To what extent can one bring classical equations to quantum mechanics

  • #1
As far as I can understand, quantization of a system is to take poisson brackets to commutators. i.e.[itex]\{something\}\to[something][/itex].

However, normally in a textbook, quantization of a system only involves commutation relations between generalized coordinates and generalized momentums. for example, [itex][q, p] = ih[/itex]

With that in mind, I have the following questions.
1. Does other poisson bracket relations(not only between generalized coordinates and generalized momentums) hold in quantum mechanics? If so, why none of them is mentioned in a textbook

2. Normally, textbooks consider other mechanical quantities as functions of generalized coordinate and generalized momentum. In this way, the classical definitions are transformed into operator relations. Generally, definitions are equations. So the question comes up naturally that whether all classical equations can be regarded as quantum operator equations.

This is definitely not true in Schrodinger picture 'cause some of the equations contain derivatives of mechanical quantities and operators are time-independent in Schrodinger picture. What if in Hisenberg picture?
 

Answers and Replies

  • #2
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As far as I can understand, quantization of a system is to take poisson brackets to commutators. i.e.[itex]\{something\}\to[something][/itex].
That's the old way promulgated by Dirac.

The correct way is found in Ballentine Chapter 3:
https://www.amazon.com/dp/9810241054/?tag=pfamazon01-20&tag=pfamazon01-20

Basically it's a requirement of the symmetries of the Principle Of Relativity.

And, equally as interestingly, the same is in fact true of Classical Mechanics - although it's not usually presented that way.

A notable exception is Landau:
https://www.amazon.com/dp/0750628960/?tag=pfamazon01-20&tag=pfamazon01-20

Basically the situation is this - the principles of QM and the symmetries of the POR imply the dynamics of QM ie Schroedinger's equation etc. For what those principles are check out a post I did a while ago - see post 137 (basically there is only one to which we apply the beautiful Gleason theorem to get two):
https://www.physicsforums.com/showthread.php?t=763139&page=8

For classical mechanics we start with the Principle of Least Action (PLA), apply the symmetries of the POR (basically Noethers Theorem) and, lo and behold, you get classical mechanics.

The neat thing though is those principles of QM actually imply the PLA. Here's why.

You start out with <x'|x> (the square is the probability of it initially being at x' and later being observed at x) then you insert a ton of ∫|xi><xi|dxi = 1 in the middle to get ∫...∫<x|x1><x1|......|xn><xn|x> dx1.....dxn. Now <xi|xi+1> = ci e^iSi (the square gives the probability of it being at xi and a very short time later observed to be at xi+1) so rearranging you get
∫.....∫c1....cn e^ i∑Si.

Focus in on ∑Si. Define Li = Si/Δti, Δti is the time between the xi along the jagged path they trace out. ∑ Si = ∑Li Δti. As Δti goes to zero the reasonable physical assumption is made that Li is well behaved and goes over to a continuum so you get S = ∫L dt.

Look familiar? Yep - S is the classical action.

What that weird integral says is in going from point A to B it follows all crazy paths and what you get at B is is the sum of all those paths. Now if the path is long compared to how fast the exponential 'turns', since the integral in those paths is complex most of the time a very close path will be 180% out of phase so cancels out. The only paths we are left with is those whose close paths are the same and not out of phase so reinforce rather than cancel.

That's why you only have paths of stationary action ie the PLA holds.

To return to your question.

First check out the following paper:
http://bolvan.ph.utexas.edu/~vadim/c...s/brackets.pdf

So we see that the algebraic properties of the classical Poisson Bracket (PB) mean, if its the same in QM, that it corresponds to the commutator in QM.

In fact its a nice exercise to show those algebraic properties also imply the usual PB definition in Classical Mechanics. You expand the functions in power series to show it. Lenny Susskinds book on Mechanics shows the detail of how you do it if you are interested:
https://www.amazon.com/dp/046502811X/?tag=pfamazon01-20&tag=pfamazon01-20

It just turns out that from the two views the algebraic structure of the PB's are exactly the same. You see this by substituting in the equations you get from the dynamics of both Classical and Quantum Mechanics.

Thanks
Bill
 
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  • #3
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For classical mechanics we start with the Principle of Least Action (PLA), apply the symmetries of the POR (basically Noethers Theorem) and, lo and behold, you get classical mechanics.
Do you know if the principle of least action can be completely derived from Newton's three laws of motion, or is there more to it?
 
  • #4
atyy
Science Advisor
14,212
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As far as I can understand, quantization of a system is to take poisson brackets to commutators. i.e.[itex]\{something\}\to[something][/itex].

However, normally in a textbook, quantization of a system only involves commutation relations between generalized coordinates and generalized momentums. for example, [itex][q, p] = ih[/itex]

With that in mind, I have the following questions.
1. Does other poisson bracket relations(not only between generalized coordinates and generalized momentums) hold in quantum mechanics? If so, why none of them is mentioned in a textbook

2. Normally, textbooks consider other mechanical quantities as functions of generalized coordinate and generalized momentum. In this way, the classical definitions are transformed into operator relations. Generally, definitions are equations. So the question comes up naturally that whether all classical equations can be regarded as quantum operator equations.

This is definitely not true in Schrodinger picture 'cause some of the equations contain derivatives of mechanical quantities and operators are time-independent in Schrodinger picture. What if in Hisenberg picture?
Probably not what you had in mind, but the Schroedinger picture can be formally made into an operator by expressing the wave function as a density matrix. This formalism is nice, becuase it deals with statistical mixtures easily. Then since quantum mechanics is a probabilistic theory, one might try to use the version of classical theory in which deterministic evolution is treated probabilitistically, ie. the canonical picture and the Liouville equation. The quantum mechanical counterpart of the classical distribution in phase space is the Wigner function (which is not a probability distribution, but whose "marginals" are). The Liouville equation is the classical limit of the equation governing the time evolution of the Wigner function. Wikipedia gives the relationship between the Wigner function and the time evolution of the density matrix http://en.wikipedia.org/wiki/Density_matrix.
 
Last edited:
  • #5
kith
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1. Does other poisson bracket relations(not only between generalized coordinates and generalized momentums) hold in quantum mechanics? If so, why none of them is mentioned in a textbook.
There are many examples of things like [xn,p] in textbooks but not all of these commutators fulfill Dirac's correspondence rule. See https://en.wikipedia.org/wiki/Canonical_quantization#Issues_and_limitations.

2. Normally, textbooks consider other mechanical quantities as functions of generalized coordinate and generalized momentum. In this way, the classical definitions are transformed into operator relations. Generally, definitions are equations. So the question comes up naturally that whether all classical equations can be regarded as quantum operator equations.
A necessary condition is that the resulting operator is self-adjoint. f(x,p)=xp for example is not self-adjoint. So you need something like f(x,p)=1/2(xp+px) in order to define such a quantity sensibly. Also QM includes additional degrees of freedom (like spin) which are not present in the classical equations.
 
  • #6
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Do you know if the principle of least action can be completely derived from Newton's three laws of motion, or is there more to it?
The PLA is implied by the usual formulation of Newtons laws (eg the 3 laws) and conversely.

Just about any advanced book on Mechanics will explain that eg - Goldstein.

My favourite however is Landau.

But from the viewpoint of QM and the path integral approach the PLA is more fundamental.

You can also base it on the algebraic properties of the PB - which is basically Diracs approach.

Thanks
Bill
 
  • #7
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0
The PLA is implied by the usual formulation of Newtons laws (eg the 3 laws) and conversely.
Are they enough to cover rotation too, or just translation?
 
  • #8
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2,643
Are they enough to cover rotation too, or just translation?
Yes - angular momentum fits the same scheme.

Thanks
Bill
 
  • #9
There are many examples of things like [xn,p] in textbooks but not all of these commutators fulfill Dirac's correspondence rule. See https://en.wikipedia.org/wiki/Canonical_quantization#Issues_and_limitations.


A necessary condition is that the resulting operator is self-adjoint. f(x,p)=xp for example is not self-adjoint. So you need something like f(x,p)=1/2(xp+px) in order to define such a quantity sensibly. Also QM includes additional degrees of freedom (like spin) which are not present in the classical equations.
Then is there a rigorous treatment of canonical quantization? It seems buggy to accept some poisson brackets and reject the others.
 
  • #10
rubi
Science Advisor
847
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Then is there a rigorous treatment of canonical quantization? It seems buggy to accept some poisson brackets and reject the others.
The process of quantization is ambiguous in general. That means that for every classical theory, there are several quantum theories that have the same classical limit. So if you want to quantize a classical theory, you have to specify some additional information that fixes these ambiguities. Replacing Poisson brackets by commutators doesn't work (there is a no-go theorem by Groenewold and van Hove). However, you can do the following replacement: ##[\hat A,\hat B] = \mathrm i\hbar\widehat{\{A,B\}} + O(\hbar^2)##. Each choice of the ##O(\hbar^2)## terms correspond to a different choice of quantization. There are two rigorous ways to do this: In deformation quantization, you start with the clasical Poisson algebra of observables and specify a so called ##\star##-product and define ##[A,B]=A\star B - B\star A##. You then try to represent this deformed algebra on some Hilbert space. The other rigorous method is geometric quantization. In GQ, the ambiguities are fixed by specifying a so called polarization of the phase space. If you express this in coordinates, it roughly means that you choose, what the position and momentum coordinates are. For these coordinates, Dirac's replacement holds exactly and all the other commutators are fixed by the choice of polarization.
 
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