Proof of Superposition Principle

[Inquiring_Mike]

Hey,
I was recently reading "If an electron can be in 2 places at once, why can't you?" in the new Discover magazine when I came across the superposition principle ( I have heard of it before, just never really looked into it). They said that evidence of the phenomena could be found in the double slit experiment, yet I always thought that the double slit experiment was proof that light is a wave (or wave-like). Is there any other evidence of this principle?

Thanks,
Mike

 

[dextercioby]

The linearity of quantum mechanics in any of its 3 well-known fomulations is a postulate.Hilbert space is a linear space.Observables are described by densly defined selfadjoint linear operators.

Daniel.

P.S.There are non linear versions of QM.

 

[masudr]

As Daniel says, the linearity of QM implies that superposition is valid. Whenever you have linear operators, it implies that the sum of two separate solutions is indeed also a solution. So if one solution involves going through one slit, and the other involves going through the other slit, the general solution is the sum of the two separate ones.

Superposition is nothing new, we've had it in wave equations before QM, in Electromagnetism, in classical mechanics etc.

 

[vanesch]

Superposition is nothing new, we've had it in wave equations before QM, in Electromagnetism, in classical mechanics etc.

The mathematical aspect of superposition is "nothing new", however, the physical aspect of the superposition principle in quantum theory is not at all comparable to the superposition property of classical systems with a linear dynamical law. Indeed, one can always assume that the linearity of classical dynamics is an (excellent) approximation, but nothing changes fundamentally (concerning the basic principles of classical physics) if this dynamics were non-linear.
The superposition principle in quantum theory, however, is fundamental: it says that just ANY system for which we know it to exist potentially in different states A and B (say, I was born in Australia, or I was born in Russia), then there are DIFFERENT states corresponding to each state A + x B, for each complex number x.
As such, it DOESN'T MATTER what the system is, we're talking about, and what are the properties (states) we're talking about. This is an extremely weird proposition, and it is the basic idea behind quantum theory. Daniel points out that the superposition principle follows from the mathematical structure of QM ; I'd rather say that the mathematical structure of QM is what it is, because it has been build up around the superposition principle!

cheers,
Patrick.

 

[Inquiring_Mike]

Ahhh... very interesting...

Thanks, it all seems clear now...

 

[CarlB]

Quantum mechanics is not, in fact, linear, except if you restrict yourself to just the wave equations. For example, the Dirac equation or Schroedinger's wave equation are linear, but if you include interactions with other stuff, those interactions are, in general, nonlinear.

The same thing is true about QFT. The propagators are linear, but the vertices are certainly not.

If you begin with a nonlinear wave equation, it is sometimes possible to extract a linear version of it, with the nonlinearity included in the form of corrections. With such a nonlinear theory, the linear part becomes the propagators, and the nonlinear part becomes the vertices. Summing up the Feynman diagrams gives you back the nonlinear theory.

Most any reasonable nonlinear theory will, if you look at very small perturbations, be linear. The first instinct of a physicist, when approaching a nonlinear problem, is to linearize. A lot of the time that works pretty well. Some of the time you can get by with a linear theory plus perturbations. And some of the time linearity just doesn't work.

At night, when a drunk man loses his wallet, he looks first under the street light because the light is better there. That's why we use linear equations for physics, not because the world is linear, but instead because we can solve linear theories. We use the linearity where we can, and the rest, the rest we push into the interactions between the linear stuff.

QED is linear enough that the perturbation series converge. QCD is so nonlinear that perturbations don't converge.

Carl

 

[vanesch]

Quantum mechanics is not, in fact, linear, except if you restrict yourself to just the wave equations. For example, the Dirac equation or Schroedinger's wave equation are linear, but if you include interactions with other stuff, those interactions are, in general, nonlinear.

The same thing is true about QFT. The propagators are linear, but the vertices are certainly not.

I think you confuse the non-linearity of the field equations (fields being operators) with interacting terms with the linearity of the operators on the state space. The linear operators are solutions to non-linear operator equations, but they are still linear operators.

For instance, from the free lagrangian, you get linear field equations ; once you introduce interacting terms (like psi-bar A^mu psi) you get a non-linear field operator equation which nobody knows how to solve except by series devellopment. But the operators themselves (the fields) remain linear, hermitean operators ! And the superposition principle remains valid (not satisfied by the field operators of course, but by the objects ON WHICH the field operators act, namely the hilbert states), also in QCD and QED.

cheers,
Patrick.

 

[CarlB]

Patrick,

> I think you confuse the non-linearity of the field equations
> (fields being operators) with interacting terms with the
> linearity of the operators on the state space. The linear
> operators are solutions to non-linear operator equations,
> but they are still linear operators.

If you go back and carefully reread what I wrote, you will discover that I never said that the linear operators of QM are not linear. What I said was that "QM is not linear". I was writing in response to the conventional wisdom as expressed on the thread, for example: "As Daniel says, the linearity of QM implies that superposition is valid." This is a false statement. Neither QM, nor the world, is linear. On the other hand, linear superposition is linear of course. But linear superposition is only a small part of QM.

> But the operators themselves (the fields) remain linear,
> hermitean operators!

This is true, but again, the operators are not QM. They're only a small part of QM. QM, taken as a whole, is simply not linear. The world is not linear. QM describes the world quite well. QM is not linear.

The world is not linear, nor does the world possesses a linear superposition principle. If the world did possesses a linear superposition principle (as opposed to a part of QM), the Pauli exclusion principle would be excluded. Since QM uses linear tools, it is forced to analyze multiple particle situations, which are inherently extremely nonlinear, with configuration space.

You can argue that configuration space is necessary, but you can't argue that it is linear. It simply isn't. Doubling a configuration space solution is not a solution with twice as many particles, or one with particles that are twice as heavy. Doubling a configuration space solution gives a new solutin that can only be interpreted as identical to the undoubled solution.

Linear is when you make a change in one variable, and something that depends on it changes to an amount proportional to the change you made. $$\Delta V_1 = \kappa \Delta V_2.$$ The wave functions in QM are not linear. If one makes a linear change to one, there is no change to the situation modeled. Multiplying a QM wave function by 1.1 gives you the same wave function. Similar for the states in QFT.

Yes, I agree that the linear parts of QM are linear. But QM, taken as a whole, is not linear, nor is the world linear. And that suggests that if one were to open God's notebook, and see the fundamental wave equation that is written there, one would discover that it was a nonlinear wave equation. We linearize it because linear tools are easier to use, not because "QM is linear".

The nonlinear nature of the world shows up in how we interpret even the simple wave function for a single particle. If linear superposition applied, then we could double that wave function and, because the new wave function also satisfies the linear wave equations, we would interpret that doubled wave function as the description of two particles.

This we cannot do. Instead, because of the inherently nonlinear nature of the world, we take all the various multiples of a state and assign them to the same real situation. That is not "QM is linear". That is linear superposition in the sense that "our mathemtatics is linear, QM is not".

Let me compare QM to a linear theory, such as electricity and magnetism. If one obtains a solution to Maxwell's equations, then 1.1 times that solution is also a solution. And the multiplied solution corresponds to a real world situation that is proportionally more intense than the unmultiplied solution. The charges are increased by 10%. The currents are increased by 10%. The electric and magnetic fields are increased by 10%. This is what linearity means. QM is not linear.

The Hilbert states are only linear to the extent that arbitrary multiples of a Hilbert state are considered to be the same state. If the world truly were linear, then the existence of a situation with one electron would imply that one can multiply that situation by 110% and end up with a 1.1 electron solution.

Carl

 

[misogynisticfeminist]

i was quite surprised when you guys jumped straight into the theoretical part of QM, i think the thread starter was looking for certain experimental proof. Firstly, the double slit experiment shows the superposition of states principle. You just have to shoot 1 electron at a time and the interference pattern goes up like normal. More fundamentally, the double-slit also shows particle with wave-like properties, shoot electrons through and you get the interference pattern.

 

[underworld]

i was quite surprised when you guys jumped straight into the theoretical part of QM, i think the thread starter was looking for certain experimental proof. Firstly, the double slit experiment shows the superposition of states principle. You just have to shoot 1 electron at a time and the interference pattern goes up like normal. More fundamentally, the double-slit also shows particle with wave-like properties, shoot electrons through and you get the interference pattern.

I have another thread with some questions about the double-slit experiment - but another that comes to mind is this:

*) if you shoot 1 electron at a time - how many electrons have to be "shot" before the interference occurs?

when stated as you have, it implies that the interference does not occur until after the first electron (maybe the 3rd or the 100th, or the 10^6th - I don't know...). in any case - the "superposition" theory seems to imply that the interference is caused not only by the "wave" nature of the object, but also because the superpositioned object interferes with itself. but if it interferes with itself, then that infers only a single electron need be "shot", right?

 

[vanesch]

The Hilbert states are only linear to the extent that arbitrary multiples of a Hilbert state are considered to be the same state. If the world truly were linear, then the existence of a situation with one electron would imply that one can multiply that situation by 110% and end up with a 1.1 electron solution.

I think that you have missed then the essential content of the superposition principle in quantum theory. It doesn't say that 1.1 times the "one electron" state is a kind of "1.1 electron" state.
No, it says, that if there exists a certain 1 electron state A, and there exists a 2 electron state B, then there exist infinitely many states which are described by |A> + c |B> with c a complex number. Now, let us take c to be, say 1.2.

Does that mean that the state as such |A> + 1.2 |B> is something like a "1 + 2.4" electron state ? Not at all. It means that if our initial state is this famous |A> + 1.2 |B> state, then if we do something to it, called U (a unitary evolution which can represent just any action), then the outcoming state is nothing else but U|A> + 1.2 U|B>. And it might be that U|A> and U|B> have something in common (a component in common); which, if we measure it, are usually called "quantum interference effects".
Most of the time, common components are NOT produced, so most of the time, this parallel evolution is not visible. But sometimes it is. And that's where quantum theory deviates from its classical analogon.

EDIT: I could add that one could artificially introduce such a principle in classical physics which says that if you have an initial state in phase space p1 and another initial state in phase space p2, each giving rise to their own evolution p1(t) and p2(t), then we could artificially introduce "new states" a |p1> + b |p2> (this is not to be confused with the state a x p1 + b x p2 which doesn't even exist in phase space if a and b are complex) which give then rise to states a |p1(t)> + b |p2(t)>. But that would be just formal overkill because we would ALWAYS start out with ONE SINGLE |p1> in practice.
The big trick in QM is the "reuse" of superpositions to make other relevant states. For instance, the superposition of position states (delta(x - x1)) to make momentum states. This is how we can make these famous "superpositions" : by changing observational basis, so that what is not a superposition in one basis is a superposition in another.

There is absolutely no CLASSICAL analogon of this quantum superposition principle. In the classical world, when we mean "superposition" we mean - as you point out, that 1.1 times a solution is also a solution (a slightly stronger E field or so). But that's NOT what the superposition principle is all about. It is about PARALLEL EVOLUTION, with (sometimes) interference effects if the parallel evolutions happen to produce common parts.

But the solutions to the evolutions of "basis states" themselves give of course rise to highly non-linear problems (which, themselves, can be solved, or not, by linearisation by parts).

cheers,
Patrick.

 

[vanesch]

If you go back and carefully reread what I wrote, you will discover that I never said that the linear operators of QM are not linear. What I said was that "QM is not linear". I was writing in response to the conventional wisdom as expressed on the thread, for example: "As Daniel says, the linearity of QM implies that superposition is valid." This is a false statement. Neither QM, nor the world, is linear. On the other hand, linear superposition is linear of course. But linear superposition is only a small part of QM.

Oops, I now only understand what you wrote
You were arguing against the idea that "quantum theory is linear" while I was trying to explain the "superposition principle".

Although I maintain everything I wrote in my previous post, I now see that you were not trying to make a point against that. You were just pointing out that not all problems in QM are "linear", like problems in passive circuit theory (resistors, selfs and capacitors) ARE linear.
Yes, of course. Granted.
Saying that quantum theory is linear, because there is a superposition principle is just as naive as saying that classical theory is "constant" because the coordinates in phase space are real numbers which are constants, or something of the kind.
I agree fully with the fact that quantum theory gives rise to complicated non-linear problems (which are much harder than their equivalent classical nonlinear problems), for instance in QCD (but even in QED).

cheers,
Patrick.

 

[CarlB]

Hi Patrick;

> You were just pointing out that not all problems
> in QM are "linear", like problems in passive circuit
> theory (resistors, selfs and capacitors) ARE linear.

The underlying issue is that the act of quantization itself is nonlinear. As far as I know, there are no fermion bound state QM problems that are linear. But most of, for example, my body's mass consists of fermions in bound states. Well, maybe there's some massive virtual bosons in there, especially around the middle.

Carl

P.S. If you want to see how a relation such as the Heisenberg uncertainty principle can be derived from the linearized solutions of the unquantized nonlinear wave equation:
$$\partial_t \psi = \nabla(\psi^2),$$
and you don't mind wading through Clifford algebra, you can try reading my paper:
http://brannenworks.com/b_nonl.pdf

 

[vanesch]

The underlying issue is that the act of quantization itself is nonlinear. As far as I know, there are no fermion bound state QM problems that are linear.

What do you mean by that ? That the problem of finding the eigensolutions is a non-linear problem ? Agreed with that. I can even say that all interacting systems give rise to nonlinear problems of that kind. That the superposition principle doesn't apply to fermions in bound states ? Don't agree with that, it does apply.

So what's the point you are trying to make ?
If it is that the linear operators themselves are subject to nonlinear equations, we already agreed upon that.
That the superposition principle holds for states, I think we agree there, no ?

There is of course a discussion possible on a possible "nonlinear act of observation" (von Neuman's first process). But you're not obliged to subscribe to it in an ontological way. Is that what you mean ?

cheers,
Patrick.