Solving Quantum Physics Problems: Wavefunction and Expectation Values

[kehler]

Homework Statement

Wavefunction = (2/L)^(1/2) * sin(2*pi*x/L) exp(−i2*pi^2*h_bar*t/mL^2)

I calculated the expectation value of momentum to be 0, and expectation value of kinetic energy to be 2*pi^2*h_bar/mL^2 (also found to be it's definite K.E)
Using the momentum operator, the wavefunction was found to have only two possible momentums with equal probability of 0.5: (2*pi*h_bar) and -(2*pi*h_bar)
The wavefunction has probability density peaks at +L/4 and -L/4
The question is:
We set up (in the lab) an ensemble of many such particles, all with the same wavefunction. Describe what you expect to be the results of measurements of their position, momentum and kinetic energy.

The Attempt at a Solution

It's a five mark question. I know that the average value of the measurements of their momentum will be 0, and the measurements of their kinetic energy will always yield 2*pi^2*h_bar/mL^2. Also, the particles will also most likely be at either +L/4 and -L/4.
Is there anything else I can conclude using the information above?

Any help with the question above would be much appreciated :). I'm not entirely sure that I got all the calculations correct either. Just started doing QM..

 

[Jhero]

I would like to congratulate you on your calculations and understanding of the wavefunction! Your results are correct and show a good understanding of the concepts involved.

Based on the given wavefunction and your calculations, we can expect the following results from measurements of position, momentum, and kinetic energy:

1. Position: As you mentioned, the particles will most likely be found at either +L/4 or -L/4. This is because the wavefunction has peaks at these positions, indicating a higher probability of finding the particles there.

2. Momentum: The average value of the momentum measurements will be 0, as you correctly calculated. This means that the particles have an equal probability of having either a momentum of +2*pi*h_bar or -2*pi*h_bar. This is consistent with the wavefunction having only two possible momentums with equal probability.

3. Kinetic energy: The measurements of kinetic energy will always yield a value of 2*pi^2*h_bar/mL^2. This is the expected value of kinetic energy for this wavefunction, and it is also the definite kinetic energy. This means that all particles in the ensemble will have the same kinetic energy, which is a unique property of this wavefunction.

In conclusion, we can expect the particles in the ensemble to have a higher probability of being found at +L/4 or -L/4, an average momentum of 0, and a kinetic energy of 2*pi^2*h_bar/mL^2. I hope this helps answer your question and gives you a better understanding of the results of measurements on the ensemble of particles. Keep up the good work in your studies of quantum mechanics!