- What equation are we solving? We are solving
the time-independent Schrödinger equation for an electron
in a one dimensional potential V(x):
.
Here Psi is the wave function, describing the state of the electron;
x is position in space, m is the mass of the electron,
is Planck's constant h divided by
two pi, and V(x) is a potential energy well.
We will be varying the energy E and looking for
well-behaved solutions.
This is a special form of the general, time dependent Schrödinger
equation
,
suitable for calculating solutions with a particular energy. In quantum
mechanics, an electron can (and often is) in a state without a fixed
energy (just like it doesn't have a fixed position): the time-dependent
equation, like the equation for waves on a string, describes the
evolution of the wave function. The time-dependent equation is very
much like our wave equation (and often is called a wave equation).
Notice that it involves i, the square root of minus one: we've
been avoiding complex numbers in this course, and by specializing to the
time-independent equation we can continue avoiding them. Also, the
time-independent equation is simpler to solve: more like
the pendulum than the wave equation. Indeed, it's very much like
the pendulum: the second derivative
is
proportional to the wave function
, just like the
second derivative of theta with respect to t was proportional
to theta.
- What do the solutions mean?
Notice we aren't explaining here how one derives the time-independent equation
from the time-dependent one: it's a bit complex.
We'll just say that the time-independent equation describes solutions of fixed
energy E: electrons in a fixed energy state keep the same
wave-function, oscillating in phase (the complex number we keep ignoring).
Solving the time-independent solution is like solving for the standing
waves on a string: it doesn't describe all the behavior (like traveling
pulses!) but it gives an excellent description of resonances and
frequency dependence.
Many of you are familiar with electron wave-functions from chemistry.
There one describes, for example, s and p
states of various shapes, which can be occupied by the electron, binding
it to the nucleus. The shapes of these wave-functions in large part
determines the chemical bonding of the atoms. The energies of these
wave-functions determines the frequencies of the light they can absorb
and emit. Unlike waves on a string, each solution does not separately
resonate with light of a particular frequency.
Instead, light is emitted and absorbed when the electron
changes from one state to another, and the frequency of the
light is determined by the difference in energies between the two
solutions:
frequency = (E2-E1)/
In this lab, we'll be determining the energies and shapes of the
solutions.
- How are we finding the solutions?
Solving the time-independent equation is easy, once you know the
boundary conditions and the energy - we use exactly the same method
we used to solve the pendulum's motion. However, for the pendulum,
both initial conditions (position Theta and velocity ThetaDot)
were known at the initial time. For us, we know that
at X=0, and (depending on the
potential) either
at X=Xshelf, or
as X goes to infinity. Because
our boundary conditions involve two different points, this is an
example of a two-point boundary value problem.
We'll solve the problem by "shooting": we pretend we know the derivative
of
at X=0, solve the equation, and
then fiddle around until it has the right value at the other boundary.
For our problem, we'll fiddle around with the energy E:
raising and lowering the energy will change how the electron wave function
acts at the right-hand boundary.
The default potential is a square well, with infinite walls. Try
varying the energy to make the wave-function zero at the right-hand
side! You'll find there are several solutions, with different numbers
of nodes, just like there are several different shapes of wave-functions
for electrons in your chemistry class.
Links Back
- Infinite Square Well
- Square Shelf
- Hydrogen