Finite Well Potential - Unbound eigenfunction

[quanticism]

Homework Statement

From qualitative arguments, make a sketch of the form of a typical unbound standing wave eigenfunction for a finite square well potential.

An unbound particle is one which has total energy E greater than the Potential V of the well

Is the amplitude of the oscillation the same in all regions?

Homework Equations

Using the Time Independent Schrödinger Equation we see that:

The Attempt at a Solution

Inside well: Solution is of form $$Asin(k_0x)$$ where $$k_0 = \sqrt{2mE/\hbar^2}$$

Outside well: Solution is of from $$Bsin(k_1x + \phi)$$ where $$k_1 = \sqrt{2m(E-V)/\hbar^2}$$ where $$\phi$$ is the phase shift which can be adjusted to satisfy the boundary conditions.

So inside the well, the wave number of the eigenfunction should be greater than the eigenfunction outside the well. This should mean that the eigenfunction's frequency inside the well is higher.

I've looked around and I found this http://vnatsci.ltu.edu/s_schneider/physlets/main/finite_sqr_well.shtml" . You can look at the unbound states by clicking and dragging your mouse on the black bar on the right with the green lines.

It suggests that the amplitude of the eigenfunction should decrease when inside the quantum well.

I thought the amplitude of the eigenfunction should give you an indication of the probability of finding the particle in that particular region. If so, shouldn't the amplitude be slightly higher inside the well?

 

[ideasrule]

The intuitive explanation is that inside the well, the particle has more energy relative to the floor of the well. It therefore moves faster, which means it spends less time inside the well than outside.

However, this intuition sometimes fails miserably (i.e. for the lower-energy states of the harmonic oscillator). It's always best to do the math and compare the amplitudes of the relevant waves.