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Why not formulate QM in terms of ψ squared? 
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#1
Nov2912, 08:22 PM

P: 239

I'm beginning to study QM, and as I understand all the information we can get out of system with 1 particle is the probability distribution function (which has position and time as variables). By knowing the wave function it's possible to know the probability distribution function, but that's an indirect way to find it. Why not formulate directly QM in terms of ψ squared?



#2
Nov2912, 08:30 PM

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#3
Nov2912, 11:14 PM

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PF Gold
P: 2,819

That's because it only applies to what are called pure states. The general formula is given an observable R the expected value of R E(R) = Trace(pR) where p is the system state which is defined as a positive operator of unit trace. Pure states are a special case being the operator u><u and the squaring rule is a further special case when the pure state is expanded in terms of eigenvectors of the observable.
Thanks Bill 


#4
Nov3012, 03:07 AM

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P: 4,612

Why not formulate QM in terms of ψ squared?
There is, however, a way to eliminate the unphysical total phase of the wave function. It can be done by reformulating QM in terms of density matrices, where each wave function is represented by a corresponding density matrix. 


#5
Nov3012, 03:56 AM

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#6
Nov3012, 05:51 AM

P: 138

The answer is the 2 slit experiment. Suppose you have that the probability of a particle entering in slit 1 is 0,3 and the probability of the same particle entering in slit 2 is 0,5, then in classical probability the probability of entering in either slit 1 or 2 is 0,8. Nevertheless in QM the answer could be more or less than that depending on the distance of the slits, the velocity of the particles and such. So, they noticed that, in order to predict such behaviour, a particle should be described by 2 real numbers (real and complex part of the wave function).
There is a paper that states more precisely this point of view: "Origin of Complex Quantum Amplitudes and Feynman's Rules" Philip Goyal, Perimeter Institute, Waterloo, Canaday Kevin H. Knuthz, University at Albany (SUNY), NY, USA John Skillingx, Maximum Entropy Data Consultants Ltd, Kenmare, Ireland Just to summarize, the AND & OR rules of probability theory does not describe the AND & OR rules of particle experiments, so they had to develope this "crazy new probability theory" 


#7
Nov3012, 10:41 AM

P: 7

Hi the_pulp,
I was always wandering why a real valued wave function can't be used in QM. The title "Origin of Complex Quantum Amplitudes and Feynman's Rules" sounds very interesting. Unfortunately it costs $25 to download :( Could you give a "light weight" version of it in this forum? 


#8
Nov3012, 12:35 PM

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P: 1,943

Decoherence (the modern link between classical and quantum mechanics) works with the density matrix, and shows why the environment turns most density matrices into nearly diagonal classical densities (in the right basis, the socalled pointer basis). 


#9
Nov3012, 12:54 PM

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#10
Nov3012, 01:03 PM

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#11
Nov3012, 01:31 PM

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#12
Nov3012, 01:47 PM

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What I wondered, probably because I don't understand qft too well: Wouldn't it be possible to consider the charge density rho as some field, something related to its time derivative as momentum and then doing second quantization with it?



#13
Nov3012, 02:02 PM

PF Gold
P: 692



#14
Dec112, 01:19 PM

PF Gold
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#15
Dec112, 09:22 PM

P: 741

1) Because you lose phase information with ψ squared. 2) Because you couldn't analyzed diffraction experiments. 3) Because linear superposition of states doesn't apply to ψ squared. 


#16
Dec212, 04:28 AM

P: 362

Why can't we forumlate f=ma in terms of x(t) since that's all we want in the end?



#17
Dec212, 04:45 AM

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PF Gold
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Thanks Bill 


#18
Dec212, 08:57 AM

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P: 1,943




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