Suppose the gravitational potential is U=mgy and the ball moves within the xy plane. Start with the Schrodinger equation
-(hbar^2/2m)\nabla^2 \psi + U\psi = i\hbar d\psi/dt.
We assume \psi= exp (i S/\hbar), then the Schrodinger equation becomes,
(1/2m)(\nabla S)^2 + U + dS/dt = i(\hbar/2m)\nabla^2 S.
If we neglect the rhs then this is just a form of the Hamiltonian-Jacobi equation in a gravitational field. The solution is,
S=(-1/3m^2g)(2mE-b^2-2m^2gy)^(3/2) + bx - Et,
where b is the momentum along x and E is the energy. The particle trajectory is given by the following conditions,
dS/dE=const.=c1
dS/db=const.=c2
From these we find (抛物线)
y=y0-(1/2)gt^2+v0*t
where v0=c1*g.
The quantum solution is obtained when we include the rhs term proportional to \hbar. For a macroscopic ball this term is very small and the solution should not be very different from the one given above.
你不就是说，用量子力学解决皮球抛物运动是杀鸡用牛刀是脱裤放屁吗？蠢货，， - @ (450 bytes) 2009-07-26, 22:27:02 (349506)
听清楚了，你的量子力学牛刀杀不了皮球抛物运动这只鸡，连一根鸡毛也割不断，就是把你全扒光了连皮也扒了你用量子力学也放不出一个皮球抛物运动的屁。你这辈子不能，再投六次胎也不能！GOT IT？以为你学过量子力学就能在老子面前忽悠了，也不
撒泡尿先照照自己。我说的话，你要能明白个零头，也就不会长期在这里丢人现眼的了。下次给老子滚远的，懒得理你这蠢货。
立此存照。