量子力学第六弹——一维定态问题

General Discussion of Schrödinger Equation

single particle equation

iℏ∂∂t|ψ(x,t)⟩=H|ψ(x,t)⟩,H=−ℏ22m∇2+V(x,t)

multi-particle

iℏ∂∂t|ψ(x1,⋯,xN,t)⟩=H|ψ(x1,⋯,xN,t)⟩,H=∑i=1N[−ℏ22m∇2+V(x,t)]+V(x1,⋯,xN,t)

Properties

∂∂tρ+∇⋅j=0

where

ρ=|ψ(x,t)2|=ψ(x,t)∗ψ(x,t)j=ℏ2mi[ψ∗(x,t)∇ψ(x,t)−ψ(x,t)∇ψ∗(x,t)]

proof
We assume V(x,t) to be real

iℏ∂∂tψ(x,t)=−ℏ22m∇2ψ(x,t)+V(x,t)ψ(x,t)⇒−iℏ∂∂tψ∗(x,t)=−ℏ22m∇2ψ∗(x,t)+V(x,t)ψ∗(x,t)⇒⎧⎩⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪ψ∗(x,t)iℏ∂∂tψ(x,t)=ψ∗(x,t)(−ℏ22m∇2ψ(x,t)+V(x,t)ψ(x,t))ψ(x,t)(−iℏ∂∂tψ∗(x,t))=ψ(x,t)(−ℏ22m∇2ψ∗(x,t)+V(x,t)ψ∗(x,t))

(1)−(2)

l.h.s=iℏ(ψ∗∂∂tψ+ψ∂∂tψ∗)=iℏ∂∂t|ψ(x,t)2|r.h.s=−ℏ22m(ψ∗∇2ψ−ψ∇2ψ∗)=−ℏ22m∇⋅(ψ∗∇ψ−ψ∇ψ∗)⇒iℏ∂∂t|ψ(x,t)2|+ℏ2mi∇⋅(ψ∗∇ψ−ψ∇ψ∗)≡0

let ρ=|ψ(x,t)2|,j=ℏ2mi[ψ∗∇ψ−ψ∇ψ∗]
thus

∂∂tρ+∇⋅j=0

Some properties of 1D stationary problem

iℏ∂∂t|ψ(x,t)⟩=H|ψ(x,t)⟩

Assume |ψ(x,t)⟩=|ψ(x)⟩⋅e−iEt/ℏ
Then

Hψ=Eψ

1. [−ℏ22m∇2+V(x)]ψ(x)=Eψ(x)
   正问题：V(x)→E,ψ(x)
   逆问题：E→V(x)
2. both ψ(x),ψ∗(x) are solutions.
3. if ψ(x) is not degenerate, it can be chosen to be real.
4. if ψ(x) is a solution and V(x)=V(−x) then ψ(−x) is another solution.
5. if V(x)=V(−x), a complete set of solutions.

Infinite Square Potential Well Problem

（上次码好第六次笔记发布之后，不小心被第七次覆盖，简直欲哭无泪。上次只备份到这里，以后有机会再补上吧，sigh……）

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Thanks to Prof. Guo Hong