How do I plot this probability density over time?

[baouba]

Here's the question: http://imgur.com/N60qRmw

I normalized the wave function and got A = sqrt(315/8L^9)

but how would I plot representative snapshots if the exponential factor will cancel when I square it? It's not a mixed state so it shouldn't depend on time as far as I can tell. Should I use a Fourier series? I'm just at a loss as to how to plot different instances in time if the complex term should cancel out when taking the square modulus. I feel like I might not have a complete understanding of the wave function or pure/mixed states. Could someone help clear this up?

Thank you

 

[fzero]

That wavefunction is not an eigenstate of the Hamiltonian, so it must evolve in time. You will need to project that wavefunction onto the wavefunctions for the energy eigenstates to determine an expression for $\psi(x,t)$.

 

[baouba]

That wavefunction is not an eigenstate of the Hamiltonian, so it must evolve in time. You will need to project that wavefunction onto the wavefunctions for the energy eigenstates to determine an expression for $\psi(x,t)$.

How can you tell if a given wave function isn't an eigenstate of the hamiltonian? I just don't get how if |Ψn(x, y)|^2 = |Anψn(x)e^-iEnt/h-bar|^22 and Ψ is a pure state, how doesn't the exponential factor cancel out in 1 = ∫ |Ψn(x, y)|2 dx ?

 

[fzero]

How can you tell if a given wave function isn't an eigenstate of the hamiltonian?

The most direct way would be to just compute $\hat{H} \psi$. If the result is not $\lambda \psi$, where $\lambda$ is a constant, then $\psi$ is not an eigenfunction of the Hamiltonian.

I just don't get how if |Ψn(x, y)|^2 = |Anψn(x)e^-iEnt/h-bar|^22 and Ψ is a pure state, how doesn't the exponential factor cancel out in 1 = ∫ |Ψn(x, y)|2 dx ?

The time-dependent wavefunction $\psi(x,t)$ will be a linear combination of the energy eigenstates. When you compute $|\psi(x,t)|^2$, you will find cross-terms where the exponents do not cancel.