- #1
Mayan Fung
- 131
- 14
- Homework Statement
- Solve the analytically the time-dependent Schrodinger equation with Hamiltonian:
$$\hat{H} = \frac{\hat{p}^2}{2} + \frac{\hat{q}^2}{2} + \alpha(t) \hat{q}$$
where ##\alpha(t) = 1-e^{-t}## for ##t \leq 0## and 0 for ##t <0##. With initial wave function in coordinate space:
$$\Psi(q, t= 0) = Nexp(-\frac{q^2}{4})$$
- Relevant Equations
- \begin{align}
E_n &= \hbar(n+\frac{1}{2})\\
|\Psi(t)> &= exp(-i\frac{\hat{H}}{\hbar}t)|\Psi(0)>
\end{align}
What I have tried is a completing square in the Hamiltonian so that
$$\hat{H} = \frac{\hat{p}^2}{2} + \frac{(\hat{q}+\alpha(t))^2}{2} - \frac{(\alpha(t))^2}{2}$$
I treat ##t## is just a parameter and then I can construct the eigenfunctions and the energy eigenvalues by just referring to a standard harmonic oscillator solution by shifting some constants. And then I used
\begin{align}
|\Psi(t)> &= exp(-i\frac{\hat{H}}{\hbar}t)|\Psi(0)> \\
|\Psi(t)> &= \sum_{n=0}^\infty exp(-i\frac{\hat{H}}{\hbar}t) |n><n|\Psi(0)> \\
|\Psi(t)> &= \sum_{n=0}^\infty exp(-i\frac{E_n}{\hbar}t) <n|\Psi(0)> |n>\\
\end{align}
where ## <n|\Psi(0)> = \int \Psi_n^*(t)\Psi(t=0) \, dx##
Because of the ##\alpha(t) = 1-e^{-t}##, I can't compute ## \int \Psi_n^*(t)\Psi(t=0) \, dx##. I think the problem can be more easily solved because the initial wave function ##\Psi(q, t= 0) = Nexp(-\frac{q^2}{4})## looks like that it is deliberately chosen. I am also not sure if I can extend the harmonic oscillator solution for a time varying constant.
$$\hat{H} = \frac{\hat{p}^2}{2} + \frac{(\hat{q}+\alpha(t))^2}{2} - \frac{(\alpha(t))^2}{2}$$
I treat ##t## is just a parameter and then I can construct the eigenfunctions and the energy eigenvalues by just referring to a standard harmonic oscillator solution by shifting some constants. And then I used
\begin{align}
|\Psi(t)> &= exp(-i\frac{\hat{H}}{\hbar}t)|\Psi(0)> \\
|\Psi(t)> &= \sum_{n=0}^\infty exp(-i\frac{\hat{H}}{\hbar}t) |n><n|\Psi(0)> \\
|\Psi(t)> &= \sum_{n=0}^\infty exp(-i\frac{E_n}{\hbar}t) <n|\Psi(0)> |n>\\
\end{align}
where ## <n|\Psi(0)> = \int \Psi_n^*(t)\Psi(t=0) \, dx##
Because of the ##\alpha(t) = 1-e^{-t}##, I can't compute ## \int \Psi_n^*(t)\Psi(t=0) \, dx##. I think the problem can be more easily solved because the initial wave function ##\Psi(q, t= 0) = Nexp(-\frac{q^2}{4})## looks like that it is deliberately chosen. I am also not sure if I can extend the harmonic oscillator solution for a time varying constant.