Schrödinger equation solutions: stability and oscillations

What is known about

1. Stability of solutions of time-independent Schrödinger equation (TISE)?

2. Existence of time dependent but oscillatory solutions (of Schrödinger equation)?

3. Stability of oscillatory solutions (if existing)?

I imagine the answer may depend on the potential (hydrogen atom, LHO,...)

By 1) I mean: If I slightly change the solution of TISE, does the modified function goes away or comes back to the original solution (stable equilibrium vs. unstable equilibrium).

The second point I consider interesting: We use to consider real atoms of hydrogen always to be in some time-independent state, therefore we take the solutions of TISE. Yet, oscillatory solutions are in some sense time independent ("stationary") and we should, maybe, think of real atoms as being at oscillatory states.

PS: Let me remind you: also solutions of TISE are actually time dependent (oscillatory) by the phase factor exp($-iET/\hbar$).

1 Answer

Regarding the stability of the Schrödinger equation, the main result is that the propagator is a unitary transformation, which therefore preserves inner products. In particular, if $|\varphi(t)\rangle$ and $|\psi(t)\rangle$ are arbitrary solutions of the time-dependent Schrödinger equation for any arbitrary hermitian (possibly) time-dependent hamiltonian $H(t)$, then $$\langle\varphi(t)|\psi(t)\rangle = \langle\varphi(0)|\psi(0)\rangle$$ for all times $t$. In particular, it means that if you perturb your initial condition $|\psi(0)\rangle$ into, say, $|\tilde\psi(0)\rangle = a|\psi(0)\rangle+|\chi(0)\rangle$, where $\langle\chi(0)|\psi(0)\rangle = 0$ and $a^2+\langle\chi(0)|\chi(0)\rangle = 1$, then the solution will retain that form: you'll have $|\tilde\psi(t)\rangle = a|\psi(t)\rangle+|\chi(t)\rangle$ for all time, where the perturbation stays orthogonal, $\langle\chi(t)|\psi(t)\rangle = 0$ and the projection on the original solution $a^2 \equiv 1-\langle\chi(t)|\chi(t)\rangle$ stays constant.

Regarding the existence of oscillatory solutions in both theory and experiment, I gave a good description in Is there oscillating charge in a hydrogen atom?. Generally, if you prepare any superposition of two eigenstates of the Schrödinger equation you will get a periodic solution. If you prepare a superposition of a finite number, you'll typically get a quasi-periodic solution (unless the energy differences happen to be exactly commensurate, which is hard to arrange, in which case you'll have an exactly periodic solution). If you've got a superposition of a continuum of eigenstates, then generically you will have no recurrences at all.

Regarding your statement

Yet, oscillatory solutions are in some sense time independent ("stationary") and we should, maybe, think of real atoms as being at oscillatory states.

that's not really how you think should think about it. Oscillatory solutions are not time-independent - they are oscillatory, period, and it doesn't really help if you try to deform the notion of stationarity to include those cases. Stationary states are really stationary: wait any arbitrary time $t$, and you get an equivalent state (i.e. equal up to an irrelevant global phase). This is not the case with oscillatory solutions.

And finally, if you want to speak about "real atoms" - that completely depends on the situation. The TISE is a tool, as are the hamiltonian's eigenstates. Sometimes real atoms are in eigenstates, sometimes they're not. Sometimes they're in pure states, sometimes they're in mixed states. It depends what kind of experiment you're describing. Trying to speak about "real atoms" in a generic way will swallow you up whole because of the wide array of different real-world situations those atoms can find themselves in.

• You are unfair: Oscillatory - time dependend - period. "Stationary" - time dependent (indeed!) - irrelevant. Feb 16, 2017 at 13:53
• @F.Jatpil I can't really understand your comment. Feb 16, 2017 at 13:54
• Any solutions we are discussing here are indeed time dependent (mathematical truth). In one case ("oscillatory" solution) you maintain this statement (you stress that they cannot be called stationnary), in a different case (solutions of TISE) you call the time dependence irrelevant (so you admit there is one). Feb 16, 2017 at 14:00
• Eigenstates of the hamiltonian have a formal time dependence as vectors on Hilbert space, but not as physical states, because time evolution only introduces a global phase which has absolutely no effect on any possible measurement. The physical state is not the Hilbert-space vector, it is the ray (or alternatively the density matrix) that it generates, and this does not change for an energy eigenstate. So, yes: the time dependence of an energy eigenstate is both formally and physically irrelevant. Feb 16, 2017 at 14:08