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\begin{document}

\csection{Mid-Term Summary}

The mid-term examination will cover the following material:

\begin{itemize}
\item
Theory of relativity: You should have a good general knowledge of the
subject, but should particularly understand relativistic four-vectors.
\item
Wave packets and group velocity:
\begin{itemize}
\item
Fourier series and Fourier transforms: These mathematical tools are
essential for understanding the evolution of wave packets.
\item
Dispersion relation: A dispersion relation for a wave is a functional
relationship between the angular frequency of the wave and its
wavenumber: $\omega = \omega (k)$.
\item
Group velocity: The group velocity is the velocity at which a wave
packet moves: $u_g = d \omega /dk$.
\item
Wave packet evolution and the propagator: This comes from assuming
that a wave packet can be represented as a superposition of plane
waves.  This expression in turn can be represented as a convolution
between the wave packet at $t = 0$ and the \emph{propagator}.
\item
Dirac delta function: The propagator invariably takes the form of
the Dirac delta function at time $t = 0$.
\end{itemize}
\item
Motion of free particles:
\begin{itemize}
\item
The Planck-Einstein relationship: Planck hypothesized and Einstein
confirmed a relationship between energy and and angular frequency, $E
= \hbar \omega$, for photons.
\item
De Broglie relationship: Louis de Broglie hypothesized that the
Planck-Einstein relation extended also to massive particles.  He also
showed that the dispersion relation for the quantum mechanical waves
associated with massive particles was forced by relativity to be
$\omega^2 = k^2 c^2 + \mu^2$ where $\mu$ is a constant to be
determined.  If the Planck-Einstein relationship is valid, then there
must also be a relationship between momentum and wavenumber: $p =
\hbar k$.  The above dispersion relation is then seen to be equivalent
to the relativistic energy-momentum relation: $E^2 = p^2 c^2 + m^2
c^4$.  The constant $\mu = m c^2 /\hbar$.
\item
Klein-Gordon equation: The wave equation which yields the above
dispersion relation is
\begin{displaymath}
	\frac{\partial^2 \psi}{\partial t^2} =
	c^2 \frac{\partial^2 \psi}{\partial x^2} -
	\frac{m^2 c^4}{\hbar^2} \psi .
\end{displaymath}
\item
Free particle Schr\"{o}dinger equation: In the non-relativistic limit
in which we consider only positive energy solutions, the dispersion
relation becomes $\omega = k^2 c^2 /(2 \mu ) + \mu = \hbar k^2 /(2m) +
mc^2 / \hbar$.  The second term is usually dropped as it represents a
constant additive contribution to the frequency and hence to the
energy.  The corresponding wave equation is
\begin{displaymath}
	i \hbar \frac{\partial \psi}{\partial t} =
	-\frac{\hbar^2}{2m} \frac{\partial^2 \psi}{\partial x^2} .
\end{displaymath}
\item
Probability interpretation of $\psi$: The relative probability of
finding a particle at position $x$ and time $t$ in non-relativistic
quantum mechanics is $P(x,t) = | \psi (x,t) |^2$.  For a wave function
$\psi$ obeying the Schr\"{o}dinger equation, there exists a
probability flux
\begin{displaymath}
	F(x,t) = \frac{i \hbar}{2m} \left(
		\frac{\partial \psi^*}{\partial x} \psi -
		\psi^* \frac{\partial \psi}{\partial x}
	\right)
\end{displaymath}
such that
\begin{displaymath}
	\frac{d}{dt} \int_a^b P dx = F(b,t) - F(a,t) .
\end{displaymath}
Letting $a \rightarrow -\infty$ and $b \rightarrow \infty$ shows that
probability is conserved overall if $\psi$ goes to zero at infinity.
\emph{This is not true for the Klein-Gordon equation} unless
consideration is restricted to just the positive (or negative) energy
states alone.
\end{itemize}
\item
Non-relativistic particle in a potential:
\begin{itemize}
\item
Schr\"{o}dinger equation: The total energy is $E = p^2 /(2m) + U(x)$
when the particle under consideration has potential energy $U(x)$.
The extended Schr\"{o}dinger equation is
\begin{displaymath}
	i \hbar \frac{\partial \psi}{\partial t} =
	-\frac{\hbar^2}{2m} \frac{\partial^2 \psi}{\partial x^2} +
		U(x) \psi .
\end{displaymath}
The solutions to this equation satisfy probability conservation just
as do solutions to the free Schr\"{o}dinger equation.
\item
Examples of solutions to the 1-D Schr\"{o}dinger equation:
\begin{itemize}
\item
Infinite square well (analytical).
\item
Finite square well (analytical development plus graphical solution).
\item
Harmonic oscillator (shooting method).
\item
One-dimensional scattering and barrier penetration (analytical and
numerical).
\end{itemize}
\end{itemize}
\item
Big ideas of quantum mechanics: We need to generalize our ideas of
how quantum mechanics works to extend it to systems which don't have
classical equivalents.
\begin{itemize}
\item
Measurement results and amplitudes: For each possible result $A_i$ of
the measurement of a physical quantity $A$, there is a complex
amplitude $\psi_i$.  The probability of finding result $A_i$ in a
measurement procedure is $| \psi_i |^2$.
\item
Expectation values:
\begin{itemize}
\item
Single variable: The expectation value of $A$ is
\begin{displaymath}
	<A> = \sum_i A_i | \psi_i |^2 =
	\sum_{i,j} \psi_i^* A_i \delta_{ij} \psi_j
	\equiv ( \psi , A \psi ) .
\end{displaymath}
The $\psi_i$ are the components of a vector $\psi$ in an abstract
vector space called Hilbert space.  The $A_i \delta_{ij}$ are the
components of a matrix which has the $A_i$ values along the diagonal
and zeros elsewhere.  $( \psi , \phi )$ is generalized notation for a
dot product between two vectors in Hilbert space.  If the Hilbert
space vectors have components $\psi_i$ and $\phi_i$, then
\begin{displaymath}
	( \psi , \phi ) = \sum_i \psi_i^* \phi_i .
\end{displaymath}
Note that $(A \psi , \phi ) = (\psi, A' \phi )$ and $(\psi , \phi ) =
(\phi , \psi )^*$.
\item
Additional incompatible variable: The expectation value of a different
physical quantity $B$ can be expressed as
\begin{displaymath}
	<B> = \sum_{i,j} \phi_i^* B_i \delta_{ij} \phi_j =
	( \phi , B \phi ) .
\end{displaymath}
The $\psi_i$ and the $\phi_i$ are related by a unitary transformation
\begin{displaymath}
	\phi_i = \sum_j q_{ij} \psi_j ~~~ \mbox{or} ~~~
	\phi = q \psi .
\end{displaymath}
Think of the $\phi_i$ and $\psi_i$ as the vector $\psi$ resolved in
two different reference frames rotated relative to each other.
Unitary means that $q^{-1} = q'$, where $q'$ is the transposed complex
conjugate of $q$ and $q^{-1}$ is the inverse.  Thus, $\psi = q^{-1} \phi
= q' \phi$.
\item
Expectation value in non-principal axis coordinate system: Note that
\begin{displaymath}
	<B> = (q \psi , B q \psi ) = ( \psi , q' B q \psi ) .
\end{displaymath}
The matrix $B_t \equiv q' B q$ is not diagonal.  However, it is
Hermitian or self-adjoint: $B_t' = B_t$.  Think of $B_t$ as the
operator $B$ resolved in a different coordinate system.
\item
If we start with the non-diagonal form of $B$, we can diagonalize
it to get the possible physical values of $B$ (the eigenvalues) and
the unit vectors in the principal axis frame (the eigenvectors).
\item
If two operators commute, i. e., if $[A,B] \equiv AB - BA = 0$,
then they share a common principal axis reference frame.  $A$ and
$B$ are called compatible.
\end{itemize}
\item
Degeneracy: If an operator is degenerate, i. e., two or more of its
eigenvalues are equal to each other, then there exist one or more
additional operators which are compatible with the original operator.
The maximal set of compatible operators has been found when each state
has a distinct set of operator values associated with it.
\item
Momentum and energy operators: The momentum and energy operators in
the spacetime representation are
\begin{displaymath}
	p = -i \hbar \delta (x - x' ) \frac{\partial}{\partial x'} ~~~~
	E = i \hbar \delta (t - t' ) \frac{\partial}{\partial t'} .
\end{displaymath}
The application of (say) the momentum operator to a state $\psi$ is
\begin{displaymath}
	\int p(x,x' ) \psi (x' ) dx' =
	- i \hbar \frac{\partial \psi}{\partial x} .
\end{displaymath}
Thus, the Dirac delta function is usually dropped and we say
\begin{displaymath}
	p = -i\hbar \frac{\partial}{\partial x} ~~~~
	E = i\hbar \frac{\partial}{\partial t} .
\end{displaymath}
\item
Time evolution: Diagonalizing the energy operator or Hamiltonian
allows us to determine the time evolution of a system.  Solving
\begin{displaymath}
	H \psi \equiv i \hbar \frac{\partial \psi}{\partial t} =
	E \psi
\end{displaymath}
leads to a set of eigenvalues $E_j$ and eigenvectors $\psi^{(j)}$,
with an arbitrary solution taking the form
\begin{displaymath}
	\psi = \sum_j a_j \psi^{(j)} \exp (-iE_j t/ \hbar ) ,
\end{displaymath}
where the $a_j$ are constants which are determined by the initial
conditions.
\end{itemize}
\end{itemize}

\end{document}
