This page is a tutorial for the 2D Fast Fourier Transform Applet: (I suggest openning it in a new window by right-clicking or control-clicking so you can read this tutorial.)

http://www.brainflux.org/java/classes/Schrodinger1D.html

In classical mechanics of a point particle, you are given a potential (say a harmonic well) and initial position & velocity for the particle. Newton's laws tell you how it evolves in time.

In quantum mechanics, you can work with a point particle in the same potential , but

• The particle has no definite position or velocity, only a single smeared out wave function .
• The wave function is a complex number which is useful to think of as a magnitude and phase rather than a real and imaginary part, which is how I'll usually plot it.
• The magnitude squared is the probability that you will find particle at if you measure its position. Of course is a continuous function (this is not where the quantum part of quantum mechanics comes in) and the probability of finding it exactly at one place is infinitesimal. Mathematically, is actually a probability density. The probability of finding the particle between and is .
• can be narrowly spiked around a particular value making the position look like that of a classical particle. It won't act like one because as we'll see below, as soon as things start evolving with time, it will spread out.
• can be peaked around two places. This doesn't mean there are two particles or many particles, but that when you measure the position of the single particle, you're half as likely to find it here as there.
• What about the velocity? There is not separate velocity function that describes the spread in velocities. Unlike classical mechanics, all the information about the particles is encoded in its wave function . Specifically, the momentum is given as the gradient of the wave function . This leads to consequences like the uncertainty principle: If you know its position well, the wave function is very spiky, so its gradient has very large opposing values. As soon as the state evolves, it will spread out.

What about the phase? Since the probability is given by the magnitude of the wave function, is the phase important and physical? Yes, but in the following ways:

• Momentum is the gradient of the wave function, and that measures the change of both magnitude and phase. For a given magnitude distribution, I can adjust the phase to give it different momentum distributions. vs
• Time Evolution: The Schrödinger Equation knows about both magnitude and phase and the phase is important for the time development.
• Interference: In situations like a double slit where the wave function is split and recombined, it's the phase difference that makes the interference pattern, not the absolute phase. Two wave packets going toward each other and interfering:
• Gauge Freedom:
  • You can add any overall constant to the phase without affecting anything
  • You can add a time-dependent constant to the phase, but this changes what you mean by the "zero" of the potential energy in a very specific way.
  • Quite confusingly, you can add a position-dependent phase too, and this corresponds to a specific change of gauge in the magnetic potential in a specific way.