How Many Bound States Exist in a Finite Potential Well?

[quasar_4]

# bound states in a given system??

Homework Statement

An electron is confined to a potential well of finite depth and width, 10^-9 cm. The eigenstate of highest energy of this system corresponds to the value ¥=3.2.

a) How many bound states does the system have?
b) Estimate the energy of the ground state with respect to the zero energy line at the bottom of the well. Express answer in eV.

Homework Equations

For the non-dimensional wavenumbers ¥=k*a, W=K*a:

We know that ¥^2 + W^2 = (2*m*|V|)/hbar^2 = p^2, which is a circle of radius p in ¥W space, and the states obey

¥*tan(¥) = W for even energy eigenstates

¥cot(¥) = -W for odd eigenstates.

The intersection of the even/odd eigenstate equation and the circle give the number of bound states, somehow (I guess).

The Attempt at a Solution

I am just confused in general. Our text says that you can locate the number of bound states by finding the intersection of the circle in ¥W space and the ¥tan¥ or ¥cot¥ term, but I don't see how to know how many there are without drawing a picture explicitly. I don't really understand the relationship here and how it translates to bound states or bands.

I thought that since we knew |V| = a = 10^-11 m we could compute the radius of the circle, but my answer just didn't seem reasonable. And I really don't see where to go to get to bound states from here. I tried setting W = sqrt(p^2 - ¥^2)= ¥tan¥, but I am not getting anywhere. Help to even know where to start would be great as I am totally lost!

 

[Jhero]

Hello there! it is important to understand the concept of bound states and how they are related to the energy levels of a system. In this case, we have an electron confined to a potential well of finite depth and width. The eigenstate of highest energy in this system corresponds to the value ¥=3.2. Now, let's address your questions.

a) To determine the number of bound states in this system, we need to look at the intersection of the circle in ¥W space and the ¥tan¥ or ¥cot¥ term. This intersection will give us the number of bound states in the system. In this case, we have a circle of radius p in ¥W space, and the states obey ¥*tan(¥) = W for even energy eigenstates and ¥cot(¥) = -W for odd eigenstates. So, to find the number of bound states, we need to find the number of intersections between the circle and the ¥tan¥ or ¥cot¥ term.

b) To estimate the energy of the ground state with respect to the zero energy line at the bottom of the well, we can use the formula ¥^2 + W^2 = (2*m*|V|)/hbar^2 = p^2. Since we know that ¥=3.2 and |V| = 10^-11 m, we can solve for W. Once we have W, we can use the formula E = hbar^2*W^2/2m to calculate the energy in eV.

I hope this helps you understand the concept of bound states and how they are related to the energy levels of a system. Let me know if you have any further questions or if you need any clarification. Keep up the good work as a scientist!