Solving the Schroedinger Equation

The funny thing about a differential equation (including the Schroedinger equation) is that its solution is, in and of itself, another equation. For example, the differential equation for the position of a macroscopic object that is moving in one dimension looks like this: . Here x(t) is the object’s position at time t, a represents how much the object is accelerating, and v₀ is the object’s original velocity. The “solution” to this differential equation is an algebraic equation that tells you the object’s position as a function of time in terms of a, v₀, and the object’s original position x₀: x(t) = x₀ + v₀t + ¹⁄₂at².

In other words, the solution to a differential equation is not just a plain old number like the solution to an algebraic equation. The solution is a mathematical function, the value of which can vary throughout time and space. What’s more, there is often more than one mathematical function that solves the differential equation, which we’ll see has particular significance in quantum physics.

The Schroedinger equation describes a certain relationship between a wave function, its derivatives, and the potential energy, plus some constants thrown in for good measure. For a given potential energy function, only certain wave functions will satisfy the requirements embodied in the Schroedinger equation. That is to say, only certain functions will be solutions to the Schroedinger equation.

The trick is to find which wave functions will work, and to find all such wave functions if possible. Mathematicians have given us a variety of techniques to use, which vary depending on the exact form of the potential energy we are working with. We won’t go into the details here. In many cases, we may only be able to get approximate solutions, using analysis methods that scientists have worked hard to develop over many years.

Sometimes, we may even resort to guessing. If you already have a wave function (e.g., a trial solution), it is relatively easy to find the derivatives of that function that are specified by the Schroedinger equation. Easier, that is, than working the other way and solving the Schroedinger equation to find all the wave functions that satisfy it.

That’s why it is often useful to make a guess and see if it works. If you make an educated guess, that means you have a trial wave function Ψ(x,t) that is a definite, complex function of x and t. It is then a simple matter to find the appropriate derivatives, and plug them in to the Schroedinger equation and either confirm your guess or throw it out if the two sides can’t be made equal. It is a little surprising how often that is exactly what physicists do when exploring new quantum mechanics.

QUANTUM LEAP

In terms of quantum wave mechanics, Schroedinger’s equation is as fundamental as it gets. It can’t be “derived” from anything more profound, and all quantum mechanical behavior can be calculated from it. In this sense, it is the quantum analogue to Newton’s laws of motion and Maxwell’s equations, which are the fundamental bases for classical mechanics and electrodynamics, respectively.

Even if we manage to find a wave function that satisfies the Schroedinger equation, we’re not necessarily out of the woods. After all this is physics, not just math, so there are often other conditions that nature imposes which are outside of the Schroedinger equation itself. If a wave function is to be a physically relevant solution, it must also obey what we call boundary conditions.

This is especially true in cases where a particle’s wave function spans an area that has different potential energy functions throughout. There will be different wave functions for each potential energy region, and the wave functions must join up seamlessly at the boundaries in between.

DEFINITION

Boundary conditions are specific conditions that must be satisfied by any solution to a differential equation in order for the solutions to be a valid description of the real physical world.

Another logical boundary condition is that the sum of the wave function squared (i.e., the probability density) added over all of space must be exactly 100 percent. This is because there is 100 percent probability that the particle is located somewhere in all of space. Other types of boundary conditions might mean that the particle has a certain average location at a given time, or that it is confined to a certain range of locations, etc.

For a given potential energy function (which means a certain configuration of forces), only particular wave functions Ψ(x,t) will satisfy the Schroedinger equation and the appropriate boundary conditions. There could be many of these functions, but if they exist, they are all different and distinguishable. We might label the different solution wave functions with subscripts, i.e., Ψ₁, Ψ₂, Ψ₃, etc.

Remarkably, it turns out that not only are these individual wave functions solutions to the Schroedinger equation, but so is any arbitrary sum of them. For example, if Ψ₁ and Ψ₂ are solutions, then so is (Ψ₁ + Ψ₂) or (Ψ₁ – Ψ₂) or even (7Ψ₁ + 42Ψ₂). Combinations of this sort made through addition and subtraction (sometimes with weighting factors thrown in) are referred to as linear combinations.