p = mv p x > h/4π λ = h p m v Ψ 2 Ψ

II

p = mv p x > h/4π λ = h p m v Ψ 2 Ψ

Ψ Ψ 2

0 x P'(x) m d 2 x = mω 2 x = kx = F(x) dt 2 x = cos(ωt + φ) mω 2 = k ω = m k v = dx = -ωsin(ωt + φ) dt = d 2 x dt 2 0 y v θ P(x,y) θ = ωt + φ ν = ω [Hz] 2π x = -ω 2 cos(ωt + φ) = -ω 2 x x - ω T = [rd/s] φ P(x,y) x = cos(ωt + φ) y = sin(ωt + φ) d2 x dt 2 2π = 2π ω = m k x m k t or d 2 x = ω 2 x dt 2 x = cos(ωt + φ) T = ω 2π = ν 1 v = ω k force constnt (1) = cos( t + φ) m k [s]

d 2 x dt 2 = ω 2 x (1) d(e x ) dx = e x e x = exp x d 2 f(x) = 2 f(x) dx 2 (1) e x = 1 + x + x 2 /2! + + x n /n! + = Σ xn (e = 2.71828183 ) n=0 n! log e e x = ln e x = x x = e y y = ln x y = cos x y = sin x dx dx = de y dy dy d(ln x) 1 sin x cos x = = dy dx dx dx x dy dx = d 2 y dx 2 = dy dx = d 2 y cos x = y dx 2 = sin x = y y = A cos x + B sin x y = C e ix + D e -ix y = e ix y = e -ix dy dx = ieix d 2 y dx 2 = e ix = y (i 2 = 1) dy dx = ie-ix d 2 y dx 2 = e-ix = y

Euler's Formulus e ix = cos x + i sin x e ix = cos x i sin x cos x = sin x = e ix + e -ix 2 e ix e -ix 2i y y z r = z 0 θ x r = (x 2 + y 2 ) 1/2 = z x = r cos θ = z cos θ y = r sin θ = z sin θ z z = x + iy 1 = x 1 + iy 1 = z 1 e iθ 1 z 2 = x 2 + iy 2 = z 2 e iθ 2 = z (cos θ + i sin θ) = z e iθ z 1 z 2 = z 1 z 2 e i(θ 1+θ 2) x z = x + iy = z e iθ z* = x iy = z e iθ z z* = z 2

y λ u t = 0 y = cos 2π x λ - y ut x(0) x t = t y = cos 2π λ (x ut) x(t) y = cos ( 2π λ x 2πνt) y = cos (kx ωt) u = λ ν ν = u λ T = 1 = ν λ = ω 2π 2π ω [s] [Hz] ω = 2π ν k = 2π λ ω = ku ψ(x,t) = cos (kx ωt) ψ(x,t) = ei(kx ωt) ψ(x,t) = cos ω( x u t) ψ(x,t) = cos 2π( x λ t T ) ψ(x,t) = sin ω(t x u ) ψ(x,t) = sin 2π( t T x λ )

ψ(x,t) 2 ψ(x,t) t 2 = u 2 2 ψ(x,t) x 2 ψ(x,t) = cos (kx ωt) ψ(x,t) t = ω sin (kx ωt) 2 ψ(x,t) t 2 = ω 2 cos (kx ωt) u u = λ ν λ ν ω = 2π ν k = 2π λ ω = ku ψ(x,t) x = k sin (kx ωt) 2 ψ(x,t) = k 2 cos (kx ωt) x 2 2 ψ(x,t) t 2 = ω 2 k 2 2 ψ(x,t) x 2 2 ψ(x,t) t 2 = u 2 2 ψ(x,t) x 2

ψ(x,y,z,t) = ψ(r,t) 2 ψ(r,t) t 2 2 ψ(r,t) t 2 Nbl: Lplcin: = u 2 2 ψ(r,t) 2 ψ(r,t) 2 ψ(r,t) ( + + ) x 2 y 2 z 2 = u 2 ( 2 2 x 2 + 2 y 2 + z 2 ) ψ(r,t) = u 2 ψ(r,t) = 2 ψ(r,t) t 2 x + + y z = 2 2 2 2 = x 2 + y 2 + z 2 = u 2 ψ(r,t) u 2 ψ(x,t) ω t 2 = 2 k 2 ψ(r,t) = cos (kr ωt) = cos (k x x + k y y + k z z ωt) 2 ψ(r,t) r 2 = 2 ψ(r,t) x 2 2 ψ(r,t) = k 2 x 2 x ψ(r,t) 2 ψ(r,t) = k 2 y 2 y ψ(r,t) 2 ψ(r,t) = k 2 z 2 z ψ(r,t) 2 ψ(r,t) + + y 2 2 ψ(r,t) z 2 = (k x 2 +k y 2 +k z 2 )ψ(r,t) = k 2 ψ(r,t) 2 ψ(r,t) t 2 = ω 2 ψ(r,t) 2 ψ(r,t) 2 ψ(r,t) 2 ψ(r,t) ( x 2 + + y 2 z 2 )

y u u y 1 = sin 2π λ (x ut) λ ψ(x,t) = 2 sin kx cos ωt = φ(x) cos ωt 2 ψ(x,t) x 2 2 ψ(x,t) t 2 = k 2 2 sin kx cos ωt = cos ωt φ(x) = 2 sin kx d 2 φ(x) dx 2 = ω 2 2 sin kx cos ωt = ω 2 cos ωt φ(x) x 2 ψ(x,t) t 2 y 2 = sin 2π λ y = y 1 + y 2 = 2 sin 2πx λ (x + ut) cos ωt = u 2 2 ψ(x,t) x 2 ω 2 cos ωt φ(x) = u 2 cos ωt ω 2 φ(x) = ω 2 k 2 d 2 φ(x) dx 2 d 2 φ(x) dx 2 (ω = ku) d 2 φ(x) dx 2 + k 2 φ(x) = 0 (k = 2π ) λ

y n = 4 ψ(x,t) = 2 sin 2πx cos ωt λ φ(x) = 2 sin 2πx λ L = n λ 2 L n = 3 n = 2 n = 1 x φ(x) = A sin n πx L ψ(r,t) = φ(r) e i ωt φ(r) + k 2 φ(r) = 0

Erwin Schrödinger

d 2 ψ(x) dx 2 + k 2 ψ(x) = 0 (k = 2π ) λ ψ(x) λ = h p E = p 2 + U(x) d 2 ψ(x) dx 2 + p x 2 ψ(x) = 0 p x 2 ψ(x) = d 2 ψ(x) dx 2 d 2 p 2 x h ψ(x) = 2 dx 2 ψ(x) p 2 d 2 x dx 2 p x i h d dx [ d 2 ψ(x) dx 2 + [E U(x)]ψ(x) = 0 d 2 ψ(x) dx 2 + U(x)ψ(x) = Eψ(x) d 2 dx 2 d 2 dx 2 + U(x)]ψ(x) = E ψ(x) + U(x) = H

p 2 x d 2 ψ(x) + U(x) = E dx 2 + U(x)ψ(x) = Eψ(x) p 2 (r) + U(r) = E h d 2 2 ψ(r) + U(r)ψ(r) = Eψ(r) dr 2 1 (px 2 + p 2 + U(x,y,z) = E y + p x 2 ) r = (x,y,z) ψ(r) = ψ(x,y,z) 2 [ 2 2 x 2 + ]ψ(r) y 2 + z 2 H ψ(r) = Eψ(r) + U(x,y,z)ψ(r) = Eψ(r) h 2 ψ(r) + U(x,y,z)ψ(r) = Eψ(r) E h [ 2 i + U(x,y,z)]ψ(r) = Eψ(r) ψ(r) i E i ψ(r) i H

Hψ(r) = Eψ(r) E i ψ(r) i ψ(r) = ψ(x,y,z) H = + U(x,y,z) 2 [ 2 = 2 x 2 + ] + U(x,y,z) y 2 + z 2 E i ψ(r) i ( p ) = 2 + U(x,y,z) = i h = i h [ + + x y 2 = ] z

ψ(r,t) = ψ(x,y,z,t) = e i(kr-ωt) = e i(k x x + k y y + k z z ωt) 2 ψ(r,t) = u 2 ψ(r,t) t 2 λ = h/p E = hν k = 2π/λ = p h ω = 2πν = E h ψ(r,t) = 2 ψ(r,t) r 2 p 2 = = k 2 ψ(r,t) = 2 ψ(r,t) = ω 2 ψ(r,t) = E2 t 2 E 2 = 2 t 2 ψ(r,t) p 2 ψ(r,t) p = i h E = i h t ψ(r,t) = e i(pr Et)/h ψ(x,t) = e i(px Et)/h

p 2 H = (r) + U(r) = E p 2 = E 2 = 2 t 2 p = i h E = i h t ψ(r,t) = e i(pr Et)/h ψ(x,t) = e i(px Et)/h h 2 ψ(r,t) + U(r)ψ(r,t) = i h ψ(r,t) t 2 { [ 2 2 x 2 + ] + U(x,y,z)}ψ(r,t) = y 2 + z 2 i h ψ(r,t) t H ψ(r,t) = E ψ(r,t) ψ(r,t) = e i(pr)/h e iet/h =ψ(r)e iet/h H ψ(r) = E ψ(r)

ψ(r) cψ(r) ψ(r)ψ(r)e iθ ψ(r) Hψ(r) = Eψ(r) ψ(r) ψ(r) Ψ(r,t) = ψ(r)e i t ψ(r) ψ(r) ψ(r) 2 = ψ(r)ψ (r) ψ(r) ψ(r) 2 dv =ψ(r) * ψ(r)dv = 1 ψ(r) ψ i (r) * ψ j (r)dv = 0 δ ij

Hψ(r) = Eψ(r) E H ψ(r)ψ(r) * = Eψ(r)ψ(r) * ψ(r) * H ψ(r) = ψ(r) * Eψ(r) ψ(r) * H ψ(r) = ψ(r) H ψ(r) * ψ(r) * H ψ(r)dv =ψ(r) * Eψ(r)dv = Eψ(r) * ψ(r)dv = E = <E> E H 1 ψ 1 (r) = E 1 ψ 1 (r) H 2 ψ 2 (r) = E 2 ψ 2 (r) <E> = ψ(r) * H ψ(r)dv H 1 + H 2 E 1 + E 2 ψ 1 (r)ψ 2 (r)

U(x) U = m U = 0 U = 0 x ψ(x)= B sin Hψ = Eψ d 2 ψ(x) dx 2 = Eψ(x) d 2 ψ(x) E dx 2 = ψ(x) ψ(0) = ψ() = 0 ψ(x) = A cos kx + B sin kx ψ(x)= (2/) 1/2 sin E nx = E k = [ ] 1/2 A = 0 k = n x π B = (2/) 1/2 8m 2 n x 2 n x πx n x πx

E nx = 8m 2 n x 2 ψ(x)= (2/) 1/2 sin n x πx E 4 = 16h2 8m 2 + + n =4 n 1 E 3 = 9h2 8m 2 + + n = 3 E 2 = 4h2 8m 2 E 1 = 8m 2 + + n = 2 n = 1 node ψ 2 (x) = 0 0 ψ(x) 0 ψ 2 (x)

Hψ = Eψ H = H x + H y E = E x + E y ψ = ψ(x)ψ(y) 10E 0 8E 0 (n x, n y ) (3,1) (1,3) (2,2) ψ(x)= (2/) 1/2 sin ψ(y)= (2/) 1/2 sin E nx,ny = n x πx n y πy 8m 2 (n x 2 + n y 2 ) 5E 0 2E 0 (2,1) (1,2) E 0 = (1,1) 8m 2 y 0 x

Hψ = Eψ H = H x + H y + H z degenerted E 3 = 9E 0 E = E x + E y + E z ψ = ψ(x)ψ(y)ψ(z) (1,2,2) (2,1,2) (2,2,1) ψ(x)= (2/) 1/2 sin ψ(y)= (2/) 1/2 sin ψ(z)= (2/) 1/2 sin E nx,ny,nz = n x πx n y πy n z πz h2 8m 2 (n x 2 + n y 2 + n z 2 ) (1,1,2) (1,2,1) (2,1,1) projection z y y 0 x 0 x (1,1,1) degenerted E 2 = 6E 0 E 1 = 3E 0

z x