Linear Potential Energy Eigenfunction for Unbound State

[Domnu]

Problem

A particle in one dimension is subjected to a constant force derivable from

$$V = \lambda x$$​

Is the energy spectrum continuous or discrete? Write down an approximate expression for the energy eigenfunction specified by $$E$$.

Solution attempt
We have an unbound state, so we would have a continuous energy spectrum. Well, I was thinking of something along the lines of $$\psi(x)=e^{-f(\lambda) x} \sin (x-e^{-h(\lambda)})}$$ for the region $$x < E/\lambda$$ (of course, for $$x > E/\lambda$$, we need an exponentially decreasing function); I chose this function based on the following:

1. The function needs to increase exponentially as we go farther left, because the energies are greater than the linear potential by a greater degree.
2. The zeroes need to be bunched together closer as $$x\rightarrow -\infty$$, because the particle is more energetic here.

Above, $$f(\lambda), g(\lambda)$$ are positive functions for $$\lambda > 0$$.

Is the above correct?

 

[Avodyne]

The energy spectrum is continuous, but your guess as to the form of the solution is wrong.

Try writing down the solution for a constant potential (in each region, V<E and V>E), and then plugging in V(x) in place of the constant V.

Also, if you've studied the WKB approximation, this is a good place to apply it. (If not, ignore this comment.)

(I'm going offline for a week, so won't reply further, good luck!)

 

[Domnu]

Oh yeah, heh, I forgot about the WKB. Thanks for the help!