Up to here, conditions have been fitted by hand to fix the wavelength. A whole number of waves fitting around a circumference, the wave reaching 0 at both ends, and so on.
Both could be counted because the place had a simple, fixed shape. In the general case, where the potential energy differs from place to place, the wavelength itself changes from place to place and there is no counting it.
What is needed is a single equation that fixes the shape of the wave once the shape of the potential energy is handed to it. It is the same role the Newtonian equation of motion played in fixing the path once the force was handed to it. That equation is the Schrödinger equation.
The function expressing the shape of the wave is called the wave function and is written \psi. \psi itself is not a quantity that can be measured. What can be measured is the probability of finding the particle there.
The probability of finding it around the position x is proportional to |\psi|^2. It is the same rule as reading the brightness of the fringes in the double slit.
The particle is somewhere, so summing |\psi|^2 over the whole gives 1. Fixing it this way is called normalization.
Written in the form that does not depend on time, it is -\dfrac{h^2}{8\pi^2 m}\dfrac{d^2\psi}{dx^2} + U\psi = E\psi.
The first term on the left is the bending of \psi. Rewriting it as \dfrac{d^2\psi}{dx^2} = \dfrac{8\pi^2m}{h^2}(U - E)\psi makes the meaning readable.
Where E > U, \psi bends toward the axis. It goes too far and comes back, which is to say it oscillates. Where E < U it bends away from the axis, taking a form that grows or shrinks exponentially. The difference between the two forms seen in tunneling was this difference of sign.
The equation has a solution for whatever E is put into it. Most of those solutions, however, grow without limit far away.
To mean anything as a probability, the solution has to fall to 0 far away. Only a few discrete values of E meet this condition.
This is how discrete energy comes out. Not as an assumption placed by hand as Bohr did, but as the result of selecting the solutions that mean something.
Try solving it for a well of width L. Inside, U = 0, so \dfrac{d^2\psi}{dx^2} = -\dfrac{8\pi^2mE}{h^2}\psi, and the solution takes the form of a sine.
For it to be 0 at both ends, the width has to be a whole number of half wavelengths. That is exactly the condition placed by hand in the previous article. This time it came out of the equation itself.
From there E_n = \dfrac{n^2h^2}{8mL^2} follows, the same answer as the previous article. A condition that was being fitted by hand comes out of a single equation together with the demand that the solution mean something. That is the role of the Schrödinger equation.