Abstract I Griffiths

January 23, 2013 I Ryuichiro Kitano Department of Physics, Tohoku University, Sendai 980-8578, Japan

Abstract I Griffiths

Contents 0 5 1 6 1.1....................... 7 1.2.............................. 8 1.3......................... 9 1.4............................. 12 1.5.................................. 15 2 16 2.1.............................. 16 2.2................................... 18 2.3......................................... 18 2.4................................... 19 2.5................................... 20 2.6................................... 20 2.7................................... 23 2.8............................ 23 3 25 3.1............................. 25 3.2.................................. 26 3.3............................... 27 3.4........................................ 28 3.5..................................... 32 3.6........................... 32 1

3.7................................ 34 3.8....................................... 35 3.9 Ehrenfest.................................. 38 3.10.................... 39 3.11.................................... 40 3.12 3..................................... 45 4 47 4.1....................................... 47 4.1.1................................... 47 4.1.2 1................ 51 4.1.3 ψ(x)........................ 53 4.1.4................................... 53 4.1.5 E V min........................ 54 4.1.6.............. 55 4.2....................... 56 4.2.1.................... 56 4.2.2................................ 61 4.2.3 c n.................................. 63 4.2.4........................ 64 4.3.................................. 67 4.4............................ 68 4.4.1.............................. 68 4.4.2................................ 70 4.4.3.......................... 74 4.4.4............................... 74 4.4.5................................ 76 4.4.6................................ 78 4.4.7.......................... 79 4.4.8............................ 83 4.5..................................... 84 4.5.1.......................... 84 4.5.2..................... 85 2

4.5.3..... 93 4.5.4................................ 97 4.5.5.......................... 104 4.5.6.............................. 108 4.6....................................... 109 4.6.1..................... 109 4.6.2................................... 113 4.6.3.............................. 115 4.6.4................. 117 4.6.5......................... 118 4.6.6................................ 118 4.7.............................. 123 4.7.1............................ 123 4.7.2............................... 125 4.7.3................................... 126 4.7.4................................... 128 4.7.5..................... 131 4.8.............................. 132 4.8.1................................... 133 4.8.2................................... 141 4.9......................... 145 4.9.1 Gamov............................. 149 5 150 5.1................................... 150 5.2..................... 152 5.2.1.............................. 152 5.2.2................................... 156 5.3............................. 157 5.3.1........................... 157 5.3.2........................... 159 5.4.............................. 162 5.5.................................... 166 3

5.5.1......................... 166 5.5.2................................... 168 5.5.3....................... 169 5.6.................................. 171 5.7 Heisenberg Heisenberg....................... 184 5.8..................................... 187 5.8.1................................ 188 A 190 4

Chapter 0 1. D. J. Griffiths, Introduction to Quantum Mechanics, Pearson Education. 2.. 3. A. Tonomura, J. Endo, T. Matsuda, T. Kawasaki and H. Ezawa Demonstration of single-electron buildup of an interference pattern, Am. J. Phys. 57 (2) (1989) 117. 4. I. 5.. [1] [2] 5

Chapter 1 A a n 6

1.1 Tonomura [3] Youtube http://www.youtube.com/watch?vzj-0pbruthc 2 (wave-particle duality) p λ h p (1.1) h h 6.626 10 34 J s (1.2) 7

[h] [ML 2 T 1 ] (1.3) h 1.2 2 (photoelectric effect) (Hertz Lenard Einstein, Richardson, Millikan Einstein ) (Taken from http://physics.info/photoelectric/) 8

K 1 2 mv2 hν W. (1.4) h W Maxwell (quanta) (E) E hν, (1.5) h ν (photon) (1.5) 1.3 9

Rutherford I[4] [2] Larmor dw dt e 2 2 3 4πc 3 v 2, (1.6) P.290 P.197 ɛ 0 1 4π e 2 V e2 4πr (1.7) e [e] [M 1/2 L 3/2 T 1 ] (1.8) r v e 2 4πm e r 2 rω2 (1.9) m e W (r) 1 2 m e v 2 e2 4πr 1 2 m e(rω) 2 e2 4πr e 2 1 2 4πr. (1.10) v rω 2, v rω (1.9) dw dt e 2 1 dr 2 4πr 2 dt. (1.11) (1.6) (1.9) (1.11) e 4 dr dt 4 3 (4π) 2 m 2 ec 3 r 2 (1.12) 10

t 0 3 4 t0 dt 0 a0 dt 0 dr dr (4π) 2 m 2 ec 3 e 4 a0 0 r 2 dr 1 (4π) 2 m 2 ec 3 a 3 0 4 e 4. (1.13) m e 9.11 10 31 kg, (1.14) c 3.00 10 8 m/s, (1.15) a 0 0.529 10 10 m, (1.16) e 2 4πα c 9.67 10 36 c kg m 2 /s. (1.17) t 0 1.56 10 11 s. (1.18) Balmer( ) ν ( 1 2 2 1 ) n 2. (1.19) ( Lyman ) (1.5) 11

r 0 1.4 h temperature T light T T ν ν + dν U(ν, T ) 8πν2 c 3 hν e hν/kt 1. (1.20) ν ν 2 Rayleigh-Jeans ν Wien 12

ε hν, (1.21) 8πν 2 c 3 (1.22) x, y, z n x, n y, n z 1 V dn xdn y dn z 1 (2π) 3 dk xdk y dk z 1 (2π) 3 4πk2 dk ( ) 1 2π 3 (2π) 3 4π ν 2 dν c ( ) 4πν 2 dν (1.23) c 3 k n x,y,z k x,y,zl 2π k 2 x + k 2 y + k 2 z 2π λ 2πν c (1.24) (1.25) 2 2 2 T E e βe (1.26) β β 1 kt (1.27) 13

E i n i hν i i ε i (1.28) n i (n i 0, 1, 2, ). ν ν i ε i n 1 0 n 2 0 n 1 0 n 2 0 ε i e β(ε 1+ε 2 + ) e β(ε 1+ε 2 + ) n i hν i e βn ihν i n i 0 n i 0 β n i 0 n i 0 β ( e βn ihν i e βn ihν i e βn ihν i 1 1 e βhν i 1 1 e βhν i hν i (1 e βhν i ) 2 1 1 e βhν i ) ( ) hν i e βhν i 1 (1.29) h E hν [ ] [ ] [h] ML 2 T 1 (1.30) h 6.626 10 34 J s (1.31) 14

1.5 15

Chapter 2 2.1 x y f(x, y) f x lim h 0 f(x + h, y) f(x, y), (2.1) h f y lim h 0 f(x, y + h) f(x, y), (2.2) h 2 2 f x 2 x 2 f y x y ( ) f, x ( ) f, x 2 f x y x 2 f y 2 y ( ) f, (2.3) y ( ) f, (2.4) y 2 f/ x y 2 f/ y x 2 f y x 2 f x y (2.5) 16

[5] 23 2 f f(x + x, y + y) f(x, y) (2.6) f f f x + x y y + ɛ 1 x + ɛ 2 y (2.7) ɛ 1 ɛ 2 x y x, y 2 p, q df f f dx + dy (2.8) x y x x(p, q), y y(p, q) (2.9) df f f dx + x y dy f ( x x dp + p q dq x ( f x x p + f y y p ) + f ( y y ) dp + y dp + p ( f x x q + f y ) q dq y q ) dq (2.10) df f f dp + dq (2.11) p q f p f x x p + f y y p, (2.12) f q f x x q + f y y q (2.13) 17

x y t x x(t), y y(t) (2.14) df f dx f dy dt + x dt y dt dt ( f dx x dt + f ) dy dt, (2.15) y dt df dt f dx x dt + f dy y dt (2.16) 2.2 x ẋ L(x, ẋ) T V (2.17) T V t [ML 2 T 2 ] 2.3 x i (t i ) x f (t f ) path (x f,t f ) (x i,t i ) 18

path( ) S[x] tf t i L(x, ẋ)dt (2.18) x(t) [ ] [ ] [ML 2 T 1 ] h 2.4 (action) path path path x ẋ 0 δs tf t i tf t i ( L dt dt L δx + x ( L x d dt ) ẋ δẋ L ẋ ) δx + L ẋ δx t f (2.19) t i δx L x d L 0. (2.20) dt ẋ T 1 2 mẋ2, V V (x) (2.21) mẍ V x F (2.22) 19

2.5 L 1 2 mẋ2 qφ + q c ẋ A (2.23) φ A E B E φ 1 c A t, B A (2.24) 0 d (mẋ + q ) dt c A + q φ q (ẋ A) c mẍ i + 1 ( ) A i c t + Ai x j ẋj + q φ x i q Aj ẋj c x i [ mẍ i q φ x i 1 A i c t + 1 ( )] A j c ẋj x i Ai x j mẍ i q [E i + 1c ] ɛ ijkẋ j B k mẍ q [E + 1c ] v B (2.25) 2 ɛ ijk B k ɛ ijk ɛ kmn A n x m (δ imδ jn δ in δ jm ) An x m Aj x i Ai x j (2.26) 2.6 L x ẋ p p L ẋ (2.27) 20

H H pẋ L (2.28) p x ẋ (2.27) p x x x, ẋ ẋ(x, p) (2.29) p p(x, ẋ), x x (2.30) dh(x, p) d(pẋ L) ẋdp + pdẋ dl ẋdp + pdẋ L L dx x ẋ dẋ ( ) d L ẋdp + pdẋ dx L dt ẋ ẋ dẋ ẋdp + pdẋ ṗdx pdẋ (p ) ( ) ẋdp ṗdx, (2.31) H(x, p) p ẋ, H(x, p) x ṗ (2.32) p 21

dh dt H dx x dt + H dp p dt ṗẋ + ẋṗ 0 (2.33) H L 1 2 mẋ2 V (x) (2.34) p L ẋ mẋ (2.35) ( ) 1 H ẋp 2 mẋ2 V (x) p2 + V (x). (2.36) 2m (2.23) p mẋ + q c A (2.37) H ẋ p L 1 2 mẋ2 + qφ 1 2m ( p q c A ) 2 + qφ (2.38) 22

2.7 A, B (x, p) t {A, B} P.B. A B x p A p B x (2.39) O do dt O dx x O x dt + O dp p dt + O t H p O H p x + O t {O, H} P.B. + O t (2.40) O x p A,B x,p {x, x} P.B. 0, {p, p} P.B. 0, {x, p} P.B. 1 (2.41) 2.8 x f t f S cl (x f, t f ) x f t f δs tf t i ( L dt x d ) L δx + L dt ẋ ẋ δx t f (2.42) t i δs cl L ẋ δx tf pδx f (2.43) 23

t f x f S cl x f p (2.44) ds cl dt f L(t t f ) (2.45) ds cl S cl + S cl ẋ f dt f t f x f S cl t f + pẋ f (2.46) (2.44) S cl t f L pẋ f H H (2.47) tf H p (2.44) S cl S t H p2 2m + V (x) 1 ( ) S 2 + V (x) (2.48) 2m x cl f 24

Chapter 3 3.1 x(t) x p V (x) x p H p2 + V (x) (3.1) 2m m Ψ(x, t) Ψ(x, t) i 2 2 Ψ(x, t) t 2m x 2 + V (x)ψ(x, t) (3.2) 2π h 2π 1.054572 10 34 J s (3.3) Ψ Ψ OK 25

[V Ψ] [ML 2 T 2 ][Ψ] (3.4) [ T 1 ][Ψ] (3.5) [ ] [ML 2 T 1 ] (3.6) [ 2 M 1 L 2 ][Ψ] [ML 2 T 2 ][Ψ] (3.7) x(t) Ψ(x, t) (3.8) 3.2 Ψ 1 (x, t) Ψ 2 (x, t) i Ψ 1 t 2 2 Ψ 1 2m x 2 + V (x)ψ 1 (3.9) i Ψ 2 t 2 2 Ψ 2 2m x 2 + V (x)ψ 2 (3.10) i t (Ψ 1 + Ψ 2 ) 2 2 Ψ 1 2m x 2 + V (x)ψ 1 2 2m 2 2m 2 Ψ 2 x 2 + V (x)ψ 2 2 (Ψ 1 + Ψ 2 ) x 2 + V (x)(ψ 1 + Ψ 2 ) (3.11) 26

Ψ 1 + Ψ 2 3.3 Ψ(x, t) x Born t x a < x < b b a Ψ(x, t) 2 dx (3.12) Ψ i Ψ(x, t) 2 Ψ Ψ (3.13) Ψ Ψ Ψ 2 27

Ψ(x,t) 2 expectation is around here! x most likely to be here! 3.4 Ψ 2 Ψ(x, t) 2 dx 1 (3.14) Ψ 1 [Ψ] [L 1/2 ] (3.15) 28

d Ψ(x, t) 2 dx dt t Ψ(x, t) 2 dx (3.16) [5] t Ψ(x, t) 2 t (Ψ Ψ) Ψ Ψ t + Ψ t Ψ (3.17) Ψ t i 2 Ψ 2m x 2 i V Ψ, Ψ t i 2 Ψ 2m x 2 + i V Ψ (3.18) V t Ψ 2 i ( Ψ 2 Ψ 2m x 2 2 Ψ ) x 2 Ψ [ ( i Ψ Ψ )] x 2m x Ψ x Ψ (3.19) (3.16) d Ψ(x, t) 2 dx i ( Ψ Ψ ) dt 2m x Ψ x Ψ 0 (3.20) [Griffiths, Problem 1.5] Ψ(x, t) Ae λ x e iωt (3.21) A, λ, ω [A] [L 1/2 ], [λ] [L 1 ], [ω] [T 1 ] 29

Ψ 2 dx A 2 e 2λ x dx 2A 2 e 2λx dx 0 2A 2 (1/( 2λ)) e 2λx A2 λ 0 (3.22) A λ (3.23) OK e iδ Ψ 2 x x 0 x 2 x 2 λ 4 λ 4 d 2 dλ 2 x Ψ 2 dx (3.24) d 2 dλ 2 1 λ x 2 Ψ 2 dx x 2 λe 2λ x dx e 2λ x dx 1 2λ 2 (3.25) σ 2 x 2 x 2 (3.26) σ 2 1 2λ 2 (3.27) 30

[σ] [L] OK 1 0.9 0.8 0.7 Ψ 2 /λ 0.6 0.5 0.4 0.3 0.2 0.1 0 3σ 2σ σ 0 x σ 2σ 3σ σ < x < σ P out 2 σ 2λ σ Ψ 2 dx + Ψ 2 dx e 2λx dx σ [ e 2λx] σ σ Ψ 2 dx e 2 0.24 (3.28) 76% σ < x < σ σ x Ψ 31

2 3.5 [Griffiths, Problem 1.14] a < x < b P ab dp ab dt d dt b a b a b a Ψ 2 dx t Ψ 2 dx [ i x 2m ( Ψ Ψ x Ψ x Ψ )] dx J(a, t) J(b, t) (3.29) J(x, t) i ) (Ψ Ψ Ψ Ψ 2m x x (3.30) (3.19) J(x, t) J(a, t) J(b, t) a b x 3.6 [Griffiths, Problem 1.15] 32

V V 0 iγ 2 (3.31) Γ ([Γ] [ML 2 T 2 ]) (3.18) Ψ t i 2 Ψ 2m x 2 i V Ψ, Ψ t i 2 Ψ 2m x 2 + i V Ψ (3.32) t Ψ 2 [ ( i Ψ Ψ )] x 2m x Ψ x Ψ Γ Ψ 2 (3.33) P (t) Ψ 2 dx (3.34) dp dt Γ P (3.35) P (t) P (t 0)e Γ t (3.36) τ Γ (3.37) [τ] [ML 2 T 1 /(ML 2 T 2 )] [T ] OK t 1/2 P (t t 1/2 ) P (t 0)e Γ t 1/2 33 P (t 0) 2 (3.38)

t 1/2 ln 2 τ (3.39) 3.7 x p p i x (3.40) p ( ) p Ψ i x Ψ dx (3.41) x (3.19) d x dt x t Ψ 2 dx i 2m i 2m i m x x ( ( Ψ Ψ x Ψ Ψ Ψ x Ψ x Ψ ) x Ψ ) dx dx Ψ Ψ dx (3.42) x 2 3 v p mv p m d x dt Ψ ( i ) Ψ dx (3.43) x 34

Ψ 2 x Ψ xψ (3.44) Ψ i x Ψ (3.45) Ψ x Ψ xψdx, p Ψ i Ψdx (3.46) x x p ( p Ψ i ( ) i x Ψ p x Ψ ) dx Ψ dx (3.47) 3.8 ˆx : Ψ(x, t) xψ(x, t), (3.48) ˆp : Ψ(x, t) i 35 Ψ(x, t), (3.49) x

ˆxΨ, ˆpΨ ˆxΨ xψ, ˆpΨ i Ψ x (3.50) ˆx ˆp ˆx(aΨ 1 + bψ 2 ) a(ˆxψ 1 ) + b(ˆxψ 2 ), (3.51) ˆp(aΨ 1 + bψ 2 ) a(ˆpψ 1 ) + b(ˆpψ 2 ), (3.52) ˆx ˆp Ψ Â ˆB Â ˆB : Ψ(x, t) Â( ˆBΨ) (3.53) Â ˆB (Â ˆB)(aΨ ( 1 + bψ 2 ) Â a( ˆBΨ 1 ) + b( ˆBΨ ) 2 ) a(â ˆBΨ 1 ) + b(â ˆBΨ 2 ) (3.54) Â ˆB Â + ˆB : Ψ ÂΨ + ˆBΨ (3.55) Â ˆB Â + ˆB (Â + ˆB)(aΨ 1 + bψ 2 ) Â(aΨ 1 + bψ 2 ) + ˆB(aΨ 1 + bψ 2 ) (aâψ 1 + bâψ 2) + (a ˆBΨ 1 + b ˆBΨ 2 ) a(â + ˆB)Ψ 1 + b(â + ˆB)Ψ 2. (3.56) ˆx ˆp Q(x, p) 36

x p 2 p 2 ˆp 2 : Ψ i ( ) x i x Ψ 2 2 Ψ x 2 (3.57) 2 x p Q(x, p) Q(x, p) Q(x, p) ( Ψ Q x, ) Ψdx (3.58) i x T p2 2m (3.59) T 2 2m Ψ 2 Ψ dx (3.60) x2 i Ψ t ĤΨ (3.61) E Ψ i Ψdx (3.62) t x E 37

3.9 Ehrenfest H p2 2m + V (3.63) dx dt H p p m, dp dt H x dv dx (3.64) d x dt p m (3.65) d p dt d Ψ dt i x Ψdx [( ) ( ) i t Ψ x Ψ + Ψ ] x t Ψ dx [ 1 ( 2 2 ) ( )] i i 2m x 2 Ψ + V Ψ x Ψ dx + [ Ψ 1 ( 2 2 )] i i x 2m x 2 Ψ + V Ψ dx ( Ψ V x Ψ ) Ψ (V Ψ) dx x Ψ dv dx Ψdx dv dx (3.66) (Ehrenfest) dv dv ( x ) dx d x (3.67) 38

3.10 Ehrenfest Ψ(x, t) e is(x,t)/ (3.68) i Ψ t S t eis/ (3.69) [ 2 2 Ψ 2m x 2 + V Ψ i 2 S 2m x 2 + 1 2m ( ) ] S 2 + V e is/ (3.70) x S t 1 2m ( ) S 2 + V i 2 S x 2m x 2 (3.71) S t 1 ( ) S 2 + V (3.72) 2m x (2.48) S(x, t) S x p, S t H. (3.73) ˆpΨ i Ψ x i S i x eis/ S x Ψ (3.74) ĤΨ i Ψ t i i S t eis/ S t Ψ (3.75) (3.73) 39

3.11 p h λ 2π λ (3.76) σ x σ p 2 (3.77) x p [Griffiths, Problem 1.9] m Ψ(x, t) Ae a[(mx2 / )+it] (3.78) 40

A a [a] [T 1 ], [A] [L 1/2 ] (3.79) Ψ 2 dx A 2 e 2amx2 / dx A 2 π 2am (3.80) A ( ) 2am 1/4 (3.81) π 41

e 1 2 ax2 dx 2π a (3.82) X 2 X 2π 0 2π 0 2π dx dθ 0 dθ 0 [( 1 a e 1 2 ax2 dx (3.83) dy e 1 2 a(x2 +y 2 ) dr re 1 2 ar2 dr d dr ) ] e 1 2 ar2 0 [( 1 ) ] e 12 ar2 a 2π a (3.84) X 2π a (3.85) a 2 V (x) 2 i t Ψ 2 Ψ + V (x)ψ (3.86) 2m x2 aψ a( 2amx 2 )Ψ + V (x)ψ (3.87) V (x) 2a 2 mx 2 (3.88) 42

x, x 2, p, p 2 x A 2 xe 2amx2 / (3.89) x 2 A 2 x 2 e 2amx2 / p A 2 ( ) e amx2 / / i x e amx2 (3.90) (3.91) p 2 A 2 ) e amx2 / ( 2 2 / x 2 e amx2 (3.92) A 2 x p x 0, p 0 (3.93) x 2 (3.82) a 1 2 x 2 e 1 2 ax2 dx 1 2 x 2 e 1 2 ax2 dx 2π a 3 (3.94) 2π a 3 (3.95) x 2 2am π 2π (4am/ ) 3 4am (3.96) p 2 p 2 A 2 ( 2 am 4a 2 m 2 x 2) e 2amx2 / am (3.97) 43

σ x x 2 x 2 4am (3.98) σ p p 2 p 2 am (3.99) σ x σ p 2 (3.100) V (x) 2a 2 mx 2 (3.101) x 0 p 0 x 2 p 2 2 4 (3.102) x 2 p 2 2 4 (3.103) x E p2 2m + 2a2 mx 2 2 8mx 2 + 2a2 mx 2 a (3.104) 44

3.12 3 Ψ(x, t) : (3.105) Ψ(x, t) 2 dx 1 : (3.106) P ab b a Ψ(x, t) 2 dx : a < x < b (3.107) Ψ(x, t) 2 x n x n Ψ(x, t) 2 dx : (3.108) σ 2 x x 2 x 2 (x x ) 2 Ψ(x, t) 2 dx : (3.109) 2 σ 2 x i Ψ t 2 2 Ψ + V (x)ψ : 2m x2 (3.110) [ ] [ ] J(x, t) i ) (Ψ Ψ Ψ Ψ : (3.111) 2m x x dp ab dt J(a, t) J(b, t) : (3.112) i x : (3.113) p Ψ i Ψdx : (3.114) x 45

( Q(x, p) Ψ Q x, ) Ψdx : (3.115) i x d x dt p m, d p dt V : Ehrenfest (3.116) x σ x σ p 2 : (3.117) 46

Chapter 4 4.1 4.1.1 V t x i Ψ t 2 2 Ψ 2m x 2 + V Ψ (4.1) Ψ(x, t) ψ(x)φ(t) (4.2) ψφ i ψ dφ dt 2 d 2 ψ φ + V ψφ (4.3) 2m dx2 i 1 dφ φ dt 2 2m 1 ψ d 2 ψ dx 2 + V (4.4) 47

t x E i dφ dt Eφ (4.5) 2 d 2 ψ + V ψ Eψ (4.6) 2m dx2 φ(t) e iet/ (4.7) 2 (4.6) E [Griffiths, Problem 2.1] Ψ(x, t) ψ(x)e iet/ (4.8) E E E 0 + iγ ψ ψe 2Γt dx 1 (4.9) E 1. 48

Ψ(x, t) ψ(x)e iet/ (4.10) Ψ 2 ( Q ψ Q x, ) d ψdx (4.11) i dx E x p 0 2. H(x, p) p2 + V (x) (4.12) 2m 2 Ĥ 2 + V (x) (4.13) 2m x2 (4.6) Ĥψ Eψ (4.14) H ψ Ĥψdx E (4.15) E Ĥ 2 ψ ĤEψ EĤψ E2 ψ (4.16) H 2 ψ Ĥ 2 ψ E 2 (4.17) σ 2 H H 2 H 2 0 (4.18) E 49

3. Ψ(x, t) c n ψ n (x)e ie nt/ n1 (4.19) c n Ψ(x, 0) ψ 1 (x), ψ 2 (x), ψ 3 (x), (4.20) E 1, E 2, E 3, (4.21) Ψ(x, 0) c n ψ n (x) (4.22) n1 c n c n Ψ(x, t) c n ψ n (x)e ie nt/ n1 (4.23) OK Ĥψ Eψ (4.24) 50

Ĥ ψ E Ĥ E ψ Ψ(x, 0) ψ E 1 ψ 1 ψ 1 ψ 1 ψ 1 ψ 1 + ψ 1 ψ 1 ψ 1 4.1.2 1 Ĥψ Eψ (4.25) E ψ ψ 1, ψ 2,, ψ n c 1 ψ 1 + c 2 ψ 2 + + c n ψ n 0 (4.26) c n c 1 c 2 c n 0 c 1 ψ 1 + c 2 ψ 2 0 (4.27) ψ 1 ψ 2 c 1 c 2 ψ 1 ψ 2 ψ Ψ ψ 2 d 2 ψ + V ψ Eψ, (4.28) 2m dx2 51

2 d 2 φ + V φ Eφ. (4.29) 2m dx2 1 d 2 ψ ψ dx 2 1 d 2 φ φ dx 2 (4.30) ( d dψ dx dx φ dφ ) dx ψ 0 (4.31) c x dψ dx φ dφ dx ψ c (4.32) ψ 0, φ 0 (4.33) c 0 1 dψ ψ dx 1 dφ φ dx (4.34) ψ c φ (4.35) c ψ φ ψ φ ψ φ c 52

4.1.3 ψ(x) ψ(x) (4.20) ψ(x) 2 d 2 ψ + V ψ Eψ (4.36) 2m dx2 2 d 2 ψ 2m dx 2 + V ψ Eψ (4.37) ψ ψ ψ cψ (4.38) 2 ψ c ψ c 2 ψ (4.39) c 2 1 c c e iδ ψ e iδ/2 ψ ψ e iδ/2 ψ c e iδ/2 ψ e iδ/2 ψ ψ ψ + ψ i(ψ ψ ) 4.1.4 V (x) V ( x) V (x) (4.40) ψ(x) 2 d 2 ψ(x) 2m dx 2 + V (x)ψ(x) Eψ(x) (4.41) 53

x x 2 d 2 ψ( x) 2m dx 2 + V ( x)ψ( x) Eψ( x) (4.42) x V ( x) V (x) 2 d 2 ψ( x) 2m dx 2 + V (x)ψ( x) Eψ( x) (4.43) ψ(x) ψ( x) ψ(x) cψ( x) (4.44) c ψ( x) cψ(x) c 2 ψ( x) (4.45) c 2 1 c ±1 ψ(x) ψ(x) ψ( x) ψ(x) ± ψ( x) 4.1.5 E V min d 2 ψ dx 2 2m (V (x) E) ψ (4.46) 2 ψ E V (x) V min ψ ψ ψ > 0 ψ < 0 x x x E V min E V E V (x) 54

4.1.6 E > V min V (x) E x E V (x) E ψ x x 0 ψ(x) x 55

E ψ 4.2 4.2.1 V (x) { 0, if 0 x a, otherwise (4.47) V (x) 0 a x ψ(x) 0 (V 0) 2 d 2 ψ Eψ (4.48) 2m dx2 d 2 ψ 2mE dx 2 k2 ψ, k (4.49) ψ(x) A sin kx + B cos kx (4.50) 56

A B ψ ψ(0) ψ(a) 0 (4.51) ψ(0) A sin 0 + B cos 0 B 0 (4.52) ψ(a) A sin ka 0 (4.53) A 0 ψ 0 sin ka 0 ka 0, ±π, ±2π, ±3π, (4.54) k 0 ψ 0 k k n nπ a, n 1, 2, 3, (4.55) A k k (4.49) k E n 2 k 2 n 2m n2 π 2 2 2ma 2 (4.56) 57

A 1 a 0 A 2 a A 2 sin 2 (kx)dx 0 [ 1 cos(2kx) x dx A 2 1 2k sin(2kx) 2 2 ] a 0 A 2 a 2 (4.57) A 2/a OK ψ n (x) 2 ( nπx ) a sin a (4.58) ψ(x) ψ 1 (x) ψ 3 (x) 0 a ψ 2 (x) x ψ 1 n 2 ψ n ψ n ψ 1 ψ 2 ψ 3 58

m n ψ m (x) ψ n (x)dx 0 (4.59) ψ m (x) ψ n (x)dx 2 a ( mπx ) ( nπx ) sin sin dx a 0 a a 1 a [ ( ) ( )] m n m + n cos πx cos πx dx a 0 a a [ ( ) ( )] 1 m n (m n)π sin 1 m + n a πx a (m + n)π sin πx a 0 1 [ ] sin[(m n)π] sin[(m + n)π] π m n m + n 0 (4.60) m n 2 3 1 sin θ eiθ e iθ, cos θ eiθ + e iθ 2i 2 (4.61) ψ mψ n dx δ mn (4.62) δ mn δ mn { 0, if m n; 1, if m n (4.63) (4.62) ψ f(x) c n ψ n (x) n1 2 a ( nπx ) c n sin a n1 (4.64) 59

ψ c n ψ m (x) f(x)dx c n ψ m (x) ψ n (x)dx c n δ mn c m. (4.65) n1 n1 c n ψ n (x) f(x)dx (4.66) ψ n E n 2 ( nπx ) Ψ n (x, t) a sin e i(n2 π 2 /2ma 2 )t a (4.67) Ψ(x, t) n1 2 ( nπx ) c n a sin e i(n2 π 2 /2ma 2 )t a (4.68) c n (4.22) Ψ(x, 0) c n ψ n (x) (4.69) n1 c n c n 2 a a 0 sin ( nπx ) Ψ(x, 0)dx (4.70) a Ψ(x, 0) c n c n (4.68) OK 60

4.2.2 Ψ(x, 0) Ax(a x), (0 x a), (4.71) Ψ 0 A 1 a 0 Ψ(x, 0) 2 dx A 2 a A OK 0 x 2 (a x) 2 dx A 2 a5 30. (4.72) 30 a 5 (4.73) (4.70) c n c n 2 a 2 15 a 3 a ( nπx ) 30 sin x(a x)dx a a5 [ a ( nπx ) a a x sin dx a 0 0 0 x 2 sin ( nπx ) ] dx a (4.74) 61

n 0, 1, 2, c2 c 1 x 2n sin(ax)dx ( 1) n d2n da 2n ( 1) n d2n da 2n c2 c 1 sin(ax)dx [ 1a cos(ax) ] c2 c 1 (4.75) c2 c 1 x 2n+1 sin(ax)dx ( 1) n d2n+1 da 2n+1 ( 1) n d2n+1 da 2n+1 c2 c 1 ( cos(ax))dx [ 1a sin(ax) ] c2 c 1 (4.76) c2 c 1 x 2n cos(ax)dx ( 1) n d2n da 2n ( 1) n d2n da 2n c2 c 1 cos(ax)dx [ 1 a sin(ax) ] c2 c 1 (4.77) c2 c 1 x 2n+1 cos(ax)dx ( 1) n d2n+1 da 2n+1 c n 2 { [ 15 ( a ) 2 ( nπx a 3 a sin nπ a [ ( a ) 2 ( nπx 2 x sin nπ a 2 15 a 3 ( 1) n d2n+1 da 2n+1 c2 c 1 ) ax ( nπx nπ cos a ) (nπx/a)2 2 (nπ/a) 3 cos sin(ax)dx [ 1a cos(ax) ] c2 ) ] a} 0 ( nπx a [ a3 nπ cos(nπ) + a3 (nπ)2 2 (nπ) 3 cos(nπ) + a 3 4 15 (cos(0) cos(nπ)) (nπ) 3 { 0, if n is even, 8 15/(nπ) 3, if n is odd, ) ] a 0 c 1 (4.78) 2 (nπ) 3 cos(0) ] (4.79) (4.68) Ψ(x, t) 30 a ( ) 2 3 1 ( nπx ) [ in 2 π n 3 sin π 2 t exp a 2ma 2 n1,3,5, ] (4.80) 62

4.2.3 c n c n Ψ (3.12) c n c n 2 E n c n 2 1 (4.81) n 1 Ψ(x, 0) 2 dx ( ) ( ) c m ψ m (x) c n ψ n (x) dx m n c mc n ψ m (x) ψ n (x)dx m n c mc n δ mn m n c n 2 (4.82) n ψ n 1 1 6 + 1 3 6 + 1 5 6 + π6 960 (4.83) c n 2 H n c n 2 E n (4.84) 63

Ĥψ n E n ψ n (4.85) H Ψ ĤΨdx ( ) ( ) c m ψ m e iemt Ĥ c n ψ n e ient dx m n c mc n E n e i(en Em)t ψmψ n dx m n c mc n E n e i(e n E m )t δ mn m n c n 2 E n (4.86) n 4.2.4 [Griffiths, Problem 2.4] n x, x 2, p, p 2 a x 2 ( x sin 2 nπx ) dx a 0 a 1 a [ ( 2nπx x 1 cos a 0 a 1 [ x 2 a 2 ax ( 2nπx 2nπ sin a a 2 )] dx ) ( a 2nπ ) ( )] 2 a 2nπx cos a 0 (4.87) x 2 2 a a2 6 a ( x 2 sin 2 nπx ) dx 0 a ( 2 3 ) n 2 π 2 (4.88) 64

p 2 a a 0 sin ( nπx ) [ d ( nπx ) ] a i dx sin dx 0. (4.89) a p 2 2 a a 0 sin ( nπx ) [ 2 d2 ( nπx ) ] a dx 2 sin dx n2 π 2 2 a a 2. (4.90) σ 2 x a2 6 ( 2 3 ) ( a 2 a n 2 π 2 2) 2 ( 1 6 ) 12 n 2 π 2, (4.91) σ 2 p n2 π 2 2 a 2 (4.92) σ x σ p 2 n 2 π 2 6 3 (4.93) n 1 /2 1.14 n [Griffiths, Problem 2.5] ψ 1 ψ 2 Ψ(x, 0) A (ψ 1 (x) + ψ 2 (x)). (4.94) Ψ(x, 0) ψ 1 (x) + ψ 2 (x) 0 a x 65

a 1 A 2 (ψ 1 (x) + ψ 2 (x)) (ψ 1 (x) + ψ 2 (x)) dx (4.95) 0 ψ 1 ψ 2 A 2 1 2 (4.96) A 1 2 (4.97) Ψ(x, t) 1 2 ( ψ1 e ie 1t + ψ 2 e ie 2t ) (4.98) Ψ(x, t) 2 E 1 π2 2 2ma 2, E 2 4π2 2 2ma 2 (4.99) Ψ(x, t) 2 1 [ψ1 2 + ψ2 2 + ψ 1 ψ 2 (e )] i(e 1 E 2 )t/ + e i(e 1 E 2 )t/ 2 1 [ ψ 2 2 1 + ψ2 2 + 2ψ 1 ψ 2 cos(3ωt) ] (4.100) ω π2 2ma 2 (4.101) x x a 0 x Ψ(x, t) 2 dx a 2 16a cos(3ωt) (4.102) 9π2 66

a/2 3ω/2π 16a/9π 2 0.18a p p a 0 Ψ(x, t) i Ψ(x, t)dx x 8 sin(3ωt) (4.103) 3a p m x E 1 E 2 1/2 H E 1 + E 2 2 (4.104) H a 0 ( Ψ(x, t) 2 2m E 1 E 2 2 ) Ψ(x, t) dx 5π2 2 x2 4ma 2 (4.105) 4.3 2 d 2 ψ + V ψ Eψ : (4.106) 2m dx2 E n, ψ n (x) (n 1, 2, 3, ) : E E n (4.107) ψ n(x)ψ m (x)dx δ nm : {ψ n } (4.108) Ψ n (x, t) ψ n (x)e ie nt/ : (4.109) 67

Ψ(x, t) n c n ψ n (x)e ie nt/ : (4.110) c n Ψ(x, 0)ψ n(x)dx : c n (4.111) c n 2 : E n (4.112) c n 2 1 : (4.113) n H ( Ψ 2 2m 2 x 2 + V ) Ψdx ( Ψ i ) t Ψ dx c n 2 E n n : (4.114) 4.4 Ψ(x, 0) c n 4.4.1 f(x) x x sin log(1 + z) z re iθ π < θ < π z 0 log(1 + z) z z2 2 + z3 3 (4.115) z 1 z 1 z e iθ log(1 + e iθ ) e iθ e2iθ 2 + e3iθ 3 (4.116) 68

Im[log(1 + e iθ )] sin θ sin 2θ 2 + sin 3θ 3 (4.117) θ/2 1 θ 1 + e iθ 1 + e iθ c ce iθ/2 θ sin 2θ sin 3θ sin θ + (4.118) 2 2 3 π < θ < π θ π θ π θ 2 sin(2π 2θ) sin(π θ) 2 sin θ + sin(2θ) + sin(3θ) 2 3 sin(nθ) n n1 + + sin(3π 3θ) 3 (4.119) 69

n 20 4.4.2 f(x) 2L f(x + 2L) f(x) f(x) a 0 2 + n1 ( a n cos nπx L + b n sin nπx ) L (4.120) 70

1. f(x) ( L, L) 2. f(x) 2L 3. f(x) f (x) ( L, L) m n L L cos mπx L dx 2L δ m0 (4.121) L L sin mπx dx 0 (4.122) L L L sin mπx L nπx cos dx 0 (4.123) L L L cos mπx L nπx cos L dx L δ mn (4.124) L L sin mπx L a n b n a n 1 L L nπx sin L dx L δ mn (4.125) L f(x) cos nπx dx, (4.126) L b n 1 L L L f(x) sin nπx dx (4.127) L c a n 1 L L+c L+c f(x) cos nπx dx, (4.128) L 71

b n 1 L L+c L+c f(x) sin nπx dx (4.129) L [ L, L+c] [ L+c, L] [ L, L+c] [L, L + c] f(x) OK f(x) { 1 ( π < x < 0) 1 (0 < x < π) (4.130) 2π f(x) 1 π π 0 2π x 1 a n b n b n 1 π 1 π 1 π π π 0 f(x) sin(nx)dx π sin(nx)dx + 1 π π 0 [ 1 ] 0 n cos(nx) + 1 π π sin(nx)dx [ 1 n cos(nx) ] π 0 2 (1 cos nπ) nπ 2 nπ [1 ( 1)n ] (4.131) 72

f(x) 4 1 π n sin(nx) n1,3,5 4 (sin x + 13 π sin 3x + 15 ) sin 5x + (4.132) ψ n Ψ(x, 0) c n 73

4.4.3 cos θ eiθ + e iθ, sin θ eiθ e iθ 2 2i (4.133) (4.120) f(x) c n e inπx/l (4.134) n c 0 a 0 2, c n a n ib n, c n a n + ib n, (n 1, 2, 3, ) (4.135) 2 2 (4.126) (4.127) c 0 1 L f(x)dx, (4.136) 2L L c n 1 L { f(x) cos nπx 2L L L i sin nπx L } dx, (4.137) c n 1 L { f(x) cos nπx 2L L L + i sin nπx L } dx, (4.138) c n 1 L f(x)e inπx/l dx, (n 0, ±1, ±2, ) (4.139) 2L L 4.4.4 b a φ m(x)φ n (x)dx δ mn (4.140) 74

ψ n [a, b] φ n f(x) c n b f(x) c n 2 n1 b a a f(x)φ n(x)dx (4.141) f(x) 2 dx, (4.142) ξ(x) b a ξ(x)φ n(x)dx 0 (4.143) n ξ(x) c n (4.142) ξ(x) 0 f(x) φ n (x) J b a f(x) c n φ n (x) n1 2 dx (4.144) 75

( b ) J f(x) 2 f(x) c n φ n (x) f(x) c mφ m(x) + c n c mφ n (x)φ m(x) dx a b n m b n1 n1 f(x) 2 dx b c n f(x) φ n (x)dx c m f(x)φ m(x)dx a n a m a + b c n c m φ n (x)φ m(x)dx n m a b f(x) 2 dx c n c n c mc m + c n c mδ nm a n m n m b f(x) 2 dx c n 2 a n 0 (4.145) J 0 (4.144) f(x) c n φ n (x) (4.146) n1 φ n f(x) c n (4.141) φ m (x) 1 2π e imx, (m 0, ±1, ±2, ) (4.147) [ π, π] 4.4.5 L f L (x) c n e inπx/l (4.148) n 76

c n 1 L f L (x)e inπx/l dx (4.149) 2L L f L (x) 2L f L (x) 1 2L n L L duf L (u)e inπu/l e inπx/l (4.150) w n nπ L, w w n+1 w n π L (4.151) f L (x) 1 2π n L w f L (u)e i(x u)w n du (4.152) L L f L (x) w 0 f(x) lim L f L(x) (4.153) f(x) 1 dw du f(u)e iw(x u) (4.154) 2π 1. f(x) f (x) 2. f(x) (, ) f(x) dx < (4.155) 77

4.4.6 F (w) 1 2π f(x) 1 2π f(u)e iwu du (4.156) F (w)e iwx dw (4.157) F (w) f(x) f(x) F (w) (4.157) f(x) F (w) a > 0 f(x) { e ax (x > 0) 0 (x < 0) (4.158) F (k) 1 2π f(x)e ikx dx 1 e ax e ikx dx 2π 0 [ ] 1 e (a+ik)x 2π a + ik 1 2π 1 a + ik 0 (4.159) 1 e ikx { e 2π a + ik dk ax (x > 0) 0 (x < 0) (4.160) 78

4.4.7 δ(x) δ(x) { 0 (x 0) (x 0) (4.161) δ(x)dx 1 (4.162) x 0 x 0 δ(x a) x a x a f(x) f(x)δ(x a)dx f(a) δ(x a)dx f(a) (4.163) ɛ > 0 e ɛ k / 2π δ ɛ (x) 1 2π [ 1 2π 1 2π e ɛ k e ikx dk ] + 1 [ 2π 1 1 ɛ + ix e ɛk+ikx 0 ɛ + ix eɛk+ikx 2ɛ x 2 + ɛ 2 (4.164) ɛ 0 x 0 δ ɛ (x)dx 1 π 1 π 1 π π/2 π/2 π/2 π/2 ɛ x 2 + ɛ 2 dx ɛ ɛ 2 tan 2 θ + ɛ 2 dθ ɛ cos 2 θ dθ 1 (4.165) ] 0 79

δ ɛ (x) ɛ 0 δ(x) 160 140 120 ǫ 0.01 δǫ(x) 100 80 60 40 20 0 ǫ 0.03 ǫ 0.1-1 -0.5 0 0.5 1 x 1 δ(x) lim e ikx ɛ k dk 1 e ikx dk (4.166) ɛ 0 2π 2π F (k) 1/ 2π ( x 0 ) δ ɛ (x) e ɛ k / 2π F (k) 1 2π δ ɛ (x)e ikx dx 1 (2π) 3/2 1 (2π) 3/2 2ɛ x 2 + ɛ 2 e ikx dx 2ɛ (x + iɛ)(x iɛ) e ikx dx (4.167) 80

z z iǫ iǫ iǫ iǫ k > 0 z iɛ F (k) 2πi 2ɛ (2π) 3/2 iɛ iɛ e ɛk 1 e ɛk (4.168) 2π k < 0 F (k) k 0 OK F (k) 2πi 2ɛ (2π) 3/2 iɛ + iɛ eɛk 1 e ɛk (4.169) 2π 2πi 2ɛ (2π) 3/2 iɛ + iɛ 1 (4.170) 2π F (k) 1 2π e ɛ k (4.171) 1, if x > 0, θ(x) 1/2, at x 0, (4.172) 0, if x < 0 f(x) x 0 f(x) dθ(x) dx dx 0 df dx θ(x)dx + [f(x)θ(x)] df dx + f(+ ) dx [f(x)] 0 + f(+ ) f(0) (4.173) 81

0 (x < 1 2n ) δ n (x) n ( 1 2n < x 1 2n ), (4.174) 0 (x > 1 2n ) δ n (x) n π e n2 x 2, (4.175) δ n (x) n π 1 1 + n 2 x 2, (4.176) δ n (x) sin(nx) πx 1 n e ikx dk. (4.177) 2π n lim δ n (x)f(x) f(0) (4.178) n f(x) x x x 0 82

δ(x) δ( x), (4.179) δ (x) δ ( x), (4.180) xδ(x) 0, (4.181) xδ (x) δ(x), (4.182) δ(ax) a 1 δ(x), (a > 0), (4.183) δ(x 2 a 2 ) (2a) 1 [δ(x a) + δ(x + a)], (a > 0), (4.184) δ(a x)δ(x b)dx δ(a b), (4.185) f(x)δ(x a) f(a)δ(x a). (4.186) 4.4.8 φ n (x) (, ) f(x) n c n φ n (x), c n f(x)φ n(x)dx (4.187) f(x) n ( ) f(y)φ n(y)dy φ n (x) ( ) f(y) φ n(y)φ n (x) dy (4.188) n φ n (x)φ n(y) δ(y x) (4.189) n 83

4.5 4.5.1 F kx m d2 x dt 2 (4.190) x(t) A sin(ωt) + B cos(ωt) (4.191) ω k m (4.192) V (x) 1 2 kx2 (4.193) 2 V (x) x 0 V (x) V (x 0 ) + V (x 0 )(x x 0 ) + 1 2 V (x 0 )(x x 0 ) 2 + (4.194) V (x 0 ) V (x 0 ) 0 V (x) 1 2 V (x 0 )(x x 0 ) 2 (4.195) x 0 V (x 0 ) > 0 V (x) 1 2 mω2 x 2 (4.196) 84

ω [ω] [T 1 ] 2 d 2 ψ 2m dx 2 + 1 2 mω2 x 2 ψ Eψ (4.197) 4.5.2 ˆp 1 [ˆp 2 + (mωx) 2] ψ Eψ (4.198) 2m ˆp i d dx (4.199) â ± 1 2 mω ( iˆp + mωˆx) (4.200) x ˆx ˆx : ψ(x) xψ(x) (4.201) â + â ˆp ˆx â â + 1 (iˆp + mωˆx)( iˆp + mωˆx) (4.202) 2 mω 1 [ p 2 + (mωx) 2] (4.203) 2 mω ( / ω) ˆxˆp ˆpˆx (4.204) 85

x x â â + 1 [ˆp 2 + (mωˆx) 2 imω(ˆxˆp ˆpˆx) ] (4.205) 2 mω ˆxˆp ˆpˆx  ˆB [Â, ˆB]  ˆB ˆB (4.206) â â + 1 ω Ĥ i [ˆx, ˆp] (4.207) 2 ˆx ˆp [ [ˆx, ˆp] f(x) x df i dx ] d i dx (xf) ( x df i dx f x df ) dx i f(x) (4.208) [ˆx, ˆp] i (4.209) â â + 1 ω Ĥ + 1 2 ( Ĥ ω â â + 1 ) 2 (4.210) (4.211) 86

â + â â + â 1 [ˆp 2 + (mωˆx) 2 + imω(ˆxˆp ˆpˆx) ] 1 2 mω ω Ĥ 1 2 (4.212) [â, â + ] â â + â + â 1 (4.213) â + â ( Ĥ ω â + â + 1 ) 2 (4.214) ( ω â ± â ± 1 ) ψ Eψ (4.215) 2 E Ĥψ Eψ (4.216) ψ â + ψ (E + ω) ( Ĥ(â + ψ) ω â + â + 1 ) (â + ψ) 2 ( ω â + â â + + 1 ) ψ ( 2â+ ωâ + â â + + 1 ) ψ 2 ( â + [ ω â â + 1 ) ] + ω ψ 2 â + (Ĥ + ω)ψ â + (E + ω)ψ (E + ω)(â + ψ) (4.217) 87

â ψ (E ω) ( Ĥ(â ψ) ω â â + 1 ) 2 ωâ ( â + â 1 2 â ψ ) ψ â (Ĥ ω)ψ â (E ω)ψ (E ω)(â ψ) (4.218) ω â + â â ± â + ω â â E V min â â ψ 0 0 (4.219) ( 1 d ) 2 mω dx + mωx ψ 0 (x) 0 (4.220) dψ 0 dx mω xψ 0, (4.221) dψ0 ψ 0 mω xdx (4.222) 88

ln ψ 0 mω 2 x2 + const. (4.223) [ ψ 0 A exp mω 2 x2] (4.224) 1 A 2 e mωx2 / dx A 2 π mω (4.225) ψ 0 ( mω ) 1/4 e mωx 2 /(2 ) π (4.226) Ĥψ 0 E 0 ψ 0 ( Ĥψ 0 ω â + â + 1 ) ψ 0 1 2 2 ωψ 0 (4.227) E 0 1 ω (4.228) 2 ω â + ψ n (x) A n (â + ) n ψ 0 (x), (4.229) E n ( n + 1 ) ω (4.230) 2 â 89

ψ n ψ 1 (x) ψ 1 (x) A 1 â + ψ 0 ( A 1 d ) (mω ) 1/4 2 mω dx + mωx e mωx 2 /(2 ) π A ( 1 mω ) 1/4 (mωx + mωx) e mωx 2 /(2 ) 2 mω π ( mω ) 1/4 2mω /(2 ) A 1 π xe mωx2 (4.231) (3.95) 1 A 1 2 mω π A 1 2 mω π A 1 1 2mω 2mω x 2 e mωx2 /( ) 2π (2mω/ ) 3 A 1 2 (4.232) â + ψ n ψ n+1 â + ψ n c n ψ n+1, â ψ n d n ψ n 1 (4.233) c n d n f g f (â ± g)dx f 1 ( (â ± g)dx f d 2 mω 1 2 mω (â f) gdx (4.234) ) dx + mωx [( ± d dx + mωx gdx ) ] f g (â f) gdx (4.235) 90

(â ± ψ n ) (â ± ψ n )dx (â â ± ψ n ) ψ n dx (4.236) â + â ψ n ( 1 ω Ĥ 1 ) 2 ( 1 ω E n 1 ) 2 [ ( 1 n + 1 ω 2 ψ n ψ n ) ω 1 2 ] ψ n nψ n (4.237) â â + ψ n (â + â + 1)ψ n (n + 1)ψ n (4.238) â + â ψ n n â + â (â + ψ n ) (â + ψ n )dx (n + 1) (â â + ψ n ) ψ n dx ψ n 2 dx n + 1 (4.239) (â + ψ n ) (â + ψ n )dx c n 2 ψ n+1 2 dx c n 2 (4.240) c n n + 1 (4.241) 91

(â ψ n ) (â ψ n )dx d n 2 n (4.242) d n n (4.243) OK â + ψ n n + 1ψ n+1, â ψ n nψ n 1 (4.244) ψ n 1 n â + ψ n 1 1 n 1 n 1 (â + ) 2 ψ n 2 1 n! (â + ) n ψ 0 (4.245) (4.229) A n A n 1 n! (4.246) A 1 1 ψ n m n ψm(â + â )ψ n dx n m ψ mψ n dx (â ψ m ) (â ψ n )dx (â + â ψ m ) ψ n dx ψ mψ n dx (4.247) ψ mψ n dx 0 (4.248) Ψ(x, 0) ψ n c n c n 2 E n 92

4.5.3 n 1 V 2 mω2 x 2 1 2 mω2 ψnx 2 ψ n dx. (4.249) ˆx ˆp â â + â + â ˆx 2mω (â + + â ), (4.250) mω ˆp i 2 (â + â ), (4.251) ˆx 2 [ (â+ ) 2 + (â + â ) + (â â + ) + (â ) 2] (4.252) 2mω V ω 4 ψn [ (â+ ) 2 + (â + â ) + (â â + ) + (â ) 2] ψ n dx (4.253) ψ n V ω 4 ψ n [(â + â ) + (â â + )] ψ n dx (4.254) (4.237) (4.238) V ω ψn (n + n + 1) ψ n dx 4 ω 4 (n + n + 1) ψnψ n dx ω ( n + 1 ) 2 2 (4.255) 93

E n ( n + 1 ) ω (4.256) 2 p 2 T 2m 1 ψn ˆp 2 ψ n dx 2m 1 mω ψ [ n (â+ ) 2 (â + â ) (â â + ) + (â ) 2] ψ n dx 2m 2 ω ψ 4 n( n n 1)ψ n dx ω ( n + 1 ) 2 2 (4.257) T V E n /2 x 0, (4.258) p 0, (4.259) x 2 V (1/2)mω 2 ( n + 1 ), (4.260) mω 2 ( p 2 2m T mω n + 1 ) 2 (4.261) ( σ x σ p n + 1 ) 2 (4.262) 94

n 0 (4.258) (4.259) [Griffiths, Problem 2.13] Ψ(x, 0) A [3ψ 0 (x) + 4ψ 1 (x)] (4.263) 1 A 2 3ψ 0 + 4ψ 1 2 dx A 2 (9 + 16) 25 A 2 (4.264) A 1 5 (4.265) Ψ(x, t) 1 5 [ 3ψ 0 e ie 0t/ + 4ψ 1 e ie 1t/ ] (4.266) E 0 1 2 ω, E 1 3 ω (4.267) 2 (3/5) 2 E 0 (4/5) 2 E 1 95

Ψ(x, t) 2 1 [ 9ψ 2 25 0 + 16ψ1 2 + 24ψ 0 ψ 1 cos((e 1 E 0 )t/ ) ] 1 [ 9ψ 2 25 0 + 16ψ1 2 + 24ψ 0 ψ 1 cos(ωt) ] (4.268) x 1 x [ 9ψ0 2 + 16ψ1 2 + 24ψ 0 ψ 1 cos(ωt) ] dx 25 24 25 cos(ωt) xψ 0 ψ 1 dx 24 25 cos(ωt) 24 25 cos(ωt) 24 25 p 1 25 12 25 12 25 12 mω 25 i 2 24 mω 25 2 2mω ( ) 2mω (â + + â ) ψ 0 ψ 1 dx ψ 1 ψ 1 dx cos(ωt) (4.269) 2mω [ ] (3ψ 0 e ie0t/ + 4ψ 1 e ie1t/ )ˆp(3ψ 0 e ie0t/ + 4ψ 1 e ie1t/ ) dx [ ] ψ 0 ˆpψ 1 e i(e 1 E 0 )t/ + ψ 1 ˆpψ 0 e i(e 1 E 0 )t/ dx ( ) mω [ψ 0 i 2 (â + â ) ψ 1 e i(e 1 E 0 )t/ ) ] mω +ψ 1 (i 2 (â + â ) ψ 0 e i(e 1 E 0 )t/ dx [ψ 0 ( 2ψ2 ψ 0 ) e i(e 1 E 0 )t/ + ψ 1 ψ 1 e i(e 1 E 0 )t/ ] dx sin(ωt) (4.270) d x /dt p /m dv/dx mω 2 x d p /dt dv/dx 96

4.5.4 2 d 2 ψ 2m dx 2 + 1 2 mω2 x 2 ψ Eψ (4.271) ξ mω x (4.272) [ ] MT 1 ML 2 T 1 L [1] (4.273) dξ d 2m dx dξ 2 ( dξ dx ) dψ 12 x2 + mω2 dξ ξ 2 ξ2 Eψ (4.274) 2 mω 2m ( d mω dξ ) dψ + 1 mω2 dξ 2 mω ξ2 Eψ (4.275) d2 ψ dξ 2 (ξ2 K)ψ (4.276) K 2E ω (4.277) K (4.276) K 97

ξ x d 2 ψ dξ 2 ξ2 ψ (4.278) ψ(ξ) Ae ξ2 /2 + Be ξ2 /2 (4.279) dψ dξ Aξe ξ2 /2 + Bξe ξ2 /2 (4.280) d 2 ψ dξ 2 A(1 ξ 2 )e ξ2 /2 + B(1 + ξ 2 )e ξ2 /2 Aξ 2 e ξ2 /2 + Bξ 2 e ξ2 /2 ξ 2 ψ (4.281) B ξ ψ(ξ) ( )e ξ2 /2, at large ξ, (4.282) h(ξ) ψ(ξ) h(ξ)e ξ2 /2 (4.283) e ξ2 /2 d 2 (h(ξ)e ξ2 /2 ) dξ 2 (ξ 2 K)h(ξ)e ξ2 /2 (4.284) d dξ ( h e ξ2 /2 ξhe ξ2 /2 ) (ξ 2 K)he ξ2 /2 (4.285) ( h e ξ2 /2 ξh e ξ2 /2 he ξ2 /2 ξh e ξ2 /2 + ξ 2 he ξ2 /2 ) (ξ 2 K)he ξ2 /2 (4.286) 98

( h 2ξh + (ξ 2 1)h ) e ξ2 /2 (ξ 2 K)he ξ2 /2 (4.287) h 2ξh + (K 1)h 0 (4.288) h(ξ) a 0 + a 1 ξ + a 2 ξ 2 + a j ξ j (4.289) h a 1 + 2a 2 ξ + 3a 3 ξ 2 + j0 ja j ξ j 1 (4.290) j0 h 2a 2 + 3 2a 3 + 3 4a 4 + j(j 1)a j ξ j 2 j0 (j + 1)(j + 2)a j+2 ξ j (4.291) j0 [(j + 1)(j + 2)a j+2 2ja j + (K 1)a j ] ξ j 0 (4.292) j0 ξ (j + 1)(j + 2)a j+2 2ja j + (K 1)a j 0, (4.293) a j+2 2j + 1 K (j + 1)(j + 2) a j (4.294) 99

a 0 a 2 1 K a 0, (4.295) 2 a 4 5 K 12 a 2, (4.296) a 1 a 3 3 K a 1, (4.297) 6 a 5 7 K 20 a 3, (4.298) a 0 a 1 2 h(ξ) h even (ξ) + h odd (ξ) (4.299) h even (ξ) a 0 + a 2 ξ 2 + a 4 ξ 4 + (4.300) h odd (ξ) a 1 ξ + a 3 ξ 3 + a 5 ξ 5 + (4.301) j (4.294) a j+2 2 j a j (4.302) ξ p e ξ2 p ξ p e ξ2 j0 1 j! ξ2j+p (4.303) 100

a 2j+p (j 1)! a 2j+p 2 1 j! j a 2j+p 2 (4.304) j 1000, p 8 a 2008 1 1000 a 2006 2 2006 a 2006 (4.305) h(ξ) ξ j e ξ2 ψ ξ e ξ2 /2 B ξ j n a n+2 0 (4.306) K 2n + 1, (n 0, 1, 2, 3, ) (4.307) h h even h odd (4.277) E n ( n + 1 ) ω, n 0, 1, 2, (4.308) 2 Mathematica 101

K 1 K 0.9 K 1.1 K 2n + 1 a j+2 2(n j) (j + 1)(j + 2) a j (4.309) 102

n 0 a 0 0 a 2 0, a 4 0, a 0 0 a 1 0 h 0 (ξ) a 0 (4.310) ψ 0 (ξ) a 0 e ξ2 /2 a 0 e mωx2 /(2 ) (4.311) n 1 a 1 0 a 0 a 2 0 a 3 a 5 0 h 1 (ξ) a 1 ξ (4.312) ψ 1 (ξ) a 1 ξe ξ2 /2 (4.313) n 2 a 0 0, a 4 2a 0 h 2 (ξ) a 0 (1 2ξ 2 ) (4.314) ψ 2 (ξ) a 0 (1 2ξ 2 )e ξ2 /2 (4.315) ψ 2 h n (ξ) n 2 n a 0 a 1 H 0 1, (4.316) H 1 2ξ, (4.317) H 2 4ξ 2 2, (4.318) H 3 8ξ 3 12ξ, (4.319) H 4 16ξ 4 48ξ 2 + 12, (4.320) H 5 32ξ 5 160ξ 3 + 120ξ, (4.321) 103

ψ n ψ n (x) ( mω ) 1/4 1 π 2 n n! H n(ξ)e ξ2 /2 (4.322) ψ n n n ψ n 4.5.5 e z2 +2zξ n0 z n n! H n(ξ) (4.323) z n z 0 H n (ξ) 104

z n z 0 H n (ξ) n n +2zξ z n e z2 z0 z n eξ2 e (z ξ)2 ξ2 n e z0 z n e (z ξ)2 z0 e ξ2 ( 1) n n ξ n e (z ξ)2 z0 z ξ ( 1) n ξ2 n e ξ n e ξ2 (4.324) n H n (ξ) ( 1) n e ξ2 ξ n e ξ2 (4.325) (4.323) z 1 ( 2z + 2ξ)e z2 +2zξ n1 z n 1 (n 1)! H n(ξ) (4.326) 2 n0 z n+1 n! H n (ξ) + 2ξ n0 z n n! H n(ξ) n1 z n 1 (n 1)! H n(ξ) (4.327) 2 n1 z n (n 1)! H n 1(ξ) + 2ξ n0 z n n! H n(ξ) n0 z n n! H n+1(ξ) (4.328) 2 n0 z z n n! nh n 1(ξ) + 2ξ n0 z n n! H n(ξ) n0 z n n! H n+1(ξ) (4.329) 105

H n+1 (ξ) 2ξH n (ξ) 2nH n 1 (ξ) (4.330) (4.323) ξ 2ze z2 +2zξ n0 z n d n! dξ H n(ξ) (4.331) 2 n0 z n+1 H n (ξ) n! n0 z n d n! dξ H n(ξ) (4.332) 2 n1 z n (n 1)! H n 1(ξ) n0 z n d n! dξ H n(ξ) (4.333) 2 n0 z z n n! nh n 1(ξ) n0 z n d n! dξ H n(ξ) (4.334) dh n (ξ) dξ 2nH n 1 (ξ) (4.335) (4.330) H n+1 (ξ) 2ξ dh n+1 (ξ) 2n d 2 H n+1 (ξ) 2(n + 1) dξ 4n(n + 1) dξ 2 (4.336) 2(n + 1)H n+1 (ξ) 2ξ dh n+1(ξ) dξ 106 d2 H n+1 (ξ) dξ 2 (4.337)

d2 H n (ξ) dξ 2 2ξ dh n(ξ) dξ + 2nH n (ξ) 0 (4.338) (n 0 OK ) (4.288) (4.307) I I e ξ2 e a2 +2aξ e b2 +2bξ dξ (4.339) e (ξ a b)2 +2ab dξ e 2ab π (4.340) I 2 n a n b n π n! n0 (4.341) I e ξ2 n0 m0 n0 m0 a n b m n!m! a n b m n!m! H n(ξ)h m (ξ)dξ e ξ2 H n (ξ)h m (ξ)dξ (4.342) a b e ξ2 H n (ξ)h m (ξ)dξ π2 n n!δ nm (4.343) ψ n ( e ξ2 /2 H n (ξ)) (4.322) 107

4.5.6 Ĥψ Eψ : (4.344) Ĥ ˆp2 2m + 1 2 mω2ˆx 2 : (4.345) â ± 1 2 mω ( iˆp + mωˆx) : (4.346) [â, â + ] 1 : (4.347) [ˆx, ˆp] i : (4.348) ( Ĥ ω â + â + 1 ) : 2 (4.349) â ψ 0 (x) 0 : ψ (4.350) ψ n (x) 1 n! (â + ) n ψ 0 (x) : n ψ (4.351) V T ω 2 ( n + 1 ) : (4.352) 2 ψ n (x) ( mω ) 1/4 1 π 2 n n! H n(ξ)e ξ2 /2, ( ) mω ξ x : (4.353) 108

4.6 4.6.1 V (x) 0 2 d 2 ψ Eψ (4.354) 2m dx2 d 2 ψ dx 2 k2 ψ, where k 2mE (4.355) k [L 1 ] ψ(x) Ae ikx + Be ikx (4.356) Ψ(x, t) ψ(x)e iet/ [ ( A exp ik x k 2m t )] [ ( + B exp ik x + k )] 2m t (4.357) k k [ )] Ψ k (x, t) A exp i (kx k2 2m t (4.358) OK Ψ k [ { ( Ψ k (x, t) A exp i k x + k ( Ψ k x + k t, t + t 2m ) 2m t k2 ) (t + t) 2m }] (4.359) 2mE k ±, with { k > 0 k < 0 109 (4.360)

λ 2π k (4.361) k E p2 2m (4.362) E 2 k 2 2m (4.363) p k (4.364) λ h/p v quantum x t k / 2m t t k 2m E 2m (4.365) E E 1 2 mv2 classical (4.366) v classical 2E m 2v quantum (4.367) 2 110

Ψ k Ψ kdx A 2 dx (4.368) ψ Ψ(x, t) 1 2π [ )] φ(k) exp i (kx k2 2m t dk (4.369) φ(k) 1/ 2π φ(k) [L 1/2 ] ω k 2 /(2m) Ψ(x, t) 2 dx 1 2π 1 2π ( ) dxdkdk φ(k )e ik x iω ( t φ(k)e ikx iωt) dkdk φ(k ) φ(k)e i(ω ω )t dkdk φ(k ) φ(k)e i(ω ω )t δ(k k ) dxe i(k k )x dk φ(k) 2 (4.370) φ(k) Ψ 111

p 1 2π 1 2π 1 2π (φ(k)e ikx iωt) ( ) dxdkdk φ(k )e ik x iω t ( ) i x dxdkdk φ(k )e ik x iω ( t k φ(k)e ikx iωt) dkdk kφ(k ) φ(k)e i(ω ω )t dkdk kφ(k ) φ(k)e i(ω ω )t δ(k k ) dxe i(k k )x dk k φ(k) 2 (4.371) φ(k) k 0 p k 0 h λ 0 (4.372) p k E dk 2 k 2 2m φ(k) 2 (4.373) φ(k) k 0 E 2 k 2 0 2m (4.374) Ψ(x, 0) Ψ(x, t) φ(k) (4.369) t 0 (4.369) Ψ(x, 0) 1 2π φ(k)e ikx dk (4.375) φ(k) φ(k) 1 2π Ψ(x, 0)e ikx dx (4.376) 112

4.6.2 Ψ(x, 0) { A, if a < x < a 0, otherwise, (4.377) a < x < a t 0 1 Ψ(x, 0) 2 dx a A 2 dx a 2a A 2 (4.378) A 1 2a (4.379) φ(k) φ(k) 1 a 2π 1 2 πa a ( 1 2a e ikx ) dx [ e ikx ] a ik a ( e ika e ika ) 1 k πa 1 sin(ka) πa k 2i (4.380) φ(k) (4.369) Ψ(x, t) 1 [ )] π sin(ka) exp i (kx k2 2a k 2m t dk (4.381) Mathematica 113

Gibbs k 20/a 20/a t ma 2 / Ψ 2 φ(k) k (4.380) k 1/k k sin k k 1/a a x 0 k a 1 φ(k) a π (4.382) 114

φ(k) k 1/a a k a k 1/a φ(k) φ(k) a π (4.383) k 1/a φ(k) 0 (4.384) a k Mathematica 4.6.3 2v quantum v classical (4.385) 115

(4.386) v classical v quantum Ψ(x, t) 1 2π ω φ(k)e i(kx ωt) dk (4.387) ω k2 2m (4.388) ω(k) ω k m 0 φ(k) k 0 ω(k) ω(k 0 ) + ω (k 0 )(k k 0 ) + (4.389) (4.387) k 0 Ψ(x, t) 1 2π 1 2π φ(k) exp [ i(kx { ω(k 0 ) + ω (k 0 )(k k 0 ) } t) ] dk φ(k) exp [ i ( k(x ω (k 0 )t) + ( ω(k 0 ) + k 0 ω (k 0 ))t )] dk 1 2π e i( ω(k 0)+k 0 ω (k 0 ))t φ(k)e ik(x ω (k 0 )t) dk 1 2π e i( ω(k 0)+k 0 ω (k 0 ))t Ψ(x ω (k 0 )t, 0) (4.390) Ψ 2 v group dω dk (4.391) 116

(4.387) v phase ω k (4.392) ω k 2 /(2m) dω dk k m, ω k k 2m (4.393) 2 4.6.4 φ(k) k E hν (4.394) (4.355) E 2 k 2 2m (4.395) ω k 2 /(2m) E ω hν (4.396) p k (4.397) k 2π λ (4.398) 117

p h λ (4.399) 4.6.5 (4.358) Ψ k (x, t) (3.30) J(x, t) i ( Ψ ) k Ψ k 2m x Ψ k Ψ k x i 2m A 2 ( ik ik) k m A 2 (4.400) k > 0 +x k < 0 x 4.6.6 Ψ(x, 0) Ae ax2 (4.401) a > 0 1 A 2 e 2ax2 dx A 2 π 2a (4.402) A ( ) 2a 1/4 (4.403) π 118

Ψ(x, t) φ(k) φ(k) 1 2π Ψ(x, 0)e ikx dx ( ) 1 2a 1/4 e ax2 ikx dx 2π π ( ) 1 2a 1/4 [ ( exp a x + ik ) ] 2 k2 dx 2π π 2a 4a ( ) 1 2a 1/4 [ ( e k2 /(4a) exp a x + ik ) ] 2 dx 2π π 2a ( ) 1 2a 1/4 +ik/(2a) e k2 /(4a) e ax 2 dx 2π π +ik/(2a) ( ) 1 2a 1/4 e k2 /(4a) e ax 2 dx 2π π ( ) 1 2a 1/4 e k2 /(4a) π 2π π a ( ) 1 1/4 e k2 /(4a) 2πa (4.404) Im x ik 2a R R Re 119

(4.369) Ψ(x, t) 1 2π ( 1 1 2π 2πa ( 1 1 2π 2πa ( 1 1 2π 2πa [ exp ( 1 1 2π ( 2a π [ φ(k) exp i (kx k2 2m t ) 1/4 ) 1/4 e k2 /(4a) exp e k2 /(4a) exp )] dk [ i (kx k2 2m t [ ( 1 4a + i 2m t )] dk ) ] k 2 + ikx dk ) 1/4 [ ax 2 ] exp 1 + 2i at/m 1 ( ) ] 2iax 2 (1 + 2i at/m) k dk 4a 1 + 2i at/m ) 1/4 [ ax 2 ] 4aπ exp 2πa 1 + 2i at/m 1 + 2i at/m ) 1/4 exp [ ax 2 /(1 + 2i at/m) ] (4.405) 1 + 2i at/m Ψ 2 Ψ(x, t) 2 2a exp [ 2ax 2 /(1 + (2 at/m) 2 ) ] π 1 + (2 at/m) 2 (4.406) a w 1 + (2 at/m) 2 (4.407) [w] [L 1 ] Ψ(x, t) 2 2 π we 2w2 x 2 (4.408) 120

x x Ψ(x, t) 2 dx 2 π wxe 2w2 x 2 0, (4.409) 121

p Ψ Ψ i x dx ) ( i 2ax 1 + 2i at/m Ψ 2 dx 0, (4.410) x 2 x 2 Ψ(x, t) 2 dx 2 wx 2 e 2w2 x 2 π 2 2π π w 4 3 w 6 1 4w 2, (4.411) t w (3.95) ( ) p 2 Ψ 2 2 Ψ x 2 dx ( ) ( ) Ψ 2 Ψ dx x x 2 2 2a 2 π 1 + 2i at/m wx 2 e 2w2 x 2 2 2 π 4aw2 wx 2 e 2w2 x 2 2 a (4.412) σ x 1 2w, σ p a, (4.413) σ x σ p 2 a w 2 1 + (2 at/m) 2 (4.414) 122

t 0 /2 t p x Φ(x, 0) Ae ax2 +ibx (4.415) p φ(k) p k 4.7 4.7.1 2 123

V (x) E x V (x) E x V (x) E x E V 2 124

{ E < V () and E < V ( ) E > V () or E > V ( ) (4.416) 4.7.2 V (x) αδ(x) (4.417) α [α] [energy][l] V (x) αδ(x) x 2 d 2 ψ αδ(x)ψ Eψ (4.418) 2m dx2 E E < 0 E > 0 125

4.7.3 x < 0 V (x) 0 d 2 ψ dx 2 2mE 2 ψ κ2 ψ (4.419) κ 2mE 2 (4.420) E < 0 κ ψ(x) Ae κx + Be κx (4.421) x ψ 0 A 0 ψ(x) Be κx, (x < 0), (4.422) x > 0 ψ(x) F e κx, (x > 0), (4.423) { 1. ψ dψ 2. dx (4.424) [ ɛ, ɛ] ɛ 2 d 2 ψ ɛ ɛ 2m ɛ dx dx + V (x)ψ(x)dx E ψ(x)dx (4.425) ɛ ɛ 126

ɛ 0 ψ(x) [ 2 dψ 2m lim dψ ] (4.426) ɛ 0 dx ɛ dx ɛ dψ/dx 2 V (x) ɛ lim ɛ 0 ɛ ( αδ(x))ψ(x)dx α lim ψ(0) αψ(0) (4.427) ɛ 0 dψ dψ 2mα ψ(0) (4.428) dx + dx 2 ψ(x) x 0 ψ ψ(x) { Be κx, (x < 0) Be κx (x 0) (4.429) dψ/dx dψ dψ Bκ Bκ dx + dx 2mα 2 ψ(0) 2mα 2 B (4.430) κ mα 2 (4.431) E 2 κ 2 2m mα2 2 2 (4.432) 1 ψ(x) 2 dx 2 B 2 e 2κx dx B 2 κ 0 (4.433) 127

B κ mα (4.434) OK ψ(x) mα e mα x / 2, E mα2 2 2 (4.435) 4.7.4 E > 0 x < 0 d 2 ψ dx 2 2mE 2 ψ k2 ψ (4.436) k 2mE (4.437) ψ(x) Ae ikx + Be ikx (4.438) A B OK x > 0 ψ(x) F e ikx + Ge ikx (4.439) ψ(x) F + G A + B (4.440) 128

dψ ik(f G) (4.441) dx + dψ ik(a B) (4.442) dx dψ dψ ik(f G A + B) (4.443) dx + dx ψ(0) A + B (4.444) (4.428) ik(f G A + B) 2mα (A + B) (4.445) 2 F G (1 + 2iβ)A (1 2iβ)B, where β mα 2 k (4.446) A, B, F, G, k +x 129

ψe iet/ x < 0 J(x, t) i (ψ ψ 2m x ( i 2m i ) ψ ψ x Ae ikx + Be ikx) ( ikae ikx ikb ikx) + c.c. ( ik A 2 + ik B 2 + ikab e 2ikx ika Be 2ikx) + c.c. 2m k ( A 2 B 2) m (4.447) t x x > 0 J(x, t) k m ( F 2 G 2) (4.448) Ae ikx : x < 0 +x Be ikx : x < 0 x F e ikx : x > 0 +x Ge ikx : x > 0 x (4.449) +x Ge ikx G 0 (4.450) J inc k m A 2, J ref k m B 2, J trans k m F 2, (4.451) J inc J ref J trans R T R J ref J inc B 2 A 2, T J trans F 2 J inc A 2 (4.452) 130

(4.440) (4.446) (4.440) (4.446) 0 2iβA + (2 2iβ)B B iβ 1 iβ A (4.453) (4.440) F 1 1 iβ A (4.454) R β2 1 + β 2, T 1 1 + β 2 (4.455) R + T 1 (4.456) β R 1 1 + (2 2 E/mα 2 ), T 1 1 + (mα 2 /2 2 E) (4.457) σ p 4.7.5 V (x) αδ(x) α > 0 α 131

V (x) αδ(x) x (4.431) β (R) (T ) β T 0 T 4.8 V (x) { V0, for a x a 0, for x > a (4.458) V 0 > 0 132

V (x) a a x V 0 E < 0 E > 0 4.8.1 x < a 2 d 2 ψ Eψ (4.459) 2m dx2 d2 ψ dx 2 κ2 ψ, κ 2mE (4.460) E < 0 κ [L 1 ] ψ(x) Ae κx + Be κx (4.461) (x ) A 0 ψ(x) Be κx, (x < a) (4.462) a < x < a 2 d 2 ψ 2m dx 2 V 0ψ Eψ (4.463) 133

d2 ψ dx 2 l2 ψ, l 2m(E + V0 ) (4.464) E > V min Section 4.1.5 l [L 1 ] ψ(x) C sin(lx) + D cos(lx), ( a < x < a) (4.465) x > a x < a ψ(x) F e κx, (x > a) (4.466) V (x) V ( x) ψ(x) Section 4.1.4 F e κx (x > a) ψ(x) D cos(lx) (0 < x < a) ψ( x) (x < 0) (4.467) ψ dψ/dx OK x a ψ F e κa D cos(la) (4.468) dψ/dx κf e κa ld sin(la) (4.469) κ l tan(la) (4.470) κ l ξ la, η κa (4.471) 134

ξ η η a ξ tan ξ (4.472) a η ξ tan ξ (4.473) κ l ξ 2 + η 2 (l 2 + κ 2 )a 2 ( 2m(E + V0 ) 2 + 2mE ) 2 a 2 2mV 0a 2 2 (4.474) (4.473) (4.474) ξ > 0, η > 0 Mathematica 135

136

V 0 2 /2ma 2 ξ η ξ nπ 2, (n 1, 3, 5 ) (4.475) 2ma 2 (E n + V 0 ) 2 n2 π 2 4 (4.476) E n + V 0 n2 π 2 2 2m(2a) 2 (4.477) n 2, 4, 6, F D Mathematica 3 V 0 30 2 /(ma 2 ) (4.478) 137

138

E 1 29.0 2 /(ma 2 ), E 3 21.4 2 /(ma 2 ), E 5 6.97 2 /(ma 2 ), (4.479) ψ x/a ψ x ±a F e κx (x > a) ψ(x) C sin(lx) (0 < x < a) ψ( x) (x < 0) (4.480) x a F e κa C sin(la), κf e κa lc cos(la) (4.481) κ l cot(la) η ξ cot(ξ) (4.482) 139

Mathematica 140

E 1 E 3 E 3 E 5 η > 0 ξ > π 2 (4.483) π/2 2mV 0 a 2 2 < ( π 2 ) 2 (4.484) V 0 < π2 2 8ma 2 (4.485) ξ nπ 2, (n 2, 4, 6, ) (4.486) 4.8.2 E > 0 x < a ψ(x) Ae ikx + Be ikx, (x < a) (4.487) k 2mE (4.488) V (x) V 0 ψ(x) C sin(lx) + D cos(lx), ( a < x < a) (4.489) l 2m(E + V0 ) 141 (4.490)

ψ(x) F e ikx (4.491) x a Ae ik( a) + Be ik( a) C sin( la) + D cos( la) (4.492) Ae ika + Be ika C sin(la) + D cos(la) (4.493) dψ/dx ikae ik( a) ikbe ik( a) lc cos(l( a)) ld sin(l( a)) (4.494) ikae ika ikbe ika lc cos(la) + ld sin(la) (4.495) x a C sin(la) + D cos(la) F e ika, (4.496) lc cos(la) ld sin(la) ikf e ika (4.497) 4 A, B, C, D, F A, B, C, D F C D (4.496) (4.497) ( (4.496)) l sin(la) + ( (4.497)) cos(la) lc sin 2 (la) + lc cos 2 (la) lf e ika sin(la) + ikf e ika cos(la) ( lc lf e ika sin(la) + ik ) l cos(la) C F e ika ( sin(la) + ik l cos(la) ) (4.498) 142

( (4.496)) l cos(la) ( (4.497)) sin(la) ld cos 2 (la) + ld sin 2 (la) lf e ika cos(la) ikf e ika sin(la) ( ld lf e ika cos(la) ik ) l sin(la) D F e ika ( cos(la) ik l sin(la) ) (4.499) A B ( (4.493)) (ik) + ( (4.495)) 2ikAe ika C( ik sin(la) + l cos(la)) + D(ik cos(la) + l sin(la)) F e ika ( sin(la) + ik l cos(la) ) ( ik sin(la) + l cos(la)) +F e ika ( cos(la) ik l sin(la) ) (ik cos(la) + l sin(la)) A 1 ] 2ik e2ika F [( ik ik) sin 2 (la) + (l + k2 + l + k2 l l ) sin(la) cos(la) + (ik + ik) cos2 (la) 1 ] 2ik e2ika F [( ik)(1 cos(2la)) + (l + k2 ) sin(2la) + (ik)(1 + cos(2la)) l e 2ika F [cos(2la) + k2 + l 2 ] sin(2la) (4.500) 2ikl ( (4.493)) (ik) ( (4.495)) 2ikBe ika C( ik sin(la) l cos(la)) + D(ik cos(la) l sin(la)) F e ika ( sin(la) + ik l cos(la) ) ( ik sin(la) l cos(la)) +F e ika ( cos(la) ik l sin(la) ) (ik cos(la) l sin(la)) B 1 [ ( k 2ik F ( ik + ik) sin 2 2 (la) + l ( k 2 1 2ik F F k2 l 2 2ikl l ) l sin(2la) ) ] l l + k2 sin(la) cos(la) + ( ik + ik) cos 2 (la) l sin(2la) (4.501) 143

R B 2 A 2 k2 l 2 2ikl cos(2la) + k2 +l 2 2ikl (k 2 l 2 ) 2 4k 2 l 2 sin(2la) 2 sin(2la) sin 2 (2la) 2 cos 2 (2la) + (k2 +l 2 ) 2 sin 2 (2la) 1 + (k 2 l 2 ) 2 ( 4k 2 l 2 (k 2 +l 2 ) 2 (k 2 l 2 ) 2 4k 2 l 2 4k 2 l 2 sin 2 (2la) ) 1 sin 2 (2la) 4k 2 l 2 sin 2 (2la) 1 + (k2 l 2 ) 2 sin 2 (2la) 4k 2 l 2 (k 2 l 2 ) 2 sin 2 (2la) 4k 2 l 2 + (k 2 l 2 ) 2 sin 2 (2la) V0 2 sin2 (2la) 4E(E + V 0 ) + V0 2 sin2 (2la) (4.502) T F 2 A 2 1 cos(2la) + k2 +l 2 2ikl sin(2la) 1 2 cos 2 (2la) + (k2 +l 2 ) 2 sin 2 (2la) 4k 2 l 2 4k 2 l 2 1 + (k 2 l 2 ) 2 sin 2 (2la) 4E(E + V 0 ) 4E(E + V 0 ) + V0 2 sin2 (2la) (4.503) R + T 1 T < 1 sin(2la) 0 T 1 2la 2a 2m(En + V 0 ) nπ (4.504) E n + V 0 n2 π 2 2 2m(2a) 2 (4.505) 144

Mathematica 4.9 V (x) { V0 for a x a 0 for x > a (4.506) V 0 > 0 x V 0 V min 0 x a ψ (4.487) (4.491) E < V 0 Eψ 2 d 2 ψ 2m dx 2 + V 0ψ (4.507) ψ(x) ic sinh(l x) + D cosh(l x), ( a x a) (4.508) l 2m(V0 E) (4.509) 145

cosh x ex + e x, sinh x ex e x. 2 2 (4.510) cosh x cos(ix), sinh x i sin(ix) (4.511) cosh 2 x sinh 2 x 1 (4.512) A, B F C C, D D, l il (4.513) OK A e 2ika F [cosh(2l a) + k2 l 2 ] 2kl i sinh(2l a), (4.514) B F k2 + l 2 2kl i sinh(2l a). (4.515) T F 2 A 2 1 cosh 2 (2l a) + (k2 l 2 ) 2 sinh 2 (2l 4k 2 l a) 2 4k 2 l 2 4k 2 l 2 (1 + sinh 2 (2l a)) + (k 2 l 2 ) 2 sinh 2 (2l a) 4k 2 l 2 4k 2 l 2 + (k 2 + l 2 ) 2 sinh 2 (2l a) 4E(V 0 E) 4E(V 0 E) + V0 2 sinh2 (2l a) (4.516) 146

E V 0 0 2 d 2 ψ 2m dx 2 (4.517) ψ(x) C x + D (4.518) A ie 2ika (i + ka)f, B ikaf (4.519) T 1 1 + (ka) 2 2 2 + 2ma 2 V 0 (4.520) (4.516) E V 0 E > V 0 T 4E(E V 0 ) 4E(E V 0 ) + V 2 0 sin2 (2la) (4.521) 147

E < V 0 148

4.9.1 Gamov E < V 0 T (4.516) l a 2m(V0 E) a 1 (4.522) sinh 2 (2l a) e4l a 4 (4.523) T 16E(V 0 E) exp e 4l a V0 2 e 4l a [ 4l a + log 16E(V 0 E) V 2 0 ] (4.524) d 2a T e 2l d (4.525) [ T exp 2 i [ exp 2 b a l i x ] 2m(V (x) E)dx ] Gamov (4.526) 149

Chapter 5 5.1 ˆx ˆp Q(x, p) (5.1) Ψ 2 dx 1 (5.2) 2 b a f(x) 2 dx < (5.3) 150

f(x) a b f (5.4) f g b a f(x) g(x)dx (5.5) f g f f g (5.6) f m f n δ mn (5.7) {f n } f(x) c n f n (x) (5.8) n1 f c n f n (5.9) n1 ( ) f n f f n c m f m m1 c m δ nm m1 c n (5.10) 151

(5.9) (5.10) f n f n f n f (5.11) 5.2 5.2.1 Q(x, p) Q Ψ ˆQΨdx (5.12) ˆQ Q Ψ ˆQΨ (5.13) Q Q (5.14) Ψ ˆQΨ ˆQΨ Ψ (5.15) f(x) f ˆQf ˆQf f (5.16) (5.17) 152

g 1 (x), g 2 (x) g 1 + g 2 ˆQ(g 1 + g 2 ) g 1 ˆQg 1 + g 1 ˆQg 2 + g 2 ˆQg 1 + g 2 ˆQg 2 (5.18) ˆQ(g 1 + g 2 ) g 1 + g 2 ˆQg 1 g 1 + ˆQg 1 g 2 + ˆQg 2 g 1 + ˆQg 2 g 2 g 1 ˆQg 1 + ˆQg 1 g 2 + ˆQg 2 g 1 + g 2 ˆQg 2 (5.19) g 1 ˆQg 2 + g 2 ˆQg 1 ˆQg 1 g 2 + ˆQg 2 g 1 (5.20) g 1, g 2 g 1 + ig 2 (5.16) g 1 + ig 2 ˆQ(g 1 + ig 2 ) g 1 ˆQg 1 + i g 1 ˆQg 2 i g 2 ˆQg 1 + g 2 ˆQg 2 (5.21) ˆQ(g 1 + ig 2 ) g 1 + ig 2 ˆQg 1 g 1 + i ˆQg 1 g 2 i ˆQg 2 g 1 + ˆQg 2 g 2 g 1 ˆQg 1 + i ˆQg 1 g 2 i ˆQg 2 g 1 + g 2 ˆQg 2 (5.22) g 1 ˆQg 2 g 2 ˆQg 1 ˆQg 1 g 2 ˆQg 2 g 1 (5.23) g 1, g 2 (5.20) (5.23) g 1 ˆQg 2 ˆQg 1 g 2 (5.24) ˆx f ˆxg f xgdx (xf) gdx ˆxf g (5.25) 153

f ˆpg f d i dx gdx [ f ] i g ( ) d i dx f gdx ( ) d i dx f gdx ˆpf g (5.26) i Ô f Ôg Ô f g (5.27) f, g Ô Ô ˆQ ˆQ (5.28) Ô f Ô g Ô g f g Ôf Ôf g (5.29) (Ô ) Ô (5.30) f Ô (a 1 g 1 + a 2 g 2 ) Ôf a 1g 1 + a 2 g 2 a 1 Ôf g 1 + a 2 Ôf g 2 a 1 f Ô g 1 + a 2 f Ô g 2 (5.31) 154

Ô Â, ˆB f ( + ˆB)g f Âg + f ˆBg  f g + ˆB f g ( + ˆB )f g (5.32) ( + ˆB)  + ˆB (5.33) f  ˆBg  f ˆBg ˆB  f g (5.34) (AB) B A (5.35) ˆQ 1 ˆQ 2 ( ˆQ 1 + ˆQ 2 ) ˆQ 1 + ˆQ 2 (5.36) ( ˆQ 1 ˆQ2 ) ˆQ 2 ˆQ1 (5.37) ˆQ 1 ˆQ2 ˆQ 2 ˆQ1 ˆQ 1 ˆQ2 ˆQ 2 1 ˆQ 3 1 ( ˆQ 2 1 )Q 1 ˆQ n 1 (n 1, 2, 3, ) Ĥ ˆp2 + V (ˆx) (5.38) 2m 155

Ĥ ˆp 2 2m + V (ˆx ) Ĥ (5.39) 5.2.2 ˆQ ˆQ ˆQ Ψ σq 2 Q 2 Q 2 Q q (Q q) 2 Ψ ( ˆQ q) 2 Ψ ˆQ ( ˆQ q)ψ ( ˆQ q)ψ 0 (5.40) ( ˆQ q)ψ 0 (5.41) ˆQΨ qψ (5.42) Ψ ˆQ q ˆQ ˆQ (5.43) ˆQ ˆQ q Ĥ ĤΨ E Ψ, (ĤΨ(x, t) EΨ(x, t) ) (5.44) 156

Ψ Ψ ψ(x)e iet/ (5.45) ψ E E 5.3 5.3.1 1. f ˆQ ˆQf qf (5.46) q f ˆQf f qfdx q f f (5.47) ˆQf f (qf) fdx q f f (5.48) f q q (5.49) 157

2. f g ˆQ ˆQf qf, ˆQg q g (5.50) f ˆQg q f g (5.51) f ˆQg ˆQf g q f g (5.52) q q f g q f g (5.53) q q f g 0 (5.54) ψ n ψ n e 1, e 2, e 3 e 1 e 1 e1 e 1, (5.55) e 2 c 2 ( e2 e 1 e 2 e 1 ) (5.56) c 2 e 1 e 3 c 3 ( e3 e 1 e 3 e 1 e 2 e 3 e 2 ) (5.57) 158

e 1 e 2 (5.58) 3. n n n (5.59) 5.3.2 f p (x) ˆp p i d dx f p(x) pf p (x) (5.60) f p (x) A p e ipx/ (5.61) p 159

p f p (x)f p(x)dx A p A p e i(p p )x/ dx A p A p2π δ(p p ) (5.62) A p 1 2π (5.63) f p (x) 1 2π e ipx/ (5.64) f p f p δ(p p ) (5.65) p 1 [p 1 ] f p (x) f(x) c(p)f p (x)dp 1 2π c(p)e ipx/ dp (5.66) c(p) f(x) c(p) c(p) 1 2π f(x)e ipx/ dx (5.67) c(p) f p f (5.68) 160

λ 2π p (5.69) e ipx/ ˆx g a (x) ˆx a ˆxg a (x) xg a (x) ag a (x) (5.70) g a (x) A a δ(x a) (5.71) A a ga(x)g b (x)dx A aa b δ(x a)δ(x b)dx A aa b δ(a b) A a 2 δ(a b) (5.72) a A a 1 g a (x) δ(x a) (5.73) g a g b δ(a b) (5.74) 161

f(x) g a (x) f(x) c(a)g a (x)da (5.75) c(a) f(a) (5.76) 5.4 Ψ(x, t) t a < x < b P ab (t) b a Ψ(x, t) 2 dx (5.77) ˆQ(x, p) q E ψ n (x) E n E n P n P n c n 2 ψn(x)ψ(x, 0)dx 2 ψ n Ψ(t 0) 2 (5.78) 162

t 0 ( Ψ(x, t) ˆQ(x, p) Q x, ) i x ˆQ ˆQf n q n f n (5.79) f n ˆQ q n t q n P n (t) P n (t) c n (t) 2, c n (t) f n Ψ(t) (5.80) ˆQ ˆQf q q f q (5.81) f q t Q a < q < b P ab (t) P ab (t) b a c q (t) 2 dq, c q (t) f q Ψ(t) (5.82) Ψ(x, t) Ψ(t) n c n (t) f n (5.83) 163

c n (t) c n (t) f n Ψ(t) (5.84) 2 2 (5.85) 1 Ψ(t) Ψ(t) c n(t)c m (t) f n f m n m c n(t)c m (t)δ nm! n m c n (t) 2 (5.86) n Q(t) Ψ (x, t) ˆQΨ(x, t)dx Ψ(t) ˆQΨ(t) c n(t)c m (t) f n ˆQf m n m c n(t)c m (t)q m f n f m n m c n(t)c m (t)q m δ nm n m c n (t) 2 q n (5.87) n 164

x a < x < b ˆx g q (x) δ(x q) (5.88) Ψ(x, t) Ψ(x, t) c q (t)g q (x)dx (5.89) c q (t) g q Ψ(t) δ(x q)ψ(x, t)dx Ψ(q, t) (5.90) P ab (t) b a b a c q (t) 2 dq Ψ(q, t) 2 dq (5.91) f p (x) 1 2π e ipx/ (5.92) f p Ψ(t) 1 2π e ipx/ Ψ(x, t)dx (5.93) Φ(p, t) 1 2π e ipx/ Ψ(x, t)dx (5.94) 165

Ψ(x, t) 1 2π p 0 < p < p 1 Φ(p, t) P 01 (t) p1 e ipx/ Φ(p, t)dp (5.95) p 0 Φ(p, t) 2 dp (5.96) 5.5 σ x σ p 2 (5.97) 5.5.1 Â ) ) σa 2 (Â A Ψ(t) (Â A Ψ(t) f f (5.98) f (Â A )Ψ ˆB σ 2 B g g (5.99) g ( ˆB B )Ψ σ 2 Aσ 2 B f f g g f g 2 (Re [ f g ]) 2 + (Im [ f g ]) 2 (Im [ f g ]) 2 ( ) 1 2 [ f g g f ] (5.100) 2i 166

f g (Â A )Ψ ( ˆB B )Ψ ÂΨ ˆBΨ A Ψ ˆBΨ B ÂΨ Ψ + A B AB A B (5.101) Â ˆB g f BA A B (5.102) σ 2 Aσ 2 B ( ) 1 2 [Â, ˆB] (5.103) 2i γ β α β α (5.104) α α γ γ ) ( ( β α β α α α β α β ) α α α β β α β 2 α α α β 2 α α + α β 2 α α β β α β 2 α α 0 (5.105) α β 2 α α β β (5.106) α α 0 167

 ˆx ˆB ˆp σ 2 xσ 2 p ( ) 1 2 [ˆx, ˆp] 2i ( ) 1 2 2i i ( ) 2 (5.107) 2 (5.108) [Â, ˆB] 0 σ A 0 σ B 0 5.5.2 σ x σ p /2 (5.100) g(x) cf(x) c f f f f g(x) iaf(x), a (5.109) g f (ˆp p )Ψ ia(ˆx x )Ψ (5.110) ( ) i x p Ψ(x, t) ia(x x )Ψ(x, t) (5.111) x Ψ(x, t) i (ia(x x ) + p ) Ψ(x, t) (5.112) 168

t t 0 x log Ψ(x, t 0 ) + C 0 i [ ( ) ] x 2 ia 2 x x + p x (5.113) Ψ(x, t 0 ) Ae a(x x )2 /2 e i p x/ (5.114) t 0 5.5.3 x p 2 (5.115) t E 2 (5.116) t (5.116) ˆQ ˆQ d dt Q d Ψ(t) ˆQΨ(t) dt Ψ(t) t ˆQΨ(t) 1 i i [Ĥ, ˆQ] + + ĤΨ ˆQΨ(t) + ˆQ t Ψ(t) ˆQ t Ψ(t) + Ψ(t) ˆQ + 1 Ψ(t) t i ˆQĤΨ(t) Ψ(t) ˆQ t (5.117) 169

d dt Q i [Ĥ, ˆQ] + ˆQ t (5.118) ˆQ ˆQ (2.40) Ĥ ˆQ ˆQ σ 2 Hσ 2 Q σ 2 Hσ 2 Q ( 1 2i i ( ) 1 2 [Ĥ, ˆQ] (5.119) 2i ) d Q 2 dt ( 2 ) 2 ( ) d Q 2 (5.120) dt σ H σ Q d Q 2 dt (5.121) E σ H t σ Q d Q /dt (5.122) t E t 2 (5.123) ˆQ t t E 0 170

5.6 Ψ(x, t) Ψ(t) x S(t) ˆQ α β (5.124) α β α α β α β α β α α (f g) α β β α, (5.125) α α 0, (5.126) 0 α α (a β 1 + b β 2 ) a α β 1 + b α β 2. (5.127) ˆQ α β α ˆQ β (5.128) 171

ˆQ γ ˆQ α γ β β γ α ˆQ γ (5.129) γ ˆQ α α ˆQ γ (5.130) α γ γ ˆQ α α ( ˆQ ) γ ( (5.130) ) ( α ˆQ γ ) α ˆQ γ ( (5.130) ) (5.131) ( ˆQ ) ˆQ (5.132) ˆQ ˆQ (5.133) { e n } e m e n δ mn (5.134) e n α n e n e n α (5.135) e m e n e n (5.136) n 172

( 1) n e n e n 1 (5.137) { e z } e z e z δ(z z ), (5.138) e z e z dz 1. (5.139) x p ˆx ˆp ˆx x x x, ˆp p p p, (5.140) x x δ(x x ), p p δ(p p ), (5.141) x x dx 1, p p dp 1. (5.142) x 173

ˆx ˆp ˆxΨ(x, t) xψ(x, t), ˆpΨ(x, t) i Ψ x (5.143) ˆx ˆp ˆQ ˆQφ n (x) q n φ n (x), Ψ(x, t) i ĤΨ(x, t) (5.144) t φ m(x)φ n (x) δ mn (5.145) q n q n P n (t) φ 2 n(x)ψ(x, t)dx. (5.146) ˆx ˆp ˆx x x ˆx S(t) x x S(t), x ˆp S(t) i x S(t) (5.147) x ˆx ˆp ˆQ i d S(t) Ĥ S(t). (5.148) dt ˆQ q n q n q n, q m q n δ mn (5.149) q n q n P n (t) q n S(t) 2. (5.150) Ψ(x, t) x S(t) (5.151) 174