,..,.,,,,,.,,.,.,. 6 1,.,,.,,.,, 1,,, ;,,,,.,,,,.,,.,,,.,.,.,,.,.,,,.,,,.,,,,.,.,,,, i
.,,,,.,,.,.,.,,.,,,., 1;,,,,,.,,,,.,,,.,.,,.,,.,,,.,,.,,.,.,.,,.,,.,..,.,,.,,,.,,,.,,,,,,.,,,,.,,????.,,,,,.,,,,., / /, GPA.,, 1.,,.,,,.,,. ii
,.., 4, 6 1 A.,..,.,..,.,,,.. LATEX.,,.,,. Email: zhpf@ustc.edu.cn Tel: 551-3663 9 9 iii
1......................................................................1 Schrödinger....................................................... 1 3.......................................................................7 4.................................................................. 76 5..................................................... 17 6.......................................................................... 19 7............................................................ 164 8...................................................17 9............................................................................... 178 1....................................................18 11.................................................................. 31 1......................................................................... 7 13......................................................................... 87 14.................................................................... 315 15...................................................... 337 16.......................................................................35 17.................................................................. 365..................................................................376................................................................................ 379..........................................................................................38 v
1 1.1,. V (x) 1 mω x m,, E = p m + 1 mω x (1.1) p = m (E 1 ) mω x (1.) E 1 mω x, 1 ω xm nh = pdx = xm = x m E m x 1 E ω m. x m = 1 E, Bohr-Sommerfeld ω m pdx x m m xm xm ( p)dx = x m (E 1 ) mω x dx = 4mω x m pdx xm x m x dx n, n = 1,,. (, x = x m sin ωt) xm nh = 4mω x m x dx = 4mω 1 [ x x m x + x m arcsin x ] x m x m = 4mω E π mω = πe ω E = nh π ω = nω., 1.4. 1.. a, b c. x, y, z, (,, ), (a, b, c). p m = 1 m (p x + p y + p z)., x, x =, x = a, () x =, x., Bohr-Sommerfeld n 1 h = p x dx = a ( E = n + 1 ) ω. p x dx a ( p x )dx = a p x dx = ap x (1.3) 1
n 1 = 1,,, p x = n 1 h/(a) = πn 1 /a. p y = n h/(b), p z = n 3 h/(c), n, n 3. E = h 8m ( n 1 a + n b + n 3 c n 1, n, n 3, n 1, n, n 3 = 1,,. 1.3 I,. I, ) (1.4) E = T = L I L. ( Bohr-Sommerfeld ) (1.5) L = n n, n = 1,, (1.5). E = n, n = 1,, 3, (1.6) I 1.4 q m, B. z, A(B = A) A = 1 B r = 1 Bye x + 1 Bxe y A x = 1 By, A y = 1 Bx. ( Gauss ) v = ṙ, a = r. mṙ a =, d dt ma = q c v B (1.7) ( ) 1 mṙ =,. (1.7) m (a 1 e x + a e y ) = q c (v 1e x + v e y ) Be z = qb c (v e x v 1 e y ) v 1 = ω = q B mc. m v 1 = qb c v, m v = qb c v 1 (1.8) ( ) qb v 1 = ω v 1, v = ω v (1.9) mc v 1 = v sin(ωt + φ) v = mc qb v 1 = mc q ω cos(ωt + φ) = v cos(ωt + φ) (1.1) qb q x x = v cos(ωt + φ), ω y y = q v sin(ωt + φ) q ω n =, ±1, ±, ±3,.
, v, ω, r = v ω. x =y =, x = v cos(ωt + φ), ω y = q v sin(ωt + φ) q ω φ =, x = v cos ωt, ω y = q v sin ωt, q ω p x = mẋ = mv sin ωt, p y = mẏ = q mv cos ωt q dx = v sin ωtdt, dy = q (1.11) q v cos ωtdt, p dl, p = mv + qc A. p x dx = mv x dx + q c A xdx. T π mv x dx = ( mv sin ωt)v( sin ωt)dt = mv sin θdθ = mv ω ω π q q c A xdx = ( 1 ) c By dx = qb ydx = q Bv c cω sin ωtd cos ωt = q Bv cω, p y dy, T p x dx = mv y dy = T q c A ydy = q c sin ωtdt = q Bv cω π sin θdθ = q Bv cω π = mv ω π q mv x dx + c A xdx = qbv cω π qbv cω π = mv ω π (1.1) π (mv cos ωt)v cos ωtdt = mv cos θdθ = mv ω ω π 1 T Bxdy = q B xdy = q Bv c cω cos ωtd sin ωt = q Bv cω p y dy = (1.1) (1.13) p dl = T q mv y dy + cos ωtdt = mv ω c A ydy = mv ω π p x dx + p y dy = mv ω π + mv cos θdθ = mv ω π π mv ω π = mv ω π (1.13) ω π = mv ω π = nh n, n = 1,,. mv ω π = nh. ω = q B mc. ( E n = n + 1 ) ω. E = T = 1 mv = nω, n = 1,, 3
C p dl = nh, p dl = (mv + q ) C c A dl = mv dl + q A dl C c C T = mv vdt + q T A ds = mv dt + q c c = mv T S dt q c BS = mv T q c BπR = mv π ω S B ds v mω π ω = mv ω π ω = q B mc (cyclotron)., mv T q c BπR = nh, E = T = 1 mv = nω n, n = 1,,.,,. 1.1 1. 1.4,.? Plank.,,. 1.5 (WKB, ), ( 1.1),. ( (a) pdx = n + 1 ) h, n =, 1,, ( (b) pdx = n + 3 ) h, n =, 1,, 4 (c) pdx = (n + 1) h, n =, 1,, pdx = nh, n = 1,, 3, p = m [E V (x)].,. 1.1 (1) ( 1.) V (x) = g x,. () {, x < V (x) = eex, x > 4
, E, e ( 1.3). 1. 1.3 (1)., E m g, Plank. E = fg α m β γ, f. g [g] = [E/r] = [E]/[r],. g m E M 1 1 1 1 L 1 T 1,. α + β + γ = 1 α + + γ = α + γ =, α = /3, β = 1/3, γ = /3. E = f 1 ( g m ) 1 3 (1.14) f 1.,, m = g = = 1,. V (x) = x. E = p + x p = (E x ) E x, E x E. x m = E., (a), n =, 1,,. ( (E x )dx = n + 1 ) π (1.15) 5
xm xm (E x )dx = E x dx = 4 E xdx x m = 4 3 (E x) 3 x m = 8 3 E 3 (1.16) (1.15) (1.16) [ ( 3π E = E n = 4 n + 1 )] 3 (1.14), ( g ) 1/3 [ ( 3π E n = n + 1 m 4 )] /3 (), E m, ee, Plank. m, (1), ee (1) g, (1), ( e E E = f m ) 1 3 (1.17) f., m = g = = 1. x >, E = p + x p = (E x) E x, < x E. x m = E., (b), ( (E x)dx = n + 3 ) π (1.18) 4 (E x)dx = xm E xdx = 3 (E x) 3 (1.18) (1.19) E = E n = 1 [ ( 3π n + 3 )] 3 4 (1.17), E n = 1 ( 9π e E ) 1/3 ( n + 3 m 4 n, n =, 1,,. ) /3 x m = 4 3 E 3 (1.19) 1.6 V (r), (?) V (r) = e (Coulomb ),. r 6 ( me V (r)dr = n + 3 ) h, n =, 1,, 4
,,,.. Coulomb E(E < ), n =, 1,,. r m = e /me. rm me + e r dr = (1.) (1.1) n =, 1,,. n = 1,,. me + e r dr = ( n + 3 ) h (1.) 4 (1.) me + e r, r e /me, rm e me 1, x = r/r m = 4e me 1 1 x dx = 1 x 1dx = 4e me 1 1 xd x 4e 1 ( x ) 1 1 x me + arcsin x = πe me (1.1) E = E n = me4 ( n + 3 ) 4 me 4 E n = (n 1/4) (n 1/4) n,. n (),, n 1,. 1.7 ( m), E = mc / 1 v /c (v ) Hamilton p = mv/ 1 v /c = Ev/c H = E = m c 4 + p c. de Broglie., c. Hamilton q i = dq dt = H p i, ṗ i = dp dt = H q i (1.) Hamilton, p = const.. v = q i, v = p m c 4 + c p = c p m c 4 + c p (1.3), s,,.,. 7
Hamilton p = mv 1 v c (1.4) E = m c 4 + p c = mc / 1 v /c (1.5) p = mv/ 1 v /c = Ev/c (1.6) de Broglie, k = p, ω = E (1.7) de Broglie v p = ω k = E p = c v v < c, v p > c. (1.5) (1.7) de Broglie (1.8) ω(k) = 1 m c 4 + c k (1.9) v g = dω dk = dω m c 4 dk (1.5) (1.7) (1.6) v g = de Broglie. + c k = c k m c 4 + c k c p m c 4 + c p = c p E = v (1.3) 1.8,, E = p c + m c 4, (1) ( pc ) ] 1/ E = mc [1 + mc = mc + p m p4 8m 3 c + mc (rest energy),. p m, 8m 3 c () (), [ ( ) mc ] 1/ E = pc 1 + = pc + 1 m c 4 + pc pc p4 (m = ) (), E = pc. 8
(1) (), v c, p = E = ( p c + m c 4) 1/ m v mv mc, pc 1 v /c (1.31) 1. mc ( pc ) (1.31) Taylor mc ( pc ) ] 1/ [ E =mc [1 + mc =mc 1 + 1 ( pc ) 1 ( pc ) 4 + ]=mc mc 8 mc + p m p4 8m 3 c + (), v c, p = m 1 v /c v = v/c mc mc, 1 v /c mc ( ) mc pc 1. (1.31) Taylor pc [ ( ) mc ] 1/ [ E = pc 1 + = pc 1 + 1 ( ) mc ] + = pc + 1 m c 4 + pc pc pc (m = ) (), (1.31) E = pc. 1.9 (1) Fermat n 1 sin α 1 = n sin α. (),, δ pdl =. p = mv, δ vdl =, p, v. n sin α 1 = n 1 sin α,.,, p = Ev/c, E,, E., δ pdl =..? L. de Broglie, Le Journal de Physique et la Radium, 7(196),1. (1) Fermat. I ( n 1 ) A ( n ) B. Fermat S P. A, B S Π, P Π., P Π P. AP P, AP AP ; 1.4 BP BP, n 1 AP + n BP < n 1 AP + n BP. AP B. P Π. Π (), ()., Fermat. A, B a, b(), α 1, α, A B l = n 1 AP + n BP = n 1 a sec α 1 + n b sec α (1.3) A, B, A, B S A, B, A B = c, 9
. Fermat (1.3) l, a tan α 1 + b tan α = c (1.33) = δl = n 1 aδ sec α 1 + n bδ sec α = n 1 a sec α 1 tan α 1 δα 1 + n b sec α tan α δα (1.34) (1.33) a sec α 1 δα 1 + b sec α δα = (1.35) (1.34) (1.35) δα 1, δα n 1 sin α 1 = n sin α,. (),. v () v p = c n (n ), v = v p. v p v g v,. Fermat,. v p v p = ω k (1.36) v g v g = dω dk (1.37), ω(k), v p = v g ; ω(k), v p v g. v = H p, ck, ω(k) = c n k, c, n.,,, n ω. v p = v g (1.38) dω = c ck dk n n dn = v pdk ck dn n dω dω v p v g = 1 + ck v dn p n dω,., de Broglie 1 p = h, E = hν (1.39) λ