1 v. 1.0.0 tongchen@ecut.edu.cn June 28, 2021
1 ct 1
2 V (x) V (x) = mgz; V (x) = G m1m2 r V (x) = e1e2 4πϵ 0r ct 1
3 F F dx = dv (x) (1) F = V x = V = ( V x, V y, V ). (2) z ct 1
4 (1805 8 3-1865 9 2 ) - - ct 1
5 dp dt p = F = V x. (3) dx = p dt m, (4) H(x, p), x p, H(x, p) = p2 + V (x). (5) 2m (3) (4) ct 1 dp dt = H x, dx dt = H p. (6)
6 dh dt = H x dx + H dt p dp = H dt x H p H p H x = 0. (7) H H p2 2m V (x) H ct 1
7 V (x 1, x 2,..., x N ) V (x 1, x 2 ) = G m1m2 x 1 x 2 H = i p 2 i 2m i + V (x 1, x 2,..., x N ). (8) ct 1
8 dp i dt = H x i, dx i dt = H p i. (9) i! (9) i F i i F i dx i = dv F i = V x i. (10) ct 1
9 x i, p i ; ; (9) F = ma ct 1
10 V (x), x V V (x) (1) E ct 1 Figure:
11 [x b, x c ] (, x a ] [x d, + ) (, x a ] [x d, + ) [x b, x c ] ct 1
12 x, p x p E = p2 + V (x). (11) 2m ct 1
13 (, x a ] [x b, x c ] [x d, + ) (2) ct 1
14 E E (3) Figure: ct 1
15 [x b, x c ] T E, T (E), dx dt = H p = p m, T (E) = E dt = E m dx, (12) p E (11), p = 2m(E V (x)) T (E) = E m 2m(E V (x)) dx. (13) ct 1
16 2πI(E) 2πI(E) = pdx = 2m(E V (x))dx. (14) E E I(E) E (14) (13), I(E) E = 1 T (E) 2m(E V (x))dx = 2π E 2π. (15) E E I ω = 2π/T, E I = ω(i). (16) ct 1
17 (16) n n 1 ω hω, E n E n 1 = hω n E n n = hω. (17) (16) n - I = n h. (18) ct 1
18 V (x) = 1 2 kx 2 ω = k/m(ω ) V (x) = 1 2 mω2 x 2. (19) p 2 2m + 1 2 mω2 x 2 = E x 2 ( 2E mω 2 ) + p2 2mE = 1. (20) 2πE/ω 2πI = 2πE/ω E = Iω. (21) (16) ω ct 1
19 - (21) E = n hω. (22) ct 1
20 ct 1
21 ct 1
22 A, B (4) Figure: ct 1
23 Figure: B () B A B x A ct 1
24 B B 0 (6) Figure: ct 1
25 B 0 ( x ) V BB = 1 2 k BBx 2 B 0 A V AB V AB A (6)(a) V AB B 0 V S = V BB + V AB (6)(a) A V S (7) ct 1
26 ct 1
27 (7)(b),(c),(d) V S B 0 (b) (d), (b) (d) V S (b) (d) A( (d) (e) ), B 0 B 0 B ct 1
28 f (x, y) A = f (x, y)dxdy, (23) D A (x, y) A = f (x, y) (x, y ) dx dy, (24) (x,y) (x,y ) = x y x y x y y x ct 1
29 dxdy dx dy dx, dy dx dy = dy dx. (25) dx dx = dx dx = 0, dy dy = dy dy = 0. (26) ct 1
30 dx dy = ( x x dx + x y dy ) ( y x dx + y y dy ) = x x y y dx dy + x y y x dy dx = ( x y x y x y y x )dx dy (x, y) = (x, y ) dx dy. (27) ct 1
31 n n dx 1 dx 2...dx n dx 1 dx 2... dx n, dx i dx j = dx j dx i, (28) f (x 1, x 2,..., x n ) dx 1 dx 2... dx n n n ω, ω = f (x 1, x 2,..., x n )dx 1 dx 2... dx n. (29) n n n ω A = ω. (30) ct 1
32 n n k- α k α 0 k n, α = 1 k! α i 1i 2...i k dx i1 dx i2... dx i k. (31) 1 n 1 n (31) α i1i 2...i k k α (31) k i 1, i 2,..., i k α n- k > n k k ct 1
33 2- α α = 1 2 α ijdx i dx j α ij α ij = α ji α ij dx i dx j = α ji dx i dx j = α ji dx j dx i = α ij dx i dx j. (32) α = α α = 0 α ij i, j 2 α i, j α ij i, j α ji = α ij k k α i1i 2...i k k ct 1
34 3 3 0,1,2,3 0 3 f (x, y, z)3- f (x, y, z)dx dy dz dx dy dz 3-0- 3 1- a 1 dx + a 2 dy + a 3 dz (a 1, a 2, a 3 ) 3 a(x) 3 1- a 1 dx + a 2 dy + a 3 dz = a(x) dx. (33) ct 1
35 3 2- a = 1 2 a ij(x)dx i dx j = a 12 dx dy + a 23 dy dz + a 31 dz dx. (34) 3 a 12 = b 3, a 23 = b 1, a 31 = b 2, dx dy dz, dy dz dx, dz dx dy( 1, 2, 3 dx, dy, dz ) a 12 dx dy + a 23 dy dz + a 31 dz dx b 1 dx + b 2 dy + b 3 dz. (35) 3 2 1 1-1 (dy dz, dz dx, dx dy) 3 ds, ds = (dy dz, dz dx, dx dy). 2- a = 1 2 a ij(x)dx i dx j = b ds. (36) ct 1
36 2 ( x, y ) 1 a = a x dx + a y dy a da da = da x dx + da y dy, (37) da da = da x dx + da y dy = ( x a x dx + y a x dy) dx + ( x a y dx + y a y dy) dy = y a x dy dx + x a y dx dy = ( x a y y a x )dx dy, (38) i = x i da ( x a y y a x ) a ct 1
37 2 (a x dx + a y dy) = ( x a y y a x )dxdy, (39) D D D D a = da. (40) D D ct 1
38 3 1 a = a dx = a x dx + a y dy + a z dz da da = da x dx + da y dy + da z dz. (41) da = ( x a y y a x )dx dy + ( y a z z a y )dy dz + ( z a x x a z )dz dx. 3 2 da 3 a a da da = ( a) ds. (42) 3 ct 1
39 3 a dx = ( a) ds (43) D D D 3 D a = da. (44) D 2 D ct 1
40 3 2 a = a 12 dx dy + a 23 dy dz + a 31 dz dx da da = da 12 dx dy + da 23 dy dz + da 31 dz dx. (45) da = ( 3 a 12 + 1 a 23 + 2 a 31 )dx dy dz. (46) (a 23, a 31, a 12 ) = (b 1, b 2, b 3 ) = b, da = ( 1 b 1 + 2 b 2 + 3 b 3 )dx dy dz = ( b)dx dy dz. (47) 3-3 ct 1
41 3 b ds = ( b)dv, (48) V V 3 dv = dxdydz V 3 V 3 2 (36) 2- a a = da. (49) (44) V V ct 1
42 (44) (49) α = dα, (50) D D α 3 k 1 D 3 D k ( D k 1 dα k- ) k = 2 k = 3 ct 1
43 n n k 1 α = 1 (k 1)! α i 1i 2...i k 1 dx i1 dx i2... dx i k 1 dα dα = 1 (k 1)! ( jα i1i 2...i k 1 )dx j dx i1 dx i2... dx i k 1. (51) dα k () (50) 3 k 3 k n (50) ct 1
44 k 1 α, d 2 = 0. (52) d 2 α = d(dα) = 0. (53) ct 1
45 d 2 α = 1 (k 1)! ( i j α i1i 2...i k 1 )dx i dx j dx i1 dx i2... dx i k 1. (54) k + 1 k + 1 i j α i1i 2...i k 1 i, j k + 1 ct 1
46 : α dα = 0 α β α = dβ α (52) de Rahm ct 1
47 F i dx i = dv. (55) i µ = 1, 2, 3,..., 3N 3 1 3 2 3 N F µ dx µ = dv (x 1,..., x 3N ). (56) ct 1
48 F µ dx µ 1 F, F = F µ dx µ, 1 F = dv, (57) V 0 0 df = 0. (58) ct 1
49 df ( µ = x µ ) df = ( µ F ν )dx µ dx ν = [ 1 2 ( µf ν ν F µ ) + 1 2 ( µf ν + ν F µ ) ] dx µ dx ν. (59 df = 0 df = 1 2 ( µf ν ν F µ )dx µ dx ν. (60) µ F ν ν F µ = 0. (61) ( 1, 2, 3) F F = 0. (62) ct 1
50 1 F 3N 1 3N D, D 3N 2 D F = df = 0. (63) D D 1 1 ct 1
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