Why is it so that probability density of eigenfunctions of time-dependent schrodinger equation are time independent while that of general wave functions (which are a combination of the eigenfunctions) are not?
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It is important to understand how the eigenfunctions are defined. In particular, here we talk about the eigenfunctions of the hamiltionian, i. e. functions that satisfy
$$\hat{H} \psi(x) = E \psi(x) , $$
where $\psi(x)$ is an eigenfunction (function of space only) and $E$ is its corresponding eigenvalue. In case of the hamiltonian the eigenvalue is also the energy of its associated quantum state.
Now, the total eigenfunction is a function of space AND time. The form of these full eigenfunctions is:
$$\psi(x, t) = \psi(x) \exp{(-iEt)}, $$
and the probability density is $P(x, t) =\psi(x, t) \psi^* (x, t) $.
Now you can see that if one takes a single eigenfunction, then the time-dependence (the exponential part) cancels with its complex conjugate in the probability density.
If one takes a linear combination, e.g. $\psi_1 + \psi_2$, then the probability density is:
$$P(x, t) = \left( \psi_1(x)\exp{(-iE_1 t) } + \psi_2(x)\exp{(-iE_2 t) } \right) \left( \psi_1(x)\exp{(-iE_1 t) } + \psi_2(x)\exp{(-iE_2 t)} \right)^*. $$ Taking the complex conjugate one obtains
$$P(x, t) = \left( \psi_1(x)\exp{(-iE_1 t)} + \psi_2(x)\exp{(-iE_2 t)} \right) \left( \psi_1(x)\exp{(iE_1 t)} + \psi_2(x)\exp{(iE_2 t)} \right). $$
Now there is no reason why the above should be time independent. The exponentials with $E_1$ do not cancel exponentials with $E_2$, therefore one expects mixing terms such as $\exp{i(E_1-E_2)t}$ in the final probability density, and thus it will no longer be time independent.
