quantum mechanics - How does the hydrogen atom know which frequencies it can emit photons at?

At university, I was shown the Schrodinger Equation, and how to solve it, including in the $1/r$ potential, modelling the hydrogen atom.

And it was then asserted that the differences between the eigenvalues of the operator were the permitted frequencies of emitted and absorbed photons.

This calculation agrees with experimentally measured spectral lines, but why would we expect it to be true, even if we accept that the electron moves according to the Schrodinger equation?

After all, there's no particular reason for an electron to be in an eigenstate.

What would make people think it was anything more than a (very suggestive) coincidence?

Answer

This calculation agrees with experimentally measured spectral lines, but why would we expect it to be true, even if we accept that the electron moves according to the Schrodinger equation? After all, there's no particular reason for an electron to be in an eigenstate.

Good question! The function $\psi$ does not need to be Hamiltonian eigenfunction. Whatever the initial $\psi$ and whatever the method used to find future $\psi(t)$, the time-dependent Schroedinger equation $$ \partial_t \psi = \frac{1}{i\hbar}\hat{H}\psi $$ implies that the atom will radiate EM waves with spectrum sharply peaked at the frequencies given by the famous formula $$ \omega_{mn} = \frac{E_m-E_n}{\hbar}, $$

where $E_m$ are eigenvalues of the Hamiltonian $\hat{H}$ of the atom.

Here is why. The radiation frequency is given by the frequency of oscillation of the expected average electric moment of the atom

$$ \boldsymbol{\mu}(t) = \int\psi^*(\mathbf r,t) q\mathbf r\psi(\mathbf r,t) d^3\mathbf r $$ The time evolution of $\psi(\mathbf r,t)$ is determined by the Hamiltonian $\hat{H}$. The most simple way to find approximate value of $\boldsymbol{\mu}(t)$ is to expand $\psi$ into eigenfunctions of $\hat{H}$ which depend on time as $e^{-i\frac{E_n t}{\hbar}}$. There will be many terms. Some are products of an eigenfunction with itself and contribution of these vanishes. Some are products of two different eigenfunctions. These latter terms depend on time as $e^{-i\frac{E_n-E_m}{\hbar}}$ and make $\boldsymbol{\mu}$ oscillate at the frequency $(E_m-E_n)/\hbar$. Schroedinger explained the Ritz combination principle this way, without any quantum jumps or discrete allowed states; $\psi$ changes continuously in time. Imperfection of this theory is that the function oscillates indefinitely and is not damped down; in other words, this theory does not account for spontaneous emission.