Clasical and quantum harmonic oscillator - correspondence principle

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SUMMARY

The discussion centers on the correspondence principle between classical and quantum harmonic oscillators, highlighting that total energy in classical oscillators is proportional to the square of frequency, while in quantum oscillators, it is directly proportional to frequency. The participants explore the relationship between the two frequencies and the transition from quantum to classical oscillators, referencing the formula W = (n + 1/2)ω, where W represents energy, ω is frequency, and n is the integer quantum number. Additional resources such as HyperPhysics and Wikipedia are suggested for further clarification on these concepts.

PREREQUISITES
  • Understanding of classical harmonic oscillators
  • Familiarity with quantum mechanics principles
  • Knowledge of the correspondence principle
  • Basic grasp of energy quantization in quantum systems
NEXT STEPS
  • Research the mathematical derivation of the quantum harmonic oscillator energy levels
  • Explore the implications of the correspondence principle in quantum mechanics
  • Study the relativistic generalization of the quantum harmonic oscillator
  • Examine practical applications of harmonic oscillators in quantum field theory
USEFUL FOR

Students and researchers in physics, particularly those focusing on quantum mechanics, classical mechanics, and the study of oscillatory systems.

exponent137
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At classical harmonic oscillator, total energy is proportional to square of frequency, but at quantum harmonic oscillator, total energy is proportional to frequency.
Are those two frequencies the same?
How it is with transition from quantum harmonic oscillator to classical harmonic oscillator? How omega^1 and omega^2 agree?

One example of this is http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hosc6.html
but where is the frequency?

Maybe I found answer:
http://en.wikipedia.org/wiki/Correspondence_principle
Is this enough?
 
Last edited:
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The next question is:
How more simply to imagine formula W = (n+1/2)ω ?
W is energy ω is frequency and n is integer quantum number.
The common calculation for quantum oscillator is too long for such short result.
 
Does relativistic generalization of quantum harmonic oscillator exist?
Maybe:
http://www.quantumsciencephilippines.com/1811/lowest-order-relativistic-energy-correction-of-1-d-harmonic-oscillator/
 
Last edited:

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