Talk:Canonical commutation relation

Question

Anyone read Dirac's book on QM? I'm missing something on this derivation. -ub3rm4th <br\> Quoted (section 22, page 93):

$({\frac {\partial }{\partial q_{r}}})^{*}\left.\psi \right\rangle ^{*}=e^{-i\gamma }{\frac {\partial }{\partial q_{r}}}\left.\psi \right\rangle =e^{-i\gamma }{\frac {\partial }{\partial q_{r}}}e^{i\gamma }\left.\psi \right\rangle ^{*}$

showing that

$({\frac {\partial }{\partial q_{r}}})^{*}=e^{-i\gamma }{\frac {\partial }{\partial q_{r}}}e^{i\gamma }$

or, with the help of ${\frac {\partial }{\partial q_{r}}}f-f{\frac {\partial }{\partial q_{r}}}={\frac {\partial f}{\partial q_{r}}}$,

$({\frac {\partial }{\partial q_{r}}})^{*}={\frac {\partial }{\partial q_{r}}}+i{\frac {\partial \gamma }{\partial q_{r}}}$

Question, more urgent!

${\frac {\partial }{\partial q_{r}}}\left.\psi \right\rangle =0$

Why?

Gauge invariant?

"The non-relativistic Hamiltonian for a particle of mass m and charge q in a classical electromagnetic field is

$H={\frac {1}{2m}}\left(p-{\frac {qA}{c}}\right)^{2}+q\phi$

where A is the three-vector potential and $\phi$ is the scalar potential. This form of the Hamiltonian, as well as the Schroedinger equation $H\psi =i\hbar \partial \psi /\partial t$, the Maxwell equations and the Lorentz force law are invariant under the gauge transformation

$A\to A^{\prime }=A+\nabla \Lambda$

$\phi \to \phi ^{\prime }-{\frac {1}{c}}{\frac {\partial \Lambda }{\partial t}}$

$\psi \to \psi ^{\prime }=U\psi$

$H\to H^{\prime }=UHU^{\dagger }$

where

$U=\exp \left({\frac {iq\Lambda }{\hbar c}}\right)$

and $\Lambda =\Lambda (x,t)$ is the gauge function."

It is not true that this H is invariant under a gauge transformation. Substituting for the transformed potentials with simple algebra H' becomes

$H^{\prime }=UHU^{\dagger }-{\frac {q}{c}}{\frac {\partial \Lambda }{\partial t}}$

not the expression given in the article. However the Schroedinger equation, the Maxwell equations and the Lorentz force law are invariant under the gauge transformation. Xxanthippe 04:02, 11 June 2007 (UTC)

Would you mind citing some references for this? I think the text (wikipedia article) is a little bit unclear about whether the Hamiltonian or the Schrödinger equation are gauge invariant. It should be nice if someone review this section. User:143.107.229.183

The Hamiltonian of semi-classical electrodynamics, given above, is, by elementary algebra (consider the potential term), not gauge invariant. The field equation that it is associated with, the Schrödinger equation is gauge invariant because of the time derivative of the wavefunction that occurs in it. On the other hand, the Hamiltonian and the field equations of quantum electrodynamics are all gauge invariant. Puzzled? The reason is that semi-classical electrodynamics is an incomplete theory because it does not take account of the internal dynamics of the (electromagnetic) field as it describes the field by externally specified potentials. Quantum electrodynamics is a complete theory in this respect but it suffers from disadvantages like giving infinite results. However, it can be shown that the probability amplitudes for transitions of the matter field in semi-classical electrodynamics are gauge invariant. So semi-classical electrodynamics is a pretty good theory for most of chemistry and much of physics, so much so that some people don't realise its shortcomings. Xxanthippe (talk) 00:34, 20 June 2010 (UTC).

Proof of Canonical Commutation Relations?

The "uncertainty principle" wikipedia page explains how to derive the uncertainty principle from these commutation relations, but how does one physically prove the canonical commutation relations?

i.e. What experiments or logic led to the proposition [x,p_x] = ih/(2*pi) being accepted as true?

I ask this because I think the answer would make a great addition to this article.

Good point. It's too bad that it has received no response.

For starters, there are two things involved in your question. One is a question of math (and not exactly of logic). I think you will find your answer in Thomas F. Jordan's Quantum Mechanics in Simple Matrix Form. The situation, as well as I can understand it, is that the relationship discussed here came out of the math. If I figure it out I'll try to remember to come back and supply here what I hoped would be provided in the article.

The other issues is whether what is mathematically true happens to describe reality. For a long time, Euclidean geometry was taken as a description, indeed the only description, of space. Euclidean geometry can be made axiomatically valid, but so can other geometries. The question then becomes whether, e.g., at huge distances the sum of the angles of a triangle are indeed equal to 180 degrees. If space itself is curved, then that shows experimentally that some other geometry must be the correct one.

There is lots of discussion about how experimental verifications would have to be made in the case of pq not being equal to qp, but I don't know whether the experiments have been pursued. There is another answer, I suspect, in that the bright line spectrum lines are not perfect geometrical lines as they would be if your laser pointer produced light at exactly 660 nanometers, or if the bright lines of the hydrogen spectrum were made entirely of photons of exactly one wavelength. There are differences in fuzziness depending on the frequencies involved, too. It's all a little fuzzy in my memory, but if and when I run onto that discussion again I'll try to remember to post it here. P0M (talk) 01:54, 22 June 2009 (UTC)

See Jordan, p. 130-132. I think this is where the proof appears. (I wish his English was a little clearer.) P0M (talk) 06:05, 24 June 2009 (UTC)

See also, Aitchison, MacManus, and Snyder, "Understanding Heisenberg's 'magical' paper of July 1925," Appendix A.

Mehra's three volume book on the history of quantum mechanics refers over and over again to the enthusiasm and stamina with which people in Heisenberg's circle had been spectrographically investigating the hydrogen spectrum -- and that over the course of several years. So it seems likely to me that the data was already there. Planck's constant is a very small number, so they would have had to prepare the experiments very carefully and make many measurements I have one book, which I can't find at the moment, that has some details (or maybe speculation) on how experiments need to be done in this regard. P0M (talk) 05:20, 25 June 2009 (UTC)

Boundedness?

"It is relatively easy to see that two operators satisfying the canonical commutation relations cannot both be bounded." - Can anyone provide a reference for this claim? —Preceding unsigned comment added by 142.151.156.48 (talk) 00:46, 5 March 2009 (UTC)

Self-evident: Trace. Cuzkatzimhut (talk) 01:15, 4 March 2013 (UTC)

Would you like to expand for the benefit of those with less intelligence than yourself? Xxanthippe (talk) 01:22, 4 March 2013 (UTC).

But under no circumstances in the article: less is more. As the aside invitation to take the trace of the commutation relation suggests, the right hand side is of order N for N-dimensional vector spaces and infinite for the Hilbert space in question. The left hand side, by the cyclicity of the trace is 0 for finite dimensional spaces, and ambiguous for Hilbert space, that is as N grows infinite, the zero and the emergent delta functions conspire to yield an infinite expression equal to the right hand side. To sum up, for finite N, the commutation relation cannot hold, as seen by thus tracing, but it tends to the one given in the article provided either x or p are unbounded, in our case both. My favorite reference is Santhanam, T. S.; Tekumalla, A. R. (1976). "Quantum mechanics in finite dimensions". Foundations of Physics. 6 (5): 583. doi:10.1007/BF00715110. where that limit emerges magnificently, but it would be overkill to foist it on the reader right here. The reader is steered to the Stone-von Neumann theorem, and might go to standard QM texts if interested. I think just the plain statement and the exhortation to take the trace suffice to let the reader in the know recall the fact, and the one who is not to shrug it off.... One could always cite the canonical derivation of Ch VIII of Reed and Simon, ISBN 0125850506, based on this commutator, but I fear this is overkill for an elementary introduction−−maybe the cite could belong in the wiki on boundedness. Cuzkatzimhut (talk) 12:59, 4 March 2013 (UTC)

Poor writing

Para 2 starts out: "In contrast..."

In contrast to what? How is the reader supposed to guess? P0M (talk) 01:40, 22 June 2009 (UTC)

? I can't guess what your question asks. "By contast" to the two conjugate variables of the preceding paragraph, of course, which fail to commute. What else? Where is the lack of clarity hiding? Cuzkatzimhut (talk) 00:56, 3 December 2012 (UTC)

Not to mention there is ZERO motivation for the result that is the topic of this article. Berrtus (talk) 19:52, 6 May 2015 (UTC)

Also, the last equatins in Generalizations are not easy to be shown. Anyone knows the proof? 88.195.251.114 (talk) —Preceding undated comment added 23:03, 14 October 2015 (UTC)