1

016 017 6 16

1

1 5 1.1............................................... 5 1................................................... 5 1.3................................................ 5 1.4............................................... 6 1.5.......................................... 6 1.6.................................................... 7 1.7.................................................. 9 10.1............................................... 10................................ 11.3.............................................. 11.4.......................... 1.5......................... 13.6............................... 13.7 E........................................... 14.8......................................... 14.9.................................. 14.10............................................. 15.11.............................................. 16.1.................................. 16.13.................................. 17 3 18 3.1.............................................. 18 3......................... 0 3.3............................................. 0 3.4..................................... 1 3.5.............................................. 3.6............................................. 3.7................................... 3 3.8.................................................. 4 3.9....................................... 4 4 7 4.1.............................................. 7 4................................................... 9 4.3............................................... 30 4.4............................................. 31 4.5........................................... 31 4.6............................................. 33

4.7......................................... 33 4.8..................................... 34 5 1 36 5.1........................................... 36 5................................ 36 5.3.............................. 40 5.4...................... 4 5.5.......................... 4 6 44 6.1.................................. 44 6..................... 45 6.3..................... 47 6.4................................ 49 7 5 7.1.......................... 5 7...................................... 53 7.3........................................... 54 7.4.................................... 55 7.5 7.8......................................... 56 7.6 7.9............................................... 59 7.7........................................... 60 8 63 8.1.......................................... 63 8.................................................... 64 8.3.......................................... 65 8.4.................................. 66 8.5.............................................. 67 8.6.................................... 68 8.7............................ 70 8.8................................ 7 9 74 9.1............................................... 74 9............................................... 74 9.3....................................... 75 9.4............................................. 76 9.5............................................... 77 10 78 10.1........................................... 78 10...................................... 79 10.3........................................... 81 10.4......................................... 8 10.5............................................... 84 10.6........................................... 85 3

11 1 86 11.1.............................................. 86 11................................................... 86 11.3.............................................. 87 11.4..................................... 89 11.5.................................. 91 11.6........................... 9 1 94 1.1.............................................. 94 1............................................ 96 1.3............................................... 97 1.4............................................. 97 1.5.................................................. 98 1.6.............................................. 99 13 101 13.1............................................. 101 13.......................................... 10 13.3......................................... 104 13.4.......................................... 105 14 108 14.1........................................ 108 14. -...................................... 108 14.3 -.................................... 109 14.4........................................ 109 14.5............................................... 11 15 115 15.1............................................. 115 15.......................................... 115 15.3................................................ 117 15.4............................................... 118 4

1 1.1 10 30 kg 1. 1.3 5

1.4 - E hν 1.1 p h λ 1. h 6.66069579 10 34 Js 1.1 1. E p ν λ - 1.1 1. - 1.1 1. - 1.5 300 m/s 1 g - 1.1 1. p mv λ ν λ h p h mv 6.6 10 34 Js 1.0 10 3 kg 300 m/s. 10 35 m 1.3 ν E 1 h mv h 1 1.0 10 3 kg 300 m/s 6.6 10 34 6.8 10 34 Hz 1.4 Js 6

100 V m 9.1 10 31 kg q e 1.6 10 19 C E ev 1.6 10 19 C 100 V 1.6 10 17 J 1.5 p E E 1 m p λ p me 9.1 10 31 kg 1.6 10 17 J 5.4 10 4 kg m/s 1.6 λ h p 6.6 10 34 Js 5.4 10 4 kg m/s 1. 10 10 m 1.7 X ν ν E h 1.6 10 17 J 6.6 10 34 Js.4 1016 Hz 1.8 1.6 X 1c t Br, t 0 1.9 1c t Er, t 0 1.10 c Br, t B 0 sink r ωt 1.11 Er, t E 0 sink r ωt 1.1 B 0 E 0 B 0 k E 0 k c k ω k π λ ω πν λ ν ν c/λ - 7

X X X - W W ν v 1 mv hν W 1.13 ν ν ϕ v θ hν hν + 1 mv 1.14 hν/c hν /c cos ϕ + mv cos θ 1.15 0 hν /c sin ϕ mv sin θ 1.16 θ m v [hν/c hν /c cos ϕ] + hν /c sin ϕ hν/c + hν /c hν/chν /c cos ϕ 1.17 mhν mhν m v 1.18 mhν mhν hν/c + hν /c hν/chν /c cos ϕ 1.19 mcc/hν mcc/hν ν/ν + ν /ν cos ϕ cos ϕ 1.0 λ c/ν, λ c/ν λ λ λ mcλ /h mcλ/h 1 cos ϕ 1.1 λ λ λ h/mc 1 cos ϕ 1. 8

1.7-1.1 1. 0 0 1/ 1 1/ 1 0 0 p mv E 1/mv 0 - p k k π 1.3 λ k p 1.4 1. p ħk 1.5 ħ h/π 1.0545717647 10 34 Js 1.1 ħ ω πν E ħω 1.6 1.5 1.6 9

.1 Ψ x Ψx, t A sin kx ωt.1 Ψx, t A cos kx ωt. A k ω kx ωt kx ωt k x ω k t.3 v ω k.4 x v k ω ω k ω vk.5 1 Ψ v t Ψ x.6 v.1..6.6 t x 10

. Ψ - p Ψx, t A sin ħ x E ħ t Ψ Ψx, t Ψ x t x v E p mv.7 E 1 mv p m.8.7 ħ p Ψx, t Ψx, t.9 m x m p mψx, t EΨx, t ħ p t Ψx, t EA cos ħ x E ħ t.10 Ψx, t { p Ψx, t A exp i ħ x E } ħ t.11 i A Ψx, t ħ p Ψx, t Ψx, t.1 m x m iħ Ψx, t EΨx, t.13 t.8 Ψ EΨx, t p Ψx, t.14 m iħ ħ Ψx, t t m Ψx, t.15 x.3 p ħ i x E iħ t.16.17 11

ˆp Ê T T p m ħ m x.18 x t ˆx ˆt Ψ Ψx, t x t ˆx Ψ x.4 U F F U F du dx T U E T + U U 0 U Ax 1 10 10 m e 4πε 0 r 1.6 10 19 C 4 3.1 8.9 10 1 F/m 1 10 10 m.3 10 8 N.19 mg 9.1 10 31 kg 9.8 m/s 8.9 10 30 N.0 - k.15 Ux, t E p + Ux, t.1 m.15 iħ { t Ψx, t ħ } + Ux, t Ψx, t. m x H H T + U. Ĥ ħ + Ux, t.3 m x iħ Ψx, t ĤΨx, t.4 t 1

.5. x p p p x r x, y, z.5 p p p x, p y, p z.6 ħ i x, ħ i y, ħ i ħ z i.7 iħ { } t Ψr, t ħ m x + y + z + Ur, t Ψr, t.8 iħ { } t Ψr, t ħ m + Ur, t Ψr, t.9.6 Ur iħ { } t Ψr, t ħ m + Ur Ψr, t.30 Ψr, t ϕrft.31.31.30.31 iħ dft 1 } { ħ ft dt ϕr m ϕr + Urϕr.3.3 t r t r E ft ϕr Eϕr iħ dft dt Eft.33 } { ħ m + Ur ϕr.34 Ĥr ħ m + Ur.35 Ĥrϕr Eϕr.36.34.36.33 C ft C exp iet ħ.37 13

.7 E Ur 0 Ψr, t.11 { p r Ψr, t A exp i ħ Et }.38 ħ Ψr, t A exp i p r ħ exp i Et ħ.39 ϕr ft ϕr exp i p r ħ ft exp i Et ħ.40.41 p E ft.37 E.8 Ψr, t i Ψr, t Ψ r, t Ψr, t Ψr, t t r dv dxdydz Ψr, t dv Ψr, t A Ψr, t dv 1.4 Ψr, t Ψr, t dv dv.4 lim Ψr, t 0.43 r Ψr, t Ψr, t ρr, t Ψr, t.9 A B A Ψ A B Ψ B 14

Ψ A B Ψ Ψ A + Ψ B.44 Ψ Ψ A + Ψ B Ψ A + Ψ B + Ψ AΨ B + Ψ A Ψ B.45 A Ψ A B Ψ B...... z 0 z L A x d/, z 0 B x d/, z 0 S x, l d A-S x + d/ + l B-S x d/ + l Ψ A C exp [ik ] x + d/ + l Ψ B C exp [ik ] x d/ + l.46.47 Ψ AΨ B + Ψ A Ψ B C cos [k x + d/ + l k ] x d/ + l l >> d x + d/ + l l 1 + x/l + d/l l [1 + 1 ] x/l + d/l [ l 1 + 1 ] x /l + dx/l.48.49 x d/ + l l 1 + x/l d/l l [1 + 1 ] x/l d/l [ l 1 + 1 ] x /l dx/l.50 [k x + d/ + l k ] x d/ + l kdx/l.51 Ψ AΨ B + Ψ A Ψ B C cos kdx/l.5 πl/kd.10 Ψr, t Ψr, t t r 15

.11 Ux, t x m Ψx, t iħ { ħ t m iħ Ψ x, t t { ħ } + Ux, t Ψx, t.53 x } m x + U x, t Ψ x, t.54 Ux, t V x, t W x, t.53 Ψ x, t.54 Ψx, t ρx, t ρx, t t Ux, t V x, t + iw x, t.55 + jx, t x W x, tρx, t.56 ħ ρx, t Ψx, t Ψ x, tψx, t.57 jx, t jx, t 1 ħ m i { Ψ Ψx, t x, t x } Ψ x, t Ψx, t x.58 Ux, t W x, t 0 ρx, t t + jx, t x 0.59 t x ρx, t x ρx, t Ux, t W x, t 0 t x ρx, t x W x, t > 0 W x, t < 0 jr, t ρr, t t + divjr, t W r, tρr, t.60 ħ jr, t 1 ħ m i {Ψ x, t Ψx, t Ψ x, t Ψx, t}.61.1 x Ux, t 0 Ψx, t A exp ipx exp iet ħ ħ.6 16

ρx, t A.63 jx, t 1 ħ m i { ip ħ A p mv p j ip } A v A.64 ħ ip r Ψr, t A exp exp iet ħ ħ.65 ρr, t A.66 jr, t p m A v A.67.13 3. x Ψx, t A exp iet exp Γt expikx.68 ħ ħ ρx, t Ψx, t A exp Γt ħ.69 jx, t ħ im ik A exp Γt ħk ρx, t.70 ħ m t ρx, t Γ ρx, t ħ.71 jx, t 0 x.7 ρx, t + t x jx, t Γ ρx, t.73 ħ W x, t Γ/ Ux, t iγ/ ħ/γ ħ/γ 17

3 3.1 Q ˆQ ˆp ħ i x 3.1 ˆp ħ i 3. Ê iħ t 3.3 ˆx x 3.4 ˆr r 3.5 ˆT ħ m x : 1 3.6 ˆT ħ m : 3 3.7 Û U 3.8 Ĥ ˆT + Û 3.9 ÊΨ ĤΨ 3.10 EΨ ĤΨ 3.11 x x 1 Ψx, t x Ψx, t t x x t x t x Ψx, t dx Ψx, t dx 3.1 1 x t < Q > t x Ψx, t dx 3.13 < Q > t Ψ x, t ˆQΨx, tdx Ψx, t dx 3.14 18

p Ψx, t Φp, t #!! Φp, t 1 πħ Ψp, t 1 πħ Ψx, t exp iħ px dx 3.15 i Φp, t exp ħ px dp 3.16 Ψx, t Φp, t 1 Ψx, t dx 3.17 { 1 } i Ψ x, t Φp, t exp πħ ħ px dp dx 3.18 { 1 } i Ψ x, t exp πħ ħ px dx Φ p, tdp 3.19 Φp, t dp 3.0 Φp, t t [p, p + dp] Φp, t dp 3.1 p < p > t p Φp, t dp 3. Φx, t < p > t p Φp, t dp Φp, t dp 3.3 Ψ x, tˆpψx, tdx 3.4 Ψ x, t ħ { d 1 } i Φp, t exp i dx πħ ħ px dp dx 3.5 { 1 } i Ψ x, t Φp, tp exp πħ ħ px dp dx 3.6 { 1 } i Ψ x, t exp πħ ħ px dx Φ p, tpdp 3.7 p Φp, t dp 3.8 < p > t Ψ x, tˆpψx, tdx Ψx, t dx 3.9 Φx, t3 Φr, t Ψp, t3 Ψp, t 19

ˆrΦr, t rφr, t r ˆpΨp, t pψp, t p ˆp ħ i ħ i x, ħ i y, ħ i z 3.30 ˆr iħ p iħ, iħ, iħ p x p y p z 3.31 Qr, p, t ˆQ ˆr, ˆp, t L r p ˆL ˆr ˆp ˆL r ħ i 3.3 3.14 < Q > t Ψ r, t ˆQΨr, tdx dy dz Ψr, t dx dy dz 3.33 3. ϕ c cϕ ϕ ϕ cϕ cϕ {ϕ n }n 1,, 3, Φ n c nϕ n ˆQ ˆQ cϕ c ˆQϕ ˆQ c 1 ϕ 1 + c ϕ c 1 ˆQϕ1 + c ˆQϕ c, c 1, c 3.34 3.35 3.34 3.35 3.3 ˆQ Φ ξ ˆQ Ψξdξ { ˆQΦξ } Ψξdξ 3.36 ˆQ ˆQ ˆQ ˆQ ˆQ 0

ˆP + ˆQ ˆP + ˆQ 3.37 c ˆQ c ˆQ 3.38 ˆQ ˆQ 3.39 ˆP ˆQ ˆQ ˆP 3.40 ˆQ < Q > { Ψ r ˆQΨrd 3 r Ψ r r} ˆQΨrd 3 3.41 { ˆQΨr } Ψrd 3 r 3.4 Ψ r ˆQΨrd 3 r Ψ r ˆQ Ψrd 3 r 3.43 ˆQ ˆQ ˆr r ˆp iħ ˆL ˆr ˆp V r Ĥ ˆp /m + V r ˆp iħ / x [ Ψ x, tˆp Ψx, tdx iħ ] Ψx, t Ψx, t 3.44 x { iħ x [Ψ x, tψx, t] dx Ψ x, t } Ψx, tdx 3.45 x iħ {[Ψ x, tψx, t] Ψ x, t } Ψx, tdx 3.46 x iħ Ψ x, t Ψx, tdx 3.47 x Ψ x, tˆpψx, tdx 3.48 ˆp ˆp 0 3.4 ˆQ Ψ ˆQΨ Φ 3.49 Φ Φ Ψ ˆQΨ λψ 3.50 Ψ λ ˆQ 1

HΨ EΨ Ψ E Ψ E 3.5 Ψξ ˆQ λ ˆQΨξ λψξ ξ Ψξ ˆQ Ψ < Q > Ψ ˆQΨdξ < Q > Ψ dξ λ 3.51 ˆQ < Q > λ ˆQ ˆQ < Q > ˆQ λ ϕ λ ξ ˆQϕ λ ξ λϕ λ ξ 3.5 ϕ λ ξ ξ ϕ λξ ξ { ϕ λξ ˆQϕ λ ξdξ ˆQϕλ ξ} ϕλ ξdξ λ λ ϕ λ ξ dξ 3.53 ϕ λξ ˆQϕ λ ξdξ ϕ λξ ˆQ ϕ λ ξdξ λ λ ϕ λ ξ dξ 3.54 ˆQ ˆQ ˆQ λ λ λ 3.6 ˆQ λ λ ϕ λ ξ ϕ λ ξ ˆQϕ λ ξ λϕ λ ξ 3.55 ˆQϕ λ ξ λ ϕ λ ξ 3.56 ϕ λ ξ ξ ϕ λ ξ { } ˆQϕλ ξ ϕλ ξdξ ϕ λξ ˆQϕ λ ξdξ λ λ ϕ λϕ λ dξ 3.57 0 λ λ ϕ λϕ λ dξ 0 3.58 λ λ λ λ ϕ λϕ λ dξ 0 3.59

ˆQϕ λ,k ξ λϕ λ,k ξ k 1,,, m 3.60 m {ϕ λ,k ξ} ϕ λξϕ λ ξdξ 1 3.61 ϕ λξϕ λ ξdξ δ λ,λ 3.6 δ λ,λ { 1 λ λ δ λ,λ 0 λ λ 3.63 µ ϕ µξϕ µ ξdξ δµ µ 3.64 3.7 ˆQ {λ} {ϕ λ ξ} ˆQϕ λ ξ λϕ λ ξ 3.65 ϕ λξϕ λ ξdξ δ λ,λ 3.66 ϕ λ ξ δ λ,λϕ λ ξ 3.67 λ [ ] ϕ λ ξ ϕ λ ξ dξ ϕ λ ξ 3.68 λ [ ] ϕ λ ξ ϕ λ ξ ϕ λ ξ dξ 3.69 λ ϕ λ ξ ϕ λ ξ δξ ξ 3.70 λ {ϕ λ ξ} {ϕ λ ξ} Ψξ Ψξ C λ ϕ λ ξ 3.71 λ 3

{ϕ λ ξ} {C λ } C λ ϕ λξψξdξ 3.7 3.8 Ψξ Ψξ C λ ϕ λ ξ 3.73 λ C λ 1 3.74 λ ˆQ < Q > Ψ ξ ˆQΨξdξ 3.75 CλC λ ϕ λξ ˆQϕ λ ξdξ 3.76 λ,λ λ,λ C λc λ λδ λ,λ 3.77 λ C λ λ 3.78 Ψ Q λ C λ C λ ϕ λ ξ 3.9 ˆp iħd/dx ˆpϕx pϕx 3.79 iħ d dx Aeikx pϕx 3.80 ϕx Ae ikx 3.81 p ħk 3.8 4

Ae ikx A Ae ikx dx A dx 3.83 ψx ψx + L ψx L L 1 n p n πħ L n 3.84 L/ x < L/ 1 ψx Ae ikx L/ L/ ψx dx 1 3.85 A 1 L e iδ 3.86 δ δ 0 p ħk p p n ψ n x 1 L e i pn ħ x 1 L e i πn L x 3.87 ψ n x n m L/ L/ L/ L/ n ψ nxψ m xdx ψ nxψ m xdx δ n,m 3.88 ψ nxψ n x δx x 3.89 1 L L/ L/ L/ L/ 1 πn i e L x 1 e i πm L x dx 3.90 L L e i πm n L x dx 3.91 1 L [e i πm n L/ L x 3.9 L πm n L/ 1 [ e iπm n e iπm n] 3.93 πm n 1 [ 1 m n 1 m n] πm n 3.94 0 3.95 n m ψnxψ n x 1 πn i e L x 1 e i πn L x 3.96 L L n n 1 e i πn L x x L n 1 π L πδ L x x 3.97 3.98 1 L π L π δx x 3.99 δx x 3.100 5

δ L/ x < L/ fx fx c n L/ n L/ c n ψ n x 3.101 ψ nxfxdx 3.10 ψ n x ħ k me/ħ d ψx Eψx 3.103 m dx ψx Ae ikx + Be ikx 3.104 Ae ikx Be ikx ˆp iħd/dx ˆpAe ikx iħ d dx Aeikx ħkae ikx pae ikx 3.105 Ae ikx p ħk Be ikx p ħk E ħ k /m L/ x < L/ fx 6

4 4.1 Ψx A exp [ x x 0 ] α d dx Ψx d x x 0 dx α Ψx 4.1 [ 1α + x x ] 0 α Ψx 4. [ ħ m d dx + [ ħ m ] ħ mα 4 x x 0 Ψx ħ Ψx 4.3 mα V x ħ mα 4 x x 0 4.4 E ħ mα 4.5 d ] dx + V x Ψx EΨx 4.6 Ψx k ħ /mα 4 V x 1 kx x 0 Ψx E ħ /mα ħ/ k/m 1/ħω A 1 Ψx dx A exp [ x x 0 ] dx A α π 4.7 A 1/α 1/ π 1/4 < x > 1 απ 1/ 1 απ 1/ α Ψ xxψxdx 4.8 x exp [ x x 0 x + x 0 exp α x α ] dx 4.9 dx 4.10 x 0 4.11 x x 0 x x 0 7

< p > Ψ x ħ d Ψxdx 4.1 i dx 1 απ ħ 1/ α x x 0 exp [ x x 0 ] α dx 4.13 x dx 4.14 1 απ ħ 1/ α x exp α 0 4.15 0 x < x > < x > x < x > x x 0 4.16 1 απ 1/ 1 απ 1/ Ψ xx x 0 Ψxdx 4.17 x x 0 exp [ x x 0 ] dx 4.18 x exp x α α dx 4.19 1 π1/ α3 1 απ1/ α 4.0 p < p > < p > < p > 4.1 Ψ x ħ d Ψxdx 4. dx ħ 1 [ 1α απ 1/ + x x 0 ] α 4 exp [ x x 0 ] α dx 4.3 [ ħ 1 α + 1 ] 1 α 4 α 4.4 ħ α α α x p 1 ħ α α ħ α 4.5 4.6 x p p x x p ħ 4.7 λ p h/λ 8

p mv v p m v x v ħ m 4.8 ħ ħ 0 0 ħ 0 4.  ˆB  ˆB  ˆB Ψ Â ˆBΨ 4.9 f ˆB  ˆB  x ˆB d dx  ˆB d Ψ x dx Ψ x dψ 4.30 dx ˆBÂ Ψ d xψ Ψ + xdψ 4.31 dx dx  ˆB ˆB ] [Â, ˆB [Â, ˆB]  ˆB ˆB 4.3  x ˆB y  ˆB ˆB [ ] Â, ˆB 0 4.33  ˆB  ˆB  x ˆB d dx  ˆB ˆB [Â, ˆB] 1 0 4.34  ˆB  ˆB  ˆx x ˆB ˆp ħ i x [ˆx, ˆp] iħ 4.35  ˆB  ˆB  ˆB + ˆB i  ˆB ˆB { Ψ ξ  ˆB + ˆB Ψξdξ ˆB +  ˆB Ψξ} Ψξdξ 4.36 Ψ ξ ˆB +  ˆB Ψξdξ 4.37 Ψ ξ  ˆB + ˆB Ψξdξ 4.38 9

{ { Ψ ξ i  ˆB ˆBÂ} Ψξdξ i ˆB  ˆB Ψξ} Ψξdξ 4.39 Ψ ξ i ˆB  ˆB Ψξdξ 4.40 Ψ ξi  ˆB ˆB Ψξdξ 4.41 4.3 Q Q ˆQ ˆQ 4.4 Ψξ ˆQ ˆQ Ψ ξ ˆQ Ψξdξ 4.43 Ψ ξ ˆQΨξdξ 4.44 x x p p ˆx ˆp x p ħ 4.45 x p x p p x p Ψx C expipx/ħ Ψx C x x λ x λ p πħ/λ x πħ/ p x p πħ 4.45 4.45 ħ x p ħ 0 p p mv m v 4.45 x v ħ m 4.46 m x v m x v 10 3 1 g 10 3 kg 9.1 10 31 kg ˆx ˆp x y ˆt Ê E iħ t 30

4.4 ˆP ˆQ ˆP λ ϕ λ [ ϕ λ ˆP, ˆQ] 0 ˆP ϕ λ λϕ λ 4.47 [ ˆP, ˆQ] ϕ λ ˆP ˆQϕ λ ˆQ ˆP ϕ λ 4.48 ˆP ˆQϕ λ λ ˆQϕ λ 0 4.49 ˆP ˆQϕλ λ ˆQϕλ 4.50 ˆQϕ λ ˆP λ ˆP ˆQϕ λ ϕ λ µ ˆQϕ λ µϕ λ 4.51 ϕ λ λ ˆP µ ˆQ ϕ λ ϕ λ,µ ˆP ϕ λ,µ λϕ λ,µ 4.5 ˆQϕ λ,µ µϕ λ,µ 4.53 U U 0 ˆp Ĥ p E ϕ λ,µ ϕ λ,µ 4.5 4.45 Â ˆB i Ĉ ] iĉ [Â, ˆB 4.54 Â ˆB Â Â Â ˆB ˆB ˆB 4.55 4.56 31

[ ] Â, ˆB  } { }] [{Â, ˆB ˆB 4.57 } { { } {  ˆB ˆB } ˆB ˆB } {  4.58  ˆB   ˆB ˆB  + ˆB  ˆB ˆB  ˆB + ˆB  4.59  ˆB ˆB 4.60 [Â, ˆB] iĉ 4.61 Ô Â + is ˆB 4.6 Ô s Ô Ô Ô Ô Â is ˆB  is  A + s Ĉ [ is + is ˆB  ˆB   ˆB + s ˆB 4.63 4.64 ˆB, ]  + s ˆB 4.65 + s B 4.66  ˆB   ˆB  A ˆB ˆB B 4.67 4.68 4.69 Ô Ô Ψ ξô ÔΨξdξ 4.70 ÔΨξ ÔΨξdξ 4.71 ÔΨξ dξ 0 4.7 A + s Ĉ + s B 0 4.73 Ĉ 0  ˆB s A B 0  x ˆx ˆB y ŷ Ĉ 0 Ĉ x p 0 4.74 0  ˆB Ĉ s D D 0 D Ĉ 4 A B 0 4.75 3

Ĉ A B 4.76 Â ˆx ˆB ˆp Ĉ ħ x p ħ 4.77 4.6 Qt < Q > t Ψ ξ, t ˆQtΨξ, tdξ 4.78 d dt < Q > t Ψ ξ, t + t Ψ ξ, t ˆQtΨξ, tdξ + Ψ ξ, t ˆQt Ψξ, tdξ t Ψξ, t ˆQt dξ 4.79 t Ψξ, t iħ Ψξ, t ĤΨξ, t 4.80 t d dt < Q > t i ħ Ψ ξ, tĥ ˆQtΨξ, tdξ + Ψ ξ, t Qt ˆ Ψξ, tdξ t + Ψ i ξ, t ˆQt ĤΨξ, tdξ 4.81 ħ ˆQt + i ] [Ĥ, ˆQt 4.8 t ħ t t ˆQt t 0 4.83 ] [Ĥ, ˆQt 0 4.84 Ĥ ˆQ Q Q U U 0 ˆp Ĥ p Ĥ U E 4.7 33

dp dt F 4.85 p m dr dt 4.86 F V r, t 4.87 r t p t d r t dt d p t dt 1 m p t 4.88 V r, t t 4.89 4.8 r x iħ [ ] t Ψr, t ħ m + V r, t Ψr, t 4.90 Ψr, t x d x t Ψ xψd 3 r 4.91 dt t v Ψ v t xψ + Ψ x Ψ d 3 r 4.9 t ħ [ Ψ xψ Ψ x Ψ ] d 3 r 4.93 im v xψ Ψ/ x + x Ψ d x t ħ [ Ψ xψ Ψ xψ ] d 3 r + ħ Ψ Ψ dt im v im v x d3 r 4.94 ħ div [ Ψ xψ Ψ xψ] d 3 r im v + 1 Ψ iħ Ψd 3 r 4.95 m x v 34

Ψ 0 div [ Ψ xψ Ψ xψ] d 3 r [ Ψ xψ Ψ xψ] ds 0 4.96 v S ˆp x iħ / x d x t dt 1 Ψ ˆp x Ψdv 1 m m p x t 4.97 v y z d r t dt 1 m p t 4.98 x d p x t Ψ ˆp x Ψd 3 r 4.99 dt t v Ψ v t ˆp xψ + Ψ Ψ ˆp x d 3 r 4.100 t [ ħ Ψ Ψ m v x Ψ ] Ψ d 3 r x [ + V Ψ Ψ ] V Ψ Ψ d 3 r 4.101 x x v Ψ 0 [ Ψ Ψ v x Ψ ] Ψ d 3 r x [ div Ψ Ψ ] Ψ Ψ d 3 r 4.10 v x x [ Ψ Ψ ] Ψ Ψ ds 4.103 S x x 0 4.104 d p x t dt [ V Ψ Ψ ] V Ψ Ψ d 3 r 4.105 v x x Ψ V v x Ψd3 r 4.106 V 4.107 x t y z d p t dt V t 4.108 35

5 1 5.1 iħ [ ] t Ψr, t ħ m + Ur Ψr, t 5.1 ] [ ħ m + Ur ψr Eψr 5. Ψr, t e i E ħ t ψr 5.3 ρr, t Ψr, t t ρr ψr iħ [ t Ψx, t ħ ] m x + Ux Ψx, t 5.4 [ ħ ] m x + Ux ψx Eψx 5.5 Ψx, t e i E ħ t ψx 5.6 ρx, t Ψx, t t ρx ψx ψr 0 5. x m { ħ d } m dx + V x ψx Eψ 5.7 1 36

V x { 0 0 x L V x 5.8 x < 0, x > L ψx 0 x < 0, x > L 5.9 ψx V x 0 ħ d ψx Eψ 5.10 m dx ψx C expikx + D exp ikx 5.11 expikx cos kx + i sin kx ψx A sin kx + B cos kx 5.1 me k ħ A B ψx 0 5.13 ψ0 0 5.14 ψl 0 5.15 B 0 k k n π n n 1,, 3, 5.16 L k k n n k n ψ n ψ n x A sin k n x 5.17 n k n 5.17 k n π L n k n π L n n 0 k 0 0 ψ 0 0 k n E n ħ kn m ħ m π L n n 1,, 3, 5.18 n E n E n+1 E n m L E n m L k n E n 37

5.17 A L 0 L 0 Aψ n x dx ψ n x dx 1 5.19 L 0 L 0 A 1 A sin k n x dx 5.0 A 1 1 cos k nxdx 5.1 [ x 1 ] L sin k n x k n 0 5. A 1 L 5.3 A e iδ /L!! δ δ δ 0 A ψ n x L sin k nx 5.4 5.4 n 1 d /dx n n n 1 E 0 n E 0 E 1 π ħ /ml n 1 L L/ ħ/l E 1 1/m ħ/l ħ /m 1/L E 1 ħ /m π /L k n nπ/l sin kxdx 1 cos kx 5.5 k x cos kxdx 1 d sin kxdx 1 1 x sin kx + cos kx 5.6 dk k 4k cos kxdx 1 sin kx 5.7 k x cos kxdx 1 d 4 dk cos kxdx 5.8 x sin kx + x 1 cos kx sin kx 5.9 k k 4k3 38

< x > n L L L 0 L 0 L sin k n xx sin k n xdx 5.30 x sin k n xdx 5.31 1 x 1 cos k n x dx 5.3 L 0 1 [ 1 L x 1 x sin k n x 1 ] L k n 4kn cos k n x 5.33 0 1 1 L L 1 L sin nπ 1 k n 4kn cos nπ 1 0 + 1 0 sin 0 + 1 k n 4kn cos 0 5.34 L 5.35 < x > n L L L 0 L 0 L sin k n xx sin k n xdx 5.36 x sin k n xdx 5.37 1 x 1 cos k n x dx 5.38 L 0 1 [ 1 L 3 x3 x sin k n x x k n kn cos k n x + 1 ] L 4kn 3 sin k n x 5.39 0 1 1 L 3 L3 L3 n cos nπ 5.40 π 1 3 1 n π L 5.41 < p > n L L 0 L iħk n 1 L iħk n 1 L iħk n sin k n x iħ d dx sin k nxdx 5.4 L 0 L 0 [ 1 sin k n x cos k n xdx 5.43 sin k n xdx 5.44 ] L cos k n x k n 0 5.45 1 iħ 1 cos 0 cos nπ L 5.46 0 5.47 < p > n L L 0 sin k n x iħ d dx sin k nxdx 5.48 L L ħk n sin k n x sin k n xdx 5.49 0 1 L L ħk n 1 cos k n xdx 5.50 0 1 L ħk n [ x 1 k n sin k n x ] L 0 5.51 1 L ħk n L 5.5 ħk n 5.53 39

x n < x > n < x > n 5.54 1 3 1 L n π L 5.55 1 L 1 1 n π 5.56 L n π L 1.136 5.57 nπ 3 nπ p n < p > n < p > n 5.58 ħk n 5.59 ħ nπ L 5.60 L n π x n p n ħ nπ nπ 3 L ħ n π 5.61 3 > ħ ħ 3 5.6 n x p ħ/ E 1 π ħ /ml 1/m p 1 E 1 p 1 0 p x > 0 x < 0 p 1 ħπ/l x > 0 x < 0 p ħπ/l 5.3 V 0 L a { 0 a x a V x V 0 x < a, x > a 5.63 0 < E < V 0 ħ d ħ d ψx Eψx a x a 5.64 m dx m dx ψx + V 0ψx Eψx x < a, x > a 5.65 d dx ψx k ψx a x a 5.66 d dx ψx λ ψx x < a, x > a 5.67 k λ me ħ 5.68 mv0 E ħ 5.69 40

x x Be λx x < a ψx Ce ikx + De ikx a x a F e λx x > a 5.70 B, C, D, F x a, a ψ dψ/dx Be λa Ce ika + De ika 5.71 Ce ika + De ika F e λa 5.7 Bλe λa Cike ika + D ike ika 5.73 Cike ika + D ike ika F λe λa 5.74 B F Cλe ika + Dλe ika Cike ika + D ike ika 5.75 C λe ika + D λe ika Cike ika + D ike ika 5.76 Cλe ika e ika + Dλe ika e ika Cike ika + e ika Cλe ika + e ika + Dλe ika + e ika Cike ika e ika +D ike ika + e ika 5.77 +D ike ika e ika 5.78 C Dλe ika e ika C Dike ika + e ika 5.79 C + Dλe ika + e ika C + D ike ika e ika 5.80 ic Dλ sin ka C Dik cos ka 5.81 C + Dλ cos ka ic + D ik sin ka 5.8 C Dλ C Dk cot ka 5.83 C + Dλ C + Dk tan ka 5.84 C Dλ + k cot ka 0 5.85 C + Dλ k tan ka 0 5.86 C D λ k C D λ k C D λ k tan ka 0 B F Ce λa coska C C cos kae λx+a x < a ψx C cos kx a x a C cos kae λx a x > a 5.87 41

x 0 ψ x ψx C D λ + k cot ka 0 B F ice λa sin ka C ic sin kae λx+a x < a ψx ic sin kx a x a ic sin kae λx a x > a ψ x ψx 5.88 5.4 λ k tan ka 0 a ξ ka, η λa η ξ tan ξ 5.89 k λ me ħ 5.90 mv0 E ħ 5.91 a E ξ a me ħ 5.9 η a mv 0 E ħ 5.93 ξ + η a mv 0 ħ 5.94 5.89 5.94 ξ-η ξ E ħ ma ξ 5.95 E ξ 0 η 0 5.89 ξ η 0 ξ π/ ξ π/ ξ nπ η 0 nπ < ξ < n + 1π/ n 1 5.94 a mv 0 /ħ 5.94 η ξ cot ξ 5.96 5.96 ξ n 1π/ η 0 n 1π/ < ξ < nπ n 1 5.5 V 0 π n 1ħ a mv 0 < π nħ 5.97 4

n V 0 0 λ 43

6 6.1 V x V x 1 Kx 6.1 F x d V x Kx 6. dx x a d V x V a + d dx V ax a + 1 dx V ax a +... 6.3 d V a dx 0 6.4 d V a dx K > 0 6.5 V x V a + 1 Kx a +... 6.6 x a x x 6.1 x A cos ωt + t 0 K mω v Aω sin ωt + t 0 mv / ma ω / sin ωt + t 0 1/Kx ma ω / cos ωt + t 0 E ma ω / ˆp iħd/dx [ 1 m ˆp + 1 ] mω x ϕx Eϕx 6.7 E E n + 1 ħω 6.8 n ħω 0 ħω/ 44

6.7 H E E H 1 ˆp + 1 m mω x 6.9 ˆp 0 x 0 ˆp ˆp ˆp p x x x x E 1 m p + 1 mω x 6.10 1 E m p mω x ω p x 6.11 p x ħ/ E 1 ħω 6.1 E ħω/ x 0 1 m p 1 mω x 1 ħω 6.13 4 p x mħω ħ mω x 6.14 6.15 ħω n n 0 ħω/ 6. 6.7 â â â â mω 1 ħ x + i ˆp 6.16 mħω mω 1 ħ x i ˆp 6.17 mħω â â fx mω 1 mω 1 â âfx ħ x i mħω ˆp ħ x + i mħω ˆp fx 6.18 mω ħ x + i 1 ħ xˆp i 1 ħ ˆpx + 1 mħω ˆp fx 6.19 mω ħ x + i 1 1 xˆp ˆpx + ħ mħω ˆp fx 6.0 xˆp ˆpx fx [x, ˆp] fx iħfx mω â âfx ħ x 1 + 1 mħω ˆp fx 6.1 45

â â + 1 ħω 1 m ˆp + 1 mω x 6. 6.7 â â + 1 ħωϕx Eϕx 6.3 6.3 â â 6.1 [â, â ] â â ââ 6.4 mω ħ x + 1 + 1 mω mħω ˆp ħ x 1 + 1 mħω ˆp 6.5 1 6.6 [â, â] ââ ââ 0 6.7 [â, â ] â â â â 0 6.8 â â â â â â 6.3 â â â âϕx k â â ϕx E ħω 1 ϕx 6.9 â âϕx kϕx 6.30 k ϕ x x ϕ xâ âϕxdx kϕ xϕxdx k 6.31 ϕ xâ âϕxdx âϕx dx 0 6.3 k 0 â ϕx âϕx â â â ϕx â ââ ϕx â â â + 1 ϕx 6.33 â k + 1 ϕx k + 1â ϕx 6.34 â â âϕx ââ 1 âϕx â â â 1 ϕx 6.35 â k 1 ϕx k 1 âϕx 6.36 k + 1 k 1 â â m [ â â â ] m ϕx [ â k + m ] m ϕx 6.37 â â [â m ϕx] k m [â m ϕx] 6.38 â m ϕx k + m â â â m ϕx k m â â m k 0 k m k n k n m n â â [â n ϕx] 0 6.39 46

m > n â n ϕx 0 k n n E ħω 1 6.40 E n n + 1 ħω 6.41 E n ħω n â â â â 6.3 âϕ 0 x 0 ϕ 0 x E n ϕ n x A n â n ϕ0 x A n 1 An A n 1 An A n 1 An A n 1 An A n 1 ϕ nxϕ n xdx 6.4 ϕ n 1xââ ϕ n 1 xdx 6.43 ϕ n 1x â â + 1 ϕ n 1 xdx 6.44 ϕ n 1x n 1 + 1 ϕ n 1 xdx 6.45 n 6.46 â â A n 1 n A n 1 6.47 A n 1 n! A 0 6.48 ϕ 0 x ξ x/α α ħ/mω â â â â mω ħ x + i 1 ˆp 1 1 mħω α x + mω ħ x i 1 ˆp 1 1 mħω α x α ħ d ħ dx 1 d α ħ d ħ dx 1 dξ + ξ d dξ ξ 6.49 6.50 âϕ 0 x 0 ϕ 0 x mω 1 ħ x + i mħω ˆp ϕ 0 x 0 6.51 1 d dξ + ξ ϕ 0 ξ 0 6.5 47

d dξ ϕ 0ξ ξϕ 0 ξ 6.53 dϕ 0 ξ ϕ 0 ξ C A 0 A 0 1 A 0 α ξdξ 6.54 log [ϕ 0 ξ] ξ + C 6.55 ϕ 0 ξ A 0 e 1 ξ 6.56 ϕ 0 x A 0 e 1 α x 6.57 ϕ 0 x dx A 0 e 1 α x dx 6.58 e ξ dξ 6.59 A 0 α π 6.60 A 0 1 1 πα 6.61 ϕ 0 ξ 1 1 e 1 ξ 6.6 πα ϕ 0 x n ϕ n x A n â n ϕ0 x ϕ n x 1 1 e 1 α πα x 6.63 1 1 â n e 1 ξ 6.64 πn!α â 1 d dξ ξ 6.65 1 e 1 ξ d dξ e 1 ξ 6.66 ϕ n x â n 1 n e 1 ξ dn dξ n e 1 ξ 1 1 e 1 [ 1 ξ n dn ξ e πn n!α 1 πn n!α 1 dξ n e ξ ] 6.67 6.68 e 1 ξ H n ξ 6.69 H n ξ ξ n e 1 ξ x 0 x H n ξ n n 1 n ħ /md /dx n 48

H n ξ H 0 ξ 1 6.70 H 1 ξ ξ 6.71 H ξ 4ξ 6.7 H 3 ξ 8ξ 3 1ξ 6.73 H 4 ξ 16ξ 4 48ξ + 1 6.74 E ma ω / E n n + 1/ħω A c n ξ Ξ c n A c n n + 1 ħ mω n + 1α 6.75 Ξ c n n + 1 6.76 n A c n Ξc n n 0 0 ω A c n x ξα ξ ħ/mω mω x / ω 6.4 { 0 x a V x V 0 x < a 6.77 ħ d m dx + V x Ψx EΨx 6.78 I x a Ψ I x Ae ikx + Be ikx 6.79 k 1 ħ me 6.80 II a < x < a Ψ II x Ce ik x + De ik x 6.81 k 1 ħ m E V0 6.8 III a x Ψ III x F e ikx + Ge ikx 6.83 49

x a x a Ae ika + Be ika Ce ik a + De ik a 6.84 Aike ika + B ike ika Cik e ik a + D ik e ik a 6.85 Ce ik a + De ik a F e ika + Ge ika 6.86 Cik e ik a + D ik e ik a F ike ika + G ike ika 6.87 e ika e ika ke ika ke ika e ik a e ik a k e ik a k e ik a A B C D e ik a e ik a k e ik a k e ik a e ika e ika ke ika ke ika C D F G 6.88 6.89 A 1 ke ika e ika e ik a e ik a C B k ke ika e ika k e ik a k e ik a D 1 k + k e ik k a k k e ik+k a C k k k e ik+k a k + k e ik k a D 6.90 6.91 C,D F,G a a k k C 1 k + k e ik k a k k e ik+k a F D k k k e ik+k a k + k e ik k a G 6.9 A,B F,G A k + k αe ik k a γ sin k a B 4kk γ sin k a βe ik k a M 11 M 1 F M 1 M G F G 6.93 6.94 k k α 1 k + k e 4ik a k k β 1 k + k e 4ik a 6.95 6.96 γ i k k k + k 6.97 G 0 A M 11 F B M 1 F T R T F A 1 M 11 1 1 + k k R B A M 1 M 11 kk sin k a k k kk sin k a 1 + k k kk sin k a 6.98 1 T 6.99 1 T + R E < V 0 E > V 0 50

E > V 0 T 1 R 0 R 0 0 T 1 n k a nπ E ħ m k + V 0 ħ m nπ + V0 6.100 a a T 1 T 1 V 0 < 0 n k a nπ E < V 0 k κ κ 1 ħ m V0 E ik 6.101 T R T R 1 1 + k +κ 6.10 kκ sinh κa k +κ kκ sinh κa 1 + k +κ 1 T 6.103 kκ sinh κa T 0 R 1 T 0 E κ sinh κa T 0 κa 1 T 4kκ k + κ e 4κa 6.104 a V 0 κa T T 0 T 51

7 F r F r V r [ ] ħ m + V r ψ Eψ 7.1 7.1 iħ [ ] t Ψr, t ħ m + Ur Ψr, t 7. ] [ ħ m + Ur ψr Eψr 7.3 Ψr, t e i E ħ t ψr 7.4 Ur { 0 0 x, y, z L Ur other cases V x { 0 0 x L V x x < 0, X > L 7.5 7.6 Ur V x + V y + V z 7.7 [ ħ m x ħ m y ħ ] m z + Ur ψr Eψr 7.8 ψr XxY yzz 1 [ ħ ] Xx + V xxx Xx m x + 1 [ ħ ] Y y + V yy y Y y m y + 1 [ ħ ] Zz + V zzz Zz m z 7.9 7.10 E 7.11 5

ħ m x Xx + V xxx E xxx 7.1 ħ m y Y y + V yy y E yy y 7.13 ħ m z Zz + V zzz E zzz 7.14 E E x + E y + E z ψr XxY yzz E k ħ nx + k ny + k nz m n x, n y, n z 7.15 ψr 1 L 3/ sin k n x x sin k ny y sin k nz z 7.16 k n πn L E ħ π n m L x + n y + n z 7.17 7.18 n x, n y, n z, 1, 1, 1,, 1, 1, 1, 7. V r V r x,y,z r,θ,φ x,y,z r,θ,φ x r sin θ cos φ 7.19 y r sin θ sin φ 7.0 z r cos θ 7.1 r x + y + z 7. θ tan 1 x + y 7.3 z φ tan 1 y x 7.4 x,y,z r 0 θ 0 π φ 0 π 7.1 [ ħ 1 m r r r + 1 ] mr ˆL + V r ψ Eψ 7.5 ˆL ˆL ħ [ 1 sin θ sin θ + 1 θ θ sin θ ] φ 7.6 53

r θ φ ψ RrQθ, φ 7.7 ˆL Qθ, φ qqθ, φ 7.8 [ ħ 1 d m r dr r + 1 ] mr q + V r Rr ERr 7.9 7.3 7.8 ˆL Qθ, φ qqθ, φ ˆL θ φ L r p ˆL ˆL x yˆp z z ˆp y iħ y z z y ˆL y z ˆp x xˆp z iħ z x x z ˆL z xˆp y yˆp x iħ x y y x ˆL x iħ ˆL y iħ ˆL z iħ φ ˆL ˆ L x + ˆLy + ˆLz 7.6 + cot θ cos φ φ sin φ θ cos φ + cot θ sin φ θ φ 7.30 7.31 7.3 7.33 7.34 7.35 L mr ω r ω T ω π/t L [ˆLx, ˆL y ] iħˆl z 7.36 [ˆLy, ˆL z ] iħˆl x 7.37 [ˆLz, ˆL x ] iħˆl z 7.38 [ˆL, ˆL ] z 0 7.39 ˆL ± ˆL + ˆL x + iˆl y 7.40 ˆL ˆL x iˆl y 7.41 54

[ˆLz, ˆL ] + [ˆLz, ˆL ] [ˆL+, ˆL ] [ˆL, ˆL ] + [ˆL, ˆL ] ħˆl + 7.4 ħˆl 7.43 ˆL z 7.44 0 7.45 0 7.46 ˆL ˆL ˆL+ + ˆL z + ˆL z 7.47 ˆL + ˆL + ˆL z ˆL z 7.48 ˆL + ˆL 7.4 ħ /m ˆp /m [ ˆp, L ˆ ] x [ˆp, yˆp z z ˆp y ] 7.49 [ˆp y, y ] ˆp z [ˆp z, z ] ˆp y 7.50 ˆp y [ˆp y, y] ˆp z + [ˆp y, y] ˆp y ˆp z ˆp z [ˆp z, z] ˆp y [ˆp z, z] ˆp z ˆp y 7.51 ˆp y iħˆp z + iħˆp y ˆp z ˆp z iħˆp y iħˆp z ˆp y 7.5 0 7.53 [ˆp, ˆL y ] [ˆp, ˆL z ] 0 7.54 0 7.55 [ V r, L ˆ ] x [V r, yˆp z z ˆp y ] 7.56 y [V r, ˆp z ] z [V r, ˆp y ] 7.57 yiħ r V r ziħ r V r z r y r 7.58 yiħ z V r ziħ y V r r r r r 7.59 0 7.60 [V r, ˆL y ] [V r, ˆL z ] 0 7.61 0 7.6 55

7.1 Ĥ ħ /m + V r [Ĥ, ˆL] 0 7.63 ˆL Ĥ Ψr, t ˆL ˆL t v t ˆL Ψ r, t ˆLΨr, td 3 r 7.64 d ˆL t dt [ Ψ ] r, t ˆLΨr, t + Ψ Ψr, t r, t ˆL d 3 r 7.65 v t t [ĤΨr, t ˆLΨr, t Ψ r, t ˆL ] ĤΨr, t d 3 r 7.66 i ħ i ħ i ħ i ħ v v v v [Ψ r, tĥ ˆLΨr, t Ψ r, t ˆLĤΨr, t ] d 3 r 7.67 Ψ r, t Ĥ ˆL ˆLĤ Ψr, td 3 r 7.68 ] Ψ r, t [Ĥ, ˆL Ψr, td 3 r 7.69 ˆL Ĥ d ˆL t 0 7.70 dt ˆL t 7.5 7.8 7.8 ˆL Qθ, φ qqθ, φ [ˆL, ˆL ] z 0 7.71 7.8 ˆL z q ħ ll + 1 s ħm Q Yl m l 0 l m l ˆL z Qθ, φ sqθ, φ 7.7 l m ˆL Yl m θ, φ ħ ll + 1Yl m θ, φ 7.73 ˆL z Yl m θ, φ ħmyl m θ, φ 7.74 l m Y l m θ, φ Yl m θ, φ l Y 0 0 θ, φ 1 4π 7.75 56

Y 0 1 θ, φ Y ±1 1 θ, φ 3 3 4π cos θ z 4π r 3 3 8π sin θ e±iφ 8π x ± iy r 7.76 7.77 5 5 Y 0 θ, φ 16π 3 cos θ 1 3 z 16π r 1 15 15 θ, φ 8π sin θ cos θ e±iφ 8π 15 Y ± θ, φ 3π sin θ e ±iφ Y ±1 7.78 zx ± iy r 7.79 15 x ± iy 3π r 7.80 7 7 Y3 0 θ, φ 16π cos θ5 cos θ 3 5 z3 16π r 3 3z 7.81 r 1 1 Y 3 ±1 θ, φ 64π sin θ5 cos θ 1e ±iφ 5 z x ± iy 64π r 1 7.8 r 105 105 Y 3 ± θ, φ 3π sin θ cos θ e ±iφ z x ± iy 3π r r 7.83 35 35 Y 3 ±3 θ, φ 64π sin3 θ e ±3iφ x ± iy 3 64π r 3 7.84 7.73 7.74 Y3 ˆL Y 3 ħ [ 1 sin θ ħ 105 3π θ [ 1 sin θ sin θ + 1 θ + 1 sin θ i sin θ cos θe iφ ħ 105 3π sin θ sin θ cos θ sin 4 θ e iφ θ ] ] 105 φ 3π sin θ cos θe iφ 7.85 [ 1 4 sin θ cos 3 θ 4 sin 3 θ cos θ 4 sin 3 θ cos θ sin θ 7.86 4 cos θ] e iφ 7.87 105 ħ [ 4 cos 3 θ 8 sin θ cos θ 4 cos θ ] e iφ 7.88 3π 105 ħ [ 4 cos θ 1 cos θ 8 sin θ cos θ ] e iφ 7.89 3π 105 ħ [ 1 sin θ cos θ ] e iφ 7.90 3π ħ 1Y3 ħ 33 + 1Y3 7.91 ˆL z Y3 iħ 105 φ 3π sin θ cos θe iφ 7.9 105 iħ 3π sin θ cos θie iφ 7.93 ħy3 7.94 Yl m θ, φ 1 m+ m l + 1 l m! l + m! P m l cos θe imφ 7.95 P m l ξ l m 57

l 3 m Y m 3 e miφ φ 0 π z φ 0 π Y m 3 e miφ + + m z m θ θ 0 π z + θ 0 π cos θ 1 1 sin θ 0 1 1 0 Y3 0 cos θ 0 θ π/ cos θ ± 3/5 θ 0 Y 3 ±1 sin θ 0 θ 0, π cos θ ± 1/5 θ 0 Y 3 ± sin θ 0 θ 0, π cos θ 0 θ π/ 0 Y 3 ±3 sin θ 0 θ 0, π 0 θ 0, π 3 m 0 x 3 m m l Yl m z m x l m z m x l m l z ˆL z ħm ˆL ħ ll + 1 m l l m l Yl m z m 0 Yl m x-y l 0 Yl m z z ˆL ħ ll + 1/mr ˆL Y m l Y m l Y lm Y m l m 0 [ 1 1 m Yl m + Y m ] l m > 0 ] 1 i [ 1 m Y m l Y m l m < 0 7.96 Y lm l Y 0,0 1 4π 7.97 Y 1,0 Y 1,1 Y 1, 1 3 z 4π r 3 x 4π r 3 y 4π r 7.98 7.99 7.100 Y,0 Y,1 Y, 1 Y, Y, 5 3 z 16π r 1 7.101 15 xz 4π r 7.10 15 yz 4π r 7.103 15 x y 16π r 7.104 15 xy 4π r 7.105 58

Y 3,0 Y 3,1 Y 3, 1 Y 3, Y 3, Y 3,3 Y 3, 3 7 5 z3 16π r 3 3z 7.106 r 1 5 z x 3π r 1 7.107 r 1 5 z y 3π r 1 7.108 r 105 z x y 16π r r 7.109 105 xyz 4π r 3 7.110 35 x 3 3xy 3π r 3 7.111 35 y 3 3x y 3π r 3 7.11 l 0 s l 1 p l d l 3 f Y 1,0 p z Y 1,1 p x Y 1, 1 p z Y,0 d z Y,1 d x z Y, 1 d y z Y, d x y Y, d xy 7.74 7.73 Y1 1 3 z 3 Y 1,1 4π r sin θ cos ϕ 7.113 4π Q Y m l [ 1 ˆL Y 1,1 ħ sin θ + 1 ] sin θ θ θ sin θ ϕ Y 1,1 7.114 [ ] 1 ħ 1 sin θ cos θ cos ϕ sin θ θ sin sin θ cos ϕ 7.115 θ [ 1 ħ sin θ + cos θ cos ϕ 1 ] sin θ sin sin θ cos ϕ 7.116 θ [ 1 ħ sin θ + cos θ 1 ] cos ϕ 7.117 sin θ ħ [ 1 sin θ sin θ ] cos ϕ 7.118 ħ Y 1,1 ħ ll + 1Y 1,1 7.119 Q Y l,m q ħ ll + 1 7.6 7.9 Q q Yl m ħ ll + 1 7.9 [ ħ 1 d ] m r dr r + ħ ll + 1 mr + V r Rr ERr 7.10 ur rrr [ ħ d ] m dr + ħ ll + 1 mr + V r ur Eur 7.11 r r > 0 Ur Ur ħ ll + 1 mr + V r 7.1 59

V r V r < 0 r 0 V r r k k > r V r 0 Ur r 0 ħ ll+1 mr l r V r Ur < 0 r Ur 0 r ur r 0 [ ħ ll + 1 ] /mr [ ħ ur r j d ] m dr + ħ ll + 1 mr ur Eur 7.13 jj 1r j ll + 1 + r r j m ħ Erj 7.14 jj 1 + ll + 1 m ħ Er 7.15 0 j l + 1 Rr r 0 r l r [ ħ d ] m dr ur Eur 7.16 B E ur exp mb/ħ r V r 7.9 ψ CurYl m θ, φ/r 7.7 V r V r e /4πε 0 r +e 7.9 E E n me4 1 3π ε 0 ħ n 7.17 n a B 4πε 0 ħ /me E n ħ 1 ma B n 7.18 a B 5.9 10 11 m 0.59 Å n 1 E.18 10 18 J 13.7 ev n l 0 n 1 m l l n n 1 l0 l + 1 n n l m 7.9 m 7.9 n l n R,0 r 3/ 1 R 1,0 r e r a B 7.19 a B 1 a B R,1 r 1 3 1 a B 3/ r a B 3/ r a B e e r a B 7.130 r a B 7.131 60

R 3,0 r 1 3 3a B R 3,1 r 9 R 3, r 4 7 10 1 3a B 1 3 3 r 3a B 3 r + r a B a B r 3 r a B 3a B e 9a B r e e r r 3a B 7.13 3a B 7.133 3a B 7.134 r 0 r R n,l r 7.9 V r e /4πε 0 r ħ /ma B r R 3, r [ ħ 1 d ] m r dr r + ħ + 1 mr + V r R 3, r 7.135 [ ħ 1 d ] 3 m r dr r + ħ 6 mr + V r 4 1 7 r 10 3a B a e r 3a B 7.136 B 3 4 [ 1 7 ħ 1 d ] 10 3a B m r dr r + ħ 6 mr + ħ r ma B r a e r 3a B 7.137 B 3 4 1 7 ħ [ 1 6r 6 r + r3 1 10 3a B m r 3a B 9a B a + 6 B a r ] B a 3 e r 3a B 7.138 B 3 4 1 7 ħ [ 1 ] r 3 1 10 3a B m r 9a B a e r 3a B 7.139 B ħ 1 ma B 3 R 3,r 7.140 E 3 R 3, r 7.141 R n,l r 3 l n l 1! r R n,l r na B n {n + l!} 3 e r na r B L l+1 n+l na B na B 7.14 L m n z E n l 0 n 1 ψ R n,l Yl m θ, φ n, l, m R n,l r r 0 R n,l r r l r R n,l r exp m E n /ħ r exp r/na B r 0 n l 1 n l r n r r l r R n,l r Yl m θ, φ Y l,m θ, φ R n,l r r 0 n l 1 Yl m θ, φ Y l,m θ, φ φ m θ l m n l 1+ m +l m n 1 n E n n n 1, l 0 1s n, l 0 s n, l 1 p n 3, l 0 3s n 3, l 1 3p n 3,l 3d n 4, l 0 4s n 4, l 1 4p n 4, l 4d n 4, l 3 4f 61

r 1 π π 0 0 0 R n,l ry l nθ, φ r sin θdrdθdφ 7.143 θ φ r P r P r π π 0 0 R n,l ry l nθ, φ r sin θdθdφ 7.144 R n,l r r 7.145 1 0 P rdr 7.146 1s P r r [ dp d ] 3 1 4r e r a B 7.147 dr dr a B 3 1 4 r r e r a B 7.148 a B a B dp/dr 0 r a B r p dp dr dp/dr 0 r 4a B 3d dp dr d r 4 e r a B 7.149 dr 4r 3 r 4 1 e r a B 7.150 a B d r 6 e r 3a B 7.151 dr 6r 5 r 6 e r 3a B 7.15 3a B dp/dr 0 r 9a B a B 1s n n a B 6

8 8.1 1000 0 0 63

0 l m n n, l, m m l l l 0 n 1 n 1 n E n 13.6/n ev l 0 s l 1 p l d l 3 f 1s s 3s, 4s, 5s, 6s,7s p 3d 4f 3p, 4p, 5p,6p,7p,4d,5d,6d,5f 8. 1/ ħ/ m +1/ m 1/ N m gµ 0 µ B 1 µ 0 µ B eħ m e g.00319 s m e g g 1 ħ m +1 m 1 0 m 0 x y m +1 z m 1 N 1 - ħ s 64

90 1 8.3 r 1, r α 1, α Ψr 1, α 1 ; r, α r 1, α 1 r, α Ψr, α ; r 1, α 1 Ψr 1, α 1 ; r, α 8.1 Ψr, α ; r 1, α 1 +Ψr 1, α 1 ; r, α 8. Ψr, α ; r 1, α 1 Ψr 1, α 1 ; r, α 8.3 + ψ 1 r, α ψ r, α Ψr 1, α 1 ; r, α 1 [ψ 1 r 1, α 1 ψ r, α ψ r 1, α 1 ψ 1 r, α ] 8.4 Ψr 1, α 1 ; r, α 1 [ψ 1 r 1, α 1 ψ r, α + ψ r 1, α 1 ψ 1 r, α ] 8.5 ψ r, α ψ 1 r, α Ψr 1, α 1 ; r, α 1 [ψ 1 r 1, α 1 ψ 1 r, α ψ 1 r 1, α 1 ψ 1 r, α ] 0 8.6 Ψr 1, α 1 ; r, α 1 [ψ 1 r 1, α 1 ψ 1 r, α + ψ 1 r 1, α 1 ψ 1 r, α ] 8.7 ψ 1 r 1, α 1 ψ 1 r, α 8.8 65

8.4 Z +Ze Z ħ m i Ze + 4πε i 0 r i i<j e Ψr 1, r, EΨr 1, r, 8.9 4πε 0 r i r j ψ i r Ψr 1, r, ψ 1 r 1 ψ r 8.10 ψ i r 8.9 ψ 1r 1 ψ k 1 r k 1ψ k+1 r k+1 dr 3 1 dr 3 k 1 dr3 k+1 ħ m Ze 4πε 0 r + V Hr k ψ k r ϵ k ψ k r 8.11 V k Hr i k e 4πε 0 r r ψ ir d 3 r 3 8.1 8.11 VH k r ψ i r 1s s p E Z 8.11 V H k r k Z 3 Li 1s 1s s p 1s r a B s p r 4a B s p +3e +e s p s R,0 r R,1 r +3Ze Li p s 1s, s, p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f, 6d, 7p p d f d m m 1 m 0 m + m 1 m m d f 8.10-66

- 47 11 1s s p 6 3s 3p 6 3d 10 4s 4p 6 4d 10 5s 1 m meħ/m e m < 0 m > 0 4d 5s m 0-8.5 H +1e 1e d 4e /d + e / d < 0 e 8e / d + e / d < 0 8 8 s p σ p π 67

8.6 H H ħ M R1 ħ M R ħ m r1 ħ m r + e + e 4πε 0 R 1 R e 4πε 0 r 1 r 4πε r 1 R 1 e 4πε r 1 R e 4πε r R 1 e 4πε r R 8.13 8.14 8.15 R i r i 1000 - R i H el ħ m r1 ħ m r e + e 4πε 0 r 1 r 4πε r 1 R 1 e 4πε r 1 R e 4πε r R 1 e 4πε r R 8.16 8.17 Ψr 1, r ψ 1 r 1 ψ r ψ k r ħ e m 4πε r R 1 e 4πε r R + V Hr k ψ k r ε k ψ k r 8.18 V k H r V Hr k ψ 1 r 1 ψ r ψ k r c 1 ϕ 1s r R 1 + c ϕ 1s r R ϕ 1s r R 1s ε 1s ϕ 1s r R 1 1 LCAO ħ e m 4πε r R 1 e 4πε r R + V Hr [c 1 ϕ 1s r R 1 +c ϕ 1s r R ] ε [c 1 ϕ 1s r R 1 + c ϕ 1s r R ] 8.19 ϕ 1sr R 1 d 3 r ε 1s + v c 1 + tc ε [c 1 + sc ] 8.0 68

v t d 3 rϕ e 1sr R 1 4πε r R + V Hr ϕ 1s r R 1 8.1 d 3 rϕ 1sr R 1 ħ e m 4πε r R 1 e 4πε r R + V Hr ϕ 1s r R 8. s d 3 rϕ 1sr R 1 ϕ 1s r R 8.3 ϕ 1sr R d 3 r t c 1 + ε 1s + v c ε [s c 1 + c ] 8.4 8.0 8.4 ε 1s + v t t ε 1s + v c 1 c ε 1 s s 1 c 1 c 8.5 [ ε 1s + v t t ε 1s + v ε 1 s s 1 ] c 1 c 0 0 8.6 ε 1s + v ε t εs t εs ε 1s + v ε c 1 c 0 0 8.7 c 1 0 c 0 0 ε 1s + v ε t εs t εs ε 1s + v ε 0 8.8 ε 1s + v εε 1s + v ε t εst εs 0 8.9 ε v 0, s 0 ε ε 1s ± t 8.30 E E ε 1s t 8.31 E ε 1s t 8.7 t c 1 + tc 0 8.3 c 1 : c t/ t : ±1 t t < 0 c 1 : c 1 : ±1 c 1 c c 1 c 69

t +1e e + 1s, s, p 1s 1s 1s n n 0 n 1 n n HOMO LUMO HOMO LUMO HOMO-LUMO HOMO LUMO HOMO-LUMO t 8.7 a 70

ħ d ψx + V xψx Eψx 8.33 m dx V x + a V x 8.34 a a ψx ψx + a 8.35 ψx + a e iδ ψx 8.36 δ π < δ π G n G n π a n 8.37 n V x n V Gn e ignx 8.38 V x + a n V Gn e ignx+a n V Gn e ignx V x 8.39 V x n V G n e ignx V x n V Gn e ignx 8.40 V G n V Gn 8.41 ψx n c Gn ke ik+gnx 8.4 k π/a < k π/a ψx + a n c Gn ke ik+gnx+a n c Gn e ik+gnx e ika e ika ψx e iδ ψx 8.43 E Gn kc Gn ke ik+gnx + V Gn c Gn e ik+gn +Gn x E c Gn ke ik+gnx 8.44 n n,n n G n G n + G n n E G k ħ m k + G 8.45 E Gn kc Gn ke ik+gnx + V Gn G n c Gn e ik+gnx E c Gn ke ik+gnx 8.46 n,n n E Gn kc Gn k + V Gn G n c Gn Ec Gn k 8.47 71

V 0 0 E G kc G k + V G G c G Ec G k 8.48 E G kc G k + V G Gc G Ec G k 8.49 E G k V G G V G G E G k c G k c G k E c G k c G k 8.50 V G V G E G k E V G G V G G E G k E c G k c G k 0 8.51 0 [E G k E] [E G k E] V G G 0 8.5 E [E G k + E G k] E + E G ke G k V G G 0 8.53 E 1 [E Gk + E G k] ± 1 4 [E Gk E G k] + V G G 8.54 k G k G E G k E G k V G G E E G k, E G k 8.55 k G k G E G k E G k E 1 [E Gk + E G k] ± 1 V G G 8.56 E G k E G k 8.8 k N ϕx + Na ϕx 8.57 ϕx + Na n n c Gn ke ik+gnx+na 8.58 c Gn ke ik+gnx e ikna 8.59 e ikna ϕx ϕx 8.60 7

k k π Na l 8.61 l 0, 1,,, N, N 1 k N k vk 1 d Ek 8.6 ħ dk Ek ħ m k 8.63 1 d ħ dk Ek 1 ħ m k ħk m p vk 8.64 m J e k occupied ħ vk e ħ k occupied d Ek 8.65 dk k E k Ek de k/dk dek/dk J 0 k < 0 k > 0 J 0 + J h e k occupied vk e k fullband vk evk h evk h 8.66 +e +e n p n p 73

9 9.1 6 10 3 9. n E n E n l 0 n 1 m l l n n L M x y z L/n x L/n y L/n z n x,n y,n z E ħ π n M L x + n y + n z n x, n y, n z, 1, 1, 1,, 1, 1, 1, n x + n y + n z 6 E E 3 9.1 74

9.3 0 1 N +m m N N N N 5 1 3 4 5 1 + 1 + 3 + 3 4 + 4 5 + 5 9. 1 + 1 + 1 + 1 + 1 + 1 9.3 N N t + t 9.4 t1 M Nm N + m N + 4mNm N M N N N N N 4 N 1 N N +N N N N/ + s N N/ s N N s 1 + 1 + + + + + + 9.5 N t N/ s + N x + y N N/ s N/ N t1 N! N t!t! xn t y t 9.6 N! N/ + s!n/ s! N/+s N/ s 9.7 N N/ + s N N/ + s N C N/+s N ms gn, s gn, s N! N/ + s!n/ s! N! N!N! 9.8 75

+ N N/ s N/ N/ s N/ N gn, s N/+s N/ s 9.9 gn, s 1 + 1 N N 9.10 A B N A A N B B N N A + N B gn, s N! N A!N B! 9.11 9.4 gn, s N 10 0 N! log g log gn, s log N! log N! log N! 9.1 1 N N 1 [ N! πn 1 N N exp N + 1 ] 1N + 9.13 log N! 1 logπ + N + 1 log N N 9.14 log gn, s 1 log 1 πn N + 1 log N N + 1 log N N N 9.15 x 1 x log1 + x x x / + N N s log gn, s 1 1 log + N log s πn N gn, s gn, 0 exp gn, 0 s N 1 1 N πn 9.16 9.17 9.18 9.17 gn, s s s 0 s 0 gn, s s s N gn, 0 s 0 gn, 0 N N!! N 9.19! 76

9.5 P s fs f f s fsp s 9.0 P s 1 9.1 s gn, s N P s gn, s/ N 9. fs s fs s 0 P s s 0 f 0 0 fs s s 1 1 N πn 1 1 πn 1 1 πn 1 N N N 3 3 π 4 1 dss exp s N/ s N 9.3 dxx exp x 9.4 9.5 1 8 N 9.6 s 1 N 1 N 9.7 N 10 0 10 10 Us B Us msb s N/ N/ Us ε Us Us+1 mb 77

10 10.1 g n P n P n 1 g 10.1 P n 0 10. 1 P n 1 10.3 X X n X n XnP n 10.4 Xn n X X X n permitted Xn g 10.5 N 5 s 1 10 gn, s g 5, 1 5! 10 10.6 3!! 10 10 10 s 5 3 78

10. S 1 S S S S 1 + S S S 1 U 1 S U S U U 1 + U U 1 U B S 1 N 1 s 1 U 1 mbs 1 S N s U mbs S U U 1 + U mbs 1 + s mbs s S N N 1 + N? S 1 gn 1, s 1 S gn, s S gn, s gn 1, s 1 gn, s S 1 S s 1 s s s 1 + s s 1 s gn, s N 1/ s 1 N 1/ gn 1, s 1 gn, s s 1 10.7 S N 1 < N s 1 S S gn 1, s 1 gn, s s 1 s 1 ŝ 1 N ŝ S S S 1 gn, s gn, 0 exp s N 10.8 gn 1, s 1 gn, s s 1 g 1 0g 0 exp s 1 s s 1 N 1 N 10.9 g 1 0 gn 1, 0, g 0 gn, 0 s 1 log gn 1, s 1 gn, s s 1 log g 1 0g 0 s 1 N 1 s s 1 N 10.10 s 1 s 1 s 1 0 s 1 s 1 log gn 1, s 1 gn, s s 1 4s 1 N 1 + 4s s 1 N 10.11 ŝ1 N 1 + s ŝ 1 N 0 10.1 ŝ 1 N 1 s ŝ 1 N 10.13 ŝ 1 N s ŝ 1 N 1 10.14 79

s 1 s 1 ŝ 1 N 1 + N sn 1 10.15 ŝ 1 s N 1 N 10.16 log gn 1, s 1 gn, s s 1 4 N 1 4 N < 0 10.17 s 1 ŝ 1 s 1 ŝ 1 ŝ s ŝ 1 ŝ 1 N 1 ŝ N s N 10.18 s 1 ŝ 1 s S 1 S S g 1 g max gn 1, ŝ 1 gn, s ŝ 1 g 1 0g 0 exp ŝ 1 exp ŝ N 1 N g 1 0g 0 exp ŝ 1 ŝ N 1 N [ ] ŝ1 ŝ g 1 0g 0 exp N 1 N [ g 1 0g 0 exp g 1 0g 0 exp g 1 0g 0 exp N 1 N s s ] N1 N N N [ s N1 + N ] N s N 10.19 10.0 10.1 10. 10.3 10.4 g 1 0g 0 g0 S 1 S S gn, s g0 exp s N 10.5 logg 1 g final logg 1 g max log g 1 0 + log g 0 s N 10.6 logg 1 g initial log g 1 0 + log g 0 s 1 N 1 s s 1 N 10.7 [ ] logg 1 g final logg 1 g initial s N s 1 s s 1 N 1 N 10.8 s N + s 1 + s 4ss 1 + s 1 10.9 N 1 N 1 s 1 1 1 4ss 1 + s 1 + 1 10.30 N N N N 1 N s N 1 N N 4ss 1 1 + s N 1 + N 1 10.31 N N 1 N N 1 N s s 1 s N 1 + s N 1 10.3 N N N 1 N 1 N N 1 N N 1 N s s 1 0 10.33 N N N 1 80

s s 1 N N 1 10.34 s 1 ŝ 1 s 1 ŝ 1 + δ gn 1, s 1 gn, s s 1 g 1 g max exp 4ŝ 1δ δ + 4s ŝ 1δ δ N 1 N 1 N N 10.35 ŝ 1 gn 1, s 1 gn, s s 1 g 1 g max exp δ δ N 1 N 10.36 N 1 N 10 δ 10 1 δ /N 1 δ /N 00 e 400 10 174 ŝ 1 N 1 / N 1 / 10 δ 10 1 /10 10 10 S 10.3 S 1 N 1 U 1 S N U S N N 1 + N U U 1 + U S gn, U g 1 N 1, U 1 g N, U U 1 10.37 U 1 U 1 U 1 U U 1 g 1 N 1, U 1 g N, U U 1 U 1 0 g1 g dg d g 1 N 1, U 1 g N, U U 1 g du 1 + g 1 du 0 10.38 U 1 N 1 U N U U 1 + U du 1 + du 0 10.39 1 g1 1 g 10.40 U 1 N 1 g U N g 1 log g1 U 1 N 1 log g 10.41 U N σn, U log gn, U 10.4 σ1 U 1 N 1 σ 10.43 U N N 1,N 81

T 1 T 10.43 T 1 σ T k B U N 10.44 k B k B 1.381 10 3 J/K 1.381 10 16 erg/k k B T 1 σ U N 10.45 gn, s gn, 0 exp s N σn, s log gn, s log gn, 0 s N B U smb 1 σn, U log gn, 0 U/mB N σn, U U s > 0 U < 0 > 0 N 4U mb N U mb N 10.46 10.47 10.48 10.49 mb N U mbn s 10.50 s U s U U s σn, U log gn, 0 U /mb N 10.51 σ g 1 σ log g 0 0 σ σ 0 B s N 1 10.5 mbn 1 mbn mbn ŝ 1 ŝ s N 1 N N ŝ 1 ŝ s ŝ 1 ŝ 10.53 10.54 10.4 σ S 1 S T U N 10.55 8

σ S k B σ 10.56 σ log g g N U N U N U 350K 10g 90K 10g T f U 0.389J/gK U 0.389 10 T f 90 0.389 10 350 T f 10.57 T f 1 35 + 90 30K 10.58 U U 1 0.389 10 30 11.7J 10.59 0.1 J 0.1 J U S 1 0.1 350.86 10 4 J/K 10.60 S 0.1 90 3.45 10 4 J/K 10.61 S S 1 + S 3.45.86 10 4 0.59 10 4 J/K 10.6 σ 0.59 10 4 1 0.59 10 4 k B 1.38 10 3 0.43 1019 10.63 exp σ exp0.43 10 19 gu U 1 g 1 U 1 g U U 1 10.64 g 1 U 10 g U U 10 U 10 S 1 1 g 1 g max g 1 Û1g U Û 10.65 g 1 U 10 g U U 10 10.66 83

σ final logg 1 g max σ initial logg 1 g 0 10.67 U 1 U σ logg 1 g final logg 1 g initial N 1 N N σ log g N s s 1 0 10.68 N 1 σ final σ initial N 1 N s s 1 0 10.69 N N N 1 S 1 U S S σ1 σ σ U + U 11 + 1 U 10.70 U 1 N 1 U N 1 > 10.5 0 A B B C A C log g1 U 1 N 1 log g 10.71 U N 1 3 1 3 1 U U δq δw 84

3 0 g0 σ log g0 g0 σ log g0 0 10.6 U δu N gn, s σn, s 1 log 1 1 N exp s πn N logπn + N log s N 10.7 10.73 N N log U s 85

11 1 11.1 R S U 0 N, V S ε s U 0 ε s S s g R+S U 0 g R+S U 0 g R U 0 ε s 1 g R U 0 ε s 11.1 S ε 1 P ε 1 ε P ε P ε 1 P ε g RU 0 ε 1 g R U 0 ε 11. P ε 1 P ε exp[σ RU 0 ε 1 ] exp[σ R U 0 ε ] exp[σ RU 0 ε 1 σ R U 0 ε ] 11.3 U 0 ε s σ R U 0 ε s σr σ R U 0 ε s σ R U 0 ε s U + 11.4 V,N σ R U 0 ε s 1 + 11.5 P ε 1 P ε exp ε 1 exp 11.6 ε S ε s exp ε s / 11. P ε s P ε s exp εs Z Z Z s exp ε s 11.7 11.8 P ε s 1 11.9 s 86

U ε U ε s ε s P ε s ε s exp εs Z s log Z 11.10 U S 0 ε Z exp 0 + exp ε 1 + exp ε 11.11 U 0 + ε exp ε ε exp ε Z 1 + exp ε 11.1 ε U 0 ε U ε/ U C V σ C V V U V 11.13 U ε ε exp ε C V [ ] 1 + exp ε 11.14 C V ε exp ε 11.15 ε ε ε C V exp 11.16 C V ε exp C V k B [ k B T 1 + exp ε k BT ε k BT ] 11.17 11.3 s ε s V V V s 87

ε s ε s V V ε s V dε s V + 11.18 dv ε ε s V V ε s V dε s V 11.19 dv A V A x + y + z 11.0 x x V x p s W s W s p s A x + y + z p s V 11.1 W s ε s p s dε s dv p p s U ε s U p V σ 11. 11.3 σ U V U V dσ σ σ du + dv 11.4 U V V U U V σ σ 0 δu σ + δv σ 11.5 U V V U δu σ σ U σ 0 + U V V V σ U V p p σ 1 U V σ σ V U 11.4 U 11.6 11.7 11.8 11.9 dσ 1 du + p dv 11.30 88

du dσ pdv 11.31 du T ds pdv 11.3 du dσ pdv 11.4 U σ F U σ 11.33 F U σ g σ U σ U σ F F U σ df du dσ 11.34 σ U V du dσ 1 11.35 df 0 11.36 F σn, s log gn, s log gn, 0 s N 11.37 U smb 11.38 F U σ F N, s smb log gn, 0 + s N 11.39 F s s ŝ F N, s mb + 4s s N 0 11.40 N 89

ŝ mbn F N, s s ŝ s σn, ŝ log gn, 0 m B N N U m B N 11.41 4 N > 0 11.4 F, N, B 11.43 11.44 F, N, B m B N m B N M log gn, 0 + m B N 11.45 log gn, 0 11.46 M ŝm V n m Bn s N 1 σs log gn, s + s log + s N 1 N s log s N 11.47 11.48 Us, B smb 11.49 F U σ N F L, s, N, B smb + + s log 1 + s N N 1 + s log s N 11.50 F L, s, N, B s F, N, B s ŝ FL mb + log N + s s N s 0 11.51 ŝ N,N,B N + ŝ mb N ŝ exp exp mb 1 exp mb mb/ 1 11.41 F L s,n,b N + s + + 1 N tanh mb N s 11.5 11.53 4N N 4s > 0 11.54 M M ŝm V nm tanh mb 11.55 90

n 1 + ŝ N exp mb exp 11.56 mb + 1 11.50 ŝ F, N, B mbn exp mb 1 exp mb 1 +N exp log mb + 1 mbn exp mb 1 1 ŝ N 1 exp 11.57 mb + 1 + N exp mb + 1 exp log mb + 1 1 exp mb exp + mbn exp mb mb + 1 exp mb + 1 [ ] mb N log exp + 1 [ ] mb mbn N log exp + 1 [ ] mb mb N log exp N log exp + 1 [ ] mb N log cosh exp mb exp mb + 1 + 1 11.58 11.59 11.60 11.61 11.6 11.5 F F U σ 11.63 df du dσ σd 11.64 du dσ pdv 11.65 df σd pdv 11.66 F F V V σ 11.67 p 11.68 11.68 F F U σ p U σ + V V 11.69 91

11.67 V 11.68 σ V p V 11.70 11.6 F Z F U σ F σ V F F U + F U F U F/ V V 11.71 11.7 11.73 11.74 U F/ [ F 1 + F 1 ] 11.75 11.76 F + F 11.77 U log Z 11.78 F log Z 11.79 F Z Z exp F 11.80 s P ε s exp εs F εs exp Z 11.81 Z F ε s msb 11.8 s ±1/ mb Z 1 exp + exp mb cosh mb 11.83 9

N F log Z [ ] N mb Z Z1 N cosh 11.84 [ ] mb F N log cosh 11.6 U F/ mb mbn tanh σ U F mb [ ] mb mb N tanh + N log cosh s U mb mb N tanh 11.85 11.86 11.87 11.88 ŝ 0 mb/ 1 tanh s N mb 11.89 s mb N U mbn s 11.90 s F M s m mb mn tanh V 11.91 M 1 V F mb mn tanh B 11.9 93

1 1.1 L 1 ħ M ψ εψ 1.1 x, y, z 0 x, y, z L ψ 0 L ψx, y, z A sin πnx L x πny sin L y πnz sin L z n x,n y,n z 1. ε n ħ π n M L x + n y + n z Z 1 n x exp n y n z [ ħ π n M L x + n y + n z ] 1 1.3 1.4 L ε n Z 1 0 dn x dn y 0 0 dn z exp [ α n x + n y + n ] z 1.5 α ħ π /ML [ Z 1 0 dn x exp α n ] 3 x 3 [ 1 dx exp x ] 3 α 0 1 α 3 3 π π3/ 8α 3 1.6 Z 1 L 3 πħ /M n Q 3/ n 1.7 n 1/L 3 1/V n Q M/πħ 3/ n Q ħ 1 n.5 10 19 cm 3 n Q 0.8 10 5 cm 3 n/n Q 3 10 6 1 n/n Q 1 Z 1 U log Z 1 3 1.8 U 3 k BT 1.9 1 94

1 N? Z 1 Z N Z1 N N! Z N 1 N! ZN 1 1 N! n QV N 1.10 N! N U log Z N 3 N 1.11 F log Z N N log Z 1 + log N! 1.1 F N log [ ] 3/ M πħ V + N log N N 1.13 F p N V V 1.14 pv N 1.15 pv Nk B T 1.16 mol n pv nrt 1.17 R 8.314 J/Kmol σ F V N log [ ] 3/ M πħ V + 3 N N log N + N 1.18 n N/V σ N [ nq log n + 5 ] 1.19 - n Q ħ U F + σ 3 N 1.0 1 3 Z 1 3 C V U V 3 N 1.1 C V 3 Nk B 1. 95

1. N t A t B gn, t N! N t!t! 1.3 σn, t log gn, t log N! logn t! log t! 1.4 σn, t N log N N N t logn t + N t t log t + t 1.5 N log N N t logn t t log t 1.6 N t log1 t/n t logt/n 1.7 x t/n σn, t N [1 x log1 x + x log x] 1.8 x < 1 A B A B F F 0 σ F 0 + N [1 x log1 x + x log x] 1.9 A B F F 0 1 x + F 0 x F 0 1.30 A B B x 1 σ xn log x 1.31 B 1 U B xn xnu F x NxU + x log x 1.3 F x x NU + log x + 0 1.33 x exp 1 exp U 1.34 U 0 96

1.3? s ε s sħω 1.4 ω ω Z s0 exp sħω 1.35 x exp ħω 1.36 Z x 1 s s s x s 1.37 s0 Z 1 1 x 1 1 exp ħω P s exp sħω Z sp s Z 1 s0 Z 1 ħ Z ω s0 1.38 s exp sħω 1.39 1.40 1.41 Z 1 exp ħω [ ] 1 exp ħω 1.4 exp ħω 1 exp ħω 1.43 1 s exp 1.44 ħω 1 97

1.5 1 ħω ε s ħω exp 1.45 ħω 1 1 ε n ω n U n ε n n ħω n exp ħω n 1.46 1 ω n L nx πx ny πy nz πz E x E x0 sin ωt cos sin sin L L L nx πx ny πy nz πz E y E y0 sin ωt sin cos sin L L L nx πx ny πy nz πz E z E z0 sin ωt sin sin cos L L L n x,n y,n z ħω n 1.47 1.48 1.49 Bt, r 0 1.50 Bt, r Et, r + t 0 1.51 Dt, r ρt, r 1.5 Dt, r Ht, r t jt, r 1.53 c Ht, r 1 µ 0 Bt, r 1.54 Dt, r ε 0 Et, r 1.55 ρt, r 0 1.56 jt, r 0 1.57 ε 0 µ 0 1 c 1.58 c Et, r Et, r 1.59 t A A A 1.60 E z c x + y + z E z t E z 1.61 n c π n L x + n y + n z ω n 1.6 n n x + n y + n z 1/ 1.63 98

ω n nπc L 1.64 U n x,n y,n z E 0 L dn x dn y dn z U 1 8 π ħc L π ħc L 0 4πn ħω n dn exp ħω n 1.65 1 4 L x 3 dx 1.66 πħc 0 exp x 1 4 L π 4 1.67 πħc 15 U V π 15ħ 3 c 3 4 1.68 4 - u ω U V dωu ω ħ π c 3 0 dω ω 3 exp ħω 1 1.69 u ω ħ ω 3 π c 3 exp 1.70 ħω 1 ħω/.8 ω 0 0 1.6 ω sω 1 exp 1.71 ħω 1 N 3 3N 3 1 3 3 4πn dn 1.7 8 n 99

3N n max n D 3 8 nmax 0 4πn dn 3N 1.73 1 πn3 D 3N 1.74 n D 1/3 6N 1.75 π U ε n n n ħω n exp ħω n 1.76 1 U 3π nd 0 dn n ħω n exp ħω n 1.77 1 c v x πħvn/l U 3π ħv L 4 L xd πħv 0 x 3 dx exp x 1 1.78 x D πħvn D L L 3 V 6π N ħv V θ ħv k B 1/3 1 k Bθ θ T 1.79 6π 1/3 N 1.80 v N/V Si 645 K 75 K Pb 105 K C V U V 3 36N k B θ k B θ 3 0 V x 3 dx exp x 1 9Nk Bθ 1 exp k Bθ 1 1.81 T θ x D 0 x 3 dx exp x 1 π4 15 1.8 4 U C V 3π4 N 5k B θ 3 4 3π4 N 5θ 3 k BT 4 1.83 U V 3 1π4 N 5k B θ 3 1.84 3 3 C V 1π4 N 5θ 3 k B T 3 1.85 100

13 13.1 1 S 1 S S 1 S N 1 N N N 1 + N F F 1 + F U 1 + U σ 1 + σ 13.1 F1 F N 1,V F dn 1 + N,V dn 0 13. dn dn 1 + dn 0 13.3 F1 F N 1 N,V,V 13.4 µ µ F N,V 13.5 µ 1 µ 13.6 101

µ 1 > µ dn 1 df dn 1 S 1 S S 1 S F F N log Z 1 + log N! 13.7 Z 1 n Q V n Q [ d dn log N! d 1 dn log N + log π + 3/ M πħ V 13.8 N + 1 ] log N N N + 1 1 N 1 log N + 1 N 13.9 log N 13.10 N N µ log Z 1 log N log n µ log n Q n N/V p n p µ log n Q Z 1 13.11 13.1 13.13 1 n/n Q 3 10 6 n/n Q 1 13. p-n S 1 S µ 1 > µ S 1 µinitial µ µ 1 q q V q V V 1 µinitial 13.14 V U F S S 1 10

N 1 µinitial N 1 q V S 1 S µ 1 final µ 1 initial + µ 13.15 µ 1 initial + [µ initial µ 1 initial] µ initial µ final 13.16 µ 0 0 µ µ tot µ int + µ ext 13.17 µ ext µ int 0 µ ext µ 1 µ µ ext µ int 13.18 µ µ F N,V F 13.19 F V N df σd pdv + µdn 13.0 F U σ U du df + σd + dσ dσ pdv + µdn 13.1 U σ V N du dσ pdv µdn µ σ µ U N σ,v 13. dσ 1 du + p dv µ dn 13.3 σ U V N µ σ µ N U,V 13.4 U,V,N σ σ σ dσ du + dv + dn 13.5 U V N V,N dv 0 dσ, du, dn 0 d 0 σ σ δσ δu + δn 13.6 U N V,N U,N U,V U,V 103

δn 1/ δσ δn µ σ U V,N δu σ + δn N U,V σ U σ + N N N U,V F N,V U σ N,V N,V 13.7 13.8 13.9 µ σ N U,V 13.30,V,N F U,V,N σ σ,v,n U µ U/ N σ,v dσ σ σ σ du + dv + dn 13.31 U V,N V U,N N U,V dσ 1 du + p dv µ dn 13.3 du dσ pdv + µdn 13.33 F F U σ df du σd dσ σd pdv + µdn 13.34 13.3 V N F F U σ 13.35 U g σ V p F G G,p,N G, p, N H σ F + pv U σ + pv 13.36 H σ, p, N Hσ, p, N U + pv 13.37 p N U g σ V 104

p > 0 G G U σ V µ dg σd + V dp + µdn 13.38 G σ 13.39 p,n G V 13.40 p,n G µ 13.41 N N G p,p GN,, p µ, pn 13.4 13.4 ε µ N R 1 S S ε s,n s R + S R gr + S gr 1 13.43 P N s, ε s gn 0 N s, U 0 ε s 13.44 P N 1, ε 1 P N, ε gn 0 N 1, U 0 ε 1 gn 0 N, U 0 ε 13.45 P N 1, ε 1 P N, ε exp [σn 0 N 1, U 0 ε 1 ] exp [σn 0 N, U 0 ε ] 13.46 ε 1 ε N 1 N U 0 N 0 σ σ σn 0 N s, U 0 ε s σn 0, U 0 N s ε s + 13.47 N 0 U 0 U 0 N 0 µ µ σn 0 N s, U 0 ε s σn 0, U 0 + N s ε 1 s 13.48 P N 1, ε 1 exp P N, ε N1µ ε 1 exp Nµ ε 13.49 105

Nµ ε exp 13.50 Nµ εs Ξµ, exp N0 sn 13.51 X P N, ε s XN, ε s P N, ε s 1 Ξ N0 sn exp Nµ εs Ξ Nµ εs XN, ε s exp N0 sn 13.5 13.53 N 1 Ξ Nµ εs N exp N0 sn Ξ Ξ µ log Ξ µ 13.54 λ exp µ/ λ Ξ N,s λ N exp ε s / 13.55 U ε 1 Ξ N λ log Ξ λ Nµ εs ε s exp N0 sn 13.56 13.57 β 1/ U µ β µ β Nµ ε N µ U log Ξ 13.58 β µ log Ξ β µ β J,V,µ log Ξ µ µ + log Ξ 13.59 J, V, µ F µn U σ µn pv 13.60 U g σ N µ < 0 J J U σ V N dj, V, µ σd pdv Ndµ 13.61 106

J 13.58 J σ J p V J N µ V,µ,µ,V 13.6 13.63 13.64 U µn log Ξ log Ξ log Ξ β β 13.65 U µn J + σ J J J/ 13.66 J log Ξ 13.67 107

14 14.1 0 1 5 3 He 1/ 6 4 He 1 N N 1 N 1 ε fε,, µ 1 0 1 0 1 N N 0 N 1 fε,, µ 14. - 1 N 0 ε 1 Ξ 1 + λ exp ε 14.1 N 0 Nε 0 N 1 ε fε Nε λ exp ε 1 + λ exp ε 108 1 λ 1 exp ε + 1 14.

µ λ expµ/ 1 fε exp ε µ 14.3 + 1 -f 0 1 ε f 1 ε f 0 0 ε < µ f 1 ε > µ f 0 µ ε F ε F µ 0 14.3-1 1 ε N Nε Ξ N0 λ N exp Nε N0 [ λ exp ε ] N 14.4 N x λ exp ε/ x < 1 Ξ N0 x < 1 x N 1 1 x 1 1 λ exp ε 14.5 fε Nε λ d log Ξ x log1 x x λ dx 1 x 1 λ 1 exp ε 14.6 1 1 fε exp ε µ 14.7 1 - - + - ε f 0 ε µ f 0 ε > µ f 0 ε µ f 14.4 - - f 1 - - 1 fε exp ε µ 14.8 ± 1 f 1 ε µ exp 1 14.9 109

- - µ ε fε exp λ exp ε 14.10 - N! N 1 N N s fε s 14.11 ε s 0 N λ s exp ε s 14.1 V 1 Z 1 N λz 1 14.13 n Q Z 1 n Q V 14.14 3/ M πħ 14.15 N λz 1 λn Q V 14.16 λ N n Q V n 14.17 n Q n N/V λ 1/n Q µ λ exp n 14.18 n Q n/n Q 1 n µ log n Q µ [log N log V 3 log + 3 ] πħ log M n Q 14.19 14.0 F N,, V N 0 µ F N,V dnµn,, V 110 N 0 14.1 dn [log N + ] 14.

dx log x x log x x 14.3 n Q F N [log N 1 log V 3 log + 3 ] πħ log M F N [ ] n log 1 n Q 14.4 14.5 N! F F p 14.6 V pv N U F + σ F p N V,N [ ] F F/ V,N V,N 14.7 14.8 U 3 N 14.9 3/ n Q 3 1 1 / k B T/ σ N F σ V,N [ n log + 5 ] n Q - C V σ V 14.30 14.31 14.3 C V 3 N 14.33 C V 3/Nk B dσ du + pdv C p σ p U V + p p p 14.34 U U/ p U/ V C p C V + N 5 N 14.35 C p C V + Nk B 5 Nk B 14.36 111

14.5? n n Q 0 πħ M n/3 14.37 σ N [ n log + 5 ] n Q 14.38 0 n Q 3/ 0 log 0 ε ε F f 1 ε F 1 ε L V L 3 ε F N ε F ħ m πnf 14.39 L n F n x,n y,n z N N 1 8 4π 3 n3 F π 3 n3 F 14.40 n F 3N/π 1/3 1/8 n n x,n y,n z ε F ħ m 3π N V /3 ħ 3π n /3 F 14.41 m n N/V F U 0 n n F ε n 1 8 4π U 0 ε n ħ m nf 0 π3 10m n5 F 3ħ 10m πn 14.4 L dnn ε n π3 m ħ nf dnn 4 14.43 L 0 πnf 3 N L 5 Nε F 14.44 1 U 0 /N 3/5 N 11

Dε dnε dε Nε ε 14.45 Nε Dε 3Nε ε N U 0 0 V 3π 3/ m ħ ε 3/ 14.46 V π ε F N U 0 3/ m ħ ε 1/ 14.47 dεdεfε,, µ 14.48 dεεdεfε,, µ 14.49 εf 0 εf 0 U N εf U Dε 0 N + 0 0 ε F 0 dεεdεfε dεfεdε dεdε 14.50 dεεdε 14.51 εf 0 εf dε ε F fεdε ε F dε ε ε F fεdε + εf 0 0 εf 0 dεεdε 14.5 dεdε 14.53 dε ε F Dε 14.54 dε ε F ε [1 fε] Dε 14.55 ε F ε > ε F ε < ε F ε F C el du d 0 dε ε ε F df Dε 14.56 d F df/d ε ε F Dε Dε F µ ε F x ε ε F / df d ε ε F ε εf exp [ exp ε εf C el Dε F ε F/ + 1 ] 14.57 e x dx x e x + 1 14.58 113

π /3 C el 1 3 π Dε F 1 π N F 14.59 C el 1 π Nk B T T F 14.60 3 114