Transcendental function and a finite well

[Taylor_1989]

Homework Statement

I am not having trouble with this question as such more trying to get to grips with the intuition of what the question is implying, also I believe there is a mistake in the question as the solutions give to not mange the given energy condition, they state that $V_0>E$ where I believe it should be $V_0<E$ for the solution that they have given.

The question is displayed below:

Homework Equations

The Attempt at a Solution

My issue with part d) is how can I say for a finite well the amount of solutions as when you find the trans-dental equation the number or solution correspond to the points of intersection which correspond to the depth/width of the well. It also dose depend on the $V_0$ as when you parametrise the transcendental the radius of the circle depends on $V_0$ so by increasing the potential you increase the radius thus increase the number of energy levels. But What I don’t understand it the regards to the infinite as there are infinite amount of energy levels due to the infinite potential.

Am I reading this question correctly or have I missied understood the context.

 

[kuruman]

Yes, the solutions given are for bound states. This would mean that $E<V_0$. In part (d) you are not asked to count how many bound states, but to discuss on what this number depends. This is implicit in the transcendental equation which you are asked to interpret and the way it is solved. For example, does a skinny but deep well have more energy levels than a broad and shallow well? What must relation between parameters must be true in order to have more bound states? Once you acquire that insight you can extend to the infinite well.