Modern 2:Overview of Chapter 3

Chapter 3. 2/24/06

• The state of a system is system is described by a wavefunction. ψ(r,t). The wavefunction evolves deterministically according to the Schrodinger equation. However, we give a probabilistic interpretation to the wave function that allows us to predict the measurement of a given physical quantity.
• On the other hand, if we perform an experiment, the system will be in some state. How do we obtain as much information about this state.
• Finally, we may wish to perform an experiment on a system in a given state; i.e., one that is prepared experimentally to have well-defined properties.

Reminder

$\langle r \rangle _ t = \int r |\psi(r,t)|^2 d^3 r$

$\langle p \rangle _ t = \int p |\phi(p,t)|^2 d^3 p$

But we can also compute the expectation of p in position space. This is essential if we want to be able to treat variables such as angular momentum, which involve both position and momentum: $L = r \times p$

Here is a fundamental result which you should prove: $\langle p _x \rangle _ t = \int \psi^*(r,t) \partial _x \psi(r,t) d^3 r$