This is not a problem that you have to do , but, surprisingly, it is not that difficult if you understand everything we have done so far. It takes our study of 1D quantum a step further by approaching the time dependence (of the state function) in a new manner. It also uses matrix formulation. Seeing time dependence in emerge in a different way may provide a new and illuminating perspective and insight into "how QM works".
So far we have also obtained time dependence by: solving the time-independent Schrodinger equation and the writing a given state function, at t=0, as a superposition on energy eigenstates each with their own time dependence. Then the time dependence of the overall state function emerges, somewhat cryptically, from the "de-phasing" of the different exp[-i E_n t/hbar] terms.
But suppose someone came along and said:
Hmm... H Psi = i hbar d Psi/dt ....
what is the difficulty? Isn't the solution just:
Psi(t) = Psi(0) exp[ i H t/hbar]
What would you say to that? Can we apply that in some concrete and useful way???
(Pause here to reflect, argue, consider and react.)
For our crystal problem, consider the state, Psi_j = phi_0 (x-ja), which corresponds to the electron being located (only) at the jth site in the crystal. Consider the set of all such states. In the basis of these states can the Hamiltonian, H, can be written as a matrix? To the same level of approximation we used for IP#3, what is that matrix? (part a))
b) What if you were told (given) that at t=0 an electron is in the ground state of the jth potential. How could you calculate what the electron does as a function of time?
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I seem to recall from 116A that we can in fact exponentiate an operator (ie exp(i H t/hbar) is defined), and that it was defined by the infinite series of the exponential. So, we'd have:
ReplyDelete1 + (i H t/hbar) + (i H t/hbar)^2/2! + ...
Does that make sense?
It makes sense to me. So then, what is H in the basis of those "spatial eigenstates" {Psi_j} ?
ReplyDeleteFrom what Zack and I talked about, one advantage to this approach is a fairly straightforward transition to a computational solution, by iterating over H Psi = i hbar dPsi/dt. For those of you also in 115, something like the Verlet method to integrate would be nice, to conserve energy. This sort of "solves" part b.
ReplyDeleteFor the matrix in part a, I remember a density matrix with beta/4 in the off-diagonals (from the integral for the bandwidth), but I don't remember what was in the main diagonal. Since it has to be Hermitian, we know the values have to be real...I'm leaning towards 1 or 0, but I'm not sure without looking it up.