Typo: you meant |<n|ψ>|2.
damn - yes. What I wrote only produces the amplitude. Must be rusty.
<n|ψ> = ∫ψ
n*ψ = Ʃ
mc
mδ
m,n=c
n but p(n)=c
n2 = |<n|ψ>|
2 ... which can look like so many magic runes to the initiate :)
Hopefully it is clear that p(n) (the probability of being in a state) is not ψ
*ψ anyway?
In physics courses, a lot of time is spent getting students used to the math: I don't think many would cover atomic wavefunctions before second year. One can get a good feel for what happens by working through solutions to simpler potentials in 1D ... infinite square well, finite square well, and harmonic are the usual starting places. It is difficult gaining an understanding from just being told stuff.
It seems to me that energy eigenfunctions are taken to be real by convention
It is difficult to know how careful to be isn't it?
The eigenvector solutions to the 1D, time-independent Schodinger equation drop out as real don't they? Technically, an energy eigenfunction can be written in the form \Psi_n(x,t)=\psi_n(x)e^{-iE_nt/\hbar} where \psi_n(x) is the nth eigenvector of the time-independent SE[1].
Of course we don't
have to represent the wavefuction that way - we can choose any representation we like, and often do. But I don't think this particular choice of representation is all that arbitrary.
OTOH: the time-dependent equation has solutions with an extra arbitrary phase which depends on when we started the clock (appearing as constants of integration). It is this factor that we choose to be zero to make \Psi_n(x,t) completely real at t=0. In that sense - the choice of
when the wavefunction is completely real is an arbitrary one... we are simply choosing to measure time from one time when the wavefunction was real.
I'm not sure this is an important arbitrariness though. Since all time-dependent equations are arbitrary in the same way ... we choose to start timing
anything from some event that we pick for whatever reason suits us. Similarly we also pick the place we measure our space coordinates from... it does not have to be the center of mass for the system, that's just convenient sometimes.
With reference to the question: the wavefunctions shown to OP are all real, likely, because they are the time-independent eigenfunctions. In context, they are probably energy eigenfunctions.
In fact, the solutions for the hydrogen potential are spherical Bessel functions aren't they?
And textbooks usually just plot the radial component.
I think bound states can always be imagined as bouncing back and forth such that the energy eigenstates don't propagate, and that's why they can be regarded as real.
That is what I remember - a stationary state, like an energy eigenstate, can always be represented as non-stationary waves interfering as they bounce back and forth. Much like how stationary waves can appear on a string. IIRC: the actual solutions to the infinite square well are complex exponentials (plane waves) - you get two for each energy level. This gives the sine and cosine solutions - which are real.
If it's only approximately an energy eigenfunction, then it will only approximately depend on time like e-iEt/h-bar
I'd actually go along with that - though, at the back of my mind is a little voice reminding me that the map is not the territory.
The answer kinda depends on what we mean by "accurate".
If the wavefunction given is an approximation only - then that particular wave-function is actually a superposition of eigenstates of the schrodinger equation ... the time evolution of the approximation would not follow that of the exact function so it is "innaccurate" wrt the exact solution[2]... but, "accurate" also means that it matches or reality (predicts the results of experiments for example.) Which is different - a very good approximation to the wave-function will result in a model that matches reality at least as closely as we can measure it.
I think this is kinda important in QM.
We have a solution to the schodinger equation, and we have the real behavior, and we have an approximate solution to the schodinger equation. The approx can be a close fit to the exact without being close to reality - is it accurate?
There are lots of places the approximation can come from - we can use a simple form of the physics of geometry, for example, or leave off the nastier parts of the calculation in the hope they don't do much. So we talk about the kind of approximation and, depending on the kind, how it compares to observations in different situations.
If you've done any philosophy, you'll have met arguments about what we mean by "reality", You'll have noticed how these discussions always seems very abstract: they don't actually affect what we do? I have sometimes said that Quantum Mechanics, these discussions about the nature of reality stop being abstract. This is why the language can get convoluted and two physicists will spend time discussing semantics like you just saw.
We have to figure out what we will agree our words mean, and also figure out how fuzzy to let our language become when we are talking to someone who is just starting out. In my reply, I let my language get very fuzzy indeed :)
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[1] which means I was being a bit glib in post #2 - it is \psi_n(x) that is usually plotted in textbooks, not the real part of \Psi_n(x,t).
[2] but you
knew that!
