Energy = eV
This SchroSolver app solves the one-dimensional time-independent Schrodinger equation for electrons subject to selected potential energy functions. Choose a potential energy function using the Configuration popup, and (optionally) adjust its characteristics using the controls below it. Then enter a value for the energy in the textbox, and press the return or enter key to display the solution. The horizontal red line on the upper graph displays the system's conserved energy. Most energies will yield physically unreasonable solutions where the solution goes off to either positive or negative infinity. By trial and error, one can find specific energy values (eigenvalues) that yield physically reasonable solutions (energy eigenfunctions) that go to zero for large values of x.
The app uses uses the method described in chapter Q12 in unit Q of the Six Ideas That Shaped Physics text (3/e, McGraw-Hill, 2017) to generate the solution. Except for the hydrogen and symmetric well cases (see below), the app initializes the calculation by setting ψE to zero at x = –5 nm (well to the left of the normal plot region), and ψE at –4.99 nm to be an arbitrary small value. The calculation terminates when ψE in the right forbidden region has a magnitude at least three times its largest magnitude in the allowed region, meaning that it is definitely heading toward plus or minus infinity. At the end of the calculation, ψE(x) is rescaled to make it as large as possible on the lower graph. To see the full solution, click the Full Solution checkbox. Note that by scanning the mouse along the solution curve you can see the values of ψE(x), so you can check by hand the validity of the computer's calculation.
After you have manually found a few energies that yield reasonable solutions, you will appreciate the Find button, which does a search to find the next higher energy eigenvalue. Please note that though the "find" algorithm and and manual searching yield very precise energy eigenvalues, the numerical method for solving the Schrodinger equation is only approximate, so the values are only accurate (that is, reflect the true energy eigenvalues for the given potential energy function) to about three significant figures.
In the Oscillator case, note that ℏω is proportional the the square root of the oscillator's spring constant, which (when the oscillating mass is an electron) is ks = (13.123 eV–1nm–2) (ℏω)2. In the Symmetric Well case, solutions are either symmetric (the solution to the left of x = 0 is the mirror image of that on the right) or antisymmetric (the solution on the left is inverted). Select which you want using the popup menu. In this case, the app initializes the calculation by setting ψE or dψE/dr to zero at x = 0, respectively. In the Hydrogen case, the solutions are spherically symmetric eigenfunctions for the –ke2/r potential energy function, and |ψE(r)|2dr gives the probability of finding the electron between r and r + dr. For reasons beyond our scope, ψE must go to zero at r = 0 in this case.