How and when to know if eigenvalue of H is energy?

[LightPhoton]

TL;DR

How to know when eigenvalue of Hamiltonian is energy when no classical analogues are available

Take a general Hamiltonian with λ's as it's eigenvalues.

How do we know a priori when λ is energy and when not? For classical analogues, we can say when classical H=E, E being energy then quantum λ=E. But we have quantum systems with no analogue in classical mechanics. In those cases as well we have time dependent conserved quantities. So do we abandon energy all together and just talk about generalized conservation quantities, whatever that maybe?

 

[PeterDonis]

Take a general Hamiltonian with λ's as it's eigenvalues.

How do we know a priori when λ is energy and when not?

Because that's the definition of the Hamiltonian in QM: it's the energy operator.

For classical analogues, we can say when classical H=E, E being energy then quantum λ=E.

This doesn't make sense. A classical model is not a quantum model. Classically the Hamiltonian is a function, not an operator.

we have quantum systems with no analogue in classical mechanics.

I'm not sure what you mean here. "Classical" and "quantum" are models, not physical things. The systems in nature don't know or care what model you're using. They're not "classical" or "quantum". They're just systems.

 

[LightPhoton]

Because that's the definition of the Hamiltonian in QM: it's the energy operator.

This is not true in general. Hamiltonian is generator of time in general. Hamiltonian Operator in a rotating frame has eigen values which are not energy eigenvalues.

This doesn't make sense. A classical model is not a quantum model. Classically the Hamiltonian is a function, not an operator.

I am not sure what you mean here. For a particle in central potential for example, In schrodinger equation we sub in Hamiltonian which is taken from classical mechanics, simply by switching position and momentum to their operators. Since in classical mechanics this hamiltonian is energy, we say that eigenvalues of it in quantum system are energy eigenvalues

I'm not sure what you mean here. "Classical" and "quantum" are models, not physical things. The systems in nature don't know or care what model you're using. They're not "classical" or "quantum". They're just systems.

Sure, but i meant, in respect to what i added above, that for systems for which we have no classical analogue, say spin and angular momentum interaction hamiltonian (S dot L), then how do we know their eigenvalues are energy or not, given that Hamiltonian by itself does not tell us that


PS: I use terms "classical analogues" and "quantum analogues" the way Sakurai does in 3rd edition section 2.2.4

 

[PeterDonis]

This is not true in general. Hamiltonian is generator of time in general.

It's the generator of time translations, but that is the energy operator, by Noether's theorem.

Hamiltonian Operator in a rotating frame has eigen values which are not energy eigenvalues.

Please be more specific. A reference would help.

For a particle in central potential for example, In schrodinger equation we sub in Hamiltonian which is taken from classical mechanics, simply by switching position and momentum to their operators.

We do that as a heuristic because we know it happens to work for that example. But that's not an actual derivation; it's just a heuristic that happens to work for that case.

For a more general discussion of how to derive the forms of the Hamiltonian and other operators from first principles in non-relativistic QM, see, for example, Ballentine; I think it's Chapter 2 of his textbook that goes into this, but I don't have my copy handy to check.

PS: I use terms "classical analogues" and "quantum analogues" the way Sakurai does in 3rd edition section 2.2.4

Ok, that reference is helpful.

 

[QuarkyMeson]

we sub in Hamiltonian which is taken from classical mechanics, simply by switching position and momentum to their operators

No you don't, it might look that way in some texts but that's just a heuristic you'll see.

The classical H is a function on phase space and the quantum $\hat{H}$ is an operator on a Hilbert space, and the classical one does not uniquely determine the quantum one.

 

[bhobba]

For a more general discussion of how to derive the forms of the Hamiltonian and other operators from first principles in non-relativistic QM, see, for example, Ballentine; I think it's Chapter 2 of his textbook that goes into this, but I don't have my copy handy to check.

It's chapter 3.

It's based on the classical POR, specifically that the probabilities are frame-independent (it breaks down relativistically).

Thanks
Bill

 

[Demystifier]

Calling something "energy" is not a matter of knowledge, it's a matter of convenience. When the Hamiltonian $H$ is conserved, classical or quantum, it is convenient to call it energy. Except in theories (such as general relativity) with a Hamiltonian constraint $H=0$, where we usually do not call it energy.

Speaking of Hamiltonian constraints, any classical theory with a conserved Hamiltonian can be rewritten as a theory with a Hamiltonian constraint. In a recent paper I used this insight to give some hints towards the solution of some conceptual problems in quantum gravity. https://arxiv.org/abs/2301.04448