + L2y + L2z ˆ ˆ ˆ ⎡ L2 , L ⎤ = 0 ˆˆ ⎢ ⎥ ⎣ ⎦ i 3
  • 50. ˆ L = −i r × ∇ ˆ ⎛∂ ∂⎞ ⎛∂ ∂⎞ ⎛∂ ∂⎞ Lx = −i ⎜ y − z ⎟ Lz = −i ⎜ x − y ⎟ L y = −i ⎜ z − x ⎟ ˆ ˆ ˆ ⎝ ∂z ∂y ⎠ ⎝ ∂y ∂x ⎠ ⎝ ∂x ∂z ⎠ x, y , z → r , θ , ϕ ⎛ ∂⎞ ∂ Lx = i ⎜ Sinϕ + ctgθ Cosϕ ˆ ⎟ ∂θ ∂ϕ ⎠ ⎝ ⎛ ∂⎞ ∂ Ly = −i ⎜ Cosϕ − ctgθ Sinϕ ˆ ⎟ ∂θ ∂ϕ ⎠ ⎝ ∂ θ Lz = −i ˆ ( ) ∂ϕ ⎛ 1 ∂⎛ 1 ∂2 ⎞ ∂⎞ ˆ Sinθ L2 = − + 2 ⎜ ⎟ ⎜ ⎟ ⎝ Sinθ ∂θ ⎝ ∂θ ⎠ Sin 2θ ∂ϕ 2 ⎠ ˆ Lz Lz Φ (ϕ ) = Lz Φ (ϕ ) ˆ d Φ (ϕ ) = Lz Φ (ϕ ) −i dϕ i Lzϕ Φ (ϕ ) = Ae Φ (ϕ ) = Φ (ϕ + 2π ) → Lz = m , m = 0, ±1, , ±∞ Φ (ϕ ) = Ae imϕ 2π 1 ∫ Φ (ϕ ) 2 dϕ = 1 → A = . 2π 0 1 imϕ Φ (ϕ ) = e . 2π ˆ L2 L2Y (θ , ϕ ) = L2Y (θ , ϕ ) ˆ Y 4
  • 51. , L = l ( l + 1) L2 = l ( l + 1) l = 0,1, 2 ∞ 2 Ylm (θ , ϕ ) = N lm Pl m ( cosθ ) eimϕ Pl m ( x ) Legendre 2l + 1 m = 0, ±1, ±2, ±l g = 2l + 1 L2 ( l − m )!( 2l + 1) 2π π Sinθ dθ dϕ Ylm (θ , ϕ ) = 1 → Nlm = ∫∫ 2 ( l + m )!4π 0 0 ˆ ˆ L2 Lz ⎡ L2 , L ⎤ = 0, Ylm (θ , ϕ ) ˆˆ θ ϕ ˆ Lz ⎢ ⎥ ⎣ ⎦ z Ylm (θ , ϕ ) ˆˆ ˆ L2 , Lz Lz L2Ylm (θ , ϕ ) = l ( l + 1) 2Ylm (θ , ϕ ) ˆ LzYlm (θ , ϕ ) = m Ylm (θ , ϕ ) ˆ l = 0,1, 2,...∞, m = 0, ±1,... ± l ±∞ m Ylm 5
  • 52. 2 V (r ) = V (r ) ⎛ ⎞ 2 ∇ 2 + V ( r ) ⎟ψ ( r ) = Eψ ( r ) − ⎜ ⎝ 2m ⎠ ⎛ ⎞ ˆ 1 ∂2 2 L2 + V ( r ) ⎟ψ ( r , θ , ϕ ) = Eψ ( r , θ , ϕ ) ⎜− r+ ⎜ 2m r ∂r 2 ⎟ 2mr 2 ⎝ ⎠ ˆ ⎡ ∂ ⎛ 2 ∂ ⎞ 2mr 2 ⎤ L2 ⎟ − 2 (V ( r ) − E ) ⎥ψ ( r , θ , ϕ ) = 2 ψ ( r ,θ , ϕ ) ⎢ ⎜r ⎣ ∂r ⎝ ∂r ⎠ ⎦ ˆ L ψ ( r ,θ , ϕ ) = R ( r ) Y (θ , ϕ ) ⎡ L2 ⎤ ˆ ⎡ d ⎛ 2 dR ⎞ 2mr 2 ⎤ − 2 (V ( r ) − E ) R ⎥ Y = ⎢ 2 Y ⎥ R ⎢ dr ⎜ dr ⎟ r ⎣⎝ ⎠ ⎢ ⎥ ⎦ ⎣ ⎦ ⎤ 1 ⎡ L2 ⎤ C ˆ 1 ⎡ d ⎛ 2 dR ⎞ 2mr 2 − 2 (V ( r ) − E ) R ⎥ = ⎢ 2 Y ⎥ = 2 ⎢⎜ ⎟ RY r R ⎣ dr ⎝ dr ⎠ ⎦ Y⎢ ⎥ ⎣ ⎦ L2Y (θ , ϕ ) = CY (θ , ϕ ) ˆ d ⎛ 2 dR ⎞ 2mr 2 ⎟ − 2 (V ( r ) − E ) R = 2 R C ⎜r dr ⎝ dr ⎠ Y (θ , ϕ ) = Ylm (θ , ϕ ) , C = l ( l + 1) 2 l = 0,1,......∞, m = 0, ±1,...... ± l d ⎛ 2 dR ⎞ 2mr 2 ⎟ − 2 (V ( r ) − E ) R = l ( l + 1) R ⎜r dr ⎝ dr ⎠ E R(r ) 2 ∫ d r ψ (r ) = 1 , 3 ∞ r 2 dr ∫ d Ω R ( r ) Ylm (θ , ϕ ) = 1 ∫ 2 2 0 ∫ dΩ Y (θ , ϕ ) = 1 2 lm 1
  • 53. ∞ R ( r ) r 2 dr = 1 ∫ 2 0 u = rR l ( l + 1) ⎞ d 2u ⎛ 2 2 − + ⎜V + ⎟ u = Eu 2 2m r 2 ⎠ 2m dr ⎝ Veff Schrödinger Veff ∞ ∫ u (r) 2 dr = 1 0 3 Ze 2 V (r ) = − r ⎛ ⎞ 2 2 ∇ 2 + V ( r1 − r2 ) ⎟ Ψ ( r1 , r2 ) = Et Ψ ( r1 , r2 ) ⎜− ∇1 − 2 2 ⎝ 2m1 ⎠ 2m2 Et m1r1 + m2 r2 r1 , r2 → r = r1 − r2 R= m1 + m2 ⎛ ⎞ ( ) ( ) 2 2 ∇ 2 + V ( r ) ⎟ Ψ r , R = Et Ψ r , R − ∇r − 2 ⎜ ⎝ 2µ R ⎠ 2M m1 im2 µ= M = m1 + m2 , m1 + m2 ( ) () Ψ r , R =ψ (r ) Φ R ⎛ ⎞ ⎛22 ⎞ () () 2 ∇ r + V ( r ) ⎟ψ ( r ) Φ R = ⎜ ∇ R + Et ⎟ Φ R ψ ( r ) − 2 ⎜ ⎝ 2µ ⎠ ⎝ 2M ⎠ 1⎛ ⎞ 1⎛22 ⎞ () 2 ∇ r + V ( r ) ⎟ψ ( r ) = − ∇ R + Et ⎟ Φ R = E 2 ⎜ ⎜ () ψ ( r ) ⎝ 2µ Φ R ⎝ 2M ⎠ ⎠ 2
  • 54. ⎧⎛ ⎞ 2 ∇ r + V ( r ) ⎟ψ ( r ) = Eψ ( r ) − 2 ⎪⎜ ⎪⎝ 2 µ ⎠ ⎨ ⎪ () () 2 ⎪ − 2M ∇ R Φ R = ( Et − E ) Φ R 2 ⎩ () ΦR Et − E V (r ) ψ (r ) E ψ ( r ) = R ( r ) Y (θ , ϕ ) Ylm (θ , ϕ ) d ⎛ 2 dR ⎞ 2 µ r 2 ⎛ Ze2 ⎞ ⎟ R = l ( l + 1) R + 2 ⎜E+ ⎜ ⎟ r dr ⎝ dr ⎠ ⎝ r⎠ R (r) r → 0, ∞ ⎧ l ⎛ 2Z ⎞⎛ ⎞ Zr − 2Z ⎪ Rnl ( r ) = N nl e r ⎟ F ⎜ − n + l + 1, 2l + 2, na0 ⎜ r⎟ ⎪ ⎝ na0 ⎠⎝ ⎠ na0 ⎪ ⎨ F (α ,γ ,ξ ) ⎪ µ Z 2e4 ⎪ En = − ⎪ ⎩ 2n 2 2 n = 1, 2, 3, ∞ l = 0,1, , n − 1 m = 0, ±1, , ±l ∞ l ∞ Rnl ( r ) r 2 dr = 1 ∫ 2 N nl 0 2 a0 = Bohr µ e2 µ e4 ψ nlm ( r ,θ ,ψ ) = Rnl ( r ) Ylm (θ , ϕ ) , En = − . 2n 2 2 µ e4 E1 = − ≠0 a) 2 2 n −1 g = ∑ ( 2l + 1) = n 2 b) l =0 c) n l Enl E 3
  • 55. {Hˆ , Lˆ , Lˆ } 2 d) z ˆ 1 ∂2 2 L2 ψ nlm ( r ,θ ,ψ ) = Rnl (r )Ylm (θ , ϕ ) ˆˆ H =− r + V (r ) + ˆ , L2 , Lz 2µ r ∂r 2µ r 2 2 1 = ∫ ψ nlm (r ) d 3 r = ∫ Rnl ( r ) Ylm (θ , ϕ ) r 2 drd Ω = ∫ Rnl ( r ) r 2 dr 2 2 2 2 e) ρ nl ( r ) = Rnl ( r ) r 2 2 r ρlm (θ , ϕ ) = Ylm (θ , ϕ ) 2 Ω f) µ e4 ⎛ 1 1 ⎞ En − Em = hν ,ν = ⎜− ⎟ Rydberg 4π 3 ⎝ n 2 m2 ⎠ 1. 1 V ( x) = mω 2 x 2 2 x0 Taylor 1 V ( x ) = V ( x0 ) + V ′ ( x0 )( x − x0 ) + V ′′ ( x0 )( x − x0 ) + 2 2 V ( x0 ) V ′ ( x0 ) = 0 1 V ( x) V ′′ ( x0 )( x − x0 ) 2 2 p2 1 + mω 2 x 2 H= 2m 2 4
  • 56. ⎧ ˆ p2 1 ˆ + mω 2 x 2 ⎪H = ˆ ⎨ 2m 2 ⎪[ x, p ] = i ⎩ˆˆ 1 ⎧⎛ ⎞ 2 d2 1 + mω 2 x 2 ⎟ψ ( x ) = Eψ ( x ) − ⎪⎜ 2 ⎨⎝ 2m dx 2 ⎠ ⎪ lim ψ ( x ) = 0, ⎩ x →±∞ ( Griffiths ) ⎧ ⎛ 1⎞ ⎪ En = ⎜ n + 2 ⎟ ω , n = 0,1, 2 ⎝ ⎠ ⎪ ⎨ ⎛ mω ⎞ − mω x ⎪ψ ( x ) = ⎛ mω ⎞ 1/ 4 1 Hn ⎜ x⎟e 2 ⎜ ⎟ ⎜ ⎟ ⎪n ⎝ 2⎠ ⎝ ⎠ n ⎩ 2 n! Hn ( x) Hermite 5
  • 57. 2 [ x, p ] = i ˆˆ ˆ = p + 1 mω 2 x 2 = mω ⎛ x − i p ⎞⎛ x + i p ⎞ ω + 1 ω ˆ2 ⎜ˆ ˆ ⎟⎜ ˆ ˆ⎟ ˆ H mω ⎠⎝ mω ⎠ 2⎝ 2m 2 2 mω mω ⎛ ⎞ ⎛ ⎞ i i a+ = a= ⎜x+ ⎜x− p⎟ p⎟ ˆ ˆ ˆ ˆ ˆ ˆ mω ⎠ mω ⎠ ⎝ ⎝ 2 2 ⎧ˆ ⎛ + 1⎞ ⎧ ˆ p2 1 ˆ ⎪H = ⎜ a a + 2 ⎟ ω + mω 2 x 2 ⎪H = ˆˆ ˆ ⎝ ⎠ → ⎨ ⎨ 2m 2 [ x, p ] = i ⎪ ⎪ ⎡ a, a ⎤ = 1 + ˆˆ ⎩ ⎣ˆ ˆ ⎦ ⎩ (a a) + + = a+a a+ ≠ a a+a ˆ ˆˆ ˆˆ ˆ ˆ ˆˆ a a+a ˆˆ a+a a+a ˆ ˆ ˆˆ ˆˆ H H a+ a n = n n ˆˆ ⎛ 1⎞ H n = ⎜n+ ⎟ ω n ˆ ⎝ 2⎠ ⎛ 1⎞ En = ⎜ n + ⎟ ω ⎝ 2⎠ n=? xn an =b ˆ 2.1 n a+ = b n a+a n = b b n nn = bb ˆ ˆˆ ∵ b b ≥ 0, n n ≥0 ∴n ≥ 0 ∵ ( a + a ) a n = ( aa + − 1) a n = a ( a + a − 1) n = ( n − 1) a n 2.2 ˆˆˆ ˆˆ ˆ ˆˆˆ ˆ (a a) a n = a + aa + n = a + ( a + a + 1) n = ( n + 1) a + n + + ˆˆˆ ˆ ˆˆ ˆ ˆˆ ˆ a+ n a+a a+a ∴ ˆ ˆ ˆˆ ˆˆ n an 1
  • 58. ˆ a†a ˆˆ H ⎛ 5⎞ (a ) +2 ⎜n+ ⎟ ω n+2 ˆ n ⎝ 2⎠ ⎛ 3⎞ ⎜n+ ⎟ ω a+ n n +1 ˆ ⎝ 2⎠ ⎛ 1⎞ ⎜n+ ⎟ ω n n ⎝ 2⎠ ⎛ 1⎞ ⎜n − ⎟ ω n −1 ˆ an ⎝ 2⎠ ⎛ 3⎞ (a) ⎜n − ⎟ ω n−2 2 ˆ n ⎝ 2⎠ n≥0 a+ a+a ˆ ˆ ˆˆ a , n − 1, n, n + 1, ⎧n0 , ⎨ n0 ≥ 0 ⎩ 2.3 n0 a + a n0 = n0 n0 ˆˆ n0 > 0 ˆ a n0 (a a) a n = ( n0 − 1) a n0 + ˆˆˆ ˆ 0 n0 − 1 < n0 n0 n0 = 0 (a a) a n 2 a + a n0 = 0 n0 a + a n0 = 0, a n0 = 0 + =0 =0 ˆˆ ˆ ˆˆ ˆ ˆˆˆ a n0 0 a+a n0 = 0 ˆ ˆˆ a n0 n0 n0 = 0 ⎧ ⎛ 1⎞ ⎪ En = ⎜ n + ⎟ ω ⎝ 2⎠ ⎨ ⎪ n = 0,1, 2, ⎩ n −1 ˆ 2.4) an a n = an n − 1 ˆ 2
  • 59. a + n = bn n + 1 ˆ ∵ n a + = n − 1 an , n a = n + 1 bn * * ˆ ˆ ∴ n a + a n = an n −1 n −1 , n = an , an = n 2 2 ˆˆ n aa + n = bn n a + a + 1 n = bn , n + 1 = bn , n +1 n +1 , bn = n + 1 2 2 2 ˆˆ ˆˆ an = n , bn = n + 1 ⎧ a n = n n −1 ⎪ˆ ⎨+ ⎪a n = n + 1 n + 1 ⎩ˆ a+a ˆ 2.5 ˆˆ H (a a) + = m a + a n = n m n = nδ mn ˆˆ ˆˆ mn ⎛ 1⎞ H mn = m H n = ⎜ n + ⎟ ωδ mn ˆ ⎝ 2⎠ amn = m a n = n m n − 1 = nδ m ,n −1 ˆ amn = m a + n = n + 1 m n + 1 = n + 1δ m, n +1 + mω (a + a ) ( a − a+ ) + x= p=i ˆ ˆˆ ˆ ˆˆ 2mω 2 ( ) (a + amn ) = nδ m ,n −1 + n + 1δ m,n +1 + xmn = 2mω 2mω mn mω ( ) nδ m,n −1 − n + 1δ m,n +1 pmn = i 2 2.6 0 ⎛ ⎞ i a 0 =0 ⎜x+ p⎟ 0 = 0 ˆ ˆ ˆ ⎝ mω ⎠ i i ∫ dx′′ x′ x + x′ x + p x′′ x′′ 0 = 0 p 0 =0 ˆ ˆ ˆ ˆ mω mω 3
  • 60. d x′′ 0 = ψ 0 ( x′′ ) , ∵ x′ x x′′ = x′δ ( x '− x′′ ) , x′ p x′′ = −i δ ( x '− x′′ ) ˆ ˆ dx′ ⎛ d⎞ ⎟ψ 0 ( x′ ) = 0 ∴ ⎜ x′ + mω dx′ ⎠ ⎝ mω 2 − ψ 0 ( x ) = Ce x 2 1 mω ⎛ mω ⎞ 4 − 2 x2 ψ 0 ( x) = ⎜ ⎟ e ⎝π ⎠ x ( a+ ) n − 2 1 1 ψ n ( x) = x n = 2 x a+ n −1 = n ( n − 1) n n 1 ⎛ mω ⎞ 2 ⎛ n ⎞ x ( a+ ) 0 = 1 i n = ⎟ x ⎜x− ⎜ p⎟ 0 ˆ ˆ mω ⎠ n! ⎝ 2 ⎠ ⎝ n! n 1 ⎛ mω ⎞ 2 i i i n! ⎝ 2 ⎠ ∫ = dxn x x − p x1 x1 x − xn −1 x − ⎜ ⎟ dx1 ˆ ˆ ˆ ˆ ˆ ˆ p x2 p xn xn 0 mω mω mω d x′ x x′′ = x′δ ( x − x′′ ) , x′ p x′′ = −i δ ( x − x′′ ) ˆ ˆ dx′ n 1 ⎛ mω ⎞ 2 ⎛ n d⎞ ⎟ ψ 0 ( x) ψ n ( x) = ⎟ ⎜x− ⎜ mω dx ⎠ n! ⎝ 2 ⎠ ⎝ a+a 2 7 ˆˆ ⎛ 1⎞ En = ⎜ n + ⎟ ω , n = 0,1, 2,3 ⎝ 2⎠ 1 ω E0 = 2 ε= ω n n n: N = a† a ˆ ˆˆ ˆ ˆ N H 4
  • 61. → → 1 2 5
  • 62. 2. ⎡ xi , p j ⎤ = i δ ij ⎡ xi , x j ⎤ = 0, ⎡ pi , p j ⎤ = 0, ⎣ˆ ˆ ⎦ ⎣ˆ ˆ ⎦ ⎣ˆ ˆ ⎦ ˆ J ⎡ J i , J j ⎤ = i ε ijk J k ˆˆ ˆ ⎣ ⎦ ˆˆ ˆ ⎡Jx, J y ⎤ = i Jz , ⎡J y , Jz ⎤ = i Jx , ⎡Jz , Jx ⎤ = i J y , J × J = i J ˆˆ ˆ ˆˆ ˆ ˆˆ ˆ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ˆ L ˆ J 2 = J x2 + J y 2 + J z2 = Ji Ji ˆ ˆ ˆ ˆˆ ˆˆ J 2, Jz ⎡J 2, J ⎤ = J ⎡J , J ⎤ + ⎡J , J ⎤ J = i ε J J + i ε J J = i ε J J + i ε J J = 0 ˆˆ ˆˆˆ ˆˆ ˆ ˆˆ ˆˆ ˆˆ ˆˆ 1 i⎣ i j⎦ ⎣ i j⎦ i ⎢ ⎥ ⎣ ⎦ j ijk i k ijk k i ijk i k kji i k ε ijk ε ijk = −ε jik ˆ ˆ J2 Jz ˆ λ, m ˆ J2 Jz ˆ ˆ J2 Jz ˆ J 2 λ, m = λ λ, m J z λ, m = m λ, m ˆ 2 λ = ?, m = ? 2 ⎧ J + = J x + iJ y ˆ ˆ ˆ ⎪ ⎪ˆ ˆ+ J x , J y , J z → ⎨ J − = J x − iJ y = J + ˆˆˆ ˆ ˆ ⎪ˆ ⎪J z ⎩ 1
  • 63. ˆ ∵ J+ J− = J 2 − J z2 + ˆˆ ˆ ˆ Jz, ˆ J− J+ = J 2 − J z2 − ˆˆ ˆ ˆ Jz, ( ) 1 ˆˆ ˆ J 2 = J+ J− + J− J+ + J z2 ˆˆ ˆ 2 ⎡ J+ , J− ⎤ = 2 J z , ⎡ J− , J z ⎤ = J− , ⎡ J+ , J z ⎤ = − J+ , ˆˆ ˆ ˆˆ ˆ ˆˆ ˆ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎡J 2, J ⎤ = ⎡J 2, J ⎤ = ⎡J 2, J ⎤ = 0 ˆˆ ˆˆ ˆˆ ⎢ ⎥⎢ ⎥⎢ ⎥ + − ⎣ ⎦⎣ ⎦⎣ ⎦ z 1 ˆ λ, m J 2 − J z2 λ, m = λ, m J+ J− + J− J+ λ, m ˆ ˆˆ ˆˆ 3 2 (λ − m ) 1 1 λ, m λ, m = λ, m J + J− λ, m + λ, m J + J+ λ, m ≥ 0 ˆ+ ˆ ˆˆ 2 2 − 2 2 ≥0 ≥0 ≥0 λ ≥ m2 ⎧ ˆˆ ˆˆ J 2 J+ λ, m = J+ J 2 λ, m = λ 2 J+ λ, m ˆ ⎪ ⎨ 4 ( ) ⎪ J z J + λ , m = J + + J + J z λ , m = ( m + 1) J + λ , m ˆˆ ˆ ˆˆ ˆ ⎩ ⎧ ˆˆ ˆˆ J 2 J− λ, m = J− J 2 λ, m = λ 2 J− λ, m ˆ ⎪ ⎨ ( ) ⎪ J z J − λ , m = − J − + J − J z λ , m = ( m − 1) J − λ , m ˆˆ ˆ ˆˆ ˆ ⎩ ˆ λ, m J+ λ, m , J− λ, m ˆ ˆ ˆ J2 Jz ˆ ˆ J2 Jz ( Jˆ ) (m + 2 ) 2 λ,m λ 2 + ( m + 1) J+ λ,m λ ˆ 2 λ,m λ 2 m (m − 1) J− λ,m λ ˆ 2 ( Jˆ ) (m − 2 ) 2 λ,m λ 2 − ˆ ˆ J− J+ ˆ λ J2 2 2
  • 64. ( j ′, , m − 1, m, m + 1, ˆ Jz j) λ ≥ j 2 , λ ≥ j '2 5 j J+ λ, j = 0 ˆ ˆ Jz j ( ) 0 = J− J+ λ, j = J 2 − J z2 − J z λ, j = (λ − j2 − j ) ˆ λ, j ˆˆ ˆ ˆ 2 λ = j ( j + 1) j′ J − λ , j′ = 0 ˆ ˆ Jz j' ( ) 0 = J + J − λ , j ′ = J 2 − J z 2 + J z λ , j ′ = ( λ − j ′2 + j ′ ) ˆ λ , j′ ˆˆ ˆ ˆ 2 λ = j′ ( j ′ − 1) j ( j + 1) = j′ ( j ′ − 1) ⎧ j +1 j′ = ⎨ ⎩ −j j′ = j + 1 > j j j' j′ = − j λ = j ( j + 1) ˆ λ J2 2 m = − j , − j + 1, , j − 1, j ˆ m Jz λ , m → j, m j=? λ, j λ, − j λ, − j ˆ ˆ 6 J− J+ 2j 2j λ, j 2 j = 0, j 13 J 2 j , m = j ( j + 1) ˆ j = 0, ,1, , 2 j, m 22 3
  • 65. J z j, m = m m = − j , − j + 1, , j − 1, j ˆ j, m 13 ˆ ˆ ˆ j= j = 0,1, 2, 3, J J L ,, 22 J 2 j , m + 1 = j ( j + 1) j, m + 1 , J z j , m + 1 = (m + 1) j, m + 1 , ˆ ˆ 2 7 J 2 J + j , m = j ( j + 1) 2 J + j , m , J z J + j , m = (m + 1) J + j, m , ˆˆ ˆ ˆˆ ˆ j, m j, m J + j , m = a jm j , m +1 ˆ J − j , m = b jm j , m −1 ˆ j , m J − = j , m + 1 a* jm ˆ j , m J + = j , m − 1 b* jm ˆ ˆ 2 j, m + 1 j, m + 1 = j, m J − J + j, m = j, m J 2 − J z 2 − J z j, m ˆˆ ˆ ˆ a jm = ( j ( j + 1) − m 2 − m ) 2 j, m j, m a jm = ( j ( j + 1) − m(m + 1) ) 2 2 j ( j + 1) − m ( m + 1) = ( j − m )( j + m + 1) a jm = ( j + m )( j − m + 1) b jm = ( j − m )( j + m + 1) ⎧ J + j, m = j, m + 1 ˆ ⎪ ⎨ ( j + m )( j − m + 1) ⎪ J − j, m = j, m − 1 ˆ ⎩ ( Jˆ ) ⎧ˆ 1 ⎪ Jx = 2 + J− ˆ + ⎪ ∵⎨ ( Jˆ ) ⎪J = 1 − J− ˆ ˆ ⎪ y 2i + ⎩ ⎧ˆ ( j − m )( j + m + 1) ( j + m )( j − m + 1) ⎪ J x j, m = 2 j, m + 1 + j, m − 1 ⎪ 2 ∴⎨ ⎪ J j, m = ( j − m )( j + m + 1) ( j + m )( j − m + 1) j, m + 1 − j, m − 1 ˆ ⎪y ⎩ 2i 2i ˆ ˆ J2 8 Jz 4
  • 66. ˆ ˆ J2 j, m Jz j D = 2 j +1 j , n J 2 j , m = j ( j + 1) 2δ nm ˆ j , n J z j , m = m δ nm ˆ ( j − m )( j + m + 1)δ n,m+1 + ( j + m )( j − m + 1)δ n,m−1 j, n J x j, m = ˆ 2 2 ( j − m )( j + m + 1)δ n,m+1 − ( j + m )( j − m + 1)δ n,m −1 j, n J y j, m = ˆ 2i 2i 9 1 j= 1 2 3 11 J 2 = j ( j + 1) = Jz = m , m = , − 2 2 4 22 11 11 1 ,− j= D=2 , 22 22 2 ⎛0 / 2⎞ ⎛0 1⎞ ⎟ = σx, Jx = ⎜ ⎟= ⎜ ⎝ /2 0⎠ 2 ⎝1 0⎠ 2 ⎛ i⎞ − ⎜0 2 ⎟= ⎛ 0 −i ⎞ ⎟ = σy Jy = ⎜ ⎟ ⎜ ⎜i 0⎟ 2⎝i 0 ⎠ 2 ⎜ ⎟ ⎝2 ⎠ ⎛ /2 0⎞ ⎛1 0 ⎞ ⎟ = σz Jz = ⎜ ⎟= ⎜ ⎝ 0 − / 2 ⎠ 2 ⎝ 0 −1⎠ 2 σ x ,σ y ,σ z Pauli ⎛a⎞ ⎜⎟ Jz ⎝b⎠ ⎛ /2 0 ⎞⎛ a ⎞ ⎛a⎞ ⎟⎜ ⎟ = m ⎜ ⎟ ⎜ ⎝ 0 − / 2 ⎠⎝ b ⎠ ⎝b⎠ ⎛1⎞ ⎛0⎞ 1 1 m= m=− ⎜⎟ ⎜⎟ ⎝0⎠ ⎝1⎠ 2 2 j =1 2 5
  • 67. J 2 = j ( j + 1) =2 Jz = m , m = 1, 0, −1 2 2 D=3 1, −1 1,1 1, 0 ⎛ 0 −i 0 ⎞ ⎛0 1 0⎞ ⎛1 0 0 ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ Jx = ⎜ 1 0 1 ⎟ , J y = 2 ⎜ i 0 −i ⎟ , J z = ⎜ 0 0 1 ⎟ 2⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ 0 0 −1⎠ ⎝0 1 0⎠ ⎝0 i 0 ⎠ ⎛1⎞ ⎛ 0⎞ ⎛ 0⎞ ⎜⎟ ⎜⎟ ⎜⎟ m = 1, ⎜ 0 ⎟ ; m = 0, ⎜ 1 ⎟ ; m = −1, ⎜ 0 ⎟ ⎜0⎟ ⎜ 0⎟ ⎜1⎟ ⎝⎠ ⎝⎠ ⎝⎠ 6
  • 68. 3. J j L 1 Stern-Gelach V = −M i B ● M =γJ ● ● S MS →S ● 1 s= 2 e e MS = − ML = − ● S L µ 2µ 2) S Dirac ˆˆ ˆ ⎡ Si , S j ⎤ = i ε ijk Sk , S × S = i S ˆˆ ˆ ⎣ ⎦ 2 S 2x = S 2 y = S 2z = Sx , S y , Sz = ± 4 2 1
  • 69. ˆ σ S= σ ⎡σ i , σ j ⎤ = 2iε ijkσ k ˆ ˆ Pauli ⎣ˆ ˆ ⎦ ˆ 2 σ 2x = σ 2 y = σ 2z = 1 σ x , σ y , σ z = ±1 (σˆ yσˆ z − σˆ zσˆ y )σˆ y + 2i σˆ y (σˆ yσˆ z − σˆ zσˆ y ) = 2i ( −σˆ zσˆ 2y + σˆ 2yσˆ z ) = 2i ( −σˆ z + σˆ z ) = 0 1 1 1 1 σ xσ y + σ yσ x = ˆˆ ˆˆ 2i {σˆ , σˆ } = 2δ i j ij { A, B} = AB + BA, ˆˆ ˆ ˆ ˆˆ ⎛ 0 −i ⎞ ⎛0 1⎞ ⎛1 0 ⎞ σz σx = ⎜ ⎟, σ y = ⎜ ⎟, σz = ⎜ ⎟ ⎝ 0 −1⎠ ⎝1 0⎠ ⎝i 0⎠ 3 r →4 3 Hilbert r , Sz D=2 {Hˆ , Lˆ , L } → {Hˆ , Lˆ , L , S } 2 2 z z z ⎛ ψ (r;t ) ⎞ ψ ( r ; t ) → ψ ( r , S z ; t ) = ⎜ 1/ 2 ⎟ ⎝ψ −1/ 2 ( r ; t ) ⎠ ψ ±1/ 2 ( r ; t ) 2 Sz = ± r 2 ψ 1/ 2 ( r ; t ) + ψ −1/ 2 ( r ; t ) 2 2 r ∫ d r ψ ±1/ 2 ( r ; t ) 2 Sz = ± 3 2 ) =1 ∫ d r (ψ (r;t ) + ψ −1/ 2 ( r ; t ) 2 2 3 1/ 2 ˆ G G = ∫ d 3 rψ + ( r , sz ; t ) Gψ ( r , sz ; t ) ˆ G12 ⎞ ⎛ ψ 1/ 2 ( r ; t ) ⎞ ⎛G = ∫ d 3 r (ψ 1/ 2 ( r ; t ) ,ψ −1/ 2 ( r ; t ) ) ⎜ 11 * * ⎟⎜ ⎟ ⎝ G21 G23 ⎠ ⎝ψ −1/ 2 ( r ; t ) ⎠ 2X2 2
  • 70. 4 Zeeman 2 ∇2 + V ( r ) , H0 = − ˆ 2µ H 0 nlm = Enl nlm ˆ ψ nlm ( r ) = r nlm = Rnl ( r ) Ylm (θ , ϕ ) , g = 2l + 1 1 V (r ) ∼ En , g = n 2 r V = −M i B 1) ˆˆ : V ∼ LiS 2) ˆˆ LiS z ( ) ( ) ( ) eB ˆ ˆ ˆ H = H 0 − M L + M S i B = H 0 − M Lz + M S z B = H 0 + Lz + 2S z B ˆ ˆ ˆ ˆ ˆ ˆ ˆ 2µ Schroedinger ⎛ ψ 1/ 2 ⎞ H ψ =Eψ , ψ =⎜ ˆ ⎟ ⎝ ψ −1/ 2 ⎠ ⎧⎛ ˆ ⎞ ( ) eB ˆ Lz + ⎟ ψ 1/ 2 = E ψ 1/ 2 ⎪ ⎜ H0 + 2µ ⎪⎝ ⎠ ⎨ ⎪⎛ H + eB L − ⎞ ψ ( ) ⎟ −1/ 2 = E ψ −1/ 2 ˆ ˆ ⎪⎜ 0 2 µ z ⎩⎝ ⎠ H 0 nlm = Enl nlm , Lz nlm = m nlm ˆ ˆ ⎛ nlm ⎞ eB ( m + 1) ψ 1/ 2 = nlm , ψ −1/ 2 = 0, ψ = ⎜ + ⎟ , Enlm = Enl + 2µ ⎝0⎠ ⎛0⎞ eB ( m − 1) ψ −1/ 2 = nlm , ψ 1/ 2 = 0, ψ = ⎜ − ⎟ , Enlm = Enl + 2µ ⎝ nlm ⎠ ± Enl → Enlm 3
  • 71. n = 1, l = 0, m = 0 eB E10 + 2µ E10 → eB E10 − 2µ Stern Gelach 5 Larmor eˆ ˆ MS= − S µ B z eˆ H = − M S i B = BS z ˆ µ Schrödinger e B ⎛ 1 0 ⎞ ⎛ c1 ⎞ ⎛ c1 ⎞ H ϕ =E ϕ ⎟⎜c ⎟ = E ⎜c ⎟ ˆ ⎜ 2µ ⎝ 0 −1⎠⎝ 2 ⎠ ⎝ 2⎠ ⎧ eB ⎧ eB E+ = , c1 = 1, c2 = 0 E− = − , c1 = 0, c2 = 1 ⎪ ⎪ 2µ 2µ ⎪ ⎪ ⎨ ⎨ ⎛ 1 ⎞ − i E+t ⎛ 0 ⎞ − i E− t ⎪ ⎪ ϕ+ = ⎜ ⎟ e ϕ− = ⎜ ⎟ e ⎪ ⎪ ⎝0⎠ ⎝1⎠ ⎩ ⎩ ψ = a ϕ+ + b ϕ− , a2 + b2 = 1 α α a = Cos , b = Sin 2 2 ⎛ α − i E+t ⎞ ⎜ Cos e ⎟ 2 ψ =⎜ ⎟ ⎜ α − i E−t ⎟ ⎜ Sin e ⎟ ⎝ ⎠ 2 Si = ψ Si ψ ˆ ˆ Si 4
  • 72. ⎛ α − i E+t ⎞ 0 ⎞⎜ ⎟ cos e ⎛ α E−t ⎞ ⎛ / 2 α i i 2 E+ t ⎜ ⎟ = cos α S z = ⎜ cos e sin e ⎟ ⎜ ⎟ 0 − / 2⎠⎜ − E− t ⎟ α ⎠⎝ i ⎝ 2 2 2 ⎜ sin e ⎟ ⎝ ⎠ 2 e Sinα Cos Sx = Bt µ 2 e Sinα Sin Sy = Bt µ 2 e ω= Larmor Larmor B µ 5
  • 73. 4. ˆˆ V ∼ L⋅S ( ) = Lˆ + Sˆ + 2Lˆ ⋅ Sˆ 2 ˆˆ L+S 2 2 L ⋅ S = ⎢( L + S ) − L − S ⎥ ˆ ˆ 1⎡ ˆ ˆ ˆ ˆ⎤ 2 2 2 2⎣ ⎦ ˆˆ L⋅S ˆˆ J1 , J 2 ⎡ J1i , J1 j ⎤ = i ε ijk J1k , ⎡ J 2i , J 2 j ⎤ = i ε ijk J 2 k , ⎡ J1i , J 2 j ⎤ = 0 ˆˆ ˆ ˆˆ ˆ ˆˆ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ J1 = j1 ( j1 + 1) , J1z = m1 , m1 = − j1 ,..., j1 2 2 J 2 = j2 ( j2 + 1) 2 , J 2 z = m2 , m2 = − j2 ,..., j2 2 ˆˆˆ J = J1 + J 2 ⎡ J i , J j ⎤ = i ε ijk J k ˆˆ ˆ ⎣ ⎦ ˆ J J 2 = j ( j + 1) ˆ , Jz = m J2 2 ˆˆ ˆˆ ˆ J J1 , J 2 J1 , J 2 j1 , j2 , m1 , m2 j, m 1 ˆ ˆ ˆ2 ˆ ˆ2 ˆ J1 J2 J 1 , J1 z , J 2 , J 2 z j1m1 j2 m2 = j1m1 j2 m2 J 1 j1m1 j2 m2 = j1 ( j1 + 1) ˆ2 J1z j1m1 j2 m2 = m1 ˆ 2 j1m1 j2 m2 j1m1 j2 m2 J 2 j1m1 j2 m2 = j2 ( j2 + 1) ˆ2 J 2 z j1m1 j2 m2 = m2 ˆ 2 j1m1 j2 m2 j1m1 j2 m2 m1 = − j1 , , j1 , m2 = − j2 , , j2 1
  • 74. ⎡ J 2 , J ⎤ ≠ 0, ⎡ J 2 , J ⎤ ≠ 0 ˆˆ ˆˆ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ 1z 2z J1 ⋅ J 2 = ⎡ J 2 − J12 − J 22 ⎤ ˆˆ 1ˆ ˆ ˆ ˆ J2 ⎢ ⎥ 2⎣ ⎦ ⎡ J 2 , J ⎤ = ⎡ J 2 , J 2⎤ = ⎡ J 2 , J 2 ⎤ = ⎡ J , J 2⎤ = ⎡ J , J 2 ⎤ = ⎡ J 2, J 2 ⎤ = 0 ˆˆ ˆˆ ˆˆ ˆˆ ˆˆ ˆˆ ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ 1 2⎥ ⎣ ⎦⎣ ⎦⎣ ⎦⎣ ⎦⎣ ⎦⎣ ⎦ z 1 2 z 1 z 2 ˆ2 ˆ2 ˆ ˆ J 1, J 2 , J 2 , J z j1 j2 jm J 1 j1 j2 jm = j1 ( j1 + 1) J 2 j1 j2 jm = j2 ( j2 + 1) ˆ2 ˆ2 2 2 j1 j2 jm j1 j2 jm J 2 j1 j2 jm = j ( j + 1) ˆ J z j1 j2 jm = m ˆ 2 j1 j2 jm j1 j2 jm ˆ ˆ J 2, Jz ˆ2 ˆ ˆ2 ˆ ˆˆ J 2, Jz J 1 , J1 z , J 2 , J 2 z ∑ j1m1 j2 m2 = 1 j1m1 j2 m2 m1 , m2 ∑ j1 j2 jm = j1m1 j2 m2 j1m1 j2 m2 j1 j2 jm m1 , m2 ∑ = j1m1 j2 m2 j1 j2 jm j1m1 j2 m2 Clebsch − Gordon m1 , m2 ( j, m ) 2 ˆ Jz ( ) ∑ J z j1 j2 jm = j1m1 j2 m2 j1 j2 jm J1z + J 2 z ˆ ˆ ˆ j1m1 j2 m2 m1 , m2 ∑ (m − m − m ) j1m1 j2 m2 j1 j2 jm j1m1 j2 m2 = 0 1 2 m1 , m2 j1m1 j2 m2 m = m1 + m2 CG =0 CG 2
  • 75. m = m1 + m2 j jmin ≤ j ≤ jmax jmax = mmax = ( m1 )max + ( m2 )max = j1 + j2 D = ( 2 j1 + 1)( 2 j2 + 1) jmax ∑ ( 2 j + 1) = j D= − jmin + 2 jmax + 1 2 2 max j = jmin ( 2 j1 + 1)( 2 j2 + 1) = ( j1 + j2 ) − j 2 min + 2 ( j1 + j2 ) + 1 2 j 2 min = ( j1 − j2 ) jmin = j1 − j2 2 ˆˆ J 2, Jz j1 , j2 J 2 = j ( j + 1) j = j1 − j2 , , j1 + j2 2 , Jz = m , m = m1 + m2 3 j1 j2 jm CG ˆˆ L⋅S ˆˆˆˆˆˆ ˆ J = L + S , J1 = L, J 2 = S 1 1 l, , j, m j l , ml , , ms 2 2 1 1 jm j = ∑ Cml ,ms l , ml , , ms l 2 2 ml , ms ⎡ 1 −1 ⎤ 11 = ∑ ⎢ Aml l , ml , , + Bml l , ml , , 22⎥ ml ⎣ ⎦ 22 ⎧m j − 1/ 2, s = 1/ 2 ⎪ m j = ml + ms , ml = m j − ms = ⎨ ⎪ m j + 1/ 2, s = −1/ 2 ⎩ 3
  • 76. 1 −1 1 11 jm j = A l , ml , , + B l , ml + 1, , l 2 22 22 A, B = ? CG J2 1 1 jm j = j ( j + 1) J2 l 2 l jm j 2 2 J2 ⎛ ˆ2 3 2 ⎞ ⎜ L + 4 + Lz ˆ ˆ L− ⎟ J = L + S + 2LiS = ⎜ ⎟ 2 2 2 ⎜ − Lz ⎟ ˆ2 3 2 L+ ˆ ˆ⎟ ⎜ L+ ⎝ ⎠ 4 L± = Lx ± iLy ˆ ˆ ˆ 1 −1 11 l , ml + 1, , l , ml , , 22 22 ⎛ l , ml ⎞ 11 11 ⎜ ⎟ l , ml , , l , ml , ⎝0⎠ 22 22 ⎛ ⎞ 1 −1 0 11 l , ml + 1, , ⎜ ⎟ l , ml 1 , ⎝ l , ml 1 ⎠ 22 22 J2 ⎧ ⎡⎛ ˆ2 3 2 ⎤ ˆ⎞ + Lz ⎟ − j ( j + 1) 2 ⎥ l , ml + B L− l , ml + 1 = 0 ⎪ A ⎢⎜ L + ˆ ⎪ ⎣⎝ ⎠ ⎦ 4 ⎨ ⎪ A L l , m + B ⎡⎛ L2 + 3 2 − L ⎞ − j ( j + 1) 2 ⎤ l , m + 1 = 0 ˆ ˆ ˆ ⎢⎜ z⎟ ⎥ ⎪ + l l ⎣⎝ ⎠ ⎦ ⎩ 4 ⎧ ⎧ ⎡⎛ ⎫ ⎤ ⎞ 3 ⎪⎨ ⎢⎜ l ( l + 1) + + ml ⎟ − j ( j + 1) ⎥ A + ( l − ml )( l + ml + 1) B ⎬ l , ml = 0 ⎪⎩ ⎣⎝ ⎠ ⎦ ⎭ 4 ⎨ ⎪ ⎧ l + m l − m + 1 A + ⎡⎛ l l + 1 + 3 − m + 1 ⎞ − j j + 1 ⎤ B ⎫ l , m + 1 = 0 ⎪⎨ ( l )( ) ⎢⎜ ( ) ( l ) ⎟ ( )⎥ ⎬ l l ⎣⎝ ⎠ ⎦⎭ ⎩⎩ 4 ⎧ ⎡⎛ ⎤ ⎞ 3 ⎢⎜ l ( l + 1) + 4 + ml ⎟ − j ( j + 1) ⎥ A + ( l − ml )( l + ml + 1) B = 0 ⎪ ⎣⎝ ⎠ ⎪ ⎦ → A, B ⎨ ⎡ ⎤ ( l + ml )( l − ml + 1) A + ⎢⎛ l ( l + 1) + − ( ml + 1) ⎞ − j ( j + 1) ⎥ B = 0 ⎪ 3 ⎜ ⎟ ⎪ ⎣⎝ ⎠ ⎦ ⎩ 4 4
  • 77. 1 1 j= l− , l+ j 2 2 1 j=l+ 2 l + ml + 1 A = l − ml B l + ml + 1 l − ml 1 −1 1 11 jm j == l , ml , , + l , ml + 1, , l 2l + 1 2l + 1 2 22 22 CG CG 1 j=l− 2 l − ml l − ml + 1 1 −1 1 11 jm j = − l , ml , , + l , ml + 1, , l 2l + 1 2l + 1 2 22 22 5
  • 78. Schroedinger 1. dF =0 dt F F =ψ Fψ ˆ ∂ ψ =H ψ ˆ Schrodinger i ∂t ∂ψ ˆ ˆ∂ψ ∂Fˆ dF Fψ +ψ ψ +ψ F = ∂t ∂t ∂t dt ∂Fˆ 1 ˆˆ ⎡F, H ⎤ = + i⎣ ⎦ ∂t ˆ t F dF 1 ˆˆ ⎡F, H ⎤ = i⎣ ⎦ dt ˆ ˆ ˆ F H F ˆ2 ˆp ⎡ p, H ⎤ = 0 H= ˆˆ a ˆ t p ⎣ ⎦ 2m ˆ2 ˆ p +V ( x) ⎡ p, H ⎤ ≠ 0 H= ˆˆ b ⎣ ⎦ 2m ˆ ∂⎛ 2∂⎞ ˆ2 2 L2 ˆ p +V (r ) = − +V (r ) ˆ H= + L2 ⎜ ⎟ c r 2mr 2 ∂r ⎝ ∂r ⎠ 2mr 2 2m ⎡ L2 , H ⎤ = ⎡ L , H ⎤ = 0 ˆˆ ˆ ˆˆ Li t ⎥ ⎣i ⎦ ⎢ ⎣ ⎦ ⎡H , H ⎤ = 0 ˆˆ ˆ d t H ⎣ ⎦ ˆ H 1
  • 79. a b ⎡F, H ⎤ = 0 ˆˆ ˆ ˆ n F H ⎣ ⎦ F n = Fn n H n = En n ˆ ˆ ψ ( t ) = ∑ C n (t ) n Cn (t ) = n ψ (t ) n 2 ˆ Fn F C n (t ) ∂ E E d 1 ψ (t ) = n H ψ (t ) = n n ψ ( t ) = n C n ( t ) C n (t ) = n ˆ ∂t dt i i i i − En t Cn (t ) = Cn (0)e Cn (t ) = Cn (0) 2 2 a b c nlm 2. ψ → ψ′ = S ψ ˆ ˆ S ∂ ψ ψ =H ψ ˆ Schroedinger i ∂t ˆ S t 2
  • 80. ∂ˆ S ψ = SH ψ = SHS −1S ψ ˆˆ ˆˆˆ ˆ i ∂t ∂ ψ′ H ′ = SHS −1 ψ ′ = H′ ψ ′ ˆˆˆ ˆ ˆ Schroedinger i ∂t H ′ = SHS −1 = H , SH = HS ˆˆˆ ˆˆ ˆˆ ˆ ˆ ˆ H ⎡S , H ⎤ = 0 ˆˆ ⎣ ⎦ ψ′ ψ Schroedinger ∂ ψ′ = H ψ′ ˆ i ∂t ˆ S ⎡S , H ⎤ = 0 S = S+ S ≠ S+ ˆˆ ˆ ˆˆ ˆ ˆˆ S S ⎣ ⎦ ⎡S , H ⎤ = 0 ⎡ ˆ ˆ⎤ F = F+ ⎣F, H ⎦ = 0 ˆˆ ˆˆ ˆˆ F S (F ) ⎣ ⎦ → 1 ψ ( r ) → ψ ′ ( r ) = Sψ ( r ) r → r′ = r + δ r ˆ ψ ( x ') ψ ( x), O: O ' : ψ '( x), ψ '( x ') ∵ψ ( r ) = ψ ′ ( r ′ ) = Sψ ( r ′ ) = Sψ ( r + δ r ) ˆ ˆ i − δr•p ˆ ψ (r ) ∴ Sψ ( r ) = ψ ( r − δ r ) = ψ ( r ) − δ r • ∇ψ ( r ) + (δ r • ∇ ) ψ ( r ) + 1 =e 2 ˆ 2 i δr•p − ˆ S =e ( p = −i ∇ ) ˆ ˆ , ˆ p 3
  • 81. ⎡S , H ⎤ = 0 ⎡ p, H ⎤ = 0 ˆˆ ˆˆ ⎣ ⎦ ⎣ ⎦ 2 ψ ( t ) → ψ ′ ( t ) = Sψ ( t ) t → t′ = t + δ t ˆ ∵ψ ( t ) = ψ ′ ( t ′ ) = Sψ ( t ′ ) = Sψ ( t + δ t ) ˆ ˆ ∂ i −δ t δ tH ˆ ∴ Sψ ( t ) = ψ ( t − δ t ) = e ∂tψ ( t ) = e ψ (t ) ˆ ∂ i δ tH ˆ ψ = Hψ ) S =e ˆ ˆ , (i ∂t ˆ H ⎡S , H ⎤ = 0 ⎡ ˆ ˆ⎤ ⎣H , H ⎦ = 0 ˆˆ ⎣ ⎦ 3 4 ψ ( r ) → ψ ′ ( r ) = Iψ ( r ) r → r ′ = −r ˆ ∵ψ ( r ) = ψ ′ ( r ′ ) = Iψ ( r ′ ) = Iψ ( − r ) ˆ ˆ ∴ Iψ ( r ) = ψ ( − r ) ˆ ψ (r ) ϕ (r ) ∫ d rψ ( r ) ( Iˆϕ ( r ) ) = ∫ d rψ ( r ) ϕ ( −r ) = ∫ d rψ ( −r ) ϕ ( r ) = ∫ d r ( Iˆψ ( r ) ) ϕ ( r ) * 3 * 3 * 3 * 3 ˆ I Iψ ( r ) = ψ ( −r ) I 2ψ ( r ) = Iψ ( −r ) = ψ ( r ) ˆ ˆ ˆ ˆ ˆ I2 1 1 -1 I ψ s ( r ) = ψ s ( −r ) ψ a ( r ) = −ψ a ( −r ) Iψ s ( r ) = ψ s ( r ) Iψ a ( r ) = −ψ a ( r ) ˆ ˆ ⎡I , H ⎤ = 0 ˆˆ ⎣ ⎦ a ˆ ˆ b I H 4
  • 82. 2 V ( −r ) = V ( r ) ∇2 + V ( r ) IHI −1 = H H =− ˆ ˆˆˆ ˆ ˆ H 2m ˆ ˆ H I ˆ L=r×p p = −i ∇ ˆ ˆ ⎡I , L⎤ = 0 ⎡ I , L2 ⎤ = 0 ˆˆ ˆ ˆˆ ˆˆ ˆˆ ILI −1 = IrI −1 × IpI −1 = r × p = L ˆˆ ˆˆ ˆ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ ˆ ˆ ˆ L2 Lz I Ylm (θ , ϕ ) ˆ ˆ L2 Lz ∵ Ylm (θ , ϕ ) = N lm Pl m ( cos θ ) eimϕ l +|m| (1 − cos2 θ ) ⎛ d cos θ ⎞ ( cos θ − 1) 1 d Pl m ( cos θ ) = |m|/ 2 l 2 ⎜ ⎟ ⎝ ⎠ l 2 l! r → −r ˆ I ϕ → π +ϕ θ → π −θ cos θ → − cos θ r →r ( cos θ ) → ( −1) ( cos θ ) eimϕ → ( −1) eimϕ = ( −1) eimϕ l+ m ∴ Pl m |m| m m Pl IYlm (θ , ϕ ) = ( −1) Ylm (θ , ϕ ) = ( −1) Ylm (θ , ϕ ) l + 2|m| l ˆ Ylm (θ , ϕ ) ˆ I = (−1)l ˆ ˆ ˆ L2 Lz I I −r r 5
  • 83. 3. 1 “ ” “ ” ψ n ( x) {ψ n ( x1 ),ψ m ( x2 )} → {ψ n ( x2 ),ψ m ( x1 )} {ψ n ( x1 ),ψ m ( x2 )} {ψ n ( x2 ),ψ m ( x1 )} 1
  • 84. 2 N ...i... j... ˆ Pij Pij ...i... j... = ... j...i... = λ ...i... j... ˆ Pij2 ...i... j... = Pij ... j...i... = ...i... j... ˆ ˆ λ = ±1 ˆ ˆ Pij2 1 Pij Pij ...i... j... = ± ...i... j... ˆ ( ) ( ) .i... j. Pij .i... j. = λ .i... j. .i... j. = .i... j. Pij .i... j. ˆ ˆ ˆ Pij ˆ H Pij H ( j...) Pij −1 = H ( )=H( ) ˆˆ ˆ ˆ ˆ i j i i j ⎡ Pij , H ⎤ = 0 ˆˆ ⎣ ⎦ ˆ ˆ Pij Pij Schroedinger 3 2
  • 85. H (1, 2 )ψ (1, 2 ) = Eψ (1, 2 ) ˆ H ( 2,1)ψ ( 2,1) = Eψ ( 2,1) ˆ H (1, 2 ) = H ( 2,1) ˆ ˆ ˆ H H (1, 2 )ψ ( 2,1) = Eψ ( 2,1) ˆ ψ (1, 2 ) ψ ( 2,1) ˆ H E ψ (1, 2 ) ψ ( 2,1) ≠ ±ψ (1, 2 ) ψ + (1, 2 ) = ψ (1, 2 ) +ψ ( 2,1) ψ − (1, 2 ) = ψ (1, 2 ) −ψ ( 2,1) ˆ E H Hψ ± (1, 2 ) = Eψ ± (1, 2 ) ˆ H (1, 2 ) = H 0 (1) + H 0 ( 2 ) ˆ ˆ ˆ H 0ϕ n ( i ) = ε nϕ n ( i ) , i = 1, 2 ˆ Hψ (1, 2 ) = Eψ (1, 2 ) ˆ ⎧E = ε n + ε m ⎨ ⎩ψ (1, 2 ) = ϕn (1) ϕm ( 2 ) ψ (1, 2 ) ⎧1 ⎡ϕ n (1) ϕm ( 2 ) + ϕ n ( 2 ) ϕm (1) ⎤ , n≠m ⎪ ψ + (1, 2 ) = ⎨ 2 ⎣ ⎦ ⎪ ⎩ ϕn (1) ϕ m ( 2 ) n=m 1 ϕ n (1) ϕ n ( 2 ) 1 ψ − (1, 2 ) = ⎡ϕ n (1) ϕ m ( 2 ) − ϕ n ( 2 ) ϕ m (1) ⎤ = ⎣ ⎦ 2 ϕ m (1) ϕ m ( 2 ) 2 ψ − (1, 2 ) = 0, → n=m Pauli → 4 1 3
  • 86. a e( 1 ) 1 ψ ( r1 , r2 ) = ϕ (k1 , r1 )ϕ (k2 , r2 ) = i k • r1 + k2 • r2 ( 2π ) 3 1 ( r1 + r2 ) r = r1 − r R= 2 ( ) 1 k= k1 − k2 K = k1 + k2 2 ψ ( R, r ) = 1 eiK • R eik •r ( 2π ) 3 r P ( r ) = ∫ ψ ( R, r ) d 3 Rr 2 d Ω = Ar 2 2 b ψ ( r1 , r2 ) ≠ ±ψ ( r2 , r1 ) (ψ ( r1 , r2 ) +ψ ( r2 , r1 ) ) 1 ψ + ( r1 , r2 ) = 2 ( ) ( ) ψ + ( R, r ) = 1 1 ik •r 1 eiK • R e + e − ik •r = eiK • R 2 cos k • r ( 2π ) ( 2π ) 3 3 2 ⎛ sin 2kr ⎞ P+ ( r ) = ∫ ψ + ( R, r ) d 3 Rr 2 d Ω = Ar 2 ⎜1 + 2 ⎟ ⎝ 2kr ⎠ c (ψ ( r1 , r2 ) −ψ ( r2 , r1 ) ) 1 ψ − ( r1 , r2 ) = 2 ⎛ sin 2kr ⎞ ( ) ψ − ( R, r ) = 1 P− ( r ) = Ar 2 ⎜ 1 − eiK • R 2i sin k • r ⎟ ( 2π ) ⎝ 2kr ⎠ 3 2 êA r2 P + 1.75 1.5 P êA r 2 1.25 1 0.75 0.5 êA r2 P - 0.25 2.5 5 7.5 10 12.5 15 17.5 20 2kr 4
  • 87. → → r →∞ 2 F ( r1 − r2 ) ˆ F = ∫ψ * ( r1 , r2 ) F ( r1 − r2 )ψ ( r1 , r2 ) d 3 r1d 3 r2 ˆ = ∫ψ ±* ( r1 , r2 ) F ( r1 − r2 )ψ ± ( r1 , r2 ) d 3 r1d 3 r2 ˆ F ± ∫ [ψ ( r1 , r2 ) F ( r1 − r2 )ψ ( r1 , r2 ) +ψ ( r2 , r1 ) F ( r1 − r2 )ψ ( r2 , r1 ) 1 = ˆ ˆ * * 2 ±ψ * ( r1 , r2 ) F ( r1 − r2 )ψ ( r2 , r1 ) ± ψ * ( r2 , r1 ) F ( r1 − r2 )ψ ( r1 , r2 )]d 3 r1d 3 r2 ˆ ˆ = F ± ∫ψ * ( r1 , r2 ) F ( r1 − r2 )ψ ( r2 , r1 ) d 3r1d 3r2 ˆ ≠F 5
  • 88. 4. ψ ( r1 , s1z ; r2 , s2 z ) ˆˆ L•S ψ ( r1 , s1z ; r2 , s2 z ) = ψ ( r1 , r2 ) χ ( s1z , s2 z ) ( Hilbert ) ψ + ( r1 , r2 ) χ − ( s1z , s2 z ) a) ψ − ( r1 , r2 ) χ + ( s1z , s2 z ) b) ˆˆ S1 • S2 ˆ Sz χ ( s1z , s2 z ) = χ ( s1z ) χ ( s2 z ) ˆ Sz ⎛1⎞ ⎛0⎞ χ1 = ⎜ ⎟ χ =⎜ ⎟ 1 ⎝0⎠ ⎝1⎠ − 2 2 4 χ 1 (1) χ 1 ( 2 ) χ 1 (1) χ ( 2) (1) χ 1 ( 2 ) (1) χ − 1 ( 2 ) χ χ 1 1 1 − − − 2 2 2 2 2 2 2 2 4 χ + ( s1z , s2 z ) = χ 1 (1) χ 1 ( 2 ) 1 2 2 1⎛ ⎞ χ + ( s1z , s2 z ) = ⎜ χ 1 (1) χ − 1 ( 2 ) + χ 1 ( 2 ) χ − 1 (1) ⎟ 2 2⎝ 2 ⎠ 2 2 2 χ + ( s1z , s2 z ) = χ (1) χ − 1 ( 2 ) 3 1 − 2 2 1⎛ ⎞ χ − ( s1z , s2 z ) = ⎜ χ 1 (1) χ − 1 ( 2 ) − χ 1 ( 2 ) χ − 1 (1) ⎟ 2⎝ 2 ⎠ 2 2 2 4 ) = Sˆ + Sˆ + 2Sˆ Sˆ ( 2 ˆ ˆˆ S 2 = S1 + S 2 + 2 S1 y S 2 y + 2 S1z S 2 z ˆˆ ˆˆ 2 2 1 2 1x 2x S z = S1z + S2 z ˆ ˆ ˆ 1
  • 89. S 2 = s ( s + 1) 2 , Sz = m , s = 0,1, m = − s,...s S =2 , S z = , 0, − 2 2 S = 0, Sz = 0 2 ˆ 1 ⎛ˆ ⎞ ⎛ˆ ⎞ ⎛ˆ ⎞⎛ ˆ ⎞ ⎛ˆ ⎞⎛ ˆ ⎞ ⎛ˆ ⎞⎛ ˆ ⎞ S 2 χ + = ⎜ S12 χ 1 (1) ⎟ χ 1 ( 2 ) + χ 1 (1) ⎜ S 2 2 χ 1 ( 2 ) ⎟ +2 ⎜ S1x χ 1 (1) ⎟⎜ S2 x χ 1 ( 2 ) ⎟ + 2 ⎜ S1 y χ 1 (1) ⎟⎜ S2 y χ 1 ( 2 ) ⎟ + 2 ⎜ S1z χ 1 (1) ⎟⎜ S2 z χ 1 ( 2 ) ⎟ ⎝ ⎠2 ⎝ ⎠⎝ ⎠⎝ ⎠⎝ ⎠⎝ ⎠⎝ ⎠⎝ ⎠ 2 2 2 2 2 2 2 2 2 ˆ 1 ⎛ˆ ⎞ ⎛ˆ ⎞ S z χ + = ⎜ S1z χ 1 (1) ⎟ χ 1 ( 2 ) + χ 1 (1) ⎜ S2 z χ 1 ( 2 ) ⎟ ⎝ ⎠2 ⎝ ⎠ 2 2 2 ⎛1 0⎞ ⎛0 1⎞ 3 ˆ ˆ ∵ S12 = S2 2 = S1x = S 2 x = ˆ ˆ 2 ⎜ ⎟ ⎜ ⎟ ⎝0 1⎠ 2 ⎝1 0⎠ 4 ⎛ 0 −i ⎞ ⎛1 0 ⎞ S1 y = S2 y = S1z = S 2 z = ˆ ˆ ˆ ˆ ⎜ ⎟ ⎜ ⎟ 2 ⎝ 0 −1⎠ 2⎝i 0 ⎠ ˆ1 ∴ S 2 χ+ = 2 2 χ+ Sz χ+ = χ+ ˆ1 1 1 ˆ2 S 2 χ+ = 2 2 χ+ Sz χ+ = 0 ˆ2 2 ˆ3 S 2 χ+ = 2 2 χ+ Sz χ+ = − χ+ ˆ3 3 3 ˆ S 2 χ− = 0 Sz χ− = 0 ˆ ˆ χ+ χ− ˆ S2 Sz χ+ ⎧ 1 ⎪ χ+ S = 1, Sz = ⎨ 0 S2 = 2 2 2 ⎪− χ+ 3 ⎩ χ− S = 0, Sz = 0 S2 = 0 4 χ− χ+ χ+ χ+ 1 2 3 5. 2
  • 90. 1 Z z⎛ Ze2 ⎞ 1 2 e2 z H = ∑⎜ − ∑ ∇ j2 − ⎟+ ˆ ⎜ 2m rj ⎟ 2 rj − rk j =1 ⎝ ⎠ j , k =1 ( j ≠ k ) 1/2 ˆˆ L•S ψ ( r1 , , rz ) χ ( s1z ,..., szz ) z=2 m ( Ze2 ) 1 2 ψ ( r1 , r2 ) = ψ nlm ( r1 )ψ n′l′m′ ( r2 ) E = ε n + ε n′ εn = − 2 2 n2 n = n ' = 1, l = l ' = m = m ' = 0, 8 −2( r1 + r2 ) / a ψ + ( r1 , r2 ) = ψ 100 ( r1 )ψ 100 ( r2 ) = E = −109 eV e π a3 ψ + ( r1 , r2 ) χ ( s1z , s2 z ) e2 −109 eV >0 r1 − r2 2 2 ˆ ˆ ˆˆˆ S = S1 + S2 S1 S2 1 1 s1 = s2 = S = 0,1 2 2 3
  • 91. ˆ ˆ ˆˆˆ L = L1 + L2 L1 L2 l1 = l l2 = l L = l1 − l2 , , l1 + l2 = 0, , 2l ˆˆˆ J = L+S S =0 ⎧L j = L−S , ,L+ S = ⎨ ⎩ L − 1, L, L + 1 S =1 L S 1 ( r1 + r2 ) R→R Φ( R) R= 2 Ψ (r ) = RNL ( r ) YLM (θ , ϕ ) r → −r P ⇔I ˆ ˆ 12 P YLM (θ , ϕ ) = ( −1) YLM (θ , ϕ ) L ˆ 12 L L S =0 L S =1 L L+S S L 6.Pauli 0 < x < Lx , 0 < y < Ly , 0 < z < Lz 2 ∇ 2ψ = Eψ − 2m ψ =0 ψ ( x, y , z ) = X ( x ) Y ( y ) Z ( z ) = ( Ax sin k x x + Bx cos k x x ) ( Ay sin k y y + By cos k y y ) ( Az sin k z z + Bz cos k z z ) 4
  • 92. 2mE y 2mEx 2mEz kx = kz = ky = E = Ex + E y + Ez Ex Ez Ey ψ ( x, 0, z ) = ψ ( x, Ly , z ) = 0 ψ ( 0, y, z ) = ψ ( Lx , y, z ) = 0 ψ ( x, y, 0 ) = ψ ( x, y, Lz ) = 0 ψ n n n ( x, y, z ) = 8 / V sin k x x sin k y y sin k z z xyz n yπ nxπ nz π kx = ky = kz = nx , n y , nz = 1, 2, 3 Lx Ly Lz V = Lx Ly Lz 2 2 (k ) = 2m k Enx ny nz = +k +k 2 2 2 2 x y z 2m kx kz ky π3 π3 ∆k x ∆ k y ∆ k z = = Lx Ly Lz V 2 N Nq 1 1 − Nq Nq / 2 2 2 1 ⎛ 4π 3 ⎞ Nq ⎛ π 3 ⎞ kF ⎟ = ⎜⎟ ⎜ 8⎝ 3 ⎠ 2 ⎝V ⎠ kx > 0 kz > 0 ky > 0 1/8 kF 3 Nq k F n= =2 3π V k dk ( 4π k 2 ) dk V 2 1 8 = 2 k dk 2 π 3 /V π 5
  • 93. 2 k2 V 2 25 k FV kF E=∫ k dk = 2m π 10π 2 m 2 0 25 kF E ε= = V 10π 2 m ε = − P + µF n µF 2 µ F = EF = 2 kF 2m 2 P= ε 3 Pauli 6
  • 94. Schroedinger ∂ ψ ( r , t ) = Hψ ( r , t ) ˆ i ∂t ˆ 1 H Hψ ( r ) = Eψ ( r ) ˆ Schroedinger ψ n (r ) En σ (θ , ϕ ) ˆ 2 H Wmn 1 H n = En n ˆ H = H ( 0 ) + H (1) ˆ ˆ ˆ H ( 0) H (1) H ( 0) H (1) ˆ ˆ ˆ ˆ ˆ 1 H ( 0) ( 0) H ( 0) n = En ( 0) n H ( 0) H ( 0) ˆ ˆ 2 n En En = En ) + En ) + En ) + ( ( ( 0 1 2 ( 0) (1) ( 2) n=n +n +n + Schroedinger ( H ( ) + H ( ) ) ( n ( ) + n ( ) + n ( ) + ) = ( E( ) + E( ) + E( ) + ) ( n ( ) + n ( ) + n ( ) + ) 0 1 2 0 1 2 ˆ ˆ 0 1 0 1 2 n n n 1
  • 95. ( 0) ( 0) H ( 0) n = En0) n ( ˆ ( 0) ( 0) En n (1) ( 0) (1) ( 0) H ( 0) n + H (1) n = En0) n ( + En1) n ( ˆ ˆ (H ( ) − E ( ) ) n ( ) = −(H ( ) − E ( ) ) n ( ) 1 0 ˆ ˆ 0 0 1 1 n n (1) En ) ( H ( 0) ˆ 1 n ( H ( ) − E( ) ) n ( ) = − ( H ( ) − E( ) ) n ( ) + E( ) n ( ) 2 1 0 ˆ ˆ 0 0 1 1 2 n n n ( 2) ( 2) H ( 0) ˆ En n …… H ( 0) H ( 0) ˆ 2 H ( 0) 1 (1) ( 0) (1) =∑ i (0) n in i ∑(H ) i ( ) ( ) i n ( ) = − ( H ( ) − E( ) ) n ( ) ˆ ( 0) ( 0) − En 0 0 1 0 ˆ 1 1 n i ( 0) m ∑ ( E( ) − E( ) )δ ( 0) (1) ( 0) (0) m H( ) n + En )δ mn = − H mn + En δ mn ( =− ˆ1 0 0 1 (1) (1) in i n mi i H (1) H ( 0) H ( 0) H mn) = ( (0) (0) ˆ ˆ ˆ 1 m H (1) n ( E( ) − E( ) ) ( ) (1) = − H mn) + En )δ mn ( ( 0 0 0 1 1 mn m n En ) = H nn) ( ( m=n 1 1 H mn) (1 ( 0) (1) = m≠n mn En ) − Em ) ( ( 0 0 2
  • 96. ( 0) (1) (1) =∑ m (1) (0)(0) nn n mn m ( )( n ) ( 0) (1) ( 0) (1) ( 0) ( 0) (1) ( 0) ( 0) (1) 1= n n = n+ +n + + n nn nn nn ( 0) ( 0) =1 nn ( 0) (1) (1) ( 0) + =0 nn nn ( ) =0 * ( 0) (1) ( 0) (1) + nn nn ( 0) (1) nn ( 0) (1) = iα nn H mn) (1 ( 0) ( 0) ( 0) ( 0) +∑ +∑ m (1) + iα n =n n=n (0)(0) m mn ( 0) ( 0) En − Em m≠ n m H mn) (1 H mn) (1 ⎛ ( 0) ⎞ ( 0) ( 0) (0) +∑ +∑ = eiα n = eiα ⎜ n ⎟ m m ⎜ ⎟ ( 0) ( 0) ( 0) ( 0) En − Em En − Em ⎝ ⎠ m≠ n m≠n ( 0) (1) α = iα nn α =0 Schroedinger ⎧ En = En0) + H nn) ( (1 ⎪ H( ) ⎨ 1 ( 0) ( 0) n = n + ∑ ( 0) mn ( 0) m ⎪ m ≠ n En − Em ⎩ H ( 0) ˆ 2 ( 2) ( 0) ( 0) ( 2) =∑ i n in i ∑ ( H ( ) − E( ) ) i ( ) ( 0) ( 0) ( 2) ( 0) ( 0) (1) ( 0) = −∑ H ( ) − En ) i (1 ( 2) + En ˆ ˆ1 0 0 in in n n i≠n i ( 0) m ∑ ( E( ) − E( ) )δ ( ) ( ) i n ( ) + E ( )δ ( 0) ( 2) = −∑ H mi) − En )δ mi ( ( 0 1 0 0 1 1 2 in i n mi n mn i≠n i ( E( ) − E( ) ) ( ) ( 2) (1) ( 0) (1) = −∑ H mi) ( + En ) ( + En )δ mn ( 0 1 (0) 0 0 1 2 mn in mn m n i≠n ( 0) (1) m=n = iα = 0 nn 3
  • 97. (1) 2 H (1) H (1) H in = ∑H = ∑ ( 0ni in( 0) = ∑ ( 0) ( 2) (1) (0) (1) E in ) ( 0) i ≠ n En − Ei i ≠ n En − Ei n ni i≠n (1) ( 2) m≠n mn Schroedinger ⎧ H in ) (1 2 ⎪ E = E ( 0) + H (1) + ∑ E ( 0) − E ( 0) ⎪n n nn i≠n ⎨ n i (1) ⎪ H in ( 0) ( 0) ( 2) ⎪ n = n + ∑ ( 0) i +n ( 0) i ≠ n En − Ei ⎩ 3 H in1) ( << 1 En0) − Ei( 0) ( H in1) << En0) − Ei( 0) ( ( H (1) H ( 0) H ( 0) H (1) H ( 0) ˆ ˆ ˆ ˆ En0) − Ei( 0) → 0 ( H ( 0) ˆ 1 ( 0) ∑ ( 0) H ( 0) ˆ 2 n En i En0) − Ei( 0) = 0 ( ε q 2 d2 1 + mω 2 x 2 − qε x H =− ˆ 2m dx 2 2 ε H = H ( 0 ) + H (1) ˆ ˆ ˆ 2 d2 1 H ( 0) = − H ( ) = − qε x + mω 2 x 2 ˆ ˆ1 2 2m dx 2 4
  • 98. ( 0) ( 0) H ( 0) n = En0) n ( ˆ ⎛ 1⎞ ( 0) En ) = ⎜ n + ⎟ ω (0 = ψ n ( x) xn ⎝ 2⎠ En ) = H nn) = −qε xnn ( ( 1 1 ⎛ n +1 ⎞ ( 0) ( 0) n En ) = 0 ( δ m ,n +1 + δ m,n −1 ⎟ xnn = 0 xmn = = 1 ⎜ ˆ mxn mω ⎜ ⎟ 2 2 ⎝ ⎠ H mn) (1 xmn (1) ( 0) ( 0) = ∑ ( 0) = − qε ∑ n m m ( 0) ( 0) ( 0) m ≠ n En − Em m ≠ n En − Em ⎛ ( 0) ⎞ n +1 ( 0) n = − qε n + 1 + ( 0) n −1 ⎟ ⎜ ( 0) 2mω ⎜ En − En0)1 ⎟ ( En − En0)1 ( ⎝ ⎠ + − ( ) qε ( 0) ( 0) = n +1 n +1 − n n −1 2m ω 3 ( ) qε (1) ( 0) ( 0) ψ n1) ( x ) = x n ( = n +1 x n +1 − n x n −1 2m ω 3 qε ( ) n + 1 n0)1 ( x ) − nψ n0)1 ( x ) ψ (+ ( = − 2m ω 3 2 ⎛ n +1 ⎞ n δ m ,n +1 + δ m ,n −1 ⎟ ⎜ H mn) (1 2 ε 2q2 ⎠ =− ε q ⎝2 2 22 En2) = ∑ ∑ ( = En0) − Em0) ( ( En0) − Em0) ( ( mω m ≠ n 2mω 2 m≠ n ε 2q2 ⎧ ⎛ 1⎞ En = ⎜ n + ⎟ ω − ⎪ 2mω 2 ⎪ ⎝ 2⎠ ⎨ ( ) q ⎪ψ ( x ) = ψ ( 0) ( x ) +ψ (1) ( x ) = ψ ( 0) ( x ) + n + 1 n0)1 ( x ) − nψ n0)1 ( x ) ψ (+ ( ⎪ − n n n n 2m ω 3 ⎩ ⎛ ⎞ 2 d2 1 + mω 2 x 2 − qε x ⎟ψ ( x ) = Eψ ( x ) − ⎜ 2 ⎝ 2m dx 2 ⎠ ⎛ qε ⎞ ⎞ q 2ε 2 ⎞ ⎛ 2 2⎛ 2 d2 1 ⎟ψ ( x ) = ⎜ E + ⎟ψ ( x ) + mω ⎜ x − ⎜− ⎟ ⎜ 2m dx 2 2 mω 2 ⎠ ⎟ 2mω 2 ⎠ ⎝ ⎝ ⎝ ⎠ 5
  • 99. qε ξ = x− mω 2 q 2ε 2 ⎞ ⎛ ⎞ ⎛ 2 d2 1 + mω 2ξ 2 ⎟ψ (ξ ) = ⎜ E + ⎟ψ (ξ ) − ⎜ ⎝ 2m d ξ 2mω 2 ⎠ 2 ⎠ ⎝ 2 ε 2q2 ⎛ ε 2q2 ⎧ 1⎞ ⎛ 1⎞ = ⎜n+ ⎟ ω En = ⎜ n + ⎟ ω − En + → ⎪ 2mω 2mω 2 ⎪ ⎝ 2⎠ ⎝ 2⎠ 2 ⎨ ⎪ψ (ξ ) = ψ ⎛ x − qε ⎞ , n⎜ ⎟ ⎪n mω 2 ⎠ ⎝ ⎩ 6
  • 100. 6.3 ( 0) ( 0) H ( 0) n, i = En0) n, i ( i = 1, 2, ˆ ,a Schmidt a ( 0) ( 0) = δ ij n, i n, j ( 0) ˆ n, i H a (0) ( 0) = ∑ ci n, i n i =1 ( 0) H ( 0) ˆ En ci ( 0) n ( H ( ) − E( ) ) n ( ) = − ( H ( ) − E( ) ) n ( ) 1 0 ˆ ˆ 0 0 1 1 n n ( H ( ) − E ( ) ) n ( ) = − ∑ c ( H ( ) − E ( ) ) n, i ( ) a 1 0 ˆ ˆ 0 0 1 1 n i n i =1 ( 0) ( j = 1,......, a) n, j ( ) a ( 0) ( 0) 0 = ∑ ci H (ji ) − En )δ ji H (ji1) = n, j H (1) n, i ( ˆ 1 1 i =1 ci (i = 1,......, a ) a a ci H11) − En ) ( ( H12) ( H1(a) 1 1 1 1 H 21) (1 H 22) − En1) (1 ( H 21a) ( =0 H a1) ( H a 2) ( H aa) − En ) ( ( 1 1 1 1 → En ) ( Eni ) ( 1 1 a ( 0) → Eni = En ) + Eni , ( i = 1,...a 0 (1) En 1
  • 101. a {ci1 , ci 2 , , cia } Eni ) ( 1 Eni ˆ H a ( 0) ( 0) = ∑ cij n, j n, i j =1 ( 0) ( 0) ( 0) 1 En n,1 n, 2 2 H11 − En (1) (1) (1) H12 = 0, H ij = (0) (0) ˆ (1) n, i H (1) n, j H 22 − En (1) (1) (1) H 21 ( E( ) ) − ( H ( ) + H ( ) ) E( ) + H ( ) H ( ) − H ( ) 2 2 =0 1 1 1 1 1 1 1 11 22 11 22 12 n n 1 ⎛ (1 (1 2 ⎞ ( ) (H( ) − H( ) ) 2 En ) = ⎜ H11) + H 22) ± (1 (1 + 4 H12) ⎟ 1 1 11 22 2⎝ ⎠ H12) = 0 ( 1 En1) = H1(1) ( En 2) = H 22) ( ( En1 = En ) + H1(1) ( En 2 = En ) + H 22) ( ( 1 1 1 1 0 1 0 1 2 2 e2 2 e2 ∇ − + eε ⋅ r = − ∇ − + eε r cos θ H =− ˆ 2 2 2µ 2µ r r ε z ε 2 e2 H (1) = eε r cos θ H( ) = − ∇2 − ˆ ˆ0 2µ r µ e4 En ) = − ( ψ nlm (r ) 0 n2 2n 2 2 ( 0) ϕ1 = ψ 200 ϕ2 = ψ 210 ϕ3 = ψ 211 ϕ 4 = ψ 21−1 n=2 4 E2 H ij1) = ∫ d 3 r ϕi* ( r ) H (1)ϕ j ( r ) ( ˆ 2
  • 102. H12) = H 21) = −3eε a0 ( ( 1 1 4 − E21) ( −3eε a0 0 0 − E2 ) ( −3eε a0 1 0 0 =0 − E2 ) ( 1 0 0 0 − E2 ) ( 1 0 0 0 E2,1) = 3eε a0 (1 (1) (1) (1) E2,2 = −3eε a0 E2,3 = E2,4 = 0 E2 ) (4 ( → E2i = E2 ) +3eε a0 , E2 ( ( 0) 3eε a0 , E2 ) (2 ( 0 0 0 ( 0) 4 E2 E2i ) ( 1 ⎛ − E21) ( 0 ⎞ ⎛ ci1 ⎞ −3eε a0 0 ⎜ ⎟⎜ ⎟ i − E2i ) ( ⎜ −3eε a0 0 ⎟ ⎜ ci 2 ⎟ 1 0 =0 ⎜ ⎟ 0 ⎟ ⎜ ci 3 ⎟ − E2i ) ( 1 ⎜0 0 ⎜⎟ ⎜ (1 ⎟ − E2i ) ⎠ ⎝ ci 4 ⎠ ⎝0 0 0 {ci1 , ci 2 , ci 3 , ci 4 } 4 ψ 20) = ∑ cijϕ j E2i = E2 ) +E2i ( ( 0 (1) i j =1 3 p2 T= 2m p2 p4 T= p 2 c 2 + m 2 c 4 − mc 2 = − 3 2+ 2m 8m c ˆ2 2 ˆ4 ˆ p −e − p H= p4 2m r 8m3c 2 ˆ2 2 p4 ˆ ˆ ( 0) = p − e H( ) = − ˆ1 H 8m3c 2 2m r ψ nlm (r ) = N nl ( r ) Ylm (θ , ϕ ) ( 0) l = 0,...n − 1, m = −l ,...l n2 En 3
  • 103. n2 ( 0) ( 0) ( 0) ( 0) 1 (1) nlm H (1) nl ′m′ nlm p 4 nl ′m′ H nlm , nl ′m′ = =− ˆ ˆ 8m 3 c 2 Shrodinger ⎛ p2 ⎞ ˆ ( 0) (0) + V ( r ) ⎟ nlm = En0) nlm ( ⎜ ⎝ 2m ⎠ ( ) ( ) ( 0) (0) = 2m En ) − V ( r ) nlm nlm p 2 = (0) nlm 2m En ) − V ( r ) ( (0 0 p 2 nlm (0) ˆ ˆ , ( ) ( 0) ( 0) 1 nlm En0) − V ( r ) 2 ( nl ′m′ H nlm ,nl ' m ' = − (1) 2mc 2 (( ) ) (0) ( 0) ( 0) ( 0) 1 nlm V ( r ) nl ′m′ nlm V 2 ( r ) nl ′m′ 2 En0) δ ll ′δ mm′ − 2 En0) ( ( =− + 2mc 2 V (r) θ ϕ e2 e2 ( 0) ( 0) nlm V ( r ) nl ′m′ = ∫ drr 2 N nl (r ) N nl ' (r )δ ll ′δ mm′ = − 2 δ ll ′δ mm′ r na e4 ( 0) (0) nlm V 2 ( r ) nl ′m′ δδ = ( l + 1/ 2 ) n3a 2 ll′ mm′ 1 ⎛ ( 0) ⎞ () 2e 2 ( 0 ) e4 H nlm ,nl ′m′ = H nl )δ ll ′δ mm′ (1) (1 2 H nl ) = − ( + En + 1 ⎜ En ⎟ 2mc 2 ⎜ ( l + 1 / 2 ) n3 a 2 ⎟ n2a ⎝ ⎠ ˆˆ (1) L2 , Lz H nlm, nl ′m′ Enl = H nl ) ( 1 (1) → Enl = En + H nl ) (2l + 1 ( 1 En (n 2 (0) (0) 4 - 1 1 dV ( r ) ˆ ˆ 2 ∇2 + V ( r ) L • S = ξ (r ) L • S H( ) = − H( ) = − 2 H = H ( 0 ) + H (1) ˆˆ ˆ0 ˆ1 ˆ ˆ ˆ 2µ 2 µ r dr H (1) ˆ {Hˆ , Lˆ , Lˆ , Sˆ } ( 0) 2 2n 2 En z z 4
  • 104. {Hˆ , Lˆ , Jˆ , Jˆ } 2 2 nlml ms nljm j z ⎡ H (1) , Lz ⎤ ≠ 0 ⎡ H (1) , S z ⎤ ≠ 0 ˆ ˆ ˆ ˆ ˆ ˆ L2 Lz ⎣ ⎦ ⎣ ⎦ H (1) H (1) ˆ ˆ ˆ ˆ ˆ ˆ L2 Sz Lz Sz nlml ms H (1) nl ′ml′ms ′ ˆ ⎡ H (1) , J 2 ⎤ = ⎡ H (1) , L2 ⎤ = ⎡ H (1) , J ⎤ = 0 ˆ ˆ ˆ ˆ ˆ ˆ ⎢ ⎥⎢ ⎥⎢ z⎥ ⎣ ⎦⎣ ⎦⎣ ⎦ H (1) H (1) ˆˆ ˆ ˆ ˆ ˆ ˆ J2 L2 , J 2 L2 Jz Jz Enlj) = nljm j H ( ) nljm j = nljm j ξ ( r ) L • S nljm j (1 ˆˆ ˆ1 ) ξ ( r ) ˆ 2 ˆ2 ˆ 2 ( = nljm j J − L − S nljm j 2 2 ∞ ⎡ j ( j + 1) − l ( l + 1) − s( s + 1) ⎤ ∫ Rnl ( r ) ξ ( r ) r 2 dr = 2 ⎣ ⎦0 2 → Enlj = En + Enlj) (1 2 j +1 En (2n 2 (0) (0) n=2 ⎧ (0) (1) ⎪ E2 + E213 ⎪ 2 ⎪ (0) (1) → Enlj = ⎨ E2 + E 1 (0) E2 (8 ⎪ 21 2 ⎪ E (0) + E (1) ⎪2 1 ⎩ 20 2 E ( )1 = 0 l =0 j=s 1 20 2 5
  • 105. 6.4 H n = En n ˆ ψ = ∑ cn n n E = ψ H ψ = ∑ cm cn m H n = ∑ cm cn Enδ mn = ∑ cn En ≥ E0 2 ˆ ˆ * * n ,m n ,m n ψ E =ψ Hψ ˆ E E0 E0 ψ (λ ) λ E (λ ) = ψ (λ ) H ψ (λ ) ˆ d E (λ ) d 2 E (λ ) =0 >0 dλ dλ2 E ( λ0 ) → λ0 E0 ≈ E ( λ0 ) 2 2 2e 2 2e 2 e2 e2 = H (0) + H =− ∇12 − ∇ 22 − − + ˆ ˆ 2µ 2µ r1 − r2 r1 − r2 r1 r2 ψ ( r1 , r2 , S1z , S2 z ) = ψ + ( r1 , r2 ) χ − ( S1z , S2 z ) H ( 0) ˆ 3/ 2 1 ⎛Z⎞ Z − r ψ + ( r1 , r2 ) = ψ 100 ( r1 )ψ 100 ( r2 ) ψ 100 ( r ) = Z =2 a0 ⎜⎟ e π ⎝ a0 ⎠ ψ 100 ( r1 ) ψ 100 ( r2 ) Z <2 ˆ H Z 1
  • 106. ψ + ( r1 , r2 , Z ) χ − ( S1z , S2 z ) ˆ H ˆ H E ( Z ) = ∫ψ +* ( r1 , r2 , Z ) Hψ + ( r1 , r2 , Z ) d 3 r1d 3 r2 ˆ 27 ⎞ −e ⎛ 2 = ⎜ −2Z 2 + Z ⎟ ⎝ 4 ⎠ 2a d E (Z ) d 2 E (Z ) =0 >0 dZ 2 dZ 27 Z0 = 16 e2 E0 ≈ E ( Z 0 ) = −2.85 a0 ψ + ( r1 , r2 , Z 0 ) E0 e2 −2.904 a0 e2 H (1) = ˆ r1 − r2 e2 5 e2 E01) = ∫ d 3 r1d 3 r2ψ +* ( r1 , r2 ) ψ + ( r1 , r2 ) = ( r1 − r2 4 a0 e2 5 e2 e2 −4 + = −2.75 E0 a0 4 a0 a0 6.5 Shrodinger 1999 Schroedinger e −α r V (r ) = −g 2 g >1 r 2
  • 107. ⎛ ⎞ 2 ∇ 2 + V ( r ) ⎟ψ ( r ) = Eψ ( r ) − ⎜ ⎝ 2µ ⎠ =1 ψ ( r ) = e− S ( r ) S (r) 1/ g 2 E E = g 4 E0 + g 2 E1 + E2 + S = g 2 S 0 + S1 + g −2 S 2 + V E0 S0 Schroedinger g ( ∇S ) 2 = −2mE0 0 ⎛ e −α r ⎞ 1 ∇S0 • ∇S1 = ∇ 2 S0 − m ⎜ + E1 ⎟ ⎝r ⎠ 2 ( ∇S1 ) + 1 ∇ 2 S1 − mE2 1 2 ∇S 0 • ∇S 2 = − 2 2 …… S0 ( r ) = −2mE0 r ψ (r) S0 ( r ) ( ) dS1 1 −2mE0 − me −α r − mE1 −2mE0 = dr r dS1 ψ ψ r =0 dr −2mE0 − me−α r lim r →0 r 3
  • 108. E0 = − m / 2, S0 ( r ) = mr , ⎛1 ⎞ S1 = ∫ dr ′ ⎜ (1 − e −α r ′ ) − E1 ⎟ r ⎝ r′ ⎠ 0 dS 2 r =0 dr E1 = α 3α 2 α 2 −2 m4 g +α g2 − E=− + g+ 4m 2m 2 2 α =0 ⎧ m4 ⎪E = − 2 g ⎨ ⎪ψ ( r ) = e − g 2 S0 ( r ) = e− mg 2r ⎩ 1. 4
  • 109. σ (θ , ϕ ) Ω σ (θ , ϕ ) ψ 1 ( r ) = Aeikz i ⎛ ∂ψ 1* ∂ψ 1 ⎞ k2 ⎜ψ 1 −ψ 1* Jz = ⎟= A 2µ ⎝ ∂z ⎠ µ ∂z dΩ dN = J zσ (θ , ϕ ) d Ω [ dΩ] = 1 1 1 ∵ [ dN ] = [Jz ] = σ L2T T ∴[σ ] = L2 σ (θ , ϕ ) σ (θ , ϕ ) 2. V (r ) → 0 ψ (r ) r →∞ r z eikr ψ ( r ) r → ∞ ψ 1 ( r ) +ψ 2 ( r ) = Aeikz + Af (θ , ϕ ) r f (θ , ϕ ) f (θ , ϕ ) θ ψ2 ψ1 ϕ 1 A 2 k k 2 f (θ , ϕ ) f (θ , ϕ ) 2 2 i ⎛ ∂ψ 2* ∂ψ 2 ⎞ ⎜ψ 2 − ψ 2* Jr = ⎟= = Jz A 2µ ⎝ ∂r ⎠ µ ∂r 2 r2 r 5
  • 110. dΩ dN = J r dS = J r r 2 d Ω = J z f (θ , ϕ ) d Ω 2 dN = J zσ (θ , ϕ ) d Ω σ (θ , ϕ ) = f (θ , ϕ ) 2 f (θ , ϕ ) σ (θ , ϕ ) ψ (r ) 1 Schroedinger eikr (r → ∞) f (θ , ϕ ) ψ ( r → ∞ ) = Aeikz + Af (θ , ϕ ) 2 r σ (θ , ϕ ) = f (θ , ϕ ) 2 3 6
  • 111. 3. Green Schroedinger ⎛ ⎞ 2 ∇ 2 + V ( r ) ⎟ψ ( r ) = Eψ ( r ) ⎜− ⎝ 2µ ⎠ 2µ E 2µ U (r ) = V (r ) k2 = 2 2 (∇ + k 2 )ψ ( r ) = U ( r )ψ ( r ) 2 U (r ) δ (r − r ') (∇ + k 2 ) G ( r , r ′) = δ ( r − r ′) 2 r ik r − r ′ 1e G ( r , r′) = − Green 4π r − r ′ (∇ + k 2 )ψ ( r ) = ∫ d 3 r ′U ( r ′ )ψ ( r ′ ) δ ( r − r ' ) 2 = ∫ d 3 r ′U ( r ′ )ψ ( r ′ ) ( ∇ r + k 2 ) G ( r , r ′ ) 2 = ( ∇ 2 + k 2 ) ∫ d 3 r ′U ( r ′ )ψ ( r ′ ) G ( r , r ′ ) ψ ( r ) = ∫ d 3 r ′U ( r ′ )ψ ( r ′ ) G ( r , r ′ ) (∇ + k 2 )ψ (0) ( r ) = 0 2 ψ (0) ( r ) = Aeik ⋅r ψ ( r ) = Aeik ⋅r + ∫ d 3 r ′U ( r ′ )ψ ( r ′ ) G ( r , r ′ ) ψ (r ) Lippmann Schwinger 1
  • 112. 1 2 4. Born V (r ) ψ ( 0) ( r ) = Aeik ⋅r ψ (1) ( r ) = Aeik ⋅r + ∫ d 3 r ′U ( r ′ )ψ ( 0) ( r ′ ) G ( r , r ′ ) V ( r′) r →∞ r ′ << r r • r′ ⎞ r • r′ 1/ 2 ⎛ = ( r + r ′ − 2r • r ′ ) ( r − r ′) 1/ 2 r − r′ = ≈ r ⎜1 − 2 2 ⎟ ≈r− 2 2 2 ⎝ r⎠ r ′ r •r 1 eikr − ik (r ) ′e r U ( r ′ ) eik ⋅r ' (1) 4π r ∫ ψ ik ⋅r r → ∞ Ae −A 3 dr r −r ' r −r ' eikr ψ ( r ) = Ae + Af (θ , ϕ ) ik ⋅r r ′ r •r − ik 1 f (θ , ϕ ) = − ′e r U ( r ′ ) eik ⋅r ' 4π ∫ 3 dr r k′ = k r i ( k − k ′ )• r ′ 1 f (θ , ϕ ) = − U ( r ′) ∫ d r ′e 3 4π r k' k q k' k r Θ k 2
  • 113. |k | σ 1 f (θ , ϕ ) = − d 3 r ′U ( r ′ ) 4π ∫ ⎧V , r ≤ a V (r ) = ⎨ 0 ⎩ 0, r > a µV µV0 4 1 f (θ , ϕ ) = − ∫ d rU ( r ) = − 2π 02 ∫ d r = − 2π π a3 3 3 4π 2 3 r ≤a 2 ⎛ 2 µV0 a 3 ⎞ σ (θ , ϕ ) = f (θ , ϕ ) = ⎜ 2 ⎟ 2 ⎝3 ⎠ 2 ⎛ 2 µV0 a 3 ⎞ = ∫ σ (θ , ϕ )d Ω = 4π ⎜ σ tot ⎟ 2 ⎝3 ⎠ V (r ) = V (r ) π 2π ∞ 1 f (θ , ϕ ) = − dr ′r ′2U ( r ' ) ∫ dθ ′ sin θ ′e − iqr ′ cosθ ′ ∫ dϕ ′ ∫ 4π 0 0 0 2µ ∞ 1∞ dr ′U ( r ') r ′ sin qr ′ = − 2 ∫ dr ′V ( r ') r ′ sin qr ′ q ∫0 =− q0 θ q θ θ q = k − k ′ = k 2 + k ′2 − 2kk ′ cos θ = 2k 2 (1 − cos θ ) = 4k 2 sin 2 = 2k sin 2 2 σ (θ ) = f (θ ) 2 Born 5. Born Lippmann Schwinger ψ ( r ) = ψ (0) ( r ) + ∫ d 3 r ′G ( r , r ′ )U ( r ′ ) ( r ′ ) ψ ψ (0) (r ) 3
  • 114. ψ (1) ( r ) = ψ (0) ( r ) + ∫ d 3 r ′G ( r , r ′ ) U ( r ′ ) (0) ( r ′ ) ψ ψ (2) ( r ) = ψ (0) (r ) + ∫ d 3r ′G ( r , r ′ )U ( r ′ ) (1) ( r ′ ) ψ = ψ (0) (r ) + ∫ d 3 r ′G ( r , r ′ ) U ( r ′ ) ⎡ψ (0) (r ') + ∫ d 3r quot; G ( r ', r quot;) U ( r quot;) (0) ( r quot;) ⎤ ψ ⎣ ⎦ = ψ (0) (r ) + ∫ d 3 r ′G ( r , r ′ ) U ( r ′ ) (0) ( r ′ ) ψ + ∫ d 3 r ′d 3 r quot; G ( r , r ′ ) U ( r ′ ) G ( r ', r quot;) U ( r quot;)ψ (0) ( r quot;) V G Ψ G G ...... Ψ0 Ψ0 Ψ0 V V Green 4
  • 115. 6. 1) { H , p, L } ˆˆˆ Aeikz z 2 k2 E= , px = p y = 0, pz = k , Lz = 0 2µ {Hˆ , Lˆ , Lˆ } L2 2 p z Hilbert {Hˆ , Lˆ , Lˆ } ˆ → ˆ L2 2 Aeikz Aeikz H z ˆ → m=0 m=0 ˆ L2 Aeikz k l Lz 2 Pl ( cos θ ) ˆ m=0 L2 Aeikz ∞ eikz = eikr cosθ = ∑ ( 2l + 1) i l jl ( kr ) Pl ( cos θ ) l =0 jl ( kr ) Bessel l ⎛ l⎞ 1 jl ( kr ) → sin ⎜ kr − π ⎟ r →∞ ⎝ 2⎠ kr ∞ ⎛ l⎞ 1 eikz → ∑ ( 2l + 1) i l sin ⎜ kr − π ⎟ Pl ( cos θ ) ⎝ 2⎠ kr l =0 eikr limψ ( r ) = Aeikz + Af (θ ) r →∞ r ∞ ⎛ l⎞ eikr l1 = A∑ ( 2l + 1) i sin ⎜ kr − π ⎟ Pl ( cos θ ) + Af (θ ) . ⎝ 2⎠ kr r l =0 3 Schroedinger m 1
  • 116. ∞ ψ ( r , θ ) = ∑ Rl ( r ) Pl ( cos θ ) l =0 Rl ( r ) l ( l + 1) ⎞ 1 d ⎛ 2 dRl ⎞ ⎛ 2 2µ ⎟ + ⎜ k − 2 V (r ) − ⎟ Rl = 0 ⎜r r 2 dr ⎝ dr ⎠ ⎝ r2 ⎠ V (r) → 0 r →∞ d 2 ( rRl ) + k 2 ( rRl ) = 0 2 dr ⎛ ⎞ Al l Rl ( r → ∞ ) = A sin ⎜ kr − π + δ l ⎟ ⎝ ⎠ kr 2 l V (r) δl −π Al A 2 ∞ ⎛ ⎞ Al l limψ ( r , θ ) = A∑ sin ⎜ kr − π + δ l ⎟ Pl ( cos θ ) ⎝ ⎠ r →∞ kr 2 l =0 ∞ ∞ ⎛ l⎞ ⎛ ⎞ eikr A 1 l ∑ ( 2l + 1) il = ∑ l sin ⎜ kr − π + δ l ⎟ Pl ( cos θ ) sin ⎜ kr − π ⎟ Pl ( cos θ ) + f (θ ) ⎝ 2⎠ ⎝ ⎠ kr r l = 0 kr 2 l =0 eix − e− ix sin x = 2i ⎛ ⎞ ∞ ∞ 0 = ⎜ 2kif (θ ) + ∑ ( 2l + 1) i l e− ilπ / 2 Pl ( cos θ ) − ∑ Al e ( l Pl ( cos θ ) ⎟ eikr i δ − lπ / 2 ) ⎝ ⎠ l =0 l =0 ⎛∞ ⎞ ∞ − ⎜ ∑ ( 2l + 1) i l eilπ / 2 Pl ( cos θ ) − ∑ Al e ( l Pl ( cos θ ) ⎟ e− ikr − i δ − lπ / 2 ) ⎝ l =0 ⎠ l =0 e −ikr eikr r Al = ( 2l + 1) i l eiδl ∞ ∞ 1 f (θ ) = ∑ ( 2l + 1) Pl ( cos θ ) eiδl sin δ l = ∑ fl (θ ) k l =0 l =0 1 fl (θ ) = ( 2l + 1) Pl ( cos θ ) eiδl sin δ l k 2
  • 117. δl δl V (r) Rl ( r ) 1 ⎛ ⎞ Al l lim Rl ( r ) δl sin ⎜ kr − π + δ l ⎟ 2 A ⎝ ⎠ r →∞ kr 2 2 ∞ ∑ f (θ ) fl (θ ) σ (θ ) = 3 l l =0 δl ⎛ l⎞ 1 sin ⎜ kr − π ⎟ ⎝ 2⎠ kr ⎛ ⎞ 1 l sin ⎜ kr − π + δ l ⎟ ⎝ ⎠ kr 2 δl 4 L ∼ pr p l ( l + 1) < ka V (r) L < pa a l E (k ) s ⎧−V r≤a V (r) = ⎨ 0 r>a ⎩0 3
  • 118. l ( l + 1) ⎞ 1 d ⎛ 2 dRl ⎞ ⎛ 2 2µ ⎟ + ⎜ k − 2 V (r ) − ⎟ Rl = 0 ⎜r r dr ⎝ dr ⎠ ⎝ 2 r2 ⎠ (l = 0) s ⎧ d 2 ( rR0 ) + k ′2 ( rR0 ) = 0 r≤a ⎪ ⎪ dr 2 ⎨2 ⎪ d ( rR0 ) + k 2 rR = 0 ( 0) r>a ⎪ dr 2 ⎩ 2µ E 2µ k ′2 = k 2 + k2 = V0 2 2 ′ ⎧ C0 ⎪ A r sin(k ′r + B0 ) ′ r≤a ⎪ R0 ( r ) = ⎨ ⎪ A C0 sin(kr + B ) r>a ⎪r ⎩ 0 R0 ( r ) ′ r =0 B0 = 0 ′ C0 = C0 ⎧ dR0 ( r ) ⎪ R0 ( r ) r=a ⎨ ⎛k ⎞ ⎪ B0 = arctan ⎜ k ′ tan k ′a ⎟ − ka dr ⎝ ⎠ ⎩ C0 lim R0 ( r ) = A sin(kr + B0 ) r →∞ r A0 lim R0 ( r ) = A sin ( kr + δ 0 ) r →∞ kr s δ 0 = B0 P0 (cos θ ) = 1 1 iδ 0 f 0 (θ ) = e sin δ 0 k f (θ ) ∼ f 0 (θ ) sin 2 δ 0 σ (θ ) = f (θ ) f 0 (θ ) 2 2 = k2 k →0 4
  • 119. 2µ ⎛k ⎞ k k ′2 → arctan ⎜ tan k ′a ⎟ → tan k ′a V0 ⎝ k′ k′ ⎠ 2 ⎛ tan k ′a ⎞ k tan k ′a − ka = ka ⎜ δ0 − 1⎟ k′ ⎝ k ′a ⎠ sin 2 δ 0 δ 02 ⎛ tan k ′a ⎞ 2 σ (θ ) − 1⎟ 2 a⎜ ⎝ k ′a ⎠ k2 k2 σ θ s ⎛ tan k ′a ⎞ 2 σ tot = ∫ σ (θ ) d Ω = 4π a 2 ⎜ − 1⎟ ⎝ k ′a ⎠ 5