For a one dimensional particle of mass m constrained to exist on the interval between x=0 and x=L, (the 1D particle in a Box), the Eigenstates of the Hamiltonian operator were shown to be

The eigenvalues of the Hamiltonian were also found.

Consider a 50/50 mixture of the first two stationary states of the system, i.e. those with n=1 and n=2:

Remember, the full, time-dependent wavefunction for the stationary states is:

What is the expectation value of the Energy in this state?

Each of the brakets on the right hand side may be easily evaluated

The expectation value of the energy in this superposition state in constant and the average of that in n=1 and n=2

Note: On each individual measurement of the energy, only E₁ or E₂ is actually observed. The average of many measurements performed on identically prepared states gives the expectation value. Such a distribution of possible outcomes has a spread or uncertainty. The RMS deviation form the mean of the variable y is

The RMS 'spread' of observed energies for this superposition state can be obtained from the expectation value of the energy and the energy squared

This, too, is easy to evaluate. (remember that H² is simple HH or H acting twice in a row)

The Deviation or Spread in energy of this state is significant:

Now, back to time dependence. Where is the particle in this superposition state?

There are four terms that we must evaluate. These terms can be obtained from the explicit integration of the wavefunctions. The first term is:

The time dependent parts of this braket vanish. The integral may be evaluted using the hint

Luckily, the result is simple and intuitive: the average position of the particle in a box in a stationary state is in the middle of the box.

But what about the cross terms?? They are clearly complex conjugates of one another, so we need only evaluate one of them.

The time dependence of the cross term does NOT vanish and the integral must be evaluated

since

The position of the particle oscillates at a frequency that corresponds to the energy difference of the stationary states (divided by hbar)!