1d potential and switching between operators

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SUMMARY

The discussion centers on the challenge of calculating the commutation relationship between position (x) and Hamiltonian (H) operators in quantum mechanics. Participants express difficulty in applying the substitution method for momentum (p) and using the commutation relationship effectively. A key suggestion is to calculate the commutator [x,H] and then evaluate in two different ways to derive the desired result. This approach is essential for understanding operator switching in quantum mechanics.

PREREQUISITES
  • Understanding of quantum mechanics principles, particularly operator algebra.
  • Familiarity with commutation relations in quantum mechanics.
  • Knowledge of Hamiltonian mechanics and its role in quantum systems.
  • Ability to perform calculations involving bra-ket notation.
NEXT STEPS
  • Study the derivation of the commutation relation [x,H] in quantum mechanics.
  • Learn about the implications of operator switching in quantum systems.
  • Explore examples of calculating for various quantum states.
  • Review the fundamentals of Hamiltonian mechanics and its applications in quantum theory.
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Students of quantum mechanics, physics researchers, and anyone seeking to deepen their understanding of operator relationships and calculations in quantum systems.

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Homework Statement


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Homework Equations





The Attempt at a Solution


As a group we're stuck on this as a result of the lecturer saying that he wouldn't help us because we should work as a group and find other ways other than asking him about it. Which is fair enough - but none of us understand what's going on!

So far I have tried substituting in for p in the middle of the bracket with -ihd/dx and then used the commutation relationship to try substituting ih for xp - px, but this seems to leave me being nowhere.

I also tried starting with the bracket having H in the middle and substituting for that - this kind of left me with a bracket with p in the middle and one with x in the middle, but whilst this kind of left things in a format that almost looked correct the two brackets were not equal to one another, only additive.

Could anyone suggest a starting point to look at or just any other helpful information that I may have missed when rereading my notes.
 
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Well, have you calculated [x,H]?

Then you can calculate <n|[x,H]|n'> in two different ways: equating the two answers gives you the desired result.
 
No I hadn't - I thought that was to do with the second part of the question.

I shall give that a go now!
 

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