25 7 18

1 1 1.1 v.s............................. 1 1.1.1.................................. 1 1.1.2................................. 1 1.1.3.................................. 3 1.2................... 3 1.3........................ 4 1.4............................ 5 2 7 2.1.................................. 7 2.1.1............................ 7 2.1.2..................... 7 2.1.3............................. 8 2.1.4............................. 13 2.2...................................... 15 2.2.1.......................... 15 2.2.2............................... 16 2.3................ 21 2.3.1........................ 21 2.3.2.......................... 22 2.3.3.......................... 22 3 25 3.1...................................... 25 3.2...................................... 26 3.3................... 27 3.4................................. 33 3.4.1 x....................... 34 4 38 4.1............................. 38 ii

iii 4.1.1................ 39 4.1.2..................... 41 4.1.3......... 42 4.1.4 Poisson............................... 44 4.1.5....................... 45 4.2................................... 46 4.2.1.............................. 46 4.2.2 Heisenberg............................ 46 4.3.............................. 48 5 50 5.1............................. 50 5.2..................................... 51 5.2.1.......................... 51 5.2.2............................... 53 5.2.3............................. 54 5.2.4............................. 54 5.3...................................... 54 5.3.1............................... 54 5.4 1................................. 58 5.4.1................................. 59 5.4.2.................... 60 5.4.3 2..................... 61 5.4.4......... 63 5.4.5 1.............. 64 5.4.6.................................. 67 5.5 L.................................. 67 5.6............................. 70 5.6.1 1............................ 71 5.6.2 1.................... 71 5.6.3............................ 75 6 77 6.1............................ 77 6.2................................. 77 6.3................................ 79 6.4.............. 81 6.4.1.................................. 81 6.4.2......................... 82

iv 6.4.3............................... 84 6.5..................................... 85 6.6................................ 87 6.7........................... 88 7 89 7.1........................... 89 7.1.1............................... 89 7.1.2......................... 89 7.1.3.............................. 91 7.2 Klein-Gordon.............................. 94 7.2.1.................................. 94 7.2.2 -...................... 95 7.3 Dirac................................... 97 7.3.1.......................... 97 7.3.2 Dirac...................... 102 7.3.3 Dirac........................ 104 7.3.4........................ 106 7.3.5.................................. 107 8 110 8.1..................................... 110 8.1.1............................. 110 8.1.2..................... 112 8.1.3 Poisson............................... 112 8.2................................... 113 8.2.1.............................. 113 8.2.2......................... 114 8.2.3......................... 114 8.2.4................................. 115 8.2.5..................... 117 8.2.6............. 120 8.2.7.............................. 121 8.2.8 1.............. 121 8.2.9................................. 123 8.3...................................... 124

1 1.1 v.s. 1.1.1 1 ψ(r) 2 ψ(r 1, r 2 ) 3 ψ(r 1, r 2, r 3 ) n ψ(r 1, r 2,, r n ) 2 2 1.1.2 2 1

2 ψ(r 1, r 2 ) 1 2 ψ(r 2, r 1 ) Cψ(r 1, r 2 ). r 1 r 2 ψ(r 1, r 2 ) Cψ(r 2, r 1 ). ψ(r 1, r 2 ) C 2 ψ(r 1, r 2 ) C 2 1, or C ±1 2 1 ψ(r 1, r 2 ) ±ψ(r 2, r 1 ) 2 1 1 1 ψ 1 (r 1 ), 2 1 ψ 2 (r 2 ) 2 ψ(r 1, r 2 ) ψ 1 (r 1 )ψ 2 (r 2 ) ψ(r 1, r 2 ) 1 2 [ψ 1 (r 1 )ψ 2 (r 2 ) ψ 1 (r 2 )ψ 2 (r 1 )] 1 ±1 +1 1 boson, fermion 2 r r 1 r 2 r ψ(r, r) r ψ(r, r) ψ(r, r) 0

3 2. 1.1.3 1.2 { 0 ( ), 1, 2, 3, } 0, p 1 p, p, p 2 p, p 0 p p, p â(p): p â (p): p 3 â(p ) p, p p, â(p) p 0, â(p) 0 0, â (p) 0 p, â (p ) p p, p, 2 n i ψ i (r i ) ψ(r 1, r 2,, r n ) 1 n! det [ψ i (r j )] i j 3 (â 0 0) p p â(p) p 0 (p p )

4 0 â (p) â(p) â(p)â (p) 0 0. (1.2.1) â (p)â(p) 0 0. (1.2.2) (â(p)â (p) â (p)â(p) ) 0 0 (1.2.3) â, â [ â(p), â (p) ] 1 (1.2.4) 4 [, ] ([A, B] AB BA) 1.3 p, p p, p â (p)â (p ) 0 p, p ± p, p ( ) â (p)â (p ) ±â (p )â (p) ( ) [ â (p), â (p ) ] 0 ( ) (1.3.1) {â (p), â (p )} 0 ( ) (1.3.2) {, } ({A, B} AB + BA) 4

5 (1.2.4) (1.2.1) (1.2.2) (1.2.3) (1.2.1) (1.2.2) (â(p)â (p) + â (p)â(p) ) 0 0 (1.3.3) â, â { â(p), â (p) } 1 (1.3.4) [ â(p), â (p ) ] δp,p (1.3.5) [ â (p), â (p ) ] 0 (1.3.6) [ â(p), â(p ) ] 0 (1.3.7) { â(p), â (p ) } δp,p (1.3.8) { â (p), â (p ) } 0 (1.3.9) { â(p), â(p ) } 0 (1.3.10) 1.4 1 2 p 1 1 ψ(r) 2 1 ψ 1 (r 1 )ψ 2 (r 2 )

6 ) 3 ψ 1 (r 1 )ψ 2 (r 2 )ψ 3 (r 3 ) 1 r p

2 2.1 2.1.1 ) q i i 1, 2,, N) L(q i, q i ) L(q i, q i ) K(q i, q i ) V (q i ) (2.1.1) q i ( ) ( q i dq i /dt) K(q i, q i ) V (q i ) (Euler-Lagrange E-L eq.): d L L. (i 1, 2,, N) (2.1.2) dt q i q i 2.1.2 q i p i p i L q i. (2.1.3) ( ) p i E-L eq. (2.1.2) p i E-L eq. dp i dt L (2.1.4) q i H H N p i q i L. (2.1.5) i1 7

8 H 1 dh N (dp i q i + p i d q i ) i1 N dp i q i i1 N i1 N i1 ( L d q i + L ) dq i q i q i L q i dq i. (2.1.6) 2 p i (2.1.3) d q i H q i p i q i p i H(q i, p i ) ( H dh(p i, p i ) N i1 H p i dp i + N i1 H q i dq i (2.1.7) (2.1.6) H p i q i, H q i L q i. E-L eq. (2.1.4) dq i dt H p i (2.1.8) dp i dt H. q i (2.1.9) 2.1.3 ( ) q i, p i A A(q i, p i ), B B(q i, p i ) (Poisson bracket; Pb) : N ( A B {A, B} Pb A ) B. (2.1.10) q i p i p i q i i1 1 2 f(x, y) 3 df(x, y) f f dx + x y dy

9 q i, p i A q i p i N da dt ( A dq i q i1 i dt + A ) dp i p i dt N ( A H A ) H q i p i p i q i i1 2 (2.1.8), (2.1.9) A H A da dt {A, H} Pb. (2.1.11) A q i, p i (2.1.8), (2.1.9) dq i dt {q i, H} Pb (2.1.12) dp i dt {p i, H} Pb (2.1.13) (2.1.10) (i) {A, B} Pb {B, A} Pb (ii) (or ) {A, cb + c B } Pb c{a, B} Pb + c {A, B } Pb c, c (iii) ( ) {ca + c A, B} Pb c{a, B} Pb + c {A, B} Pb {A, BC} Pb {A, B} Pb C + B{A, C} Pb {AB, C} Pb {A, C} Pb B + A{B, C} Pb

10 (iv) {A, {B, C} Pb } Pb + {B, {C, A} Pb } Pb + {C, {A, B} Pb } Pb 0. (v) {q i, q j } Pb {p i, p j } Pb 0, { 1 (i j) {q i, p j } Pb {p i, q j } Pb δ ij 0 (i j) (2.1.10) (i) (ii) (iii) (iv) (v) 2.1 (2.1.10) (i) (iii) 2.2 (2.1.10) (v) 2.3 (2.1.10) (iv) 2.4 A D A D A X {A, X} Pb. (ii) (iii) D A 2.5 D A (iv) ( ) D A {B, C} Pb {D A B, C} Pb + {B, D A C} Pb. 2.6 2.4 2.5 or (i) (v) (i) (v) (2.1.10)

11 (i) (iii), (v) (2.1.10) 2 1 (N 1) ( (1) {A, 1} Pb 0. (iii) C 1 {A, B 1} Pb {A, B} Pb 1 + B{A, 1} Pb. B {q, 1} Pb 0, {p, 1} Pb 0. {A, 1} Pb 0 (2) {A, q n } Pb {A, q} Pb nq n 1, {A, p n } Pb {A, p} Pb np n 1 n n 0 0 (1) 0 n 1 n n + 1 (iii) 2, {A, q n+1 } Pb {A, q q n } Pb {A, q} Pb q n + q {A, q n } Pb {A, q} Pb q n + q {A, q} Pb nq n 1 {A, q} Pb (n + 1)q n n+1 {A, q n } Pb {A, q} Pb nq n 1 q n {A, p n } Pb {A, p} Pb np n 1 2 Q i F i (q j, p j ), P i G(q j, p j ) (Q i, P i ) (v) Poisson {A, B} Pb N ( A i1 Q i B A P i P i B Q i ).

12 (3) {A, B} Pb {A, p} Pb B/ p + {A, q} Pb B/ q B B(p, q) B n, m c nmp n q m (ii), (iii) {A, B} Pb n, m c nm {A, p n q m } Pb n, m c nm ({A, p n } Pb q m + p n {A, q m } Pb ) (2) ( c nm {A, p}pb np n 1 q m + p n {A, q} Pb mq m 1) n, m ( ) (p n q m ) (p n q m ) c nm {A, p} Pb + {A, q} Pb p q n, m B {A, p} Pb p + {A, q} B Pb q. (4) {A, p} Pb A/ q, {A, q} Pb A/ p (3) A p B A A {p, A} Pb {p, p} Pb p + {p, q} A Pb q A q 2 (v) {p, p} Pb 0, {p, q} Pb 1 (i) {A, p} Pb {p, A} Pb A q. (3) A q B A A {q, A} Pb {q, p} Pb p + {q, q} A Pb q A p. 2 (v) {q, p} Pb 1, {q, q} Pb 0 (i) {A, q} Pb {q, A} Pb A p. (5) {A, B} Pb A q (3) (4) B p A B p q B {A, B} Pb {A, p} Pb p + {A, q} B Pb q A q B p A p B q.

13 ( ) (4) A/ p 0 A/ q 0 A(p, q) f(q) A(p, q) g(p) {q, f(q)} Pb {f(q), q} Pb 0, (2.1.14) {p, g(p)} Pb {g(p), p} Pb 0. (2.1.15) 2.1.4 m 1 V (x) L(x, ẋ) m 2 ẋ2 V (x). (2.1.16) E-L eq. m d2 x dt 2 (x) V x. (2.1.17) p p L ẋ mẋ. (2.1.18) H(p, x) pẋ L ( m ) mẋẋ 2 ẋ2 V (x) m 2 ẋ2 + V (x) p2 + V (x). (2.1.19) 2m p x

14 {A(p, x), B(p, x)} Pb p, x A(p, x) B(p, x) x p A(p, x) p B(p, x). (2.1.20) x {x, x} Pb 0, {p, p} Pb 0 (2.1.21) {x, p} Pb {p, x} Pb 1. (2.1.22) (2.1.12), (2.1.13) (2.1.20) (2.1.21), (2.1.22) (i) (iii) dx dt {x, H} Pb { } p 2 x, 2m + V (x) Pb 1 2m {x, p2 } Pb + {x, V (x)} Pb (H ) 1 2m ({x, p} Pbp + p{x, p} Pb ) + 0 1 2m 2p p m. ((2.1.22) ) ( (ii) ) ( (iii) {x, f(x)} Pb 0 ) (2.1.23) 4 {x, f(x)} Pb 0 (i) (iii) (2.1.21) (2.1.14) dp dt {p, H} Pb { } p 2 p, 2m + V (x) Pb (H ) 1 2m {p, p2 } Pb + {p, V (x)} Pb ( (ii) ) V (x) 0 ((2.1.21), (2.1.22) (i) (iii) ) x V (x) x (2.1.24) 4 {p, V (x)} Pb V (x)/ x 12 (4) (i) (iii) (v)

15 2.2 2.2.1 A, B Â, ˆB Â ˆB ˆBÂ, Â ˆB ˆBÂ 0. 0 (or ) [Â, ˆB] Â ˆB ˆBÂ (2.2.1) Â ˆB ˆBÂ [Â, ˆB] 0 (2.2.2) Â ˆB ˆBÂ + [Â, ˆB] (2.2.3) (i) (ii) ( ) [Â, ˆB] [ ˆB, Â] [Â, c ˆB + c ˆB ] c[â, ˆB] + c [Â, ˆB ] [câ + c Â, B] c[â, ˆB] + c [Â, ˆB] c, c (c ) (iii) ( ) [Â, ˆBĈ] [Â, ˆB] Ĉ + ˆB[Â, Ĉ] [Â ˆB, C] [Â, Ĉ] ˆB + Â[ ˆB, Ĉ] (iv) [Â, [ ˆB, Ĉ]] + [ ˆB, [Ĉ, Â]] + [Ĉ, [Â, ˆB]] 0.

16 2.7 (2.2.1) (i) 2.8 (2.2.1) (ii) 2.9 (2.2.1) (iii) 2.10 (2.2.1) (iv) ( [Â, [ ˆB, Ĉ]] [Â, [ ˆB, Ĉ]] [Â, ˆBĈ] [Â, Ĉ ˆB] (iii) 2.11 Â L A ˆX L A ˆX [ Â, ˆX]. (ii) (iii) L A 2.12 L A (iv) L A L A [ ˆB, Ĉ] [L A ˆB, Ĉ] + [ ˆB, L A Ĉ] 2.13 L A 2.2.2 A, B Â, ˆB i {A, B} [Â, ˆB] (i) (iv) (i) (iv)

17 (i) (iii) (v) q i p i (i 1, 2,, N) ˆq i, ˆp i (Canonical Commutation Relation; CCR) [ˆq i, ˆq j ] [ˆp i, ˆp j ] 0 [ˆq i, ˆp j ] [ˆp i, ˆq j ] i δ ij (2.2.4) i i dˆq i dt [ˆq i, Ĥ] i dˆp i dt [ˆp i, Ĥ] (2.2.5) Ĥ H(p i, q i ) Ĥ H(ˆp i, ˆq i ) V (x) 1 1 H(p, x) p2 2m + V (x) x p x, p [ˆx, ˆp] i ˆx, ˆp Ĥ ˆp2 2m + V (ˆx)

18 i dˆx dt [ˆx, Ĥ] [ ] ˆp 2 ˆx, 2m + V (ˆx) 1 (Ĥ ) 2m [ˆx, ˆp2 ] + [ˆx, V (ˆx)] ( (ii) ) ( ) [ˆx, ˆp] ˆp + ˆp [ˆx, ˆp] + 0 ( (iii) [ˆx, V (ˆx)] 0 ) 1 2m 1 2m 2i ˆp i ˆp m ( ) 4 [ˆx, V (ˆx)] 0 [ˆx, ˆx n ] 0 (iii) [ˆx ˆx] 0 i dˆp dt [ˆp, Ĥ] [ ] ˆp 2 ˆp, 2m + V (ˆx) (Ĥ ) 1 2m [ˆp, ˆp2 ] + [ˆp, V (ˆx)] ( (ii) ) 1 0 + [ˆp, V (ˆx)]) 2m ( ) i V (ˆx) ( ) V (ˆx) x V (x) V (x) V (x)/ x x ˆx V (x) c n x n n0 V (x) V (x) x c n nx n 1 n1

19 x ˆx V (ˆx) 3 V (ˆx) c n n ˆx n 1. [ˆp, V (ˆx)] i V (ˆx) n1 [ˆp, ˆx n ] i nˆx n 1 (iii) [ˆp, ˆx] i dˆx dt ˆp m dˆp dt V (ˆx) 2.14 ˆx, ˆp 1 1 [ˆx, ˆp] i 3 V (ˆx) [ˆx, ˆx n ] 0, [ˆx, f(ˆx)] 0, [ˆx, ˆp n ] i nˆp n 1, [ˆx, g(ˆp)] [ˆx, ˆp] g p (ˆp), [ˆp, ˆp n ] 0, [ˆp, g(ˆp)] 0, [ˆp, ˆx n ] i nˆx n 1, [ˆp, f(ˆx)] [ˆp, ˆx] f x (ˆx). V (ˆx) c n ˆx n n0

20 f(x) n f n x n, g(p) n g n p n f(ˆx) g(ˆp) f nˆx n, n0 g nˆp n, n0 f x (ˆx) nf n ˆx n 1, n1 g p (ˆp) ng n ˆp n 1. n1

21 2.3 ) 2.3.1 (Schrödinger picture) Ĥ ψ(t) S i d dt ψ(t) S Ĥ ψ(t) S. (2.3.1) ÔS d dtôs 0. (2.3.2) ˆq ˆp 4 ψ(t) S e iĥt/ ψ(0) S. (2.3.3)  etâ d dt etâ Âet (2.3.4) 2.15  ( ) t tâ e tâ e tâ n0 t n n! Ân 1 + tâ + t2 2! Â2 +. (2.3.5) 4 ˆq, ˆp ÔS ˆq S, ˆp S Ĥ S

22 1 1 ψ ψ d dt etâ Âet e tââ. (2.3.6) 2.3.2 (Heisenberg picture) ψ H ψ H ψ(t) S (2.3.3) ψ H e itĥ/ ψ(t) S. (2.3.7) ψ H ψ(0) S (2.3.8) ψ H ÔS ÔH(t) Ô H (t) e itĥ/ Ô S e itĥ/ (2.3.9) i d dtôh(t) i d ) (e itĥ/ Ô S e itĥ/ dt (i ddt ) eitĥ/ Ô S e itĥ/ + e itĥ/ Ô S (i ddt ) e itĥ/ ĤeitĤ/ Ô S e itĥ/ + e itĥ/ Ô S e itĥ/ Ĥ ĤÔH(t) + ÔH(t)Ĥ [ÔH(t), Ĥ] (2.3.10) ÔH(t) 2.3.3 Ĥ H (t) e itĥ/ Ĥ e itĥ/ e itĥ/ e itĥ/ Ĥ Ĥ ( ĤS).

23 t 0 ψ(0) S ψ H (2.3.11) t 0 Ô H (0) ÔS (2.3.12) ψ(t) S ψ H ÔS ÔH(t) Ô S ψ(t) S ÔH(t) ψ H ) ) Ô H (t) ψ H (e itĥ/ Ô S e (e itĥ/ itĥ/ ψ(t) S e itĥ/ Ô S e itĥ/ e itĥ/ ψ(t) S ( ) e itĥ/ Ô S e itĥ/ e itĥ/ ψ(t) S ) e (ÔS itĥ/ ψ(t) S (2.3.13) ÔS ψ(t) S ÔS Ô S ÔH(t) Ô H (t) ÔSÔ S Ô H (t)ô H (t) ) Ô H (t)ô H(t) (e itĥ/ Ô S e (e itĥ/ itĥ/ Ô Se ) itĥ/ e itĥ/ Ô S e itĥ/ e itĥ/ Ô Se itĥ/ ( ) e itĥ/ Ô S e itĥ/ e itĥ/ e itĥ/ Ô Se itĥ/ ) e (ÔS itĥ/ Ô S e itĥ/ (2.3.14) ÔSÔ S ÔS f(ˆp S, ˆq S ) ÔH(t) f(ˆp H (t), ˆq H (t)) Ĥ H(ˆp S, ˆq S ) H(ˆp H (t), ˆq H (t)). ÔS Ô S ÔH(t) Ô H (t) [ÔS, Ô S ] [ÔH(t), Ô H (t)] [ÔH(t), Ô H(t)] e itĥ/ [ÔS, Ô S]e itĥ/ (2.3.15)

24 [ˆq Si, ˆq Sj ] 0, [ˆp Si, ˆp Sj ] 0 [ˆq Si, ˆp Sj ] [ˆp Si, ˆq Sj ] i δ ij (2.3.16) [ˆq Hi (t), ˆq Hj (t)] 0, [ˆp Hi (t), ˆp Hj (t)] 0 [ˆq Hi (t), ˆp Hj (t)] [ˆp Hi (t), ˆq Hj (t)] i δ ij. (2.3.17) ψ(t) S ψ H ÔS ÔH(t) : H ψ ÔH(t) ψ H S ψ(t) ÔS ψ(t) S. (2.3.18) { ) } { } } S ψ(t) (e itĥ/ e itĥ/ Ô S e {e itĥ/ itĥ/ ψ(t) S S ψ(t) e itĥ/ e itĥ/ Ô S e itĥ/ e itĥ/ ψ(t) S S ψ(t) ÔS ψ(t) S ψ(t) S ψ H ψ (t) S ψ H 2.16 (2.3.12) 2.17 (2.3.15) 2.18 (2.3.16) (2.3.17) 2.19 (2.3.19) H ψ ψ H S ψ(t) ψ (t) S. (2.3.19) 2.20 ÔS {ˆp S } n {ˆq S } m ÔH(t) {ˆp H (t)} n {ˆq H (t)} m f(p, q) n,m c nmp n q m f(ˆp S, ˆq S ) f(ˆp H (t), ˆq H (t))

3 3.1 m ω L(x, ẋ) m 2 ẋ2 mω2 2 x2 (3.1.1) m d2 x dt 2 mω2 x (3.1.2) x A sin(ωt α) (3.1.3) p L ẋ mẋ. (3.1.4) H pẋ L mẋẋ m 2 ẋ2 + mω2 2 x2 1 2m p2 + mω2 2 x2 ( ) m 2 ẋ2 mω2 2 x2 H(p, x) p2 2m + mω2 2 x2. (3.1.5) x p {x, x} Pb 0, {p, p} Pb 0 (3.1.6) {x, p} Pb {p, x} Pb 1. (3.1.7) 25

26 3.2 x, p ˆx, ˆp x ˆx p ˆp [ˆx, ˆx] 0, [ˆp, ˆp] 0 (3.2.1) [ˆx, ˆp] [ˆp, ˆx] i. (3.2.2) H(p, x) Ĥ H(ˆp, ˆx) Ĥ ˆp2 2m + mω2 2 ˆx2. (3.2.3) : i d ψ(t) Ĥ ψ(t). (3.2.4) dt Ĥ Ĥ E n E n E n (3.2.5) E n E n (3.2.4) ψ(t) c n e ient/ E n (3.2.6) n c n (3.2.5) 3.1 (3.2.6) ψ(t) (3.2.4)

27 3.3 Ĥ ˆp2 2m + mω2 2 ˆx2. (3.3.1) Ĥ E n E n E n (3.3.2) Ĥ ˆx ˆp [ˆx, ˆp] i (3.3.3) H p2 2m + mω2 2 x2 ( ) ( ) mω 2 1 mω 2 x + i 2m p 2 1 2 x i 2m p Â,  mω 2  2 mω  2 2 1 ˆx + i ˆp, (3.3.4) 2m 1 ˆx i ˆp. (3.3.5) 2m   [ mω [Â, 2 1  ] ˆx + i 2 2m ˆp, mω 2 2 ] 1 ˆx i 2m ˆp i mω 2 2 i ω 2 i + iω 2 ( i ) 1 [ˆx, ˆp] + i 2m 1 mω 2 2m 2 [ˆp, ˆx] ω. (3.3.6) 1 4 2 ˆx ˆp 2

28 Â Â Â ω â, Â ω â â â mω 1 â 2 ˆx + i ˆp, (3.3.7) 2mω mω 1 â 2 ˆx i ˆp (3.3.8) 2mω [â, â ] 1 (3.3.9) (3.3.7), (3.3.8) (3.3.1) ˆx 2mω (â + â ), (3.3.10) ˆp 1 mω (â â ) (3.3.11) i 2 Ĥ ˆp2 2m + mω2 2 ˆx2 1 2m ( 1)mω (â â ) 2 + mω2 2 2 ω { (â + â ) 2 (â â ) 2} 4 ω 4 2mω (â + â ) 2 {â2 + ââ + â â + â 2 (â 2 ââ â â + â 2 ) } ω 2 (ââ + â â) (3.3.12) ( ω â â + 1 ) (3.3.13) 2 [â, â ] 1 ( Ĥ ω â â + 1 ), (3.3.14) 2 [â, â ] 1 (3.3.15) Ĥ E n E n E n (3.3.16)

29 Ĥ Ĥ â â ˆN ˆN â â (3.3.17) ( Ĥ ω ˆN + 1 ) 2 (3.3.18) Ĥ ˆN ˆN ν ν ν (3.3.19) ˆN â â ν (ν 0) ν (3.3.19) ν ν ˆN ν ν ν ν ν. φ â ν ν ˆN ν ν â â ν ν â â ν φ φ φ φ 0 (3.3.20) φ â ν 0 ˆN â, â ˆN â ν â ν â ν 0 ν > 0, (3.3.21) â ν 0 ν 0 (3.3.22) [ ˆN, â ] â, (3.3.23) [ ˆN, â] â. (3.3.24) ˆN ( â ν ) ˆNâ ν (â ˆN + [ ˆN, â ]) ν (â ˆN + â ) ν (â ν + â ) ν (ν + 1) ( â ν ). (3.3.25)

30 ˆN (â ν ) (â ˆN + [ ˆN, â]) ν (â ˆN â) ν (ν 1) (â ν ). (3.3.26) â ν â ν ˆN ν +1 ν 1 â ν ν + 1, â ν ν 1. â ˆN 1 â 1 ν â ν â ν + 1 â ν + 2 â ν + 3 â â ˆN (3.3.21), (3.3.22) â (n + 1 ) ν â ν 1 â ν 2 â â ν n â 0. n + 1 â ν n 0 â 0 â â ν n ˆN ν n ν n 0 ν n ν n ν n ˆN ν ˆN ν ν ν ν 0, 1, 2, 3,. (3.3.27) ˆN (number operator) ˆN (ν )n n n 0 0 â 0 0 (3.3.28)

31 0 â 0 â 1 â 2 â 3 â n n n (â ) n 0 0 1 c â 0 1 c 1 1 1 c c 0 â â 0 0 0 1. (3.3.29) c 2 0 (â â + [â, â ]) 0 c 2 0 [â, â ] 0 ( â 0 0) c 2 0 1 0 ( [â, â ] 1) c 2. c 1 1 â 0. (3.3.30) 2 c â 1 2 c 1 2 2 c c 1 â â 1 c 2 1 (â â + [â, â ]) 1 c 2 1 (1 + [â, â ]) 1 ( â â 1 ˆN 1 1 1 ) c 2 1 2 1 ( [â, â ] 1) 2 c 2. c 1/ 2 2 1 2 â 1 1 2 (â ) 2 0. (3.3.31)

32 n n 1 n! (â ) n 0 (3.3.32) ˆN n n n, (3.3.33) n n 1 (3.3.34) n m δ nm, (3.3.35) â n n n 1, (3.3.36) â n n + 1 n + 1 (3.3.37) 3.2 n (3.3.32) (3.3.33) (3.3.37) ˆN Ĥ Ĥ ( Ĥ ω ˆN + 1 ) 2 ˆN n Ĥ E n Ĥ n ω(n + 1/2) n E n n (3.3.38) E n ( n + 1 ) ω (n 0, 1, 2, ) (3.3.39) 2 0 ( E 0 1 ω (3.3.40) 2

33 3.4 ψ(x) ˆp i / x ψ(x) 2 ˆx n ˆx n ˆx ˆx 2 x x 2 (ˆx ˆx ) 2 ˆx 2 ˆx 2. ˆx n ˆx n (3.3.10) ˆx 2mω (â + â ) n ˆx n 2mω n (â + â ) n ( n n n 1 + n ) n + 1 n + 1 2mω 0. n ˆx 2 n 2mω n (â + â ) 2 n 2mω n â2 + (â ) 2 + ââ } {{ + â â } n 2 ˆN+1 n 0 + 0 + (2n + 1) n 2mω (2n + 1). 2mω 3.3 (3.3.11) ˆp, ˆp 2 3.4 n ˆp 2 mω2 n n ˆx 2 n ω 2m 2 2 ( n + 1 ) 2

34 3.4.1 x ψ ψ(x) ψ(x) x ψ (3.4.1) x x ψ(x) ψ x 1 [ x ] ψ(x) 2. x x ˆx x x 2 ˆx x x x. (3.4.2) ˆx x x δ(x x ). (3.4.3) ψ x x ψ c(x) x ψ c(x ) c(x) x dx. c(x) x x dx c(x)δ(x x) dx 1 ψ φ φ ψ 2 φ ψ 2 2  ψ  ψ a ψ. ψ  a  A

35 c(x) ψ x ψ(x) ψ ψ(x) x dx. (3.4.4) ψ ψ(x) x ψ ˆp ψ x ˆp ψ x ψ ψ(x) (3.4.4) x ˆp ψ x ˆp ψ(x ) x dx ψ(x ) x ˆp x dx (3.4.5) x ˆp x ˆp ˆx [ˆx, ˆp] i x x x [ˆx, ˆp] x i x x. x ˆxˆp x x ˆpˆx x i δ(x x ). x ˆx x x, ˆx x x x 3 (x x ) x ˆp x i δ(x x ) (3.4.6) (x x ) δ (x x ) δ(x x ) (3.4.7) x ˆp x i δ (x x ) (3.4.8) x ˆp x (3.4.5) x ˆp ψ i x ψ(x ) ( i δ (x x )) dx ψ(x ) δ(x x ) dx 3 x δ(x) 0 δ(x) i ψ(x) (3.4.9) x δ(x) + x δ (x) 0

36 ψ x ˆp ψ i x ψ (3.4.10) x ψ ψ(x) ˆp ψ ( /i) ψ(x)/ x p i x x 0 (ψ 0 (x) x 0 ) 0 ψ 0 (x) x 0 â 0 0 0 x â 0 ( ) mω 1 x 2 ˆx + i 2mω ˆp, 0 ( ) mω 1 2 x + i x 0 2mω i x ( mω 2mω x + ) x 0. x ψ 0 (x) x mω x ψ 0(x) (3.4.11) ( ψ 0 (x) C exp mω ) 2 x2 (3.4.12) C

37 ψ 1 (x) x 1 ψ 1 (x) x â 0 ( ) mω 1 x 2 ˆx i 2mω ˆp 0 ( ) mω 1 2 x i x 0 2mω i x ( mω 2mω x x C x exp ( mω 2 x2 ) ) ( C exp mω ) 2 x2 (3.4.13) n ψ n (x) 1 x (â ) n 0 n! ( mω x ) n ( exp mω ) x 2 x2 (3.4.14)

4 1 u(x, t) x x u(x, t) t u(x, t) 4.1 x L (tension)t ) σ (x 0 x L) u(x, t) u(0, t) u(l, t) 0. N N N u n u n+1 x x 0 0 x 1 x 2 x n x n+1 x N L x N x x n x n nl N (n 0, 1,, N) L N x x x n+1 x n L N (4.1.1) 38

39 δm δm σ x (4.1.2) δm ( ) x n N 1 1 x n u(x, t) u(x n, t) u n u n u(x n, t). (4.1.3) u n ( ) u n u n t u(x n, t). (4.1.4) 4.1.1 L(u 1, u 2, ; u 1, u 2, ) K U n K δm 2 u2 n σ 2 u2 n x N 1 n1 N 1 δm σ 2 u2 n 2 u2 n x (4.1.5) n1 u n u n+1 u n+2 x n x n+1 x n+2 1 ( )

40 n n + 1 n l x n + 1 u n u n+1 u n x l T x l U n T (l x) x l ( x u n ) ( u n x l l ( x) 2 + ( u n ) 2 [ ( ) ] 2 1/2 un x 1 + x [ x 1 + 1 ( ) ] 2 un 2 x x + 1 2 ( ) 2 un x x 2 n n + 1 U n U n T 2 ( ) 2 un x x U N 1 n0 T 2 ( ) 2 un x (4.1.6) x L K U [ σ 2 u2 n T 2 n ( ) ] 2 un x (4.1.7) x 2 x 1 (1 + x) α 1 + αx

41 N x 0 u n u(x, t)/ t, u n / x u(x, t)/ x n x dx L L 0 [ σ 2 ( ) 2 u T t 2 ( ) ] 2 u dx (4.1.8) x L L L σ 2 L 0 L dx, (4.1.9) ( ) 2 u T t 2 ( ) 2 u. (4.1.10) x 4.1.2 S t2 t 1 t2 t 1 t 1 t2 Ldt dtdxl L dt dx 0 L dt dxδ 0 [ σ 2 ( ) 2 u T t 2 ( ) ] 2 u. (4.1.11) x u(x, t) u(x, t) + δu(x, t) 3 S [ t2 ( ) 2 σ u δs T ( ) ] 2 u 2 t 2 x t 1 t2 t 1 dt dt L 0 L 0 [ dx σ u [ dx σ u t ( ) u t δ t u δu T t x T u x δ x δu ( )] u x ] (4.1.12) δu(x, t 1 ) δu(x, t 2 ) 0 δu(0, t) δu(l, t) 0 [ ] δs dtdx σ 2 u t + T 2 u δu(x, t). (4.1.13) 2 x 2 3 t t 1, t 2 u δu(x, t 1 ) δu(x, t 2 ) 0.

42 δu(x, t) δs 0 u(x, t) σ 2 u t T 2 u 0, (4.1.14) 2 x2 c T/σ 1 2 u c 2 t 2 u 0, (4.1.15) 2 x2 c u(x, t) f(x ct) + g(x + ct) (4.1.16) (D Alembert ) f(x), g(x) 1 f(x ct) c (x ) g(x + ct) c (x ) 4.1 u(x, t) (4.1.16) (4.1.15) 4.1.3 (4.1.7) u n p n p n L u n σ u n x (4.1.17) N x 0 p n 0 p n x π n π n σ u n, (4.1.18) p n π n x (4.1.19) N π n π n σ u(x, t)/ t

43 p n u n H n n n n p n u n L σ u 2 n x [ σ 2 u2 n T 2 n [ σ 2 u2 n + T ( ) ] 2 un x 2 x [ ) 2 x] p 2 n 2σ x + T 2 ( un x ( ) ] 2 un x x (4.1.20) N p n u n π n u n H [ πn 2 2σ + T ( ) ] 2 un x (4.1.21) 2 x n N [ L π 2 H 2σ + T ( ) ] 2 u dx (4.1.22) 2 x 0 H H L 0 H π2 2σ + T 2 H dx, (4.1.23) ( ) 2 u. (4.1.24) x π π n π n π(x, t) π n σ u n u(x, t) π(x, t) σ t π n x p n L/ u n L π L u (4.1.25) u u/ t

44 4.1.4 Poisson p n π n {u n, u m } Pb 0, (4.1.26) {p n, p m } Pb 0, (4.1.27) {u n, p m } Pb δ nm, (4.1.28) {u n, u m } Pb 0, (4.1.29) {π n, π m } Pb 0, (4.1.30) {u n, π m } Pb 1 x δ nm, (4.1.31) 1 2 3 N ( x 0) x n x, x m x lim N 1 x δ nm { 0 (x x ) + (x x ) (4.1.32) x, x Dirac lim N 1 x δ nm C δ(x x ). C x Cδ(x x ) dx C. (n ) [ ] 1 dx lim N x δ nm [ ] 1 lim x N x δ nm lim N 1 C 1 lim N n n δ nm 1 x δ nm δ(x x ). (4.1.33)

45 (π(x, t), u(x, t)) Poisson {u(x, t), u(x, t)} Pb 0, (4.1.34) {π(x, t), π(x, t)} Pb 0, (4.1.35) {u(x, t), π(x, t)} Pb δ(x x ). (4.1.36) 4.1.5 u n (n 1, 2, ) q n n n 1 1 x n 2, 3 1 y z n 4, 5, 6 2 x, y, z ) u n n x x n u n u(x n, t). x (x ) u(x) (u(x, t)) ( ) φ(x, y, z; t) φ x (x, y, z) x (x, y, z) x (x, y, z) q n n. 4 q n (t) φ(x, t) ( ) ( ) 4 n d 3 x δ mn δ(x x )

46 4.2 ) 4.2.1 Poisson i u(x, t) û(x, t) π(x, t) ˆπ(x, t) ( ) [û(x, t), û(x, t)] 0, (4.2.1) [ˆπ(x, t), ˆπ(x, t)] 0, (4.2.2) [û(x, t), ˆπ(x, t)] i δ(x x ), (4.2.3) 4.2.2 Heisenberg Ĥ Ĥ Ĥ L 0 Ĥdx, ˆπ(x, t)2 2σ + T 2 ( ) 2 û(x, t) (4.2.4) x (Heisenberg picture Ô(t) Heisenberg i dô(t) dt [Ô(t), Ĥ] (4.2.5)

47 Heisenberg û(x, t) ˆπ(x, t) û(x, t) i t [û(x, t), Ĥ] [ û(x, t), L 0 L 0 1 2σ 0 L 0 ] H(y) dy dy [û(x, t), Ĥ(y)] [ ˆπ(y, t)2 dy û(x, t), + T ( ) ] 2 û(y, t) 2σ 2 y L dy [û(x, t), ˆπ(y, t) 2 ] + T [ L ( ) ] 2 û(y, t) dy û(x, t),. (4.2.6) 2 y [û(x, t), ˆπ(y, t) 2 ] [û(x, t), ˆπ(y, t)]ˆπ(y, t) + ˆπ(y, t)[û(x, t), ˆπ(y, t)] 2i δ(x y) ˆπ(y, t), (4.2.7) [ ( ) ] 2 [ ] [ ] û(y, t) û(y, t) û(y, t) û(y, t) û(y, t) û(x, t), û(x, t), + û(x, t), y y y y y ˆπ(x, t) i t L 0 1 2σ T 2 û(y, t) [û(x, t), û(y, t)] + y y 0 û(y, t) y [û(x, t), û(y, t)] y 0. (4.2.8) û(x, t) i t 1 2σ i σ dy [ˆπ(x, t), Ĥ(y)] L 0 L 0 L 0 û(x, t) t ˆπ(x, t). dy 2i δ(x y)ˆπ(y, t) dy [ˆπ(x, t), ˆπ(y, t) 2 ] + T 2 dy 2 û(y, t) y ˆπ(x, t). (4.2.9) σ L 0 [ ( ) ] 2 û(y, t) dy ˆπ(x, t), y (4.2.10) [ˆπ(x, t), û(y, t)] (4.2.11) y i T 2 û(x, t) x 2. (4.2.12)

48 ˆπ(x, t) t (4.2.9) t (4.2.13) 2 û(x, t) t 2 T 2 u(x, t) x 2. (4.2.13) 1 ˆπ(x, t) σ x T 2 û(x, t) σ x 2 σ 2 û(x, t) t 2 T 2 û(x, t) x 2 (4.2.14) 4.2 (4.2.10) (4.2.11) 4.3 (4.2.11) (4.2.12) 4.3 φ(x, t) φ L L(φ, φ, φ ) (4.3.1) φ φ/ t, φ φ/ x δs δ dt dx L 0 φ L φ π π L φ (4.3.2)

49 H π φ L (4.3.3) H H(π, φ) H dx H. (4.3.4) φ, π {φ(x, t), φ(y, t)} Pb 0, (4.3.5) {π(x, t), π(y, t)} Pb 0, (4.3.6) {φ(x, t), π(y, t)} Pb δ(x y). (4.3.7) (φ ˆφ, π ˆπ) (CCR) [ ˆφ(x, t), ˆφ(y, t)] 0, (4.3.8) [ˆπ(x, t), ˆπ(y, t)] 0, (4.3.9) [ ˆφ(x, t), ˆπ(y, t)] i δ(x y). (4.3.10) H(π, φ) Ĥ H(ˆπ, ˆφ) (4.3.11) H Ĥ dx Ĥ. (4.3.12) Heisenberg picture Heisenberg i dô(t) dt [Ô(t), Ĥ] (4.3.13) φ(x, t) 3 φ(x, y, z, t) φ(x, t)

5 5.1 1 ψ(x, t) ( ) ψ(x, t) i t [ 2 2m ] 2 x + V (x) ψ(x, t) (5.1.1) 2 ψ(x, t) ψ(x, t) h ( x, x) 1 h ( x, x) 2 2m 2 + V (x). (5.1.2) x2 h ( x, x) ψ(x, t) i t h ( x, x) ψ(x, t) (5.1.3) (Remark) (N ) 1 ψ(x, t) 1 V ψ(x, t) 2 d 3 x 1 h ( x, x) 1 50

51 1 V N N N ) V N 100 0.02 2 V ρ(x, t) ψ(x, t) 2 ψ N ψ(x, t) 2 d 3 x (5.1.4) ρ(x, t) ψ(x, t) 2 (5.1.5) ψ(x, t) (5.1.4) h ψ (x, t)h ( x, x) ψ(x, t) d 3 x (5.1.6) p ψ (x, t) i x ψ(x, t) d3 x (5.1.7) 1 h N 1 N 5.2 5.2.1 (5.1.3) Euler-Lagrange δs 0, S L dt L d 3 xdt (5.2.1) (5.1.3) L [ L ψ ψ(x, t) (x, t) i h ( ] t x, x) ψ(x, t). (5.2.2)

52 L ψ δs 0 ψ ψ 1, ψ 2 ψ ψ 1 + iψ 2 ψ ψ 1 iψ 2 δs {A δψ 1 + B δψ 2 } d 3 x dt 0 A 0, B 0. (5.2.3) A 0 B 0 Euler-Lagrange ψ 1 1 2 (ψ + ψ ) ψ 2 1 2i (ψ ψ ) δs A δψ 1 + B δψ 2 A 1 2 (δψ + δψ ) + B 1 2i (δψ δψ ) 1 2 (A ib) δψ + 1 (A + ib) δψ 2 δψ δψ δs 0 { { A ib 0 A 0 (5.2.4) A + ib 0 B 0 ψ ψ ψ ψ ψ ψ ψ ψ (5.2.2) S S Ld 3 xdt ψ [i t h( x, x)] ψ(x, t)d 3 xdt (5.2.5)

53 [ 0 δs δψ (x, t) i t h( x, x)] ψ(x, t)d 3 xdt [ + ψ (x, t) i t h( x, x)] δψ(x, t)d 3 xdt. (5.2.6) δψ [ i t h( x, x)] ψ(x, t) 0 (5.2.7) (5.1.3) δψ (5.2.6) δψ [ (5.2.6) ψ i ( )] t δψ 2 2m 2 δψ + V δψ d 3 xdt )] [ i ( ψ t 2 2m 2 ψ + V ψ δψ d 3 xdt ] [ i ψ t h( x, x) ψ δψ(x, t) d 3 xdt. (5.2.8) 0 δψ i ψ (x, t) h ( t x, x) ψ (x, t) 0 (5.2.9) (5.2.7) ψ 5.2.2 (5.2.2), L i ψ (x, t) ψ(x, t) ψ (x, t) h ( x, x) ψ(x, t) (5.2.10) π L ψ i ψ. (5.2.11) ψ ψ π ψ ψ L ψ L ψ 0 ψ π i ψ ψ ψ 2 2

54 5.2.3 H π ψ L i ψ ψ ψ (i ψ h ( x, x) ψ) ψ h ( x, x) ψ. (5.2.12) H H d 3 x ψ (x, t) h ( x, x) ψ(x, t) d 3 x (5.2.13) h 5.2.4 {ψ(x, t), π(y, t)} Pb δ 3 (x y), {ψ(x, t), ψ(y, t)} Pb {π(x, t), π(y, t)} Pb 0, (5.2.14) π(x, t) i ψ (x, t) π ψ i {ψ(x, t), ψ (y, t)} Pb δ 3 (x y), {ψ(x, t), ψ(y, t)} Pb {ψ (x, t), ψ (y, t)} Pb 0, (5.2.15) 5.3 5.3.1 ψ(x, t) ˆψ(x, t), (5.3.1) π(x, t) ˆπ(x, t). (5.3.2) ψ ψ ψ (x, t) ˆψ (x, t). (5.3.3) π i ψ ˆπ(x, t) i ψ (x, t)

55 ˆπ ˆψ [ ˆψ(x, t), ˆπ(y, t)] i δ 3 (x y), [ ˆψ(x, t), ˆψ(y, t)] [ˆπ(x, t), ˆπ(y, t)] 0. (5.3.4) [ ˆψ(x, t), ˆψ (y, t)] δ 3 (x y), [ ˆψ(x, t), ˆψ(y, t)] [ ˆψ (x, t), ˆψ (y, t)] 0 (5.3.5) H Ĥ, Ĥ ˆψ (x, t) h ( x, x) ˆψ(x, t)d 3 x. (5.3.6) Ô i Ô [Ô, Ĥ] (5.3.7) ˆψ ˆψ i ˆψ(x, t) [ ˆψ(x, t), Ĥ] [ ˆψ(x, t), ˆψ (y, t) h ( ] y, y) ˆψ(y, t) d 3 y d 3 y [ ˆψ(x, t), ˆψ (y, t) h ( y, y) ˆψ(y, t) ] d 3 y { [ ˆψ(x, t), ˆψ (y, t) ] h ( y, y) ˆψ(y, t) + ˆψ (y, t) [ ˆψ(x, t), h ( y, y) ˆψ(y, t) ] }

56 { [ d 3 y ˆψ(x, t), ˆψ (y, t) ] h ( y, y) ˆψ(y, t) + ˆψ (y, t) h ( y, y)[ ] ˆψ(x, t), ˆψ(y, } t) d 3 y δ 3 (x y) h ( y, y) ˆψ(y, t) h ( x, x) ˆψ(x, t). (5.3.8) i ˆψ (x, t) [ ˆψ (x, t), Ĥ] [ ˆψ (x, t), (5.3.8) (5.3.9) ˆψ (y, t) h ( ] y, y) ˆψ(y, t) d 3 y d 3 y [ ˆψ (x, t), ˆψ (y, t) h ( y, y) ˆψ(y, t) ] d 3 y { [ ˆψ (x, t), ˆψ (y, t) ] h ( y, y) ˆψ(y, t) + ˆψ (y, t) [ ˆψ (x, t), h ( y, y) ˆψ(y, t) ] } d 3 y { [ ˆψ (x, t), ˆψ (y, t) ] h ( y, y) ˆψ(y, t) + ˆψ (y, t) h ( y, y)[ ˆψ (x, t), ˆψ(y, t) ]} d 3 y ˆψ (y, t) h ( y, y) δ 3 (x y) { d 3 y h ( } y, y) ˆψ (y, t) δ 3 (x y) h ( x, x) ˆψ (x, t) h ( x, x) ˆψ (x, t). (5.3.9) i ˆψ(x, t) t i ˆψ (x, t) t h ( x, x) ˆψ(x, t), (5.3.10) h ( x, x) ˆψ (x, t) (5.3.11) (5.2.7), (5.2.9) ψ ψ ˆψ ˆψ 3 3 ψ L/ ψ 0 ψ ψ ψ ( i ) Heisenberg

57 (Remark) ˆψ ˆψ Heisenberg (5.3.8) (5.3.8) 2 1 2 Ĥ (5.3.6) x y 3 y 4 [A, BC] [A, B]C + B[A, C] 5 h ( y, y) 6 (5.3.5) 7 δ(x y) 5.1 (5.3.8) (5.3.9) (5.1.4) (5.1.7) ˆN, ˆP ˆN ˆψ (x, t) ˆψ(x, t) d 3 x, (5.3.12) ˆP ˆψ (x, t) i x ˆψ(x, t) d 3 x. (5.3.13) 1

58 1 A ( x, x) A A ψ (x, t)a ( x, x)ψ(x, t) d3 x (5.3.14) ψ, ψ ˆψ, ˆψ Â Â ˆψ (x, t)a ( x, x) ˆψ(x, t) d 3 x. (5.3.15) 5.4 1 1 V (x) 0 ˆψ(x, t), ˆψ (x, t) [ ˆψ(x, t), ˆψ (y, t)] δ(x y), [ ˆψ(x, t), ˆψ(y, t)] [ ˆψ (x, t), ˆψ (y, t)] 0 (5.4.1) Ĥ ˆψ (x, t)h ( x, x) ψ(x, t) dx (5.4.2) h ( x, x) h ( x, x) 2 2m 2 x 2 (5.4.3) ˆψ i ˆψ(x, t) t 2 2 ˆψ(x, t), (5.4.4) 2m x 2 ˆψ x ψ(x, t) L ψ(x + L, t) ψ(x, t) (5.4.5)

59 L/2 < x L/2 (5.4.2) Ĥ L/2 L/2 ˆψ (x, t)h ( x, x) ψ(x, t) dx (5.4.6) 1 i ˆψ(x, t) t 1 2 ˆψ(x, t), (5.4.7) 2m x 2 5.4.1 (5.4.7) ψ(x, t) e i(px Et) 4 1 2m ψ(x, t) i t E p i( ie)e i(px Et) Eψ(x, t), 2 ψ(x, t) 1 x 2 2m ( p2 )e i(px Et) p2 ψ(x, t), 2m E p2 2m (5.4.8) (5.4.5) p (5.4.5) e i[p(x+l) Et] e i(px Et) e ipl 1. p p n 2πn L (n ) (5.4.9) 4 e i(kx ωt) k ω 1 p k k E ω ω k ω p E

60 p ( n ϕ n (x, t) 1 L e ie nt+ip n x (n ) (5.4.10) p n (5.4.9) E n E n p n 2 2m (5.4.11) ϕ n (x, t) ψ(x, t) ψ(x, t) i 1 2 ψ(x, t) (5.4.12) t 2m x 2 ϕ n(x, t) ψ (x, t) i ψ (x, t) 1 2 ψ (x, t) (5.4.13) t 2m x 2 ϕ n (x, t) L/2 L/2 ϕ n(x, t)ϕ l (x, t) dx δ nl (5.4.14) n l n l ( ) 1 L L/2 L/2 L/2 L/2 1 L ei(en E l)t 1 L ei(e n E l )t ϕ n(x, t)ϕ l (x, t) dx e ie nt ip n x e ie lt+ip l x dx L/2 L/2 L/2 L/2 e i(p l p n)x dx e i2π(l n)x/l dx 0 ( ). (5.4.15) 5.4.2 (5.4.12) : ψ(x, t) a n ϕ n (x, t). (5.4.16) n

61 a n ( ) ψ a n : a n L/2 L/2 ( ) L/2 L/2 L/2 L/2 l l ϕ n(x, t)ψ(x, t) dx. (5.4.17) ϕ n(x, t)ψ(x, t) dx ϕ n(x, t) l a l ϕ l (x, t) dx L/2 a l ϕ n(x, t)ϕ l (x, t) dx a l δ nl L/2 a n. (5.4.18) ψ (5.4.16) n l 5.4.3 2 ψ ˆψ (5.4.16) a n â n ˆψ a n â n. ˆψ(x, t) n â n ϕ n (x, t) (5.4.19) ˆψ ˆψ (x, t) n â n ϕ n(x, t). (5.4.20)

62 ˆψ â n â n L/2 L/2 ϕ n(x, t) ˆψ(x, t) dx (5.4.21) â n â n L/2 L/2 ϕ n (x, t) ˆψ (x, t) dx (5.4.22) ˆψ(x, t), ˆψ (x, t) (5.4.1) â n, â n (5.4.21), (5.4.22) [ L/2 [â n, â l ] ϕ n(x, t) ˆψ(x, t) dx, L/2 L/2 L/2 L/2 L/2 ϕ l (y, t) ˆψ(y, t) dy dx dy ϕn(x, t) ϕ l (y, t) [ ˆψ(x, t), ˆψ(y, t)] L/2 L/2 }{{} 0 0. (5.4.23) [â n, â l ] 0. (5.4.24) ] [ L/2 [â n, â l ] ϕ n(x, t) ˆψ(x, t) dx, L/2 L/2 L/2 L/2 L/2 dx L/2 L/2 L/2 L/2 ϕ l (y, t) ˆψ (y, t) dy dy ϕ n(x, t) ϕ l (y, t) [ ˆψ(x, t), ˆψ (y, t)] }{{} δ(x y) ϕ n(x, t) ϕ l (x, t) dx δ nl. (5.4.25) ] [â n, â l ] 0, [â n, â l ] δ nl, (5.4.26) [â n, â l ] 0.

63 n l 1 [â 1, â 1 ] [â 1, â 1] 0 [â 1, â 1] 1, (3.3.9) 5.2 â n, â n (5.4.26) (5.4.19), (5.4.20) ˆψ(x, t), ˆψ (x, t) (5.4.1) e i2πn(x y)/l Lδ(x y). n 5.4.4 Ĥ â n, â l (5.4.20) Ĥ (5.4.6) (5.4.19), Ĥ L/2 L/2 L/2 L/2 ˆψ (x, t)h ( x, x) ψ(x, t) dx dx n l n â n ϕ n(x, t) h ( x, x) â n â l L/2 L/2 l â l ϕ l (x, t) dx ϕ n(x, t) h ( x, x) ϕ l (x, t) h ( x, x) ϕ l (x, t) p2 l 2m ϕ l(x, t) E l ϕ l (x, t) n l n L/2 â n â l E l dx ϕ n(x, t) ϕ l (x, t) L/2 } {{ } δ nl E n â n â n. (5.4.27) Ĥ E n â n â n (5.4.28) n

64 â n, â l ˆN ˆP ˆN ˆP L/2 L/2 n L/2 L/2 n ˆψ (x, t)ψ(x, t) dx â n â n, (5.4.29) ( 1 ˆψ (x, t) i ) ψ(x, t) dx x p n â n â n, (5.4.30) 5.3 ˆN ˆP 5.4.5 1 0 â n 0 0 for all n (5.4.31) Ĥ 0 0 (5.4.32) 0 Ĥ 0 ˆP 0 0 (5.4.33) 0 ˆP 0 0 0

65 1 0 p n Ĥ p n l l p n â n 0 (5.4.34) E l â l âl â n 0 E l â l E n â n 0 (â n â l 0 ) + [â l, â }{{ n] 0 } δ ln E n p n (5.4.35) ˆP p n p n p n (5.4.36) p n E n p 2 n/(2m) p n p n m 1 p n ˆN p n ˆN p n p n, (5.4.37) 1 5.4 (5.4.30), (5.4.29) (5.4.36) (5.4.37) 1 â n p n, p l â n â l 0 p n, p l p n, p l 2 p n + p l Ĥ p2 n + p2 l 2m 2m ˆN 2

66 Ĥ, ˆN, ˆP â n, â n [ ] [Ĥ, â n] E l â l a l, â n l l l E l [â l âl, â n] ) E l ([â l, â n] â l + â l [â l, â n] l E l (0 + â l δ ln) E n â n. (5.4.38) [Ĥ, â n] E n â n, (5.4.39) [ ˆP, â n ] p n â n, [ ˆP, â n] +p n â n, (5.4.40) [ ˆN, â n ] â n, [ ˆN, â n] +â n (5.4.41) E Ĥ E Ĥ E E E. Ĥ ( â n E ) ( â nĥ + [Ĥ, ) â n] E E E n â n (E + E n ) ( â n E ) (5.4.42) â n E Ĥ E + E n â n E n (5.4.40) (5.4.41) ˆP ˆN â n p n 1 p n, p l p n +p l Ĥ p2 n + p2 l 2m ˆN 2 p n1, p n2,..., p nl â n 1 â n 2 â n l 0 2m

67 Ĥ p n1,..., p nl (E n1 + + E nl ) p n1,..., p nl, ˆP p n1,..., p nl (p n1 + + p nl ) p n1,..., p nl, ˆN p n1,..., p nl l p n1,..., p nl 5.5 (5.4.40), (5.4.41) 5.4.6 p n, p n â nâ n 0 [â n, â n ] 0 p n, p n p n, p n (5.4.43) 1 2 l p n1, p n2,..., p nl â n 1 â n 2 â n l 0 (5.4.26) 5.5 L x L p n 2πn L (n 0, ±1, ±2, ) (5.5.1)

68 L p n p n+1 p n 2π L 0 (5.5.2) p ˆψ(x, t) â n ϕ n (x, t) 1 â n e i(pnx Ent) n L n n p p n 2π/L dp L ˆψ(x, t) â n e i(p nx E n t) 2π 2π L n 1 ( ) L e i(p nx E n t) p n (5.5.3) 2π L ( p n 0) n ˆψ(x, t) 1 2π 2π ân â(p) e i(px E pt) dp (5.5.4) E p p 2 /(2m) â(p) p n p L 2π ân â(p) (5.5.5) ˆψ (x, t) ˆψ (x, t) 1 2π p n p â (p) e i(px E pt) dp (5.5.6) L 2π â n â (p) (5.5.7) p â(p) â (p) [ L [â(p), â (p )] lim L 2π ân, ] L 2π â n lim L L 2π δ nn δ(p p ) (5.5.8)

69 p n p, p n p 1 δ nn L ( ) L 2π δ nn p n n n 2π δ nn dp L 2π δ nn δ(p p ) [â(p), â (p )] δ(p p ), [â(p), â(p )] [â (p), â (p )] 0 (5.5.9) (5.5.9) (5.5.4), (5.5.6) ˆψ ˆψ ] [ ˆψ(x, t), ˆψ (y, t)] [ 1 2π 1 2π 1 2π 1 2π dp dp dp e ip(x y) â(p) e i(px E pt) dp, 1 2π dq e i(px E pt) i(qy E q t) [â(p), â (q)] dq e i(px Ept) i(qy Eqt) δ(p q) â (q) e i(qy E qt) dq δ(x y) (5.5.10) L (5.4.28) (5.5.7) Ĥ n E n â nâ n n n E n L 2π â n E n L 2π â n L 2π ân E p â (p) â(p) dp 2π L L 2π ân p n Ĥ E p â (p)â(p) dp. (5.5.11)

70 (5.4.30) (5.4.29) ˆP ˆN p â (p)â(p) dp., (5.5.12) â (p)â(p) dp. (5.5.13) (5.5.9) Ĥ, ˆP, ˆN â(p), â (p) p 1 [Ĥ, â(p)] E p â(p), [Ĥ, â (p)] + E p â (p), (5.5.14) [ ˆP, â(p)] p â(p), [ ˆP, â (p)] + p â (p), (5.5.15) [ ˆN, â(p)] â(p), [ ˆN, â (p)] + â (p), (5.5.16) p, p p â (p) 0 (5.5.17) p p δ(p p ) (5.5.18) p, p â (p) â (p ) 0 (5.5.19) p, p p, p (5.5.9) 5.6 1 0 â (p) 1 1 1 1

71 5.6.1 1 1 1 ˆx ˆp [ˆx, ˆp] i (5.6.1) ( 1 ) x, p ˆx x x x, x y δ(x y), (5.6.2) ˆp p p p, p p δ(p p ) (5.6.3) (5.6.1) x p x p 1 2π e ipx (5.6.4) (3.4.1 ) x ˆp y 1 i ϕ x ˆp ϕ 1 i ϕ x ϕ(x) δ(x y) (5.6.5) x x ϕ (5.6.6) x ϕ(x) x ϕ (5.6.7) (5.6.6) ϕ ˆp x ϕ ϕ(x) ˆp ϕ 1 i x ϕ(x). 5.6.2 1 1 p â (p) 0

72 ˆP ˆψ (x, t) 1 i x ˆψ(x, t) dx (5.6.8) p â (p)â(p) dp (5.6.9) ˆP p p p (5.6.10) [â(p), â (p )] δ(p p ) p p δ(p p ) (5.6.11) 1 p p x 1 x p 1 2π e ipx x dp p p x 1 dp p e ipx 2π x 1 2π 1 2π e ipx p dp e ipx â (p) 0 dp ˆψ (x, 0) (5.5.6) ˆψ (x, 0) 0 (5.6.12) ˆψ (x, t) 1 2π t 0 â (p) e i(px E pt) dp

73 1 ˆψ (x, 0) t 0 ÔH(t 0) ÔS ˆψ (x, 0) ˆψ (x) ˆψ ˆψ(x) ˆX ˆP x ˆψ (x) 0 (5.6.13) ˆψ (x) 1 i x ˆψ(x) dx ˆX ˆψ (x) x ˆψ(x) dx (5.6.14) [ [ ˆX, ˆψ ] (x)] ˆψ (y) y ˆψ(y) dy, ˆψ (x) dy [ ˆψ (y) y ˆψ(y), ˆψ (x)] ( dy [ ˆψ (y), ˆψ (x)] y ˆψ(y) + ˆψ (y) y [ ˆψ(y), ˆψ (x)] dy ˆψ (y) y δ(y x) x ˆψ (x) (5.6.15) ˆX x ˆX ˆψ (x) 0 ( ˆψ (x) ˆX + [ ˆX, ˆψ (x)] 0 x ˆψ (x) 0 x x (5.6.16) ˆX 0 0 ˆψ(x) 0 0 1 x ˆX )

74 x x y 0 ˆψ(x) ˆψ (y) 0 0 [ ˆψ(x), ˆψ (y)] 0 δ(x y) 0 0 δ(x y) (5.6.17) x 1 ˆψ (x) 0 ϕ ϕ(x) (5.6.7) ϕ(x) x ϕ x 0 ˆψ(x) 1 ϕ ϕ(x) ϕ(x) 0 ˆψ(x) ϕ (5.6.18) ϕ(x) ϕ ϕ ϕ(x) ˆψ (x) 0 dx (5.6.19) 5.6 (5.6.15) 5.7 (5.6.15) ˆN [ ˆX, ˆψ(x)] x ˆψ(x) (5.6.20) ˆN [ ˆX, ˆP ] i ˆN (5.6.21) ˆψ (x) ˆψ(x) dx 1 ˆN 1 ˆX ˆP 1

75 5.8 [ ˆψ(x), ˆP ] (1/i) ˆψ(x)/ x ϕ ϕ(x) x ϕ x ˆP ϕ 1 i ϕ(x) x x x 0 ˆψ(x). 5.9 ϕ ϕ ϕ, t e iĥt ϕ ϕ(x, t) 0 ˆψ(x) ϕ, t ϕ(x, t) 0 ˆψ(x, t) ϕ ϕ(x, t) 5.6.3 ˆψ (x) 0 x 1 Ĥ Ĥ(x) ˆψ (x) h ( x, x) ˆψ(x). (5.6.22) x ˆψ (x) 0 x y Ĥ(x) y 0 ˆψ(y) ˆψ (x) h ( x, x) ˆψ(x) ˆψ (y ) 0. ˆψ(x) ˆψ (y ) 0 [ ˆψ(x), ˆψ (y )] 0 δ(x y ) 0 0 ˆψ(y) ˆψ (x) y Ĥ(x) y 0 δ(y x) h (, x) δ(x y ) 0 x δ(y x) h ( { 0 (x y y, x) δ(x y ) ) x 0 (otherwise) (5.6.23) y y

76 h (, x) x (5.6.19) Ĥ(x) ϕ Ĥ(x) ϕ dy y ϕ (y) Ĥ(x) dy ϕ(y ) y dy dy ϕ (y)ϕ(y ) y Ĥ(x) y dy dy ϕ (y)ϕ(y )δ(y x)h (, x) δ(x y ) x dy ϕ (x)ϕ(y )h (, x) δ(x y ) x ϕ (x)h ( x, x) dy ϕ(y )δ(x y ) ϕ (x)h ( x, x) ϕ(x). ϕ(x) x x 0 x x 0 ˆψ (x) x ˆψ(x) x x, y ˆψ (x) ˆψ (y) 0 x y 1 2 1 3 ˆψ (x) x ˆψ(x) x (Remark) g ˆψ (x) ˆψ(x) ˆφ(x). x φ ψ ψ φ ψ x φ ψ g g [ ] [ ]

6 6.1 p, p p, p p, p p, p â (p)â(p ) 0 [â (p), â (p )] 0 3 p, p, p â (p) â (p ) â (p ) 0 (5.5.9) 6.2 (5.5.9) ˆψ(x, t) ˆψ (x, t) (5.4.1) (5.4.1) 77

78 ˆψ(x, t) ˆψ (x, t) { ˆψ(x, t), ˆψ (y, t)} δ(x y), { ˆψ(x, t), ˆψ(y, t)} { ˆψ (x, t), ˆψ (y, t)} 0. (6.2.1) {Â, ˆB} Â ˆB + ˆBÂ (6.2.2) (5.4.1) â(p), â (p ) (5.5.9) (6.2.1) â(p), â (p ) ˆψ(x, t) ˆψ (x, t) (5.5.4), (5.5.6) ˆψ(x, t) 1 2π ˆψ (x, t) 1 2π â(p) e i(px E pt) dp, (6.2.3) â (p) e i(px Ept) dp, (6.2.4) â(p) 1 2π â (p) 1 2π ˆψ(x, t) e i(px E pt) dx, (6.2.5) ˆψ (x, t) e i(px Ept) dx, (6.2.6) { 1 {â(p), â (p )} ˆψ(x, t) e i(px Ept) 1 } dx, ˆψ (y, t) e i(p y E t) p dy 2π 2π 1 dx dy e i(px Ept) e i(p y E t) p { 2π ˆψ(x, t), ˆψ (y, t)} 1 dx dy e i(px Ept) e i(p y E t) p δ(x y) 2π 1 dx e i(p p )x e i(e p E p )t 2π e i(ep E p )t δ(p p ) δ(p p ) {â(p), â (p )} δ(p p ), {â(p), â(p )} {â (p), â (a )} 0. (6.2.7)

79 â(p), â (p ) (6.2.7) p, p â (p) â (p ) 0 p, p p, p p, p 1. ˆψ(x, t) 2. â (p) 0 p E p p 2 /2m 3. 1 6.3 Â ˆB {Â, ˆB} (6.2.2) {Â, ˆB} Â ˆB + ˆBÂ. (6.2.2) Â ˆB Â ˆB ˆBÂ Â ˆB Â ˆB ˆBÂ {Â, ˆB} 0 (6.3.1) Â ˆB ˆBÂ + {Â, ˆB} (6.3.2)

80 (i) (ii) ( ) {Â, ˆB} { ˆB, Â} {Â, c ˆB + c ˆB } c{â, ˆB} + c {Â, ˆB } {câ + c Â, B} c{â, ˆB} + c {Â, ˆB} c, c (c ) (iii) ( ) [Â, ˆBĈ] {Â, ˆB} Ĉ ˆB{Â, Ĉ} [Â ˆB, C] Â{ ˆB, Ĉ} {Â, Ĉ} ˆB (iv) [Â, { ˆB, Ĉ}] + [ ˆB, {Ĉ, Â}] + [Ĉ, {Â, ˆB}] 0. (i) (ii) (6.2.2) (iii) 1 (iii) [Â, ˆBĈ] [Â, ˆB] Ĉ + ˆB[Â, Ĉ] (6.3.3) (iii) [Â, ˆBĈ] Â ˆBĈ ˆBĈÂ Â ˆBĈ + ˆBÂĈ ˆBÂĈ ˆBĈÂ (Â ˆB + ˆBÂ)Ĉ ˆB(ÂĈ + ĈÂ) {Â, ˆB} Ĉ ˆB{Â, Ĉ}. [Â, ˆBĈ] {Â, ˆB} Ĉ ˆB{Â, Ĉ}. (6.3.4) (6.3.3) 1 (iv) (iii)

81 1. Â, ˆB, Ĉ 2. ˆBĈ Â ˆBĈ 2 3. (6.3.4) Â ˆBĈ 4. (6.3.4) A, B, C ABC ABC BAC A B Â, ˆB, Ĉ 1 5. (iii) [Â ˆB, Ĉ] Â{ ˆB, Ĉ} {Â, Ĉ} ˆB. (6.3.5) 6.1 (6.3.5) 6.2 (iii) [Â, { ˆB, Ĉ}] [Â, ˆBĈ] + [Â, Ĉ ˆB] (6.3.6) (iv) 6.4 1 6.4.1 ψ(x, t) ψ (x, t) ψ(x, t) ˆψ(x, t) ψ (x, t) ˆψ (x, t).

82 { ˆψ(x, t), ˆψ (y, t)} δ(x y), { ˆψ(x, t), ˆψ(y, t)} { ˆψ (x, t), ˆψ (y, t)} 0. (6.4.1) Ĥ Ĥ ˆψ (x, t)h ( x, x) ˆψ(x, t) dx (6.4.2) ˆP ˆN ˆP ˆψ (x, t) 1 i x ˆψ(x, t) dx, (6.4.3) ˆN 6.4.2 Ô(t) ˆψ (x, t) ˆψ(x, t) dx. (6.4.4) i [Ô(t), Ĥ] tô(t) ˆψ(x, t) (6.4.1) i ˆψ(x, t) t [ ˆψ(x, t), Ĥ] [ ˆψ(x, t), ˆψ (y, t)h ( y, y) ˆψ(y, t) dy ] dy [ ˆψ(x, t), ˆψ (y, t)h (, y) ] ˆψ(y, y t) ( dy { ˆψ(x, t), ˆψ (y, t)}h (, y) ˆψ(y, y t) ˆψ (y, t) { ( ˆψ(x, t), h, y) } ) ˆψ(y, y t) ( dy { ˆψ(x, t), ˆψ (y, t)}h (, y) ˆψ(y, y t) ˆψ (y, t)h ( ), y) { ˆψ(x, t), ˆψ(y, t)} y dy δ(x y)h ( y, y) ˆψ(y, t) h ( x, x) ˆψ(x, t). (6.4.5)

83 i ˆψ (x, t) t [ ˆψ (x, t), Ĥ] [ ˆψ (x, t), ˆψ (y, t)h ( y, y) ˆψ(y, t) dy ] dy [ ˆψ (x, t), ˆψ (y, t)h (, y) ] ˆψ(y, y t) ( dy { ˆψ (x, t), ˆψ (y, t)}h (, y) ˆψ(y, y t) ˆψ (y, t) { ˆψ (x, t), h (, y) } ) ˆψ(y, y t) ( dy { ˆψ (x, t), ˆψ (y, t)}h (, y) ˆψ(y, y t) ˆψ (y, t)h ( ), y) { ˆψ (x, t), ˆψ(y, t)} y dy ˆψ (y, t)h (, y) δ(x y) y ( dy h (, y) ˆψ ) (y, t) y δ(x y) h ( x, x) ˆψ (x, t) h ( x, x) ˆψ (x, t). (6.4.6) i ˆψ(x, t) t i ˆψ (x, t) t h ( x, x) ˆψ(x, t), (6.4.7) h ( x, x) ˆψ (x, t) (6.4.8) 6.3 57 (5.3.8) (6.4.5), (6.4.6)

84 6.4.3 ˆψ(x, t), ˆψ (x, t) ˆψ(x, t) 1 2π ˆψ (x, t) 1 2π â(p) e i(px E pt) dp, (6.4.9) â (p) e i(px E pt) dp, (6.4.10) E p p 2 /(2m) (6.2.5), (6.2.6) â(p), â (p ) {â(p), â (p )} δ(p p ), {â(p), â(p )} {â (p), â (a )} 0. (6.4.11) Ĥ, ˆP, ˆN â(p), â (p) Ĥ (6.4.9), (6.4.10) (6.4.2) Ĥ 1 2π ˆψ (x, t)h ( x, x) ˆψ(x, t) dx dx 1 2π dx dp dp dp â (p) e i(px Ept) h (, x) 1 dp â(p ) e i(p x E p t) x 2π dp E p â (p)â(p) dp. dp â (p) â(p ) e i(px Ept) E p e i(p x E p t) dp E p â (p) â(p ) e i(e p Ep)t 1 2π dp E p â (p) â(p ) e i(e p E p)t δ(p p) dx e i(p p)x â(p), â (p) ˆP ˆN Ĥ ˆP E p â (p)â(p) dp, (6.4.12) p â (p)â(p) dp, (6.4.13)

85 ˆN â (p)â(p) dp. (6.4.14) â(p), â (p) Ĥ â(p), â (p) [ ] [Ĥ, â(p)] E p â (p )â(p ) dp, â(p) dp E p [â (p )â(p ), â(p)] dp E p dp E p E p â(p) ( ) â (p ){â(p ), â(p)} {â (p ), â(p)} â(p ) ( ) â (p ) 0 δ(p p)â(p ) [Ĥ, â(p)] E p â(p) (6.4.15) [Ĥ, â (p)] E p â (p) (6.4.16) [ ˆP, â(p)] p â(p), [ ˆP, â (p)] p â (p) (6.4.17) [ ˆN, â(p)] â(p), [ ˆN, â (p)] â (p) (6.4.18) 6.5 0 â(p) 0 0 for all p, 0 0 1. (6.5.1)

86 Ĥ 0 0, (6.5.2) ˆP 0 0, (6.5.3) ˆN 0 0 (6.5.4) 0 1 p â (p) 0 (6.5.5) Ĥ p Ĥâ (p) 0 ( ) â Ĥ + [Ĥ, â (p)] 0 ( ) â Ĥ + E p a (p) 0 E p â (p) 0 E p p (6.5.6) ˆP p p p, (6.5.7) ˆN p p (6.5.8) p p E p p 2 /(2m) m 1 p p 0 â(p) â (p ) 0 0 â(p) â (p ) + â (p ) â(p) 0 0 {â(p), â (p )} 0 0 δ(p p ) 0 δ(p p ) (6.5.9)

87 2 p, p â (p) â (p ) 0 (6.5.10) ( ) p Ĥ p, p 2 2m + p 2 p, p, 2m (6.5.11) ˆP p, p (p + p ) p, p, (6.5.12) ˆN p, p 2 p, p (6.5.13) p p 1 p, p p, p p 1, p 2,..., p n â (p 1 )â (p 2 ) â (p n ) 0 Ĥ p 1, p 2,..., p n (E p1 + E p2 + E pn ) p 1, p 2,..., p n (6.5.14) ˆP p 1, p 2,..., p n (p 1 + p 2 + p n ) p 1, p 2,..., p n (6.5.15) ˆN p 1, p 2,..., p n n p 1, p 2,..., p n (6.5.16) n p j j 1, 2,..., n) 6.6 1 1 1 ϕ ϕ(x) ϕ(x) 0 ˆψ(x) ϕ

88 1 x x ˆψ (x) 0 ˆψ (x) x 2 φ 2 φ(x 1, x 2 ) φ(x 1, x 2 ) 0 ˆψ(x 1 ) ˆψ(x 2 ) φ ˆψ φ(x 1, x 2 ) φ(x 2, x 1 ) 6.7 0 1 2, 1/2, 3/2,

7 7.1 7.1.1 m step 1) E p E p2 2m. (7.1.1) step 2) E p E i t, (7.1.2) p i x. (7.1.3) step 3) step 1) i ( 1 ψ(x, t) t 2m i ) 2 ψ(x, t). (7.1.4) x 7.1.2 2 ψ(x, t) 2 d 3 x (7.1.5) 89

90 t x d 3 x dxdydz ) 1 ψ(x, t) 2 d 3 x 1. (7.1.6) ψ(x, t) t 1 d dt ψ(x, t) 2 d 3 x 0 (7.1.7) ρ(x, t) ρ(x, t) ψ(x, t) 2. (7.1.8) d ρ(x, t) d 3 x 0 dt j(x, t) j(x, t) ( ) ψ ψ(x, t) (x, t) ψ (x, t) ψ(x, t). (7.1.9) 2mi x x ρ(x, t) j(x, t) ρ t + divj 0. (7.1.10) d ρ(x, t) d 3 ρ(x, t) d 3 x dt V V t divj(x, t) d 3 x V j(x, t) ds (7.1.11) S 3 V ds S ( ) V S ψ(x, t) j(x, t) S d dt S ρ(x, t) d 3 x 0 (7.1.12)

91 7.1.3 1/2 up down 2 ψ(x, t) ( ψ (x, t) ψ (x, t) ) (7.1.13) ψ (x, t) 2 x ψ (x, t) 2 x x ψ (x, t) 2 + ψ (x, t) 2 (7.1.14) ) ψ (x, t) (ψ (x, t) ψ (x, t) (7.1.15) ψ (x, t) ψ (x, t) ψ(x, t) (7.1.16) ( ) ( ) ψ (x, t) ψ(x, t) ψ (x, t) ψ (x, t) ψ (x, t) ψ (x, t) ψ (x, t)ψ (x, t) + ψ (x, t)ψ (x, t) ψ (x, t) 2 + ψ (x, t) 2 (7.1.17) (7.1.14) ψ (x, t) ψ(x, t) d 3 x (7.1.18) d ψ (x, t) ψ(x, t) d 3 x 0 (7.1.19) dt i ψ(x, t) Ĥψ(x, t) (7.1.20) t

92 Ĥ 2 Ĥ 2 + V (x) (7.1.21) 2m x2 2 V (x) 2 2 ψ(x, t) V V V V V ( ) ( ) ( ) V 11 V 12 V V V11 V21. V 21 V 22 V12 V22 V ρ(x, t) ψ (x, t)ψ(x, t) (7.1.22) j(x, t) j(x, t) 2mi ( ψ ψ(x, t) (x, t) x ) ψ (x, t) ψ(x, t) x (7.1.23) ρ t + div j 0 (7.1.24) ρ t ( ψ ψ ) t ψ ψ ψ + ψ t t 1 ) ( 2 2 ψ i 2m x + V ψ ψ + ψ 1 ) ( 2 2 ψ 2 i 2m x + V ψ 2 ( ) 2 ψ 2mi x ψ 2 ψ 2 ψ 1 ( (V ψ) ψ ψ V ψ ) x 2 i ( ) 2 ψ 2mi x ψ 2 ψ 2 ψ x 2 ( ) ψ 2mi x ψ ψ ψ x x div j (7.1.25)

93 4 5 V (V ψ) ψ V ψ V ŝ (ŝ x, ŝ y, ŝ z ) 1/2 2 2 ŝ x 2 σ 1, ŝ y 2 σ 2, ŝ z 2 σ 3. (7.1.26) σ 1, σ 2, σ 3 ( ) ( ) 0 1 0 i σ 1, σ 1, σ 1 1 0 i 0 ( ) 1 0 0 1 (7.1.27) 3 ŝ x, ŝ y, ŝ z [ŝ x, ŝ y ] i ŝ z, [ŝ y, ŝ z ] i ŝ x, [ŝ z, ŝ x ] i ŝ y, (7.1.28) [σ 1, σ 2 ] 2iσ 3, [σ 2, σ 3 ] 2iσ 1, [σ 3, σ 1 ] 2iσ 2, (7.1.29) σ 1 σ 2 σ 2 σ 1 iσ 3, σ 2 σ 3 σ 3 σ 2 iσ 1, σ 3 σ 1 σ 1 σ 3 iσ 2. (7.1.30) (σ 1 ) 2 (σ 2 ) 2 (σ 3 ) 2 1. (7.1.31) 1 2 2 1 σ i 2 {σ i, σ j } 2δ ij (7.1.32) {, } 7.1 (7.1.27) (7.1.30), (7.1.31) 1 I 1 1 δ ij i j δ ij 1 1

94 7.2 Klein-Gordon m E p (c ) E E 2 p 2 c 2 + m 2 c 4 (7.2.1) E ± p 2 c 2 + m 2 c 4 (7.2.2) E p 2 c 2 + m 2 c 4 (7.2.3) 7.2.1 (7.2.3) step 1) E p 2 c 2 + m 2 c 4. step 2) E p E i t, p i x. step 3) step 1) i t ψ(x, t) 2 c 2 2 + m 2 c 4 ψ(x, t). (7.2.4) 2 ψ(x, t) 2 Einstein 2 2

95 t x 7.2.2 - (7.2.1) step 1) E 2 p 2 c 2 + m 2 c 4. step 2) E p E i t, p i x. step 3) step 1) ( i t) 2 ( ) ] 2 i ψ(x, t) [c 2 + m 2 c 4 ψ(x, t). [ 1 c 2 2 t 2 2 + ( mc ) ] 2 ψ(x, t) 0. (7.2.5) Klein-Gordon 3 1 c 2 2 t 2 2 (7.2.6) Klein-Gordon [ + ( mc ) ] 2 ψ(x, t) 0 (7.2.7) 2 4 Klein-Gordon 3 Klein-Gordon

96 t x ψ(x, t) 2 ρ j i ( ψ ψ ) 2mc 2 t ψ t ψ ( ) ψ ψ ( ψ ) ψ 2mi Klein-Gordon ρ j (7.2.8) (7.2.9) ρ t + j 0 (7.2.10) ( ψ 2 ρ(x, t) d 3 x (7.2.11) ρ(x, t) ρ ρ ( ψ 2 ) ψ(x, t) e iωt u(x) ρ ω mc 2 u(x) 2 ω ρ ρ Klein-Gordon 2 ρ t 1 Klein-Gordon ψ e i(et p x)/ Klein-Gordon E2 2 c + p2 2 + m2 c 2 0 2 2

97 E 2 p 2 c 2 + m 2 c 4 E E ± p 2 c 2 + m 2 c 4 4 or E p 2 c 2 + m 2 c 4 p E Klein-Gordon Klein-Gordon 2 Klein-Gordon 2 7.3 Dirac 7.3.1 - Klein-Gordon 2 ρ 1 ρ ψ ψ 0 Dirac 1 (7.2.4) 1 1 4

98 E E 2 p 2 c 2 + m 2 c 4 (7.2.1) E α p c + βmc 2 (7.3.1) p 1 5 α (α 1, α 2, α 3 ) β 1 α, β (7.3.1) 2 (7.2.1) c 1 p (p x, p y, p z ) (p 1, p 2, p 3 ) (p i ) E 2 (α p + βm) 2 (7.2.1) 6 1 (7.3.2) (α p) 2 + (α p)βm + βm(α p) + β 2 m 2 (α p) 2 p 2 (7.3.2) (α p)β + β(α p) 0 (7.3.3) β 2 1 (7.3.4) ( ) (α p) 2 (α 1 p 1 + α 2 p 2 + α 3 p 3 + mβ) 2 α 1 2 p 1 2 + α 2 2 p 2 2 + α 3 2 p 3 2 + α 1 p 1 α 2 p 2 + α 2 p 2 α 1 p 1 + α 2 p 2 α 3 p 3 + α 3 p 3 α 2 p 2 + α 3 p 3 α 1 p 1 + α 1 p 1 α 3 p 3 (α 1 ) 2 p 1 2 + (α 2 ) 2 p 2 2 + (α 3 ) 2 p 3 2 + (α 1 α 2 + α 2 α 1 )p 1 p 2 + (α 2 α 3 + α 3 α 2 )p 2 p 3 + (α 3 α 1 + α 1 α 3 )p 3 p 1 5 α β c ) 6 1 3 2 0

99 p ( ) p 2 p 2 1 + p 2 2 2 + p 3 (α 1 ) 2 (α 2 ) 2 (α 3 ) 2 1 α 1 α 2 + α 2 α 1 0 α 2 α 3 + α 3 α 2 0 α 3 α 1 + α 1 α 3 0 (7.3.5) 7 2 (7.3.3) (α 1 p 1 + α 2 p 2 + α 3 p 3 )β + β(α 1 p 1 + α 2 p 2 + α 3 p 3 ) 0. (α 1 β + βα 1 )p 1 + (α 2 β + βα 2 )p 2 + (α 3 β + βα 3 )p 3 0 p α 1 β + βα 1 0 α 2 β + βα 2 0. (7.3.6) α 3 β + βα 3 0 3 (7.3.4) (i j 1, 2, 3 ) α i α j + α j α i 2δ ij α i β + βα i 0. (7.3.7) β 2 1 4 β, α 1, α 2, α 3 2 H(p, x) α p + βm α i, β 7 1 1

100 (7.1.32) 3 2 2 4 4 4 4 4 4 2 2 4 ( ) 0 σ α, β σ 0 ( ) 1 0 0 1 (7.3.8) α ( ) ( ) ( ) 0 σ 1 0 σ 2 0 σ 3 α 1, α 2, α 3 σ 1 0 σ 2 0 σ 3 0 (7.3.9) α i β (7.3.7) 7.2 A, B, C, D 4 2 2 4 4 X ( ) A B X C D X X ( X A C ) ( ) B A C D B D (7.3.10) 7.3 4 4 X X 2 2 A, B, C, D ( ) ( ) A B X X A B C D C D ( XX A C ) ( ) ( ) B A B AA + BC AB + BD D C D CA + DC CB + DD (7.3.11) 7.4 (7.3.11)

101 7.5 (7.3.8) (7.3.9) 4 α i, β (7.3.10) 7.6 (7.3.8) (7.3.9) 4 α i, β (7.3.7) (7.3.11) (7.1.32) 7.7 (7.3.7) {, } {α i, α j } 2δ ij {α i, β} 0 (7.3.12) {β, β} 2 7.8 α i (7.3.7) 1 or (7.3.12) 1 ) (α p) 2 p 2 ( 3 (α p) 2 i1 3 α i p i i1 3 i1 3 i1 3 i1 3 i1 3 i1 α i p i ) 2 3 α j p j j1 3 α i α j p i p j j1 3 α j α i p j p i j1 3 α j α i p i p j j1 3 j1 1 2 (α iα j + α j α i ) p i p j 3 δ ij p i p j j1 3 p i p i i1 3 (p i ) 2 i1 p 2

102 (7.3.8) α i, β E p x H(p, x) p 1 : H(p, x) α p + βm c H(p, x) α p c + βmc 2 (7.3.13) step 1) E H(p, x) α p c + βmc 2. step 2) E p E i t, p i x i. step 3) step 1) i t ψ(x, t) ( i c α + βmc 2) ψ(x, t) (7.3.14) (7.3.14) α β 4 4 ψ(x, t) 4 ψ ψ 1 ψ 2 ψ 3 ψ 4. (7.3.15) 7.3.2 Dirac 1 1 ρ(x, t) ψ (x, t)ψ(x, t) (7.3.16)

103 ρ/ t Dirac ψ i t ψ i c α ψ + βmc2 ψ (7.3.14) i ψ t i c ψ α + mc 2 ψ β (7.3.17) α β ρ t t (ψ ψ) ψ t 1 i ψ ψ + ψ t ( i c ψ α + mc 2 ψ β ) ψ + 1 ( i ψ i c α ψ + βmc 2 ψ ) c( ψ αψ + ψ α ψ) + 1 i (mc2 ψ βψ ψ βmc 2 ψ) (ψ αcψ). (7.3.18) j(x t) j(x, t) ψ (x, t)αc ψ(x, t) (7.3.19) ρ t + j 0 (7.3.20) d ρ(x, t) d 3 x 0 dt ρ ( ) ρ ψ ψ ψ1 ψ2 ψ3 ψ4 ψ 1 ψ 2 ψ 3 ψ 4 ψ 1ψ 1 + ψ 2ψ 2 + ψ 3ψ 3 + ψ 4ψ 4 ψ 1 2 + ψ 2 2 + ψ 3 2 + ψ 4 2 0.

104 7.3.3 Dirac 1, c 1 Dirac i ψ iα ψ + βmψ (7.3.21) t ψ(x, t) u t x 4 ψ(x, t) e i(et p x) u (7.3.22) u (7.3.22) (7.3.21) u u 1 u 2 u 3 u 4 E e i(et p x) u α p e i(et p x) u + βm e i(et p x) u. H(p)u Eu, (7.3.23) (7.3.22) ψ Dirac H(p) H(p) α p + βm (7.3.24) 4 4 H(p) (7.3.23) p 0 H(p) H(0) m 0 0 0 0 m 0 0 H(0) βm (7.3.25) 0 0 m 0 0 0 0 m E ±m 1 0 0 0 u (1) 0 0, 1 u(2) 0, 0 u(3) 1, 0 u(4) 0 0 0 0 1 (7.3.26) u (1) u (2) E +m u (3) u (4) E m p 0 Dirac

105 p 0 p 0 4 4 H(p) 2 2 ( ) m σ p H(p) (7.3.27) σ p m ( ) m σ p u Eu (7.3.28) σ p m E ±Ep ± p 2 + m 2 (7.3.29) 2 χ (s) (s 1, 2) ( ) ( ) χ (1) 1, χ (2) 0 0 1 (7.3.30) u (s) (p) N χ (s) σ p Ep + m χ(s), (7.3.31) σ p u (s+2) (p) N Ep + m χ(s) (7.3.32) u (1) (p) u (2) (p) E +Ep u (3) (p) u (4) (p) E Ep N N Ep + m (7.3.33) χ (s) u (s) (p) u (s ) (p) 2Ep δ ss, (s, s 1, 2, 3, 4) (7.3.34) 7.9 (7.3.31), (7.3.32) (7.3.28) u (1) (p) u (2) (p) E +Ep u (3) (p) u (4) (p) E Ep 7.10 N Ep + m (7.3.31), (7.3.32) (7.3.34)

106 7.3.4 1 Klein-Gordon Diraqc equation No! Yes No! No (?) (E ± m 2 c 4 + p 2 c 2 ) E mc 2 ( ) 0 mc 2 ( ) 7.1: Dirac Klein-Gordon Dirac sea)

107 E mc 2 0 γ mc 2 γ γ 7.2: 1 (hole) e +e p E Ep p +Ep Dirac sea Dirac positron) Dirac 7.3.5 Klein-Gordon Dirac Klein-Gordon

108 E mc 2 0 mc 2 7.3: Dirac sea: E mc 2 0 γ mc 2 7.4: (hole) Dirac Dirac sea Klein-Gordon Dirac equation No! Yes No! Dirac sea 1 N.A. OK OK

109 1 1 Dirac Klein-Gordon

8 0 0 4 0 1 Klein-Gordon φ(x) φ(x, t) φ Klein-Gordon ( + m 2 )φ 0 (8.0.1) c 1 2 / t 2 2 m 8.1 8.1.1 Klein-Gordon Euler-Lagrange φ L φ φ φ φ m 2 φ φ (8.1.1) φ φ/ t L S L d 3 x dt L d 4 x φ 110

111 φ δφ 0 δs δl d 4 x { δ φ } φ δφ φ m 2 δφ φ d 4 x δφ ( φ ) + 2 φ m 2 φ d 4 x δφ (x)( + m 2 )φ(x) d 4 x δφ (x) φ ( + m 2 )φ(x) 0 δφ φ (x) Klein-Gordon ( + m 2 )φ (x) 0 φ, φ Klein-Gordon S L d 4 x φ ( + m 2 )φ d 4 x (8.1.2) ( + m 2 )φ φ d 4 x (8.1.3) δφ φ Klein-Gordon δφ φ Klein-Gordon φ φ 1 (φ 1 + iφ 2 ) (8.1.4) 2 φ 1 (φ 1 iφ 2 ) (8.1.5) 2 φ 1 φ 2 φ L 1 { } φ2 2 1 ( φ 1 ) 2 m 2 2 φ 1 + 1 { } φ2 2 2 ( φ 2 ) 2 m 2 2 φ 2

112 1 ϕ L 1 { ϕ 2 ( ϕ) 2 m 2 ϕ 2} 2 Euler-Lagrange K-G 8.1.2 (8.1.1) φ π φ φ π φ (8.1.1) π φ L φ φ (8.1.6) π φ L φ φ. (8.1.7) H π φ φ + πφ φ L φ { } φ + φ φ φ φ φ φ m 2 φ φ φ φ + φ φ + m 2 φ φ. H π φ π φ + φ φ + m 2 φ φ (8.1.8) (πφ H H d 3 x π φ + φ φ + m 2 φ φ ) d 3 x. π φ φ ( φ) (π φ ) H H (H 0) 8.1.3 Poisson φ, φ π φ π φ {φ(x, t), π φ (y, t)} Pb δ 3 (x y), (8.1.9) {φ (x, t), π φ (y, t)} Pb δ 3 (x y), (8.1.10)

113 {φ(x, t), φ(y, t)} Pb {φ(x, t), φ (y, t)} Pb {φ (x, t), φ (y, t)} Pb 0, (8.1.11) {φ(x, t), π φ (y, t)} Pb {φ (x, t), π φ (y, t)} Pb 0, (8.1.12) {π φ (x, t), π φ (y, t)} Pb {π φ (x, t), π φ (y, t)} Pb {π φ (x, t), π φ (y, t)} Pb 0. (8.1.13) 8.2 8.2.1 φ, φ π φ π φ φ(x, t) ˆφ(x, t), (8.2.1) φ (x, t) ˆφ (x, t), (8.2.2) π φ (x, t) ˆπ φ (x, t), (8.2.3) π φ (x, t) ˆπ φ (x, t). (8.2.4) π φ (π φ ) ˆπ φ (x, t) ˆπ φ (x, t) (8.2.5) i ( ) [ ˆφ(x, t), ˆπ φ (y, t)] iδ 3 (x y), (8.2.6) [ ˆφ (x, t), ˆπ φ (y, t)] iδ 3 (x y), (8.2.7) [ ˆφ(x, t), ˆφ(y, t)] [ ˆφ(x, t), ˆφ (y, t)] [ ˆφ (x, t), ˆφ (y, t)] 0, (8.2.8) [ ˆφ(x, t), ˆπ φ (y, t)] [ ˆφ (x, t), ˆπ φ (y, t)] 0, (8.2.9) [ˆπ φ (x, t), ˆπ φ (y, t)] [ˆπ φ (x, t), ˆπ φ (y, t)] [ˆπ φ (x, t), ˆπ φ (y, t)] 0. (8.2.10) ˆφ ˆπ φ ˆφ ˆπ φ

114 8.2.2 (8.2.5) H Ĥ ˆπ φˆπ φ + ˆφ ˆφ + m 2 ˆφ ˆφ, (8.2.11) Ĥ Ĥ d 3 x. (8.2.12) ˆπ φ ˆπ φ H 8.2.3 d i[ĥ, Ô(t)] (8.2.13) dtô(t) ˆφ(x, t) i[ĥ, ˆφ(x, t)] i[ Ĥ(y, t)d 3 y, ˆφ(x, t)] i d 3 y[ĥ(y, t), ˆφ(x, t)] i d 3 y[ˆπ φ (y, t)ˆπ φ (y, t) + ˆφ (y, t) ˆφ(y, t) + m 2 ˆφ (y, t) ˆφ(y, t), ˆφ(x, t)] i d 3 y[ˆπ φ (y, t), ˆφ(x, t)]ˆπ φ (y, t) i d 3 y( i)δ 3 (y x)ˆπ φ (y, t) ˆπ φ (x, t). (8.2.14) ˆφ (x, t) i[ĥ, ˆφ(x, t)] ˆπ φ (x, t), (8.2.15) ˆπ φ (x, t) i[ĥ, ˆπ φ(x, t)] ( 2 m 2 ) ˆφ (x, t), (8.2.16) ˆπ φ (x, t) i[ĥ, ˆπ φ (x, t)] ( 2 m 2 ) ˆφ(x, t). (8.2.17)

115 ˆφ(x, t) ( 2 m 2 ) ˆφ(x, t), (8.2.18) ˆφ (x, t) ( 2 m 2 ) ˆφ (x, t), (8.2.19) ˆφ, ˆφ Klein-Gordon ( + m 2 ) ˆφ(x, t), (8.2.20) ( + m 2 ) ˆφ (x, t), (8.2.21) 8.1 (8.2.15) (8.2.21) 8.2.4 Klein-Gordon ψ(x, t) e i(p x Et) Klein-Gordon [ ] 2 t 2 2 + m 2 ψ(x, t) 0 [ E 2 + p 2 + m 2] e i(p x Et) 0 E p E 2 p 2 + m 2 Ep + p 2 + m 2 (8.2.22) E E ±Ep (8.2.23) ψ(x, t) exp [ i(p x Ep t) ], p p E p Ep

116 ψ(x, t) exp [ i(p x Ep t) ], 4 p (p 0, p) (Ep, p) 4 x (x 0, x) (t, x) p x 4 px p 0 x 0 p x Ep t p x (8.2.24) Klein-Gordon ψ(x) e ipx (8.2.25) x, y, z L p p 2πn L 2π L (n x, n y, n z ), (n x, n y, n z ) (8.2.26) p ϕp(x) 1 V 2E p e ipx (8.2.27) ϕ p(x) 1 V 2E p e+ipx (8.2.28) V V L 3 ( ) ϕ p(x, t) ϕp (x, t) d3 x 1 2Ep δ pp (8.2.29) ϕp(x, t) ϕp (x, t) d3 x 1 2Ep δ p, p exp( 2iE p t) (8.2.30) ϕ p(x, t) ϕ p (x, t) d3 x 1 2Ep δ p, p exp(+2ie p t) (8.2.31) V L 3 (x, y, z L/2 +L/2) 8.2 (8.2.29) (8.2.31)

117 (8.2.29) (8.2.31) exp(±2iep t) Klein-Gordon i ϕ 1(x) 0 ϕ 2 (x) d 3 x (8.2.32) 0 / t 0 0 0 ϕ 1(x) 0 ϕ 2 (x) ϕ 1(x) 0 ϕ 2 (x) 0 ϕ 1(x) ϕ 2 (x) ϕ 1(x) ϕ 2 (x) ϕ 1(x) ϕ 2 (x). (8.2.33) i i i ϕ p(x, t) 0 ϕp (x, t) d3 x δpp, (8.2.34) ϕp(x, t) 0 ϕp (x, t) d3 x 0, (8.2.35) ϕ p(x, t) 0 ϕ p (x, t) d3 x 0, (8.2.36) ϕp(x, t) 2Ep 8.3 ϕ 1 (x) ϕ 2 (x) Klein-Gordon ( + m 2 )ϕ j (x) 0 (j 1, 2) (8.2.32) t ( x 0 ) 8.4 (8.2.34) (8.2.36) 8.2.5 Klein-Gordon ( L ) ψ(x) ψ(x, t) ϕp(x) ϕ p(x) ψ(x) p { } a(p) ϕp(x) + b (p) ϕ p(x). a(p) b (p)

118 b b ψ (8.2.34) (8.2.36) i ϕ p(x) 0 ψ(x) d 3 x i d 3 x ϕ p(x) } 0 {a(p )ϕp (x) + b (p )ϕ p (x) {a(p ) i p p p { a(p ) δpp + b (p ) 0 } ϕ p(x) 0 ϕp (x) d3 x + b (p ) i ϕ p(x) } 0 ϕ p (x) d3 x a(p) (8.2.37) ˆφ(x) ˆφ (x) Klein-Gordon ϕp(x) ϕ p(x) ˆφ(x) { â(p) ϕp(x) + ˆb } (p) ϕ p(x), (8.2.38) p ˆφ (x) { â (p) ϕ p(x) + ˆb(p) } ϕp(x). (8.2.39) p ˆφ(x) ˆφ (x) â(p) ˆb(p) â (p) ˆb (p) (8.2.34) (8.2.36) â(p) i ϕ p(x) 0 ˆφ(x) d 3 x, (8.2.40) ˆb(p) i ϕ p(x) ˆφ 0 (x) d 3 x, (8.2.41) â (p) i ˆφ (x) 0 ϕp(x) d 3 x, (8.2.42) ˆb (p) i ˆφ(x) 0 ϕp(x) d 3 x. (8.2.43) 0 ˆφ(x) ˆπφ ˆφ (x) ˆπ φ { â(p) i ϕ p(x)ˆπ φ (x) ϕ p(x) ˆφ(x) } d 3 x, (8.2.44) { ˆb(p) i ϕ p(x)ˆπ φ (x) ϕ p(x) ˆφ } (x) d 3 x, (8.2.45) { } â (p) i ˆφ (x) ϕp(x) ˆπ φ (x)ϕp(x) d 3 x, (8.2.46) { } ˆb (p) i ˆφ(x) ϕ p(x) ˆπ φ (x)ϕp(x) d 3 x (8.2.47)

119 â(p) ˆb(p) â (p) ˆb (p) [â(p), â (p )] [ { i ϕ p(x)ˆπ φ (x) ϕ p(x) ˆφ(x) } { } ] d 3 x, i ˆφ (y) ϕp (y) ˆπ φ(y)ϕp (y) d 3 y, { d 3 xd 3 x ϕ p(x) ϕp (y)[ˆπ φ (x), ˆφ (y)] + ϕ p(x)ϕp (y)[ ˆφ(x), } ˆπ φ (y)] { } d 3 xd 3 x +ϕ p(x) ϕp (y)( i)δ3 (x y) + ϕ p(x)ϕp (y)iδ3 (x y) } i d 3 x {ϕ p(x) ϕp (x) ϕ p(x)ϕp (x) i d 3 xϕ p(x) 0 ϕp (x) δpp. (8.2.48) x y x (x, t) y (y, t) ( 8.3 ) 3 4 2 [â(p), ˆb(p )] [ { i ϕ p(x)ˆπ φ (x) ϕ p(x) ˆφ(x) } } i d 3 x {ϕ p(x) ϕ p (x) ϕ p(x)ϕ p (x) i d 3 xϕ p(x) 0 ϕ p (x) { d 3 x, i ϕ p (y)ˆπ φ(y) ϕ p (y) ˆφ } ] (y) d 3 y, { d 3 xd 3 x ϕ p(x) ϕ p (y)[ˆπ φ (x), ˆφ (y)] ϕ p(x)ϕ p (y)[ ˆφ(x), } ˆπ φ (y)] { } d 3 xd 3 x ϕ p(x) ϕ p (y)( i)δ3 (x y) ϕ p(x)ϕ p (y)iδ3 (x y) 0. (8.2.49) [â(p), â (p )] [ˆb(p), ˆb (p )] δpp, (8.2.50) [â(p), ˆb (p )] [â (p), ˆb(p )] 0, (8.2.51) [â(p), â(p )] [ˆb(p), ˆb(p )] [â(p), ˆb(p )] 0. (8.2.52) a(p) b(p) a (p) b (p)

120 8.5 (8.2.50) (8.2.52) 8.2.6 (8.2.12) â(p) ˆb(p) â (p) ˆb (p) Ĥ Ĥ {ˆπ φˆπ φ + ˆφ ˆφ } + m 2 ˆφ ˆφ d 3 x { ˆφ (x) ˆφ(x) ˆφ (x)( 2 m 2 ) ˆφ(x) } d 3 x { ˆφ (x) ˆφ(x) ˆφ (x) ˆφ(x) } d 3 x ˆφ (x) 0 ˆφ(x) d 3 x (8.2.53) 1 2 ˆπ φ ˆφ, ˆπ φ ˆφ 3 ˆφ Klein-Gordon ˆφ (8.2.38) ϕp iep ϕp { â(p) ϕp(x) + ˆb } (p) ϕ p(x) ˆφ(x) p p { ( iep) â(p)ϕp(x) ˆb } (p)ϕ p(x). (8.2.54) (8.2.53) Ĥ ˆφ (x) 0 ( iep) p { Ep i p p p { â(p)ϕp(x) ˆb (p)ϕ p(x) ˆφ (x) 0 ϕp(x) d 3 x â(p) i { Ep â (p)â(p) + ˆb(p)ˆb } (p) { Ep â (p)â(p) + ˆb } (p)ˆb(p) + 1 } d 3 x } ˆφ (x) 0 ϕ p(x) d 3 x ˆb (p) (8.2.55) ˆb(p) ˆb (p ) Ĥ Ĥ { Ep â (p)â(p) + ˆb } (p)ˆb(p) + E 0, (8.2.56) p E 0 p Ep. (8.2.57)