1 Time-Dependent Wave Functions
Recall that the Schrodinger equation is
\( \hat{H} = i\hbar \frac{\partial}{\partial t} \)
(1)
$\phi_1(x)$ and $\phi_2(x)$ are solutions to the time-independent Schrodinger equation. The eigenenergy $E_1$ corre-
sponds to eigenfunction $\phi_1(x)$ while $E_2$ is the eigenenergy for eigenfunction $\phi_2(x)$.
a. Write the time-dependent eigenfunctions $\phi_1(x,t)$ and $\phi_2(x, t)$ in terms of time t and the eigenenergies
$E_1$ and $E_2$
b. Express the orthogonality conditions using integrals
c. Some other state $\psi(x, t)$, which is not an energy eigenfunction can be written for t = 0 as
$\psi(x, 0) = \frac{1}{\sqrt{4}} \phi_1(x) - i\sqrt{\frac{3}{4}}\phi_2(x)$
(2)
Write $\psi(x, t)$.
d. Express the probability that the state at some later time $\psi(x,t)$ is found to be in the initial state
$\psi(x,0)$. (Hint: Your answer should be a real-valued function never greater than 1)
