the final exam
...will be an oral examination concentrating on the material since the mid term.
Time dependence of quantum states
- Solving the time-dependent Sch. equation
- "Tacking on" the time dependence $e^{-iEt/\hbar}$ works with energy eigenstates
The EPR paradox
- What was the challenge to quantum theory?
- The assumptions behind hidden variable theories: Realism and locality.
- The entangled singlet state $\ket \psi = \frac1{\sqrt 2}\ket{\uparrow\downarrow}-\frac1{\sqrt 2}\ket{\downarrow\uparrow}$.
- What Bell's theorem, and tests of that theorem have shown.
Particles and wave functions
- The time-independent Sch equation. Solutions? (oscillating and non-oscillating)
- Solutions for the infinite and finite square well and delta function well.
- Parallels with matrix mechanics: Dot products, probabilities, expectation values, time dependence, normalization: Know how to set up integrals to calculate these things.
- Sketching solutions: Solving the Sch time independent eqn in terms of a second derivative ($\Rightarrow$ curvature), $V-E$, curving towards or away from the $x$ axis: $$\frac{\del^2 \varphi(x)}{\del x^2}=\frac{2m(V(x)-E)}{\hbar^2}\varphi(x).$$
- Approach to finding exact solutions:
- Finding general solution to Sch equation (differential equation):
- particular solutions in different regions, depending on $V-E$
- apply appropriate boundary conditions
- apply normalization constraint.
- Most general solution is superposition of eigenstates.
- Free particle solutions and new approach to normalization