Electron moving in a constant force field

The time dependent system of a particle experiencing a constant force is more difficult to treat theoretically than the case of zero potential. You can obtain exact solutions to Schrodinger's equation when the t=0 wave function is proportional to a Gaussian. Hints for obtaining exact solutions are given at the bottom of the page.
In this movie the wave function for an electron at t=0 is proportional to exp{-[(x-100)/10]^2}exp(-ix) which represents an electron localized at 100 a.u. of distance with a velocity of 1 a.u. to negative x. One frame from the movie is given below.


1.1 Mb MPEG movie of an electron moving in a constant force field

When watching the movie, note how the packet behaves like you might expect from classical mechanics. The packet initially moves to negative x; its speed slows with time until it stops; after this it accelerates to positive x with its speed increasing with time. This is how a classical particle behaves when it experiences a constant force in the positive x direction. In fact, it is possible to show that the packet moves exactly the same as a classical distribution (if they have the same position and momentum distribution at t=0). (Hints for showing this are given below.) You should notice how the packet's height decreases and width increases with time. Again this is from dispersion of the packet caused by the high energy components of the packet moving faster than the lower energy components; also the higher energy components can move to smaller x than the lower energy components. A peculiar aspect from launching the packet up potential is that the high energy components are at the back of the packet after it begins moving to positive x; this should make sense by analogy with classical mechanics. The higher the initial velocity the longer it takes the particle to return to the initial position. Finally, you should ask yourself if the movie is roughly consistent with the information I've given. For example, to how small an x could a classical electron travel if it starts with a velocity of -1 at x=100 atomic unit and how does this compare with the smallest x that the peak of the packet reaches.

There is one final feature you might want to ponder. On the previous page, the packet showed an interference type of structure while reflecting from the infinite wall. Why is this interference missing for the packet reflecting from the linear potential? Don't you still have the interference between the parts of the packet with positive and negative velocity?

There are bunch of methods for obtaining the time dependent wave function for a linear potential. One method is to use momentum space wave functions; instead of using p=-i hbar d/dx in Schrodinger's equation use x=i hbar d/dp. Another method is to guess that if you start the wave function as a Gaussian it will stay a Gaussian for all times. So you guess psi(x,t) = exp[ a_2(t) x^2 + a_1(t) x + a_0(t)] and substitute this into Schrodinger's equation; the time dependent coefficients a_2, a_1, a_0 are then the solution of first order, nonlinear differential equations. You might also guess that the Gaussian moves like a classical particle and try psi(x,t) = exp{ b_2(t) [x-x_cl(t)]^2 + b_1(t) [x-x_cl(t)] +b_0(t)] where the x_cl(t) is the classical trajectory of a particle with the initial position and momentum (maxima) from the initial packet.

To show that the quantum distribution moves exactly like the classical distribution, show that the differential equation for the time dependence of the probability distribution and current distribution is equivalent to the differential equations for the corresponding classical quantities. If the differential equations are the same and the initial conditions are the same then they must be equivalent for all time.