About wavefunctions, their complex nature and their approximations?

[darfunkel]

Hello forum, I'm new here. Please bear in mind I'm not a physicist, I'm sure you hear this a lot but anyway. I've recently taken an interest into quantum chemistry. So my question is the following:

I understand electronic and nucleic wave functions are complex. When they are multiplied by their complex conjugate, one obtains the probability of the particle being at that state. I've learned from computational chemistry sources that wave functions can be aproximated by using things such as Slater Type Orbitals and Gaussian Orbitals. However, these are real valued functions. Why is it that complex wavefunctions can be approximated using real valued functions?

Thanks in advance and excuse me if the answer is elementary.
D.

 

[Simon Bridge]

When they are multiplied by their complex conjugate, one obtains the probability of the particle being at that state.

Nope - that would give you a probability-density function. The probability of being in an eigenstate |n> would be <n|ψ> since |ψ> = Ʃa_n|n>

Why is it that complex wavefunctions can be approximated using real valued functions?

The approximations are for the real part of the wavefunction only.

 

[Ken G]

Nope - that would give you a probability-density function. The probability of being in an eigenstate |n> would be <n|ψ> since |ψ> = Ʃa_n|n>.

Typo: you meant |<n|ψ>|².

The approximations are for the real part of the wavefunction only.

It seems to me that energy eigenfunctions are taken to be real by convention, and this carries over into the approximations of them also. They can always be chosen to be exactly real at t=0 because E and V are real, so each (d/dx)²|ψ> must be also, but that means the phase of |ψ> must not vary with x for an energy eigenfunction. Thus its phase can only vary with t, so is real to within a global phase factor that is taken to be zero at t=0 by convention. But maybe you are saying that if one applies this real convention to the approximation of the wave function, it's no longer guaranteed that the exact wavefunction being approximated will satisfy that convention as well.

 

[darfunkel]

Thank you for your answers. Would that mean that the behavior in time of the wavefunction approximation is inaccurate?

 

[Ken G]

Would that mean that the behavior in time of the wavefunction approximation is inaccurate?

If it's only approximately an energy eigenfunction, then it will only approximately depend on time like e^-iEt/h-bar. By the way, the reason I gave for the energy eigenfunction being real isn't right, there's more to it than that. The (d/dx)² operator can give a negative real result, as for a momentum eigenstate, and those cannot be purely real, but they are also propagating. 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.

 

[Simon Bridge]

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!

 

[Ken G]

Yes, I wasn't really correcting anything, just trying to get my head around the realness of a wavefunction whose most bizarre attribute is that it must more generally be complex!

 

[Simon Bridge]

It takes a while - as you found out, it is a large and subtle thing.
And that is without having to reconcile it with relativity.
After all that OP should be well and truly confused xD

BTW: complex numbers are not so bizarre in physics - Foucaults pendulum and AC circuits are understood best in terms of complex functions for example.
Admittedly not as extreme - but any wave-like behavior is easiest to handle in complex notation.

 

[Ken G]

True enough, but the way complex numbers are used is rather different. In classical applications, complex numbers just give a convenient way to track the phase of anything periodic, but the initial conditions of the problem will eventually impose taking the real part of the solution. But in quantum mechanics, the realness of the initial conditions appears in the form of real matrix elements, there's no need to take the real part of anything. So the way realness is imposed is more fundamental in quantum mechanics than in classical mechanics-- which is a bit ironic if you think about it!

 

[Simon Bridge]

It's the role as probability amplitudes that does that - you can get rid of the complex part of the others by premultiplying the complex conjugate to give two terms ... in the model of Foucaults pendulum, for eg, both those terms have a real meaning.

I'll concede it feels different.

Usually we explicitly stick the complex notation in at the start.
Still - the classical wave equation has complex solutions.

It is the statistical nature of the wavefunction that leads to the rather odd-looking treatment.

The complex part in QMwf also encodes phase information btw.
Try writing the complex amplitude out in real and imaginary components.

 

[Ken G]

The complex part in QMwf also encodes phase information btw.
Try writing the complex amplitude out in real and imaginary components.

It's different though. When you take the real part, as you do when fitting to boundary conditions in classical wave equations, you end up with a sinusoidally varying magnitude, like an E field. But in QM, you don't take the real part, you take the modulus, and that does not sinusoidally vary. That's the fundamental difference.

 

[Simon Bridge]

And why do you take the square modulus?
Because, that way, you get a probability distribution.

Which is more fundamental - the statistical nature of quantum phenomena or the rule to take the square-modulus of the wave-function?

 

[Ken G]

Which is more fundamental - the statistical nature of quantum phenomena or the rule to take the square-modulus of the wave-function?

I would say they are both equally fundamental-- and equally different from how complex numbers are used in classical physics. But the analogies certainly are useful, no question-- if one understands how the algebra of complex numbers helps in one application, it will help the other applications also.

 

[darfunkel]

Thanks for your answers and insight Ken G and Simon Bridge. I am confused indeed! But I guess I'll have to keep on reading and thinking about it.

 

[Simon Bridge]

Confusion is normal when you start out on this - after a while, I'm not sure if you get less confused or just used to it. Mostly, though, you end up being able to use the tools to make decent predictions that seem to pan out OK ... if you ever get to the edge of things, you may even be able to get modified solutions that predict something previously unsuspected. (You'll get to that edge real fast, but doing something novel there takes a bit more effort.)

 

[bhobba]

And why do you take the square modulus?
Because, that way, you get a probability distribution.

Which is more fundamental - the statistical nature of quantum phenomena or the rule to take the square-modulus of the wave-function?

Well actually there is a rather neat theorem called Gleasons Theorem that shows the only way you can define a probability on the complex functions used in QM is to take the square.

The real question in QM is why you use complex numbers. It has to do with a technical issue with what are known as pure states. In standard probability theory there are only a finite number of such states and you can't continuously go from one pure state to another. A bit of math shows if you want that you must go to complex numbers. So basically you have two choices - and if you want continuous transformations you must go to QM. For more detail check out:
http://www.scottaaronson.com/democritus/lec9.html
http://arxiv.org/pdf/quant-ph/0111068v1.pdf
http://arxiv.org/pdf/quant-ph/0101012v4.pdf

Thanks
Bill

 

[akhmeteli]

The real question in QM is why you use complex numbers. It has to do with a technical issue with what are known as pure states. In standard probability theory there are only a finite number of such states and you can't continuously go from one pure state to another. A bit of math shows if you want that you must go to complex numbers. So basically you have two choices - and if you want continuous transformations you must go to QM. For more detail check out:
http://www.scottaaronson.com/democritus/lec9.html
http://arxiv.org/pdf/quant-ph/0111068v1.pdf
http://arxiv.org/pdf/quant-ph/0101012v4.pdf

Thanks
Bill

Let me note, on the other hand, that you can do with real functions (not pairs of real functions to replace complex functions) much more than one would typically believe. For example, Schroedinger noted that in a general case, every solution of the Klein-Gordon equation can be made real by a gauge transform. A similar thing can be done for the Dirac equation (see an article in Journ. Math. Phys. http://akhmeteli.org/wp-content/uploads/2011/08/JMAPAQ528082303_1.pdf , there is also a reference to the Schroedinger's note there).

 

[Ken G]

So basically you have two choices - and if you want continuous transformations you must go to QM. For more detail check out:
http://www.scottaaronson.com/democritus/lec9.html

I really like Scott Aaronson's article, it uses a very intuitive perspective. It sounds like he's saying that if you want your pure states to evolve both unitarily and continuously, you need to use algebraically complete combinations of basis states, i.e., complex amplitudes. So it's more than just the ability of complex numbers to have phase and exhibit interference, you also need them for unitarity, which is related to time reversibility. That also sounds like it has to do with how using complex numbers algebraically to produce real observables always seems to induce an ambiguity in the sign of i, which maps into an ambiguity in the direction of time, which appears to be an ambiguity that the fundamental laws need to be built to respect (at least until you get into more bizarre things outside of quantum mechanics).

 

[Simon Bridge]

Which provides a nice segway into Gauge theories which are all about symmetry.