Quantum Well Levels: Energy Changes with Depth

[dacs]

When the depth of a quantum well increases, the first level increases or decreases its energy with respect of the bottom of the quantum well?

 

[Meir Achuz]

For a square well, E-V increases as V gets more negative.

 

[dacs]

Then, I understand that the first energy level increases its energy when the well is more profound. What happens with the number of energy levels inside the well? When the well is more profound, the number of levels increases?

 

[jtbell]

The classic infinitely-deep well that students of QM always learn about first, has an infinite number of energy levels.

 

[dacs]

Yes, you are right jtbell, thank you. But my concern is about wells with finite depth, i.e. square wells.

 

[cmos]

Then, I understand that the first energy level increases its energy when the well is more profound. What happens with the number of energy levels inside the well? When the well is more profound, the number of levels increases?

For constant well width, yes, increasing the depth of the well will increase the energy level of the first eigenstate with respect to the bottom of the well. However, this increase is asymptotic to $$\pi^2\hbar^2/2mw^2$$, where w is the width of the well. I'm sure you can reason out why.

And yes, as the well becomes deeper, you will increase your number of bound states (in quantized steps of course).

 

[dacs]

Yes! When the well is more and more profound, the first level of the square well tends to the first level of an infinitely-deep well. For constant width, obviously.
Thanks, cmos.

 

[jtbell]

But my concern is about wells with finite depth, i.e. square wells.

When a well with finite depth becomes deeper and deeper, it becomes more and more like an infinite square well.