Allowed energy for a potential in quantum mechanics

[happyparticle]

Homework Statement

I have to find the different energies allowed for a specific potential

Relevant Equations

$\sqrt{\frac{2m(E + V_0)}{\hbar^2}} = tan(ka)k$

Hi,
I'm working on a problem where I need to find the different energies allowed for a potential, and I found this link https://quantummechanics.ucsd.edu/ph130a/130_notes/node151.html,
which is similar of what I'm doing. I'm using mathematica to find the values of E.

However, I'm not sure how to find the value of E.
For a bound state $E < V(\infty)$. I'm wondering how to pick a value for E.
If $V = 0.1$ should I pick all E from 0 to 0.1? Then without using a graph, how will I know if this is a bound state for a specific value of E?

Thank you

 

[vela]

What specific potential are you looking at? What's $k$ defined as?

 

[happyparticle]

I'm looking for $V_0 = 0.1$ and $k = \sqrt{\frac{2m(E+V_0)}{\hbar^2}}$, $\kappa = \sqrt{\frac{-2mE}{\hbar}}$

$\kappa = tan(ka)k$
$\kappa = -cot(ka)k$

 

[vela]

I find it easier to solve for $k$ numerically, rather than $E$. Also, I prefer to multiply the last two equations by $a$, so everything is in terms of $ka$ and $\kappa a$.

Note that $(ka)^2 + (\kappa a)^2 = R^2$ where $R^2=\frac{2mV_0a^2}{\hbar^2}$. That's the equation of a circle in the $(ka, \kappa a)$-plane.

What you want to do is find where the graph of $\kappa a = (ka) \tan ka$ (or $\kappa a = -(ka) \cot ka$) intersects the circle. The intersections correspond to the allowed values of $ka$.

 

[happyparticle]

What the intersections means physically? I mean, at the intersection I will have a value for $E$. What this value means?

 

[vela]

What the intersections means physically? I mean, at the intersection I will have a value for $E$. What this value means?

The intersection corresponds to a solution of the equations. The corresponding value of $E$ would then be one of the allowed energies for the particle in the well.

 

[happyparticle]

Thank you! I know a bit more what I'm doing.
I have another question though.

We can see on the page I gave you. "There is always one even solution for the 1D potential well"
I'm not sure to know why and why there is not always an odd solution? I'm not sure to see the conditions.

Edit:
I'm trying to solve this equation with mathematica for some values, but I got only one imaginary value for E.
I'm not sure if the problem is my code, mathematica or my equation.
I opened a post in mathematica section, I hope is fine.

 

[DrClaude]

We can see on the page I gave you. "There is always one even solution for the 1D potential well"
I'm not sure to know why and why there is not always an odd solution? I'm not sure to see the conditions.

The ground state will always be even (adding nodes to the wave function always increases the energy), and one can show that such a finite square well has always at least one bound state, no matter how narrow or shallow it is.

 

[happyparticle]

I tried with the even function for a NO molecule. I should have at least one solution, but again I don't have one. I fixed 2 errors with the code, for Cot and h. v and h are in eV. I'm not sure what's wrong.

 

[DrClaude]

I tried with the even function for a NO molecule. I should have at least one solution, but again I don't have one. I fixed 2 errors with the code, for Cot and h. v and h are in eV. I'm not sure what's wrong.

I don't get what you are doing. NO is not a square well potential.

 

[happyparticle]

I should have few solutions for a NO molecule with the corresponding value for the constants. Am I clear?

 

[vela]

I tried with the even function for a NO molecule. I should have at least one solution, but again I don't have one. I fixed 2 errors with the code, for Cot and h. v and h are in eV. I'm not sure what's wrong.

I seem to recall the even solutions correspond to the relation involving $\tan$, not $\cot$.

 

[DrClaude]

I should have few solutions for a NO molecule with the corresponding value for the constants. Am I clear?

I still don't get how a square well can approximate a NO molecule, but lets accept it for a moment.

As I wrote in the other thread, with the given parameters there doesn't appear to be any bound solution.

Looking a bit more at what you have in the other thread, I see a couple of problems. If I get it right, you use eV for energy, m for length, and kg for mass. These units are incompatible, as kg m² s^-2 ≠ eV.