Expectation value of momentum

 For example, for a free particle Y = eikz.

 

 

For a bound particle in a box of length L, Y = Ö(2/L)sin(npz/L)

Let u = npz/L and du = np/L dz, then dz = L/np du

 Note that this makes sense since the particles spends an equal amount of time traveling in the +x and –x direction.

 

Expectation value of energy

For a bound particle

and using the normalized wavefunction

 Let u = npz/L and du = np/L dz, then dz = L/np du

 

Expectation value of position

 For the bound particle in a box.

 Let u = npz/L and du = np/L dz, then dz = L/np du

 

 

Therefore, <z> = L/2. This is logical since the average position of the particle is in the middle of the box.