What is the Born Interpretation in Quantum Mechanics?

[sponsoredwalk]

Hi, I've started to watch some lectures on quantum mechanics & they're going well except for the fact that some of it makes no sense. Basically I just don't see how |ψ|²dτ represents the probability of finding a particle described by ψ in the volume element dτ. Most likely it's due to me having missed something in the development of the material thus far or maybe it's because the answer to this question hasn't been fully given yet - or maybe I'm just missing some elementary logic. I think it's best to briefly state exactly what's been done in the course:

History
Defined Schrödinger Equation
Mentioned, without justification, the Born Interpretation:
- The element |ψ|²dτ represents the probability of finding the particle described by ψ in the volume element dτ
Derives the 1-Dimensional Schrödinger equation & momentum operators
Evaluates some Gaussian integrals
Gives four representations of the dirac delta function
- via a Rectangle function
- via the Heaviside step function
- via a Gaussian function
- via the sinc function (using this to give an integral representation)
Goes through Fourier's integral theorem
Derives Parseval's identity (Also known as the normalization condition)
Inserts the Debroglie relation into the integral representation of the Dirac delta function & uses Parseval's identity to derive a relationship between position & momentum
Derives a solution to the 1-D Schrödinger equation for a free particle
Shows how the equation of continuity can be derived from the Schrödinger equation
Mentioned the commutator [a,b] = ab - ba
Just started to derive material related to expected values
(If any of this is too short I'll expand on it)

So I can see how normalizing the integral ∫|ψ|²dτ due to the linearity of the Schrödinger equation is a way to get ∫|ψ|²dτ = 1 & that this obviously relates things to probability but I mean to make such a definitive claim about what |ψ|²dτ stands for seems a bit much.

 

[DrewD]

Experimental evidence. I don't think that this is a conclusion that one would arrive at through mathematical logic alone. I could certainly be wrong, but that was how it was presented to me.

 

[Ken G]

Yes, there is no logic behind the Born rule that is generally accepted, it's usually just given as a postulate of the theory. Think of it in the reverse direction-- let there be some function that tells you the probability of getting a given outcome of an experiment (say getting an outcome means finding the particle in the volume element you mention). This function must be positive real, because it is a probability. So call it a complex amplitude times its complex conjugate, since that will always be positive real, and will also allow us to have destructive interference given the superposition principle for those amplitudes. This is all pretty general so far, it's just a system that gives us the flexibility we need for it to work. Now all we have to do is give a name to that complex amplitude function (psi), and an equation for its evolution (the Schroedinger equation), and we're well on our way. So if you just describe things in that order, the Born rule seems more like a starting point than something you should be inferring from something else.

 

[tom.stoer]

There is some work mainly by Zurek which focusses on a derivation of Born's rule based on decoherence. I have to check the literature, but I think the following paper may serve as a reference:
W. H. Zurek, “Environment-assisted invariance, entanglement,and probabilities in quantum physics,” Phys. Rev. Lett. 90(12), 120404 (2003).

 

[bhobba]

It follows from a very important theorem called Gleasons Theorem:
http://kof.physto.se/theses/helena-master.pdf

Seeing its a theorem and all that you may be inclined to think it must be true - and if you accept its premises of course it is. But there is a hidden assumption - that it does not depend on whatever other states that may be possible outcomes - this assumption is known as non contextuality and as the theorem shows is actually a very powerful assumption.

This is the reason Ken is correct in saying:

Yes, there is no logic behind the Born rule that is generally accepted, it's usually just given as a postulate of the theory.

Yes Gleasons Theorem proves it - but you have to accept non contextuality which is itself an assumption so why not just state it without worrying about the theorem. I prefer Gleasons Theorem personally because I find contextuality ugly - but that's just me.

Thanks
Bill

 

[tom.stoer]

Gleason's theorem is not sufficient but only necessary to proof Born's rule.

It says (very roughly speaking!) that Born's rule formulated is the only probability interpretation on a Hilbert space of sufficiently large dimension.

 

[bhobba]

Gleason's theorem is not sufficient but only necessary to proof Born's rule. It says (very roughly speaking!) that Born's rule formulated is the only probability interpretation on a Hilbert space of sufficiently large dimension.

A newer version based on POVM's removes that restriction. BTW that sufficiently large dimension is greater than two - but the POVM version has no such restriction.

Thanks
Bill

 

[tom.stoer]

A newer version based on POVM's removes that restriction. BTW that sufficiently large dimension is greater than two - but the POVM version has no such restriction.

Thanks
Bill

Which restriction? The dimension? Maybe yes, but that's certainly not enough to let Gleason's theorem become sufficient to derive Born's rule.

Or do I miss something else? And what's POVM?

 

[bhobba]

Which restriction? The dimension? Maybe yes, but that's certainly not enough to let Gleason's theorem become sufficient to derive Born's rule.

Or do I miss something else? And what's POVM?

Positive Operator Valued Measure:
http://en.wikipedia.org/wiki/POVM

Yea - it fixes up the dimension thing (its tricky how it does it - some may say it doesn't really - but no need to go into the details here) but you are correct it is not sufficient - theories exist that break the assumptions of the theorem but obeys Born Rule.

Thanks
Bill

 

[tom.stoer]

Thanks
Tom

 

[Ken G]

Yea - it fixes up the dimension thing (its tricky how it does it - some may say it doesn't really - but no need to go into the details here) but you are correct it is not sufficient - theories exist that break the assumptions of the theorem but obeys Born Rule.

That sounds like Gleason's theorem is not necessary for the Born rule, sufficiency requires that there be no theories that satisfy Gleason's theorem without having a Born rule. If we hold the assumptions needed for Gleason's theorem, we seem to get that we must adopt a Born rule to get a theory that works the way we want (but that's not quite the same as saying we get the Born rule from Gleason's theorem). So I think Gleason is neither necessary nor sufficient, but taking Gleason's theorem plus some basic assumptions of what we want quantum mechanics to be able to do does seem to be sufficient (though not necessary) for the Born rule. I agree with bhobba that given that situation, and the non-contextuality issue, it seems like we might do better just treating Born like a postulate, though I haven't read Zurek (which will probably involve some other assumptions that may seem more reasonable than the Born rule, but are still not a priori requirements for reality).

 

[bhobba]

Yea that's correct - the postulates of Gleasons Theorem (most notably non contextuality - the other ones are not really in doubt eg the probability of an element of the vector space being the outcome of an observation is one, are additive etc) is not necessary for Borns rule - but obviously is sufficient since if its assumptions are correct then Borns rule follows. Its purely a matter of personal taste if you prefer and find more intuitive the assumption of non contextuality or Borns Rule - I find non contextuality very intuitive.

Thanks
Bill

 

[Ken G]

I think the value of noncontextuality is not so much that we can tell it is true, but rather, it is a signpost to what needs to not be true if the Born rule were to need replacing. That's of value.