问一道量子力学的题Consider a particle of mass in the state |n> of a b

来源：学生作业帮编辑：作业帮分类：英语作业时间：2024/11/05 18:31:06

问一道量子力学的题
Consider a particle of mass in the state |n> of a box of length L.Find the force F=-dE/dL（偏微分,那个符号打不出来用d代替了）encountered when the walls are slowly pushed in ,assuming the particle remains in the nth state of the box as its size changes.Consider a classical particle of energy En in this box.Find its velocity,the frequency of collision on a given wall,the momentum transfer per collision,and hence the average force.Compare it to -dE/dL computed above.

粒子的初始能量是
En=h^2n^2/(8mL^2)
因为势垒是缓慢移动的,所以满足绝热近似条件,因此粒子仍处于该本征态上,所以
F=-dE/dL=h^2n^2/(4mL^3)=2En/L
考虑经典粒子
因为En=mv^2/2
所以v=根号下(2En/m)
p=mv=根号下(2mEn)
粒子与势垒碰撞后以相同的速度反弹回去,所以与一个势垒两次碰撞的间隔是
t=2L/v=2L/(根号下2En/m）
则碰撞频率是
f=1/t=根号下(2En/m)/2L
在一次碰撞中改变的动量是2p,所以单位时间内动量改变量即F是
F=2pf=2根号下(2mEn)*根号下(2En/m)/2L=2En/L
和前面的结果是相同的.

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