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The Energy Representation: Eigenstate Expansions

 

Background: In lecture we explored the representations for quantum state of the system corresponding to the observables of position x and momentum p. We used these representations to derive the forms for operators for position and momentum in both the position and momentum representations. There are more than these two representations for a quantum state. By the quantum principle of superposition, we expect there to be a representation for each physical observable. The observable of energy is no exception to this. In this problem, you will practice the skills developed in lecture for dealing with different representation, but now in a new context.

You will also find that the energy representation is useful for proving many theorems about the time development of quantum states. This is a result of the close relationship between energy and time.

This problem will discuss both general considerations and a specific example. For the general considerations in this problem, we begin by using the principle of superposition to write any quantum state, or , in terms of the pure states of energy as

Here, the pure states are given by and are properly normalized so that .

For the specific considerations, we use use the energy states of the Infinite Square Well of length L, which are given by

with energies

where here the Hamiltonian operator is just . At times we will also use the (properly normalized) wavefunction

Assignment:





Prof. Tomas Alberto Arias
Sat Apr 6 15:33:22 EST 1996