# Energy levels potential well

#### Cerkit

Joined Jan 4, 2009
287
When using a numerical method to solve for the energy levels of a finite potential well I am using the method of checking to see when the wavefunctions match inside and outside the well. When they are equal then there is an energy level. Do I need to equate the two functions or do I need to equate their derivatives in order to evaluate the energy levels?

#### steveb

Joined Jul 3, 2008
2,436
When using a numerical method to solve for the energy levels of a finite potential well I am using the method of checking to see when the wavefunctions match inside and outside the well. When they are equal then there is an energy level. Do I need to equate the two functions or do I need to equate their derivatives in order to evaluate the energy levels?
Both the wave function and its derivative must be continuous throughout space, including across the boundary.

#### Cerkit

Joined Jan 4, 2009
287
How am I to check that both those are satisfied. If I am to do Eq1-Eq2 with Eq1 being the wavefunction inside the well and Eq2 outside to see when they are equal am I also meant to to have a scan alongside that to that ensure the dervatives are equal as well?

#### steveb

Joined Jul 3, 2008
2,436
How am I to check that both those are satisfied. If I am to do Eq1-Eq2 with Eq1 being the wavefunction inside the well and Eq2 outside to see when they are equal am I also meant to to have a scan alongside that to that ensure the dervatives are equal as well?
The method is usually to write the equations so that the wave equation is forced to be equal. Then check the derivative. Is the inside greater or less than the outside? This tells you whether your guessed energy level is too high or too low.

#### Cerkit

Joined Jan 4, 2009
287
I believe we have discussed this in a previous thread. I have an equation for the even parity energy levels √((V-E)/E)=tan((√2mE)L/hbar). This was derived taking into account continuity of the wavefunction and continuity of its derivative. Therefore if I find values of E that satisfy this equation then surely I have located an energy level? Does this sound right?

#### steveb

Joined Jul 3, 2008
2,436
I believe we have discussed this in a previous thread. I have an equation for the even parity energy levels √((V-E)/E)=tan((√2mE)L/hbar). This was derived taking into account continuity of the wavefunction and continuity of its derivative. Therefore if I find values of E that satisfy this equation then surely I have located an energy level? Does this sound right?
It sounds right, but I can't be sure unless I dive into the details and check it over carefully. I don't have time to do this and it shouldn't be necessary.

You have a way to check whether or not you end up with the right method and correct values. Just plot out the wavefunction with your calculated energy levels and look at it. The boundary should show continuity and no kink in the slope. Your eyes will be pretty good at noticing a problem. If you want to check to high precision, simply print the values of the wavefunction and the derivative of the wavefunction on each side of the boundary. They should match. If they dont' match, then there is a mistake in your theoretical equations, or in your computer program.