1. ## Quantum 1

I was given a few Quantum questions, and they're all math related. Not sure how many understand this, but it was worth a shot.

Given that a free electron is contained in a box described by the wave function $\displaystyle \psi(x, 0) = \left(\frac{1}{2}\right)\left(\frac{2}{a}\right)^{ \frac{1}{2}}\sin{\frac{2\pi x}{a}} + \frac{3^{\frac{1}{2}}}{2} \left(\frac{2}{a}\right)^{\frac{1}{2}}\sin{\frac{4 \pi x}{a}}$ and with walls located at $\displaystyle x=0$ and $\displaystyle x=a$, answer the following:

1.) Find the wave function $\displaystyle \psi(x,t)$ at any time $\displaystyle t > 0$.

2.) Determine all possible energy measurements and determine the probability at which they occur.

3.) Find the wave function after a measurement of energy given that that energy gives ground state energy.

4.) Determine $\displaystyle <\hat{H}>$ using the results from 2.). Then, using the result from 1.), find $\displaystyle <\hat{H}>$

5.) Suppose suddenly that the wall moved from $\displaystyle x=a$ to $\displaystyle x=3a$, determine the probability that the electron is located in the 3rd energy state $\displaystyle E_3$ of this box.

2. Originally Posted by DiscreteW
I was given a few Quantum questions, and they're all math related. Not sure how many understand this, but it was worth a shot.

Given that a free electron is contained in a box described by the wave function $\displaystyle \psi(x, 0) = \left(\frac{1}{2}\right)\left(\frac{2}{a}\right)^{ \frac{1}{2}}\sin{\frac{2\pi x}{a}} + \frac{3^{\frac{1}{2}}}{2} \left(\frac{2}{a}\right)^{\frac{1}{2}}\sin{\frac{4 \pi x}{a}}$ and with walls located at $\displaystyle x=0$ and $\displaystyle x=a$, answer the following:

1.) Find the wave function $\displaystyle \psi(x,t)$ at any time $\displaystyle t > 0$.
For all of the following answers look at the normalization of the wavefunction you are using.

Use the time evolution operator $\displaystyle U(t, t_0) = e^{-iH (t - t_0)}$ where H is the Hamiltonian. (I'll leave it to you to find the Hamiltonian. Hint: It doesn't matter what kind of particle it is, if it's in a box the Hamiltonian can only have one form.)

Originally Posted by DiscreteW
2.) Determine all possible energy measurements and determine the probability at which they occur.
The different possible energy measurements are just the energy eigenvalues of the wavefunction. Hint: Apply the Hamiltonian to each separate term of the wavefunction.

The probability for which they occur can be found by
$\displaystyle P_n = \int \psi \psi _n~dx$
where $\displaystyle \psi _n$ is the eigenfunction for the eigenvalue $\displaystyle E_n$.

Note: Properly speaking this should be $\displaystyle \int \psi ^* \psi _n~dx$ but the wavefunction is real anyway, so I won't worry about this detail.

Originally Posted by DiscreteW
3.) Find the wave function after a measurement of energy given that that energy gives ground state energy.
You know that the state is in the ground state. So the wavefunction is the ground state eigenfunction of the overall wavefunction. (What part of the given wavefunction is that?) Now apply time evolution to that eigenfunction.

Originally Posted by DiscreteW
4.) Determine $\displaystyle <\hat{H}>$ using the results from 2.). Then, using the result from 1.), find $\displaystyle <\hat{H}>$
You have the probability for each energy state to occur. So
$\displaystyle < \hat{H} > = \sum_{n} P_n E_n$

To do it more directly:
$\displaystyle < \hat{H} > = \int \psi H \psi~dx$

Originally Posted by DiscreteW
5.) Suppose suddenly that the wall moved from $\displaystyle x=a$ to $\displaystyle x=3a$, determine the probability that the electron is located in the 3rd energy state $\displaystyle E_3$ of this box.
I'm going to take my best shot at this now. I might post later if I change my mind after I've given it some more thought.
Write out the eigenfunction for the third energy level for the expanded box. Then the probability of the particle being in this state is
$\displaystyle P_3 = \int \psi \psi_3~dx$
where $\displaystyle \psi$ is your original wavefunction.

The reason I am not certain about this is I can't think of what to do about the normalization at the moment. So your $\displaystyle P_3$ calculated in this manner is going to be off by a constant factor.

-Dan

3. Originally Posted by topsquark
Use the time evolution operator $\displaystyle U(t, t_0) = e^{-iH (t - t_0)}$ where H is the Hamiltonian. (I'll leave it to you to find the Hamiltonian. Hint: It doesn't matter what kind of particle it is, if it's in a box the Hamiltonian can only have one form.)
I owe you an apology. I'm used to working Quantum Physics in the Heaviside-Lorentz system of units, where $\displaystyle \hbar = c = 1$. The time evolution operator is
$\displaystyle U(t, t_0) = e^{-iH (t - t_0)/ \hbar}$
so that the argument of the exponent, $\displaystyle Ht/ \hbar$, is unitless as it should be.

-Dan