Introducing QM via commutators

[MichPod]

I understand there exists some way of teaching QM via postulating commutation relation between coordinate and momentum operator. May be even not simply postulating but bringing some reasons why such a commutator should be equal to "i"?

Could you recommend some good book or article which teaches QM that way? Better at some undergraduate level.

I am personally interested not only in understanding of how to develop the QM theory that way formally, but also in understanding what arguments and reasons may be brought into justify assigning "i" value to such a commutator and justifying other postulates through that path of QM introduction/teaching (to myself or others).

 

[kith]

The commutation relations can be motivated by symmetry principles. Momentum is the generator of spatial translations, angular momentum is the generator of rotations and energy is the generator of translations in time. Sakurai explains nicely how this is related to the commutation relations in his book "Modern Quantum Mechanics". A more in-depth description can be found in Ballentine's book "Quantum Mechanics: A Modern Development".

There's also an upcoming book from @A. Neumaier who seems to teach QM the way you like but probably not at your intended level.

 

[Swamp Thing]

You know what, MichPod, I have been trying to understand in a simple way (and trying to learn how to explain to someone else in a simple way) exactly this kind of idea for a long time, without making much progress.

Actually, what I had been focusing on was to explain the "hbar" part rather than the "i" part. Perhaps the "hbar" part is trivial compared to the "i", but here are some thoughts anyway... I have put it down in a personal, anecdotal form because of the strange coincidence of my reading your question a minute ago.

So here goes... Well, as I said, this kind of thing is something I've been trying to understand for a while now. Over the last couple of days, for some reason, I have felt an "aha" moment coming on, but it never quite took shape.

Until this morning, a couple of hours ago, I woke up and became aware of the following notions swirling in my brain:

Imagine some process that involves frequency and time. For an example of this, I always like to think of a sound. A very sharp click, in the frequency domain, doesn't sound like any particular musical note. You can't quite say it's a C or a D or an E. If you capture it as a WAV file its Fourier transform will be all over the place in terms of fequency.

On the other hand, a not-so-sharp "pop" or "ping" "or "pip" does carry a sense of being lower down or higher up the musical scale. Its Fourier transform will be reasonably localized as well.

Now, imagine taking the WAV file and doing an FFT in Python or MATLAB or even in Excel (my favorite). The FFT is agnostic about the meaning of the signal, but it takes N points and produces N points. It is up to us to attach a scale to the output. If the WAV file holds N samples per second, we can calibrate the FFT's X axis (up to some factor) in terms of N hertz per sample.

Planck's constant is merely a part of that scaling that you need to apply going from the FFT's input to the FFT's output. In the case of sound it is obvious, it's as simple as f=1/T. But in the case of position and momentum it's empirical, it's something that Planck fitted to the blackbody problem so many years ago.

Another little thought:
If you have a particle in a box, or waves in a cavity, then the wavelengths are constrained to integer values. So the DFT / FFT analogy is particularly relevant here. In the discrete Fourier world, what goes round must come round. It is a reversible world, in fact it is a world that is programmed to repeat again and again. This feature connects rather nicely (at least to my mind) with wavefunctions sloshing around forever inside those boxes, cavities and infinite wells.

 

[bhobba]

I understand there exists some way of teaching QM via postulating commutation relation between coordinate and momentum operator. May be even not simply postulating but bringing some reasons why such a commutator should be equal to "i"?.

That was Dirac's old Q number formulation after he read a pre-print of Heisenberg's Matrix Mechanics paper given to him by, if I remember correctly, his adviser Fowler. It impressed the hell out of Heisenberg who recognized it was a lot better than his effort. It was subsumed into his later transformation theory he came up with in 1926.

Read Dirac - Principles of QM if you like that approach - he develops the idea there.

However Ballentines derivation of the dynamics is much much better being based solely on symmetry, specifically the probabilities are frame independent - its such an obvious assumption you almost say - how could it be otherwise. Strictly speaking you are invoking the POR - but it seems so obvious you don't even realize that what's being invoked. From that the commutation relations easily follow.

Thanks
Bill

 

[kith]

Planck's constant is merely a part of that scaling that you need to apply going from the FFT's input to the FFT's output. In the case of sound it is obvious, it's as simple as f=1/T. But in the case of position and momentum it's empirical, it's something that Planck fitted to the blackbody problem so many years ago.

I'm not sure if you realize this, so let me emphasize it: the numerical value of dimensionful constants like Planck's doesn't have much physical content. It depends on the units and units simply reflect a convenient choice of what's considered to be big and small in certain situations. So if we leave the perspective of everyday experiences with its meters, seconds, kg, etc., and take the perspective of fundamental physics, it seems much more natural to make Planck's constant equal to one (see Planck units).

 

[Swamp Thing]

numerical value of dimensionful constants like Planck's doesn't have much physical content. It depends on the units and units simply reflect a convenient choice of what's considered to be big and small in certain situations.

Thanks for pointing that out.

So when we apply a Fourier transform to go from position to momentum, we actually need to "unscale" the conversion just to remove the arbitrary factor that our system of units has introduced.

 

[A. Neumaier]

There's also an upcoming book from @A. Neumaier who seems to teach QM the way you like but probably not at your intended level.

This will be a revised version of my somewhat unpolished online book. The revision process goes slower than expected, so the publication is now expected early 2018.

 

[hilbert2]

It's useful to note that any transformation $(\hat{x'},\hat{p'})=(\alpha \hat{x} + x_0 , \alpha^{-1}\hat{p} + p_0 )$, where $\alpha$, $x_0$ and $p_0$ are constants, keeps the commutation relation intact: $[\hat{x'},\hat{p'}] =[\hat{x},\hat{p}]$. This corresponds to Galilean invariance and freedom of choice with the length and time units.