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Let us first look at how we specify the state for a classical system.
Once again, we use the ubiquitous billiard ball. As any player knows, there
are three important aspects to its motion: position, velocity and spin
(angular momentum around its centre). Knowing these quantities we can in
principle (no friction) predict its motion for all times. We have argued
before that quantum mechanics involves an element of uncertainty. We
cannot predict a state as in classical mechanics, we need to predict a
probability. We want to be able to predict the outcome of a measurement
of, say, position. Since position is a continuous variable, we cannot
just deal with a discrete probability, we need a probability density, To
understand this fact look at the probability that we measure x to be
between X and
X +
X. If
X is small enough, this
probability is directly proportional to the length of the interval
P(X < x < X + X) = (X) X |
(3.1) |
Here
(X) is called the probability density. The standard statement
that the total probability is one translates to an integral statement,
dx (x) = 1 |
(3.2) |
(Here I am lazy and use the lower case x where I have used X before;
this a standard practice in QM.) Since probabilities are always positive,
we require
(x)
0.
Now let us try to look at some aspects of classical waves, and see
whether they can help us to guess how to derive a probability density from
a wave equation. The standard example of a classical wave is the motion
of a string. Typically a string can move up and down, and the standard
solution to the wave equation
can be positive as well as negative. Actually the square of the wave
function is a possible choice for the probability
(this is proportional to the intensity for radiation). Now let us try to
argue what wave equation describes the quantum analog of classical
mechanics, i.e., quantum mechanics.
The starting point is a propagating wave. In standard wave problems this
is given by a plane wave, i.e.,
= A exp(i(kx - t + )). |
(3.4) |
This describes a wave propagating in the x direction with wavelength
= 2
/k, and frequency
=
/(2
). We interpret this
plane wave as a propagating beam of particles.
If we define the probability as the square of the wave function,
it is not very sensible to take the real part of the exponential:
the probability
would be an oscillating function of x for given t. If we take the
complex function
Aexp(i(kx -
t +
)), however, the probability,
defined as the absolute value squared,
is a constant (| A|2) independent of x and t,
which is very sensible for a beam
of particles.
Thus we conclude that the wave function
(x, t) is complex,
and the probability density is
|
(x, t)|2.
Using de Broglie's relation
p = / , |
(3.5) |
we find
p = k. |
(3.6) |
The other of de Broglie's relations can be used to give
One of the important goals of quantum mechanics is to generalise
classical mechanics. We shall attempt to generalise the relation
between momenta and energy,
E = mv2 =  |
(3.8) |
to the quantum realm.
Notice that
p (x, t) |
= |
k (x, t) =   (x, t) |
|
E (x, t) |
= |
  (x, t) =  (x, t) |
(3.9) |
Using this we can guess a wave equation of the form
Actually using the definition of energy when the problem includes a
potential,
E = mv2 + V(x) =
+ V(x) |
(3.11) |
(when expressed in momenta, this quantity is usually called a
"Hamiltonian") we find the time-dependent Schrödinger equation
-   (x, t) + V(x) (x, t) =  (x, t). |
(3.12) |
We shall only spend limited time on this equation. Initially we are
interested in the time-independent Schrödinger equation, where the
probability
|
(x, t)|2 is independent of time. In order to reach
this simplification, we find that
(x, t) must have the form
(x, t) = (x)e-iEt/ . |
(3.13) |
If we substitute this in the time-dependent equation, we get
(using the product rule for differentiation)
- e-iEt/   (x) + e-iEt/ V(x) (x) = Ee-iEt/ (x). |
(3.14) |
Taking away the common factor
e-iEt/
we have an equation for
that no longer contains time:
  (x) + V(x) (x) = E (x). |
(3.15) |
Next: 3.2 Operators
Up: 3. The Schrödinger equation
Previous: 3. The Schrödinger equation