For now we will focus on how the character of the wavefunction ( tells us the amount at each location x) describes certain aspects of quantum states. The first characteristic we explore is the momentum for a given quantum state . We look for a state that doesn’t change under the action of the momentum operator and we need to introduce the operator for momentum.

Eigenvalue equation

Momentum operator in position space

Momentum eigenstates

Note: states cannot be normalized in the usual manner

The momentum eigenstate is a state that has an amplitude associated with every location x. The above formula is written as a discrete sum to emphasize that we are simply summing up the contribution from every contributing state. The need for an integral is based solely on the continuity of the position x. The student is encouraged to think of the sum in a similar fashion as a sum over 3-d basis vectors. Addition so envisioned leads to very important properties. Often these characteristics become clouded when the integral replaces the sum.

The relationship between p and x will be basically the same as variable that are related via a Fourier transform or Fourier series.

Do you know the difference between the two?

Energy operator

Note: Energy eigenstates are also eigenstates of the momentum operator.

Dirac Delta

Expectation value

At this point we need to introduce the expectation value ad hoc.

Let us return to the state

We can calculate this for a genral state as

Text book works through the case of a Gaussian wavefunction.

Schrodinger equation and time evolution

We realized that translation in space or time can be generated by

The translation throught a step where

For time we identify the generator g as the Hamiltonian or energy operator.

States in egenstates of energy have a simplified time evolution.

A general state can be written as a linear combination of energy eigenstates.

REVIEW

We are going to ask to describe a quantum state by asking the question:

Where is the particle?

Our answer is that we can prepare a quantum state so that the particle is in a particular location x1

or in a particular state of definite momentum

or a state of definite energy

We can choose a state which is in a linear combination of these states above

Whatever the state we prepare we describe it by providing the wavefunction which answers the question: where is the particle? by giving us the amplitude for the particle to be at any location x.

Once we know what the state of the particle is we can evaluate the state by asking questions about what measurements of the state might reveal.

Once we know what state we have prepared we can also ask how it evolves in time. Quantum statest evolve forward in time based on the time evolution operator U

where

By requiring the Hamiltonian to be proportional to the time derivative of the state you are giving this operator a special role in the time evolution. Knowing how the state is changing in time one can evaluate what the state will be at some time later through U(t,t0) as defined above.