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 *ψ _{E}* to zero at

*x*= –5 nm (well to the left of the normal plot region), and

*ψ*at –4.99 nm to be an arbitrary small value. The calculation terminates when

_{E}*ψ*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,

_{E}*ψ*(

_{E}*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

*ψ*(

_{E}*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 ℏ*ω*
is proportional the the square root of the oscillator's spring constant, which (when the
oscillating mass is an electron) is *k _{s}* = (13.123 eV

^{–1}nm

^{–2}) (ℏ

*ω*)

^{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

*ψ*or

_{E}*dψ*to zero at

_{E}/dr*x*= 0, respectively. In the

**Hydrogen**case, the solutions are spherically symmetric eigenfunctions for the –

*ke*

^{2}

*/r*potential energy function, and |

*ψ*(

_{E}*r*)|

^{2}

*dr*gives the probability of finding the electron between

*r*and

*r*+

*dr*. For reasons beyond our scope,

*ψ*must go to zero at

_{E}*r*= 0 in this case.

Written (in javascript) by Tom Moore, using the open-source
Plotly.js library. Please
send bug notices via the
*Six Ideas* website.
(Please note that changes in various algorithms mean that energy eigenvalues will not be precisely
the same as in previous versions.)