[Quantum Mechanics] Anharmonic oscillator

Thread starter aiaiaial
Start date Sep 18, 2014
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Mechanics Oscillator Quantum mechanics

In summary, an anharmonic oscillator in quantum mechanics is a system with a more complex potential energy function, leading to different energy levels and behavior compared to a harmonic oscillator. Applications of anharmonic oscillators include modeling diatomic molecules, studying solids, and understanding chemical reactions. The energy spectrum of an anharmonic oscillator is calculated using perturbation theory, and its behavior can be studied experimentally using techniques such as spectroscopy.

Sep 18, 2014

aiaiaial

how to compute <n|ξ^4|n>?

The problem is above
I guess the ladder operator becomes some very ugly term. There should be a trick to compute <n|ξ^4|n>. Could anyone tell me?

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Sep 18, 2014

aiaiaial

I found a solution : )

just start with ξ = (a+ + a-) * sqrt(mw/2h)
and keep dealing with (a+)φ and (a-)φ.

1. What is an anharmonic oscillator in quantum mechanics?

An anharmonic oscillator is a system in quantum mechanics that does not follow the simple harmonic motion of a harmonic oscillator. It has a more complicated potential energy function, which leads to more complex behavior and energy levels.

2. How is an anharmonic oscillator different from a harmonic oscillator?

The main difference between an anharmonic oscillator and a harmonic oscillator is in their potential energy functions. A harmonic oscillator has a potential energy function that is directly proportional to the square of its displacement, while an anharmonic oscillator has a more complex potential energy function that includes higher order terms. This leads to different energy levels and behavior in the two systems.

3. What are the applications of anharmonic oscillators in quantum mechanics?

Anharmonic oscillators have various applications in quantum mechanics, such as in modeling the behavior of diatomic molecules, studying the properties of solids, and understanding the dynamics of chemical reactions. They are also used in the design of quantum computers and in the study of quantum chaos.

4. How is the energy spectrum of an anharmonic oscillator calculated?

The energy spectrum of an anharmonic oscillator is calculated using perturbation theory, which involves expanding the Hamiltonian of the system in terms of a small parameter and solving for the energy levels using a series of approximations. This allows for an accurate calculation of the energy levels in anharmonic systems.

5. Can the behavior of an anharmonic oscillator be studied experimentally?

Yes, the behavior of an anharmonic oscillator can be studied experimentally using various techniques, such as spectroscopy, neutron scattering, and laser spectroscopy. These experiments can provide insights into the energy levels and dynamics of anharmonic systems, and help validate theoretical predictions.