Oscillator Model with Eigenfunctions

In summary, the conversation discusses the Oscillator Model in the Milonni and Eberly book "Laser." The Hamiltonian of the process is H=H0+HI, where the first term is the "undisturbed" Hamiltonian and the second term is the interaction produced by an applied field. The eigenfunctions Φn are solutions to the Schrödinger equation with HI=0, but it is assumed that they are also solutions when there is an applied field. The rule of thumb for two problems sharing eigenfunctions is that if two operators share the same set of eigenfunctions, they necessarily commute. This can be proven in an introductory quantum mechanics course.
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Vajhe
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Hi, I have been reading the Milonni and Eberly book "Laser": in one of the chapters they discuss the Oscillator Model. The treatment is quite straightforward, the Hamiltonian of the process is

H=H0+HI

where the first term is the "undisturbed" hamiltonian, and the second one is the interaction produced by an applied field.

Φn are the eigenfunctions that solve the Schrödinger equation with HI=0 (i.e no applied field), but to continue you have to assume that the same functions (Φn) are also solutions to the problem when there is an applied field. How can you say that? What do you lose? Is there some kind of rule of thumb to be able to say that two problems (not necessarily these) will share eigenfunctions?
 
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  • #2
Vajhe said:
Is there some kind of rule of thumb to be able to say that two problems (not necessarily these) will share eigenfunctions?

You've almost certainly seen this in an introductory quantum mechanics course: if two operators share the same set of eigenfunctions, they necessarily commute. So for your case, ##[H_0,H_I] = 0##. Be sure you know how to prove this statement!
 
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Oh, I was looking the answer in the wrong place: I will have to remove the dust from my Sakurai. Thanks a lot!
 

1. What is an oscillator model with eigenfunctions?

An oscillator model with eigenfunctions is a mathematical model that describes the behavior of a system that oscillates or vibrates. It uses eigenfunctions, which are solutions to the oscillator's equation of motion, to represent the different possible states of the system.

2. How is an oscillator model with eigenfunctions used in science?

An oscillator model with eigenfunctions is used in various fields of science, such as physics, chemistry, and engineering, to study the behavior of oscillating systems. It can help predict the frequencies and amplitudes of vibrations, as well as the stability and energy of the system.

3. What are the advantages of using an oscillator model with eigenfunctions?

One advantage of using an oscillator model with eigenfunctions is that it provides a more accurate and detailed description of the system's behavior compared to simpler models. It also allows for the analysis of complex systems with multiple oscillators, as each oscillator can be described by its own set of eigenfunctions.

4. Are there any limitations to the oscillator model with eigenfunctions?

Yes, there are limitations to the oscillator model with eigenfunctions. It assumes that the system is linear, meaning that the oscillations are proportional to the applied force. It also assumes that the system is in a steady state, with no external forces acting on it.

5. Can the oscillator model with eigenfunctions be applied to real-world systems?

Yes, the oscillator model with eigenfunctions can be applied to real-world systems, as many physical systems exhibit oscillatory behavior. However, the model may need to be modified or combined with other models to account for non-linearities or external forces that may affect the system's behavior.

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