HUP and good quantum numbers that commute

[Salman2]

I have a question about the HUP. As I understand the HUP, it only applies to conjugate attributes that do not commute, such as position and momentum. However, many good quantum numbers do commute, so does this mean that the HUP does not apply to simultaneous measurement of such good quantum numbers ?

Also, for the hydrogen atom, is it not true that the total energy Hamiltonian, and angular momentum commute, thus the HUP would not apply to their simultaneous measurement ?

 

[DrClaude]

The set of good quantum numbers uniquely designate a particular quantum state. Therefore, you have to be able to know them simultaneously. By the way, quantum numbers themselves do not "commute", it is the operators relate to them that do (or do not...).

So yes, you can simulatneously know the total energy of the hydrogen atom and its total angular momentum. But that does not mean that you can also know individual angular momenta, such as electronic orbital angular momentum. That will depend on what terms you actually consider in the Hamiltonian (spin-orbit coupling, hyperfine interaction, ...)

 

[Salman2]

The set of good quantum numbers uniquely designate a particular quantum state. Therefore, you have to be able to know them simultaneously. ..So yes, you can simultaneously know the total energy of the hydrogen atom and its total angular momentum.

Thanks.

I would like to follow-up your reply to a question about the double slit experiment. At the moment in time a quantum state enters the 3D volume of each slit, would it be correct to say that we can 'know' simultaneously all good quantum numbers for the state at each 3D slit space ? If yes, can we then know the combined set of good quantum numbers at both slits, simultaneously ?