||By selecting the buttons on top or typing in the text boxes, you can
specifiy a one dimensional potential and an initial wavefunction
which will go into the time-dependent Schrödinger Equation:
After you type in you initial expressions, hit the reset button (the red square)
to sample them into the data area.
|Meaning of the Wavefunction|
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
- 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
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
- 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
- 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
- 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
- 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.