[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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