Step operators for harmonic oscillator

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SUMMARY

The discussion centers on the calculation of energy shifts in a one-dimensional harmonic oscillator (H.O.) due to a weak perturbation proportional to x⁴. The user initially claims the energy shift is proportional to (3+4n+6n²), while the answer sheet states it is (3+6n+6n²). After further analysis, the user acknowledges an error in their calculations regarding the prefactors associated with the creation (a⁺) and annihilation (a⁻) operators, confirming that the answer sheet is correct.

PREREQUISITES
  • Understanding of quantum mechanics, specifically harmonic oscillators
  • Familiarity with perturbation theory in quantum mechanics
  • Knowledge of creation (a⁺) and annihilation (a⁻) operators
  • Ability to perform calculations involving matrix elements in quantum states
NEXT STEPS
  • Study the derivation of energy shifts using first-order perturbation theory
  • Learn about the properties and applications of creation and annihilation operators
  • Explore the mathematical techniques for calculating matrix elements in quantum mechanics
  • Review examples of perturbations in quantum systems, particularly in harmonic oscillators
USEFUL FOR

Students and researchers in quantum mechanics, particularly those focusing on harmonic oscillators and perturbation theory, will benefit from this discussion.

TorKjellsson
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Hi!

Info:
This is a rather elementary question about the creation a(+) and annihilation (a-) operators for the 1D H.O.

The problem is to calculate the energy shift for a given state if the weak perturbation is proportional to x⁴.

Using first order perturbation theory for the non-degenerate case we expect the shift to be proportional to:

< n| x⁴ |n >.

x is proportional to (a(+) + a(-)) which means that x⁴ will give terms that are multiples of these. Since the states of HO are orthogonal to each other, only terms that contain two (a+) and two a(-) will give a contribution (since all other terms lead to <m|n> where m=/=n )

Question:
My problem here is that I claim this shift to be proportional to (3+4n+6n²) whereas the answer sheet states that it is 3+6n+6n². My teacher said that he'll solve the problem on monday but since we've got the exam for this course on tuesday I'd rather know it straight away. So if there's anyone who'd like to either confirm my answer or explain why the answer sheet is correct it would be most welcome!

Regards
Tor
 
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At a quick guess, you might have forgotten to carefully calculate the prefactors (sqrt n, etc.) which occur as you apply the operators.
 
sorry guys, I was wrong. I thought I had checked all steps very carefully but I see that I made an error in the signs of some terms...

so the answer sheet was correct =P
 

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