Commutator Relations vs. Schrodinger Equation

[Enjolras1789]

Some books begin QM by postulating the Schrodinger equation, and arrive at the rest.

Some books begin QM by postulating the commutator relations, and arrive at the rest.

Which do you feel is more valid? Or are both equally valid? Is one more physical/mathematical than the other?

I would like some commentary on the following proposal:

Given that the uncertainty principle is a mathematical notion from Fourier analysis, and given that experiments imply that there is a wave/particle duality, starting from the commutator relations is perhaps less of a logical leap, and a better starting point, than the Schrodinger equation. To me, it makes sense to start from a mathematical property and an experimental fact, and arrive at the Schrodinger equation as a result. This is in contrast to starting from a rather weird equation, which has no particular theoretical motivation (other than that it is similar to the wave equation, but seemingly arbitrarily has first order time dependence).

Most books do not, however, do this...I assume there is a good reason and a flaw in my logic. I solicit criticism.

 

[alxm]

You don't need to postulate either (except the time-evolution of the S.E.) See e.g. http://hyperphysics.phy-astr.gsu.edu/Hbase/quantum/qm.html" .

The commutation relations come from the operators, which in turn can be derived from symmetries.
I know that if you look at e.g. Griffiths, he starts with the S.E. immediately. But more advanced textbooks (Landau-Lifgarbagez, Messiah, to mention a few) are more rigorous.

 

[Enjolras1789]

alxm:

I don't think our communication is clear. You say that you don't need to postulate the Schrodinger equation, and at the same time you need to postulate the time evolution of the Schrodinger equation. How can you postulate the time evolution of something you haven't accounted for? That doesn't make sense. We must be speaking on very different levels.

Shankar and Sakurai are considered "rigorous" by most graduate schools and they do start from the Schrodinger equation.

 

[alxm]

Nobody said it's not accounted for. It follows from the other postulates: Momentum follows from translational invariance of the wave function, and the Hamiltionian follows from replacing the momentum variable in the classical Hamiltonian with the operator version. The time-independent S.E. is then the eigenvalue equation corresponding to this operator.

But the time evolution cannot be derived from the other postulates, so it is a separate one.

 

[Enjolras1789]

So you propose (or someone other than Griffiths, Shankar, and Sakurai, anyway) that the beginning postulates are

1.) Translational invariance is assumed, giving us the momentum operator'
2.) Use the classical notion of the Hamiltonian, just modifying the momentum
3.) Time evolution of Hamilton's equation is first order in time, not second order
4.) Postulate Born interpretation of probability
5.) Postulate spin or just say it comes from QFT

Is that an accurate summary? If so, I am a bit perplexed. It seems to put spatial invariance on "higher footing" than time invariance or rotational invariance. That seems funny.

If we don't explicitly postulate time independence, we can have a non-conserved Hamiltonian. Can QM work without a non-conserved Hamiltonian? ie, just following Hamilton's equations of motions?

 

[alxm]

Well the list of postulates are given on the hyperphysics page, it doesn't include coordinate-transform postulates. I think those would probably have to be counted as fundamental postulates of physics in general.

From rotational invariance you have the angular momentum operator and from time-invariance you could also get the Hamiltonian (Landau-Lifgarbagez does this, starting with the time-evolution expression - without knowing the Hamiltonian, by using an expression for the action at the classical limit).

There are equivalent sets of postulates, and it seems everyone lists them slightly differently, or substitutes some equivalent expression, or merges them or splits them. (E.g. The wave function is normalized and square-integrable. Is that two or one?)

Personally I'm partial to Landau's derivation, as he makes the connection to classical mechanics quite clear and mostly works from very basic postulates. (That said I'm not sure I can recommend the textbook as a whole.)

 

[Enjolras1789]

1.) What parts of the book do you *not* recommend, for the sake of saving me time, if it can be briefly said without too much of your time?

2.) I respectfully disagree that most people list the postulates the same way. The books I mentioned above do not start with a principle-of-least-action logic, which to me is MUCH more intuitive, MUCH less arbitrary, than what those books do: postulate the Schrodinger equation, which is a rather bizarre, out of left field equation in my mind (a wave equation would make sense, but not one with first order time dependence). Do you not think that's a big deal? To me it's a huge deal; to follow from action minimization heuristically, intuitively, reasonably follows from classical mechanics. I really can't rationalize, a priori, the Schrodinger equation. You've obviously put some major thought into this, can you show me your logic for me to appreciate?

3.) As a matter of priority, what would you recommend doing first: learning QFT, or reading Landau and Lifgarbagez QM book with the derivation via least action?