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\begin{document}

\section{Probability Conservation}

The conventional interpretation of the wavefunction, $\psi$,
associated with a particle is that $| \psi (x) |^2 = \psi^{*} (x) \psi
(x)$ expresses the relative probability of finding the particle at
point $x$.  (The superscripted asterisk indicates complex
conjugation.)  If particles aren't created or destroyed, then the
probability integrated over the $x$-domain must satisfy
\begin{equation}
	P = \int_{-\infty}^{\infty} \psi^{*} \psi dx = 1,
\label{tprob}
\end{equation}
and should not change with time.  In order to test whether the theory
satisfies this condition, we first need to derive the time-dependent
form of the Schr\"{o}dinger equation.  We can then take the time
derivative of $P$ and see if it is indeed zero.

\subsection{Time-Dependent Schr\"{o}dinger Equation}

The time independent Schr\"{o}dinger equation in one space dimension
for a particle with total (non-relativistic) energy $E$ is
\begin{equation}
	\frac{\partial^2 \psi_E}{\partial x^2} +
	\frac{2m (E - U)}{\hbar^2} \psi_E = 0 .
\label{notime}
\end{equation}
(Note that we have dropped the prime on $E$ for convenience.)  Using
$E = \hbar \omega$, the time-dependent wavefunction is simply
$\psi (x,t) = \psi_E (x) \exp (-iEt/ \hbar )$.

Since the principle of superposition applies to quantum mechanical
waves, the wavefunction for an arbitrary mix of possible energy states
is obtained as a weighted sum of such expressions:
\begin{equation}
	\psi (x,t) = \sum_E W_E \psi_E (x) \exp (-iEt/ \hbar ) ,
\end{equation}
where $W_E$ is the (possibly complex) weight for the wavefunction
associated with energy $E$.  Normally, the time-independent wavefunctions
are normalized such that
\begin{equation}
	\int_{-\infty}^{\infty} | \psi_E (x) |^2 dx = 1 .
\end{equation}
With this normalization, $| W_E |^2$ just becomes the probability of
the particle having energy $E$.  Note that the sum is over just the
{\em possible} energies, i. e., those energies for which equation
(\ref{notime}) and its associated boundary conditions are satisfied.

We now rearrange equation (\ref{notime}) and multiply by $\exp (-iEt/
\hbar )$ to incorporate the time dependence of the wavefunction,
\begin{equation}
	E \psi_E \exp (-iEt/ \hbar ) = -\frac{\hbar^2}{2m}
	\frac{\partial \psi_E \exp (-iEt/ \hbar )}{\partial x^2} +
	U \psi_E \exp (-iEt/ \hbar ) ,
\end{equation}
and then note that the right side can be written $i \hbar \partial
\psi_E \exp (-iEt/ \hbar ) / \partial t$.  Multiplying by $W_E$ and
summing over $E$ finally results in the time-dependent Schr\"{o}dinger
equation:
\begin{equation}
	i \hbar \frac{\partial \psi}{\partial t} =
	-\frac{\hbar^2}{2m} \frac{\partial^2 \psi}{\partial x^2} +
	U \psi .
\label{ytime}
\end{equation}
This partial differential equation governs the fully time-dependent
behavior of non-relativistic particles in one dimension.  (The
three-dimensional version is obtained by replacing $\partial^2 \psi /
\partial x^2$ with $\nabla^2 \psi$.)

\subsection{Probability Flux}

Let us now ask a somewhat more complex question than the one posed
in the introduction.  Suppose we want to know how the probability
of a particle being within some interval $[a,b]$ on the $x$-axis
changes with time.  This probability is obtained by replacing the
infinite limits in equation (\ref{tprob}) with $a$ and $b$:
\begin{equation}
	P_{ab} = \int_a^b \psi^{*} \psi dx .
\end{equation}
Taking the time derivative of this yields
\begin{equation}
	\frac{d P_{ab}}{dt} = \int_a^b \left(
		\frac{\partial \psi^{*}}{\partial t} \psi +
		\psi^{*} \frac{\partial \psi}{\partial t}
	\right) dx .
\end{equation}
The Schr\"{o}dinger equation can now be substituted into the above
formula, yielding
\begin{equation}
	\frac{d P_{ab}}{dt} =
	\frac{\hbar}{2im} \int_a^b \left(
		\frac{\partial^2 \psi^{*}}{\partial x^2} \psi -
		\psi^{*} \frac{\partial^2 \psi}{\partial x^2}
	\right) dx .
\label{intermed1}
\end{equation}
Note that the equation for $\psi^{*}$ is obtained by taking the
complex conjugate of equation (\ref{ytime}), which changes the sign of
$i$ on the left side, resulting in the minus sign in the above
equation.

Further progress comes from using the identity
\begin{equation}
	\frac{\partial}{\partial x} \left(
	\frac{\partial \psi^{*}}{\partial x} \psi \right) =
	\frac{\partial^2 \psi^{*}}{\partial x^2} \psi +
	\frac{\partial \psi^{*}}{\partial x}
	\frac{\partial \psi}{\partial x}
\end{equation}
and its complex conjugate to eliminate the second derivatives in
equation (\ref{intermed1}),
\begin{equation}
	\frac{d P_{ab}}{dt} =
	\frac{\hbar}{2im} \int_a^b \frac{\partial}{\partial x}
	\left(
		\frac{\partial \psi^{*}}{\partial x} \psi -
		\psi^{*} \frac{\partial \psi}{\partial x}
	\right) dx ,
\end{equation}
which can then be easily integrated to yield
\begin{equation}
	\frac{d P_{ab}}{dt} = \frac{\hbar}{2im}
	\left. \left(
		\frac{\partial \psi^{*}}{\partial x} \psi -
		\psi^{*} \frac{\partial \psi}{\partial x}
	\right) \right|_a^b .
\label{final}
\end{equation}

The quantity 
\begin{equation}
	F(x,t) = -\frac{\hbar}{2im} \left(
		\frac{\partial \psi^{*}}{\partial x} \psi -
		\psi^{*} \frac{\partial \psi}{\partial x}
	\right)
\end{equation}
is called the {\em probability flux}.  Notice that $F$ is real even
though $\psi$ is in general complex.  (Prove this by showing that
$F^{*} = F$.)  Equation (\ref{final}) can be written
\begin{equation}
	\frac{d P_{ab}}{dt} = F(a,t) - F(b,t) ,
\end{equation}
which in words says that the rate of change of the probability of
finding a particle within the interval $[a,b]$ equals the probability
flux into the left side of the interval minus the flux out of the
right side.

Notice that where $\psi = 0$, the flux is zero.  It is often the case
that $\psi = 0$ at $\pm \infty$, which means that $d P_{-\infty
\infty} /dt = 0$.  This constitutes a proof of the hypothesis posed
in the introduction; the probability of finding a particle {\em
somewhere} doesn't change with time in non-relativistic quantum
mechanics.  Also note that if $\psi$ is real, so that $\psi^{*} =
\psi$, then $F = 0$.  This explains why the quantum mechanical
wavefunction must in general be complex --- if it weren't, nothing
would ever go anywhere!

\subsection{The Relativistic Case}

Oddly enough, the above proof does not extend to the relativistic
case.  To see this, note that the relativistic form of the wavenumber
squared is
\begin{equation}
	k^2 = \frac{p^2}{\hbar^2} =
	\frac{E^2 - m^2 c^4}{\hbar^2 c^2}
\end{equation}
when no potential energies are involved.  The total energy, $E$,
appears quadratically in this equation as opposed to linearly in the
non-relativistic form, leading inevitably to a second time derivative
of $\psi$ in the relativistic wave equation:
\begin{equation}
	\frac{1}{c^2} \frac{\partial^2 \psi}{\partial t^2} =
	\frac{\partial^2 \psi}{\partial x^2} -
	\frac{m^2 c^2}{\hbar^2} \psi .
\label{kleingordon}
\end{equation}
This is called the {\em Klein-Gordon} equation, and it resembles
the ordinary wave equation of electromagnetism with an additional
term proportional to $\psi$ added on.

The interpretation of $| \psi |^2$ as the probability of finding a
single particle doesn't work, because the spatially integrated value
of $| \psi |^2$ can change with time.  To see this, note that $\psi =
\cos (kx) \cos( \omega t)$ is a solution to equation
(\ref{kleingordon}) if $\omega^2 = k^2 c^2 + m^2 c^4 /\hbar^2$.  The
probability of finding a particle at point subject to this wave
equation oscillates between zero and $\cos^2 (kx)$ with a frequency $2
\omega$.  Thus, probability is not conserved for this solution to the
Klein-Gordon equation.

What does this mean?  It turns out that equations of this type have a
different interpretation in relativistic quantum mechanics --- the
wavefunction represents an {\em ensemble} of zero or more particles of
a given type.  In the relativistic case, all particles can be created
and destroyed by various processes, so it is not surprising that the
probability of finding a particle is not generally conserved over
time.  {\em Quantum field theory} governs the rules for particle
creation and destruction, and it arises as an inevitable consequence
of the collision of quantum mechanics with special relativity.

\end{document}
