*Revised* Possible bound states of a one-dimensional square well

In summary, to find the number of bound states for a potential with p(max) = 4, we need to determine the number of different values of n that satisfy the condition k=nπ/a within the range of 0 to 4. This will give us the total number of bound states for the potential.
  • #1
messedmonk18
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Homework Statement


Find the solutions of even and odd parity from the transcendental equations then find the number of bound states that are possible for a potential such that p(max) = 4?


Homework Equations


p=ka/2 & p(max)^2 = (u(not)a[tex]^{2}[/tex]/4), u(not) = [tex]\overline{2m(not)}[/tex][tex]\underline{\hbar^{2}}[/tex]V(not)

I've found that for Even parity: p tan(p)= [tex]\sqrt{p(max)^{2}-p^{2}}[/tex]

Odd: -p cot(p)= [tex]\sqrt{p(max)^{2}-p^{2}}[/tex]



The Attempt at a Solution



So after I've found the Even and Odd solutions from a lot of algebra I'm completely lost on how to find the number of bound states. I assume that this has to do with integers of k but I'm not sure what this all means and how to derive a "bound" state from the information given. I need a lot of help... or at least some just to get started!
 
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  • #2


Thank you for your post. It seems like you have made good progress in finding the solutions for even and odd parity from the transcendental equations. To find the number of bound states, we need to understand what a bound state is and how it relates to the equations you have found.

A bound state is a state in which a particle is confined to a finite region of space due to the presence of a potential. In other words, the particle is "bound" to the potential and cannot escape. In terms of the equations you have found, a bound state occurs when the value of p is restricted to certain values, as determined by the potential. This can be seen by setting the right-hand side of your equations to a constant value, which represents the potential.

To find the number of bound states, we can use the fact that the value of p is related to the wave vector k. In general, k can take on any real value, but for a bound state, it must be restricted to certain values. This means that k must be quantized, or take on discrete values. The number of bound states is then determined by the number of allowed values of k.

In your equations, you have the parameter p(max), which represents the maximum value of p for a given potential. This means that the allowed values of p are restricted to a range from 0 to p(max). To determine the number of bound states, we can use the fact that the allowed values of k must satisfy the condition p=ka/2. This means that the allowed values of k are given by k=nπ/a, where n is an integer. The number of bound states is then given by the number of different values of n that satisfy this condition within the range of 0 to p(max).

I hope this helps to clarify the concept of bound states and how they relate to the equations you have found. Good luck with your further analysis!
 

1. What is a one-dimensional square well?

A one-dimensional square well is a theoretical model used in quantum mechanics to describe the behavior of a particle confined to a one-dimensional space, such as a particle moving along a line or a particle in a narrow tube. It is characterized by a potential energy function that is constant within a certain range and infinite outside of that range.

2. What are bound states in the context of a one-dimensional square well?

Bound states refer to the energy levels at which a particle can exist within the potential well. These are states in which the particle is confined to the well and cannot escape, as opposed to unbound states where the particle can freely move in and out of the well. Bound states in a one-dimensional square well are typically quantized, meaning they can only take on certain discrete energy values.

3. How does the potential depth of the square well affect the bound states?

The depth of the potential well determines the number and energy levels of the bound states. A deeper potential well will have more bound states, while a shallower potential well will have fewer bound states. The energy levels of the bound states also increase with increasing potential depth.

4. Can a particle in a one-dimensional square well have multiple bound states?

Yes, a particle in a one-dimensional square well can have multiple bound states. The number of bound states depends on the potential depth, with a deeper well allowing for more bound states. However, there is a maximum number of bound states that can exist for a given potential well depth, which is determined by the size of the well.

5. Are there any applications for the concept of bound states in a one-dimensional square well?

Yes, the concept of bound states in a one-dimensional square well has important applications in fields such as quantum computing, where confinement of particles is crucial for manipulating and storing information. It is also used in various areas of physics, such as studying the behavior of electrons in semiconductors and understanding the properties of atoms and molecules.

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