Commutators, operators and eigenvalues

  • #1
dyn
773
61
Hi
I just wanted to check my understanding of something which has come up when first studying path integrals in QM. If x and px are operators then [ x , px ] = iħ but if x and px operate on states to produce eigenvalues then the eigenvalues x and px commute because they are just numbers. Is that correct ?
Thanks
 
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  • #2
Operators ##x## and ##p_x## don't have eigenstates in common.
 
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  • #3
Yes . i know that but x can act on a position eigenstate and p can act on a momentum eigenstate. I just want to check that once the operators act on states to produce eigenvalues , that eigenvalues commute because they are just numbers ?
 
  • #4
dyn said:
the eigenvalues x and px commute because they are just numbers. Is that correct ?
No, it makes no sense, because you don't multiply eigenvalues in QM. You could of course measure the position of one quantum system and the momentum of another and then multiply the results you get, but what sense would it make?
 
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  • #5
dyn said:
something which has come up when first studying path integrals in QM
I'm struggling to see how the question you are asking could arise in that context. Can you give a reference?
 
  • #6
dyn said:
. I just want to check that once the operators act on states to produce eigenvalues , that eigenvalues commute because they are just numbers ?
Numbers and operators commute. It does not matter from where the numbers come. 2024 commutes with operators. We do not care whether it is a measured value of position of some state or new calendar year. Happy new year!
 
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  • #7
PeterDonis said:
I'm struggling to see how the question you are asking could arise in that context. Can you give a reference?
The derivation of the path integral in QM on P211-212 of "QFT for the gifted amateur" by Lancaster & Blundell
 
  • #8
dyn said:
The derivation of the path integral in QM on P211-212 of "QFT for the gifted amateur" by Lancaster & Blundell
Yes, that is correct. All the ##p##'s and ##q_n##'s here,

1704165225828.png


are just real numbers and of course commute.
 
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