The best strategy is to read through my web notes from General
Uncertainty Principle and Energy-Time Uncertainty Principle up to and
including Angular Momentum and Orbital Eigenfunctions:
2-D case.

Review all the answers to homework problem sets 7,
8, 9, 10 and read your own notes from class.

**Generalized Uncertainty Principle**

Be able to derive the Generalized Uncertainty Principle:
know the definition of expectation value and of the root mean square deviation
in a set of measurements on identically prepared systems.* *

**Energy-Time Uncertainty Principle**

Be able to explain why the alpha particle emitted in alpha decay of a nucleus does not have a perfectly defined energy, but an energy spread related to the lifetime of the nucleus.

**Simple Harmonic Oscillator**:
know

_{}

and be able to use these in evaluating expectation values.

Know the definition of the **propagator**,
and be able to derive the free particle propagator.

Be familiar with the **Heisenberg
representation**, know how to find the equation of motion for an
operator in that representation, and know the connection with Ehrenfest’s theorem.

**Coherent States**

Know how a coherent state of a simple harmonic oscillator develops in time, what it is an eigenstate of, how it can be expressed in energy eigenstates, how they form a complete set.

Memorize the result _{}, know for what operators it is valid, and be able to use it
for normalizing coherent states, etc.

We won’t cover squeezed states on this Midterm.

**Path Integrals**

Know Feynman’s formulation of a sum over paths, know the phase factor for a path, and be able to prove that for a particle in one dimension, it is equivalent to Schrödinger’s equation. Know how to evaluate the integral for the free particle case.

**Angular Momentum**

*From class notes*: be able to show, if the
operator _{} is defined by _{} under a rotation
defined by _{}, then from our knowledge of *classical* rotations, _{}, and know this formula by heart! Be able to derive from this the commutation
relations among _{} Be able to derive the
matrix elements of these operators for the common eigenkets of *J*^{2},
*J _{z}*, and know how to prove that 2

_{}

Be able to use these to construct matrix representations of
the components of angular momentum for given *j*.

**2-D Orbital Angular Momentum**

Be able to derive the angular part of the wave function in 2D, the allowed form of the radial function at the origin (at end of lecture), and the points covered in the homework #9 questions.