1. Families of finite square wells. A particle of mass m has N quantized energy levels in a one-dimensional square well of depth V_0 and width L. N is more than 10.

a) Approximately how many bound states does the particle have in a well of the same depth but width 2L?
b) Approximately how many states does the particle have in a width L but depth 2V_0?
c) Approximately how many states does the particle have in a depth 2V_0 and width 2L?
d) How many bound states does the particle have in a well of depth V_0 / 4 and width 2L? In a well of depth 4V_0 and width L/2?

Solution Preview

You can solve this problem by using the results for an infinite square well. This is because is there are many bound states in this finite square well, therefore the square well must be very deep. To first approximation, you can then treat it as an infinite square well, as far as the physics inside the well is concerned. However, this approximaton will then start to break down for the bound state with the highest energy. Now, you can estimate using the infinite square well model, at what energy this will happen. The energy eigenfunctions for a particle in an infinite square well of width L can be written as:

psix) = A sin(k x) (1)

where we take the boundaries of the well at x = 0 and x = L. The cos(k x) term is then absent due to the boundary ...

Solution Summary

It is explained in detail how to estimate the number of bound states.

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