QMI_10.dvi

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1 Erwin Schrödinger σ τ x u u x t ux, t) u 3.1 t x P ux, t) Q θ P Q Δx x + Δx Q P ux + Δx, t) Q θ P u+δu x u x σ τ P x) Q x+δx) P Q x 3.1: θ P θ Q P Q equation of motion P τ Q τ σδx

2 u/ 2 x σδx 2 u 2 = τ sin θ Q τ sin θ P θ Q θ P sin θ Q sin θ P = tan θ Q tan θ P = σ τ 2 ) u x Q ) u x P = x ) u Δx = 2 u x x 2 Δx 2 ux, t) = ux, t) 3.1) 2 x2 wave equation 2 u/ ) x [ x ux, t) = A sinkx ωt) =A sin 2π λ t )] T 3.2) sine wave x t A amplitude k wave number ω angular frequency λ t = wave length T x = period x = frequency ν ν = 1 T, k = 2π λ, ω =2πν. 3.3) 3.2) φ = kx ωt phase kδx ωδt =0 v ph = Δx Δt = ω k phase velocity x = λν 3.4) 3.2) sin cos 3.1) f u = fkx ωt)

3 φ = kx ωt φ fkx ωt) =dfφ) x dφ x = kf kx ωt), 2 x 2 fkx ωt) =k2 f kx ωt) φ t φ fkx ωt) =dfφ) dφ = ωf kx ωt), 2 2 fkx ωt) =ω2 f kx ωt) 1 v 2 ph 2 2 fkx ωt) = fkx ωt) 3.5) 2 x2 ω = v ph k ux, t) =fkx ωt) gkx + ωt) fkx ωt) x gkx + ωt) x ux, t) =fkx ωt)+gkx + ωt) f g f g f + g superposition sinkx ωt) sin cos coskx ωt) ux, t) =e ikx ωt) = coskx ωt)+i sinkx ωt) ux, t)

4 m p E E = p2 3.6) E ν ω E = hν = hω. 3.7) k p k = p h. 3.8) hω = E = p2 = h2 k 2 ψkx ωt) ψ 3.9) h ψ = h2 2 ψ x ) t 3.10) ψx, t) = A sinkx ωt). 3.11) h ψ = hω A coskx ωt), h 2 2 ψ x 2 = h2 k 2 A sinkx ωt) 3.12) 3.10) ψ??) sin cos ψx, t) = A e ikx ωt) 3.13) 3.10) h ψ = i hω A e ikx ωt) h 2 2 ψ x 2 = h2 k 2 A eikx ωt) 3.14)

5 ) i h h2 ψx, t) = 2 ψx, t) 3.15) x2 3.13) Schrödinger equation 3.13) t x h 3.7) 3.8) E p i h ψx, t) = hω ψx, t) =Eψx, t), 3.16) i h ψx, t) = hk ψx, t) =pψx, t). 3.17) x p E x k = p/ h ω = E/ h x 3.13) ψx, t) wave function x p Hamiltonian H x, t) m H = p2 + x, t) 3.18) H i h ψx, t) = Hψx, t) 3.19) x x

6 probability density ρx,t) ρx,t) = ψ x,t) ψx,t)= ψx,t) ) t x t t d 3 xρx,t) = ) 3.21) 3.21) conservation of probability x, t) m i h ψ i h ψ = h2 2 ψ x 2 + ψ = h2 2 ψ x 2 + ψ 3.22) x [ a, b ] d b dxρx, t) = dt a b a dx ψ ψ ) + ψ ψ. 3.23) 3.22) d b dxρx, t) dt a = h 1 b i = h i = h i 1 1 a b a ) dx ψ 2 ψ x 2 2 ψ x 2 ψ dx ψ ψ ) x x ψ x ψ ψ ψ x ψ x ψ ) x=b + h i 1 ψ ψ ) x ψ x x=a ψ 3.24)

7 ) j x x, t) = h i 1 ψ ψ ) x ψ x ψ 3.25) d b dxρx, t) = j dt x x = b, t)+j x x = a, t) 3.26) a [ a, b ] ) x = b x = a [ a, b ] 3.25) x probability current density [ x, x + Δx ] Δx 0 lim Δx 0 d 1 dt Δx = lim Δx 0 x+δx x dxρx, t) j x x + Δx, t)+j x x, t) Δx ρ x,t) lim Δx 0 1 x+δx dxρx, t) =ρx, t) Δx x j x ρx, t) + j xx, t) x = ) a j x a,t) b j x b,t) 3.2: x 3.25) jx,t) = h i 1 ) ψ ψ) ψ )ψ 3.27) ρx,t) 3.28) + jx,t) = ) 3.29) ρ j 3.29) equation of continuity

8 ψ 1 x,t) ψ 2 x,t) ψ 1,ψ 2 ) ψ 1,ψ 2 ) = d 3 ψ2 x ψ 1 x,t)) x,t). 3.30) 0 ψ 1 x,t) ψ 2 x,t) ψ 1,ψ 2 ) = ) ψx,t) ) ψx,t)= ψ, ψ ) = d 3 x ψx,t) d 3 x ψx,t) 2 > ) norm ψx,t) = ψ, ψ ) 3.33) 1 a, b )=a b = a x b x + a y b y + a z b z 0 0 a 1 ψ 1 + a 2 ψ 2 = 0 a 1 = a 2 = ) { ψ i } 1 1 orthonormal system ψj ψ i,ψ j ) = d 3 x ψ i x,t)) x,t) = δ ij 3.35) δ ij δ ij = 1 i = j 0 i j 3.36)

9 ψx,t) a k t) { u k x) } ψx,t) = k a k t)u k x), 3.37) complete system E 3.16) p 3.17) operator p x,p y,p z ) E i h, 3.38) h i x, h i y, h i ) z 3.39) O O expectation value x p x = ψ, p x ψ )= x x = ψ, xψ )= O = ψ, Oψ ). 3.40) ) d 3 h x ψx,t) i x x ψx,t) 3.41) x xψx,t) d 3 x ψx,t)) 3.42) x, y, z ) x, y, z ) 3.43) ψ O ψ O ψ = ωψ 3.44)

10 34 3 ω O eigenvalue ψ eigen function 3.44) eigenvalue equation O O ψ k = ω k ψ k, 3.45) spectrum discrete spectrum continuous spectrum O ψ φ Λ Oφx,t) ψ, Oφ ) = d 3 x ψx,t)) φx,t) = d 3 x Λψx,t)) = Λψ, φ ). 3.46) Λ O conjugate operator Λ = O 3.47) ψ, Oφ ) = O ψ, φ ). 3.48) O ) = O 3.49) Hermitian operator self-conjugate operator O = O. 3.50) O = O ω k ) = ω k. 3.51) 3.45) ω k d 3 x ψ k x,t)) x,t) Oψk = d 3 x ψ k x,t)) x,t)= d 3 x O ψ k x,t)) x,t) = d 3 x Oψ k x,t)) x,t)= d 3 x ωψ k x,t)) x,t) = ω k ) d 3 x ψ k x,t)) x,t) 3.52)

11 ) 3.50) ψl ω k ω l ψ k,ψ l )= d 3 x ψ k x,t)) x,t)= ) O ω k ω l ω k d 3 x ψ l x,t)) x,t) Oψk = d 3 x ψ l x,t)) x,t)= d 3 x Oψ l x,t)) x,t) ) = d 3 x ω l ψ l x,t) x,t)=ω l d 3 x ψ l x,t)) x,t). 3.54) ω k ω l ) d 3 x ψ l x,t)) x,t) = ) O Λ O Λ Oψ k = ω k ψ k, Λψ k = λ k ψ k. 3.56) [ O, Λ] = OΛ ΛO = ) [, ] commutator commutation relation O Λ φ = k a k ψ k. 3.58) O Λ OΛφ = OΛ k a k ψ k = k a k OΛψ k = k a k Oλ k ψ k = k a k ω k λ k ψ k 3.59)

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