- The Wave Function
- The Double Slit Experiment Again
- What is an Electron
- Schrödinger's Cat
- The Implications of the Quantum

The state of a physical system is described by a "wave function", usually
denoted by the symbol**.**.
In particular cases, we will know more or less exactly what function this
is - e.g a sine or a cosine, a quadratic expression, etc. This wave function
can depend on time, spatial coordinates, etc. Quantum Theory tells us that
to make calculations about real measurements that could be made on the
system, we must **take the square of the wave function.** The value
so obtained will give us the **probability** of obtaining, through measurement
on the system, a particular value of the quantity we are interested in.
We would expect a theory from Classical Physics to give us an **exact**
value of the quantity we were interested in; here, however, the best we
can do is calculate a probability of obtaining the value. The wave function
is also called a Probability Amplitude, for this reason.

For example, if, on the basis of our knowledge of conditions in which
a particle might find itself (in a box, with a magnetic field, for example)
we knew how to write down the particle's wave function; and let's say this
wave function depended on its position (call that *x*) and the time
measured from some starting time (call that *t*). In that case, we
would write its wave function as **( x,t)**.

Now let us look at the odd way in which Quantum Theory does its calculations
about the world. Suppose we have an experiment about a physical process
which can happen in more than one way, and we know the Probability Amplitude
(or wave function) for each way. To calculate what results we would expect
in an experiment which does not distinguish which way actually happens,
we have to **first** add the Probability Amplitudes; **then** we
square the result of this addition to get the answer to compare to measurement.
If, on the other hand, the experiment does distinguish which way actually
happens, we **square** the Probability Amplitudes **before** adding
them. To see how this works, let's look at the Double Slit Experiment for
electrons.

Suppose we have a set-up which has equal size slits, located at the
same distance from the source of electrons, then the probability that the
electron goes through slit number 1 is equal to the probability that it
goes through slit number 2. We express this fact by writing **1^{2}**
=

Then :
EITHER**1**
= +**2**
and the result is **1** (or 100%);

OR**1** =^{
-}**2**,
and the result is **0** (or 0%).

For the Double Slit Experiment, this is obviously (??) a calculation
of the interference pattern, with its maxima (** 1**, in some arbitrary
units) and minima (** 0**) which we observe. However, if our experiment
has some means for detecting, even in principle, which hole the electron
goes through (Experiment **II**), the result of **this** experiment
must be written as **II^{2}**
=

Thus the Quantum Theory has managed to come up with a recipe to give calculations which agree with the observations we make on this weird world in which we live.

What can we say about the wave function (Probability Amplitude) of the
electron after it has gone through the slit system, but** just before**
we look at it to decide which slit it went through? In this case, Nature
tells us we must write its wave function as **.**
=**1** +**2
**,
as explained above. But if we make a measurement to determine which slit
the electron did go through, we know we must get the result **1**
(if it went through slit number 1) OR**2**
(if it went through slit number 2). Then we say that the wavefunction has
**collapsed**
on to its final value.

According to Schrödinger, the electron can be
represented by a wave-function, which contains all the information we can
know about the particle. If an electron looks like anything we are familiar
with (and it doesn't!!), it comes closest to a small "packet" of waves
confined to a region of space **x**.
This wavefunction obeys a wave equation first written down by Schrödinger.
The **square** of the wavefunct ion gives the probability of finding
it at a given place (and time).

(MATH NOTE: To represent such a function,
we need a superposition of many wave forms, with a "spread" of wavelengths.
Since **p** = **h/**
this implies a corresponding spread in momentum; this can be calculated
to be **p
= h/x **-
as we might have expected from Heisenberg's Uncertainty Principle).