[領域] (題目相關領域)
量子力學考題
[來源] (課本習題、考古題、參考書...)
參考書
[題目]
In early period of development of quantum mechanics, a few physicists
(including Schrodinger himself) tried to develop so-called
hidden-variable theories in which quantum fluctuations are attributed to
some unknown variables in classical mechanics. In 1952, Bohm succeeded
in identifying such kind of random forces in the Schrodinger equation:
Suppose ψ satisfies the Schrodinger equation with the the
potential V (r).
Let ψ =∣ψ∣exp(iS/h), show that
(a) S is real
(b) when ψ is a plane wave, ▽S =hk is the momentum.
In general, ▽S has the meaning of momentum. Let p=▽S.
Here p is a function of r and is the momentum of the particle that arrives
at r. This is in the Euler description. In general,
we want to follow the same particle in the so-called Lagragian description.
In this case, show that
(c)dp/dt=-▽(V+Vq),where Vq=(-h^2/2m∣ψ∣)▽^2∣ψ∣
[瓶頸] (寫寫自己的想法，方便大家為你解答)
(a)小弟的看法是如果不是實數的話，那麼波函數在無窮遠處會發散，無法進行歸一化
所以不可以為實數，不知道這樣有無錯誤??
(b)小弟則是利用平面波特質ψ =∣ψ∣exp(iS/h)，令S=h(Kr-ωt)
→▽S=kh
(c)小題小弟幾乎毫無頭緒，只能從第二題的提示出發
▽S=p, ▽ψ=exp(iS/h)▽∣ψ∣+∣ψ∣(i/h)exp(iS/h)▽S
→▽S=[▽ψ-exp(iS/h)▽∣ψ∣]/∣ψ∣(i/h)exp(iS/h)
→▽S=(h/i)[(▽ψ/ψ)-(▽∣ψ∣/∣ψ∣)]
=(h/i)▽(lnψ/∣ψ∣)
帶入c小題原式
又E=ihd/dt=(-h^2/2m)▽^2+V
→d/dt=(1/ih)(-h^2/2m)▽^2+V
→dp/dt=→d▽S/dt=[(1/ih)(-h^2/2m)▽^2+V]*(h/i)▽(lnψ/∣ψ∣)
=-▽[(-h^2/2m)▽^2+V]*(lnψ/∣ψ∣)
=-▽[(-h^2/2m)▽^2(lnψ/∣ψ∣)+V(lnψ/∣ψ∣]
到這邊就卡住了= = 回不去c小題的原式
請問小弟有哪裡做錯，或者根本就不是這樣做= =
請版上高手指正~非常感謝
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