Superposition and Beats

A single energy eigenstate is frozen. Combine two of them with different energies and the system comes alive: probabilities oscillate at a frequency set by the energy gap. This is the quantum origin of beats.

Two-state evolution

Mix two eigenstates equally:

|\psi(0)\rangle = \frac{1}{\sqrt 2}\big(|E_0\rangle + |E_1\rangle\big).

Each component winds at its own rate, $e^{-iE_n t/\hbar}$ :

|\psi(t)\rangle = \frac{1}{\sqrt 2}\big(e^{-iE_0 t/\hbar}|E_0\rangle + e^{-iE_1 t/\hbar}|E_1\rangle\big).

Factor out the common phase $e^{-iE_0 t/\hbar}$ (it is global and unobservable):

|\psi(t)\rangle \simeq \frac{1}{\sqrt 2}\big(|E_0\rangle + e^{-i\omega_{10} t}|E_1\rangle\big), \qquad \omega_{10} \equiv \frac{E_1 - E_0}{\hbar}.

Only the energy difference survives. The quantity $\omega_{10}$ is the Bohr frequency (or beat frequency) of the pair.

The beating probability

Project the evolved state back onto the initial superposition $|+\rangle = \tfrac{1}{\sqrt2}(|E_0\rangle + |E_1\rangle)$ :

\langle +|\psi(t)\rangle = \frac{1}{2}\big(1 + e^{-i\omega_{10} t}\big),

so the survival probability is

P_{+}(t) = \big|\langle +|\psi(t)\rangle\big|^2 = \frac{1}{4}\big|1 + e^{-i\omega_{10} t}\big|^2 = \cos^2\!\frac{\omega_{10}\, t}{2}.

It oscillates between $1$ and $0$ with angular frequency $\omega_{10}$ . The probability beats in and out of the original state purely because the two components drift out of phase and back into phase. This is the same mathematics as two slightly detuned tuning forks producing an audible beat — here the "pitches" are energies and the beat is in probability.

Spectroscopy in one line

The Bohr frequency $\omega_{10} = (E_1 - E_0)/\hbar$ is exactly the frequency of light a system emits or absorbs when transitioning between these two levels — the Bohr frequency condition $\Delta E = \hbar\omega$ . The beating of a superposition and the color of a spectral line are two faces of the same energy gap.

Try it

This is a numerical exercise — your code should return a number. With energy gap $E_1 - E_0 = 2.0$ (units $\hbar = 1$ ) and $t = \pi/4$ , compute the survival probability $P_{+} = \cos^2(\omega_{10}\,t/2)$ .

Run your code to see the quantum state.