How small for perturbation theory to be valid?

In summary, the conversation discussed the addition of a small perturbation to a particle in the ground state of a harmonic oscillator potential, and the question of how small the perturbation must be for perturbation theory to be valid. It was determined that the first order correction to the energy must be much smaller than the unperturbed energies for perturbation theory to be applicable.
  • #1
czaroffishies
15
0

Homework Statement



A particle of mass m is in the ground state in the harmonic oscillator potential

V(x) = [tex]\frac{1}{2}Kx^{2}[/tex]

A small perturbation [tex]\beta x^{6}[/tex] is added to this potential.

How small must [tex]\beta[/tex] be in order for perturbation theory to be valid?

Homework Equations



All here:
http://en.wikipedia.org/wiki/Pertur...chanics)#Time-independent_perturbation_theory

The Attempt at a Solution



Well, this is kind of a conceptual question, and I'm not sure how to start. It feels like I am guessing.

All I know is that:

[tex]\beta x^{6} << \frac{1}{2}Kx^{2}[/tex]

I would really appreciate a pointer in the right direction... thanks =)
 
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  • #2
czaroffishies said:

Homework Statement



A particle of mass m is in the ground state in the harmonic oscillator potential

V(x) = [tex]\frac{1}{2}Kx^{2}[/tex]

A small perturbation [tex]\beta x^{6}[/tex] is added to this potential.

How small must [tex]\beta[/tex] be in order for perturbation theory to be valid?

Homework Equations



All here:
http://en.wikipedia.org/wiki/Pertur...chanics)#Time-independent_perturbation_theory

The Attempt at a Solution



Well, this is kind of a conceptual question, and I'm not sure how to start. It feels like I am guessing.

All I know is that:

[tex]\beta x^{6} << \frac{1}{2}Kx^{2}[/tex]

I would really appreciate a pointer in the right direction... thanks =)

Well, these are operators so it is not well-defined to write an inequality between operators.
What you must do is impose that the first order correction to the energy due to the perturbation must much smaller than the unperturbed energies.
 

FAQ: How small for perturbation theory to be valid?

What is perturbation theory and how does it relate to small perturbations?

Perturbation theory is a mathematical method used to approximate the behavior of a complex system by breaking it down into simpler, solvable parts. It can be applied to systems where a small change or disturbance (known as a perturbation) is introduced, and the resulting changes in the system can be calculated. Therefore, perturbation theory is most valid for small perturbations, where the changes can be accurately approximated.

How small does a perturbation have to be for perturbation theory to be valid?

There is no exact measurement for how small a perturbation must be for perturbation theory to be valid, as it depends on the specific system and the desired level of accuracy. Generally, a perturbation is considered small if it results in small changes to the system's behavior and can be approximated using first- or second-order calculations in perturbation theory.

3. Can perturbation theory be applied to large perturbations?

In general, perturbation theory is only valid for small perturbations. This is because as the perturbation gets larger, the resulting changes in the system become more unpredictable and cannot be accurately approximated using perturbation theory. However, there are some cases where higher-order calculations in perturbation theory can be used to approximate larger perturbations.

4. Are there any limitations to perturbation theory for small perturbations?

While perturbation theory can be a powerful tool for approximating the behavior of systems with small perturbations, it does have its limitations. One limitation is that it may not accurately predict the behavior of the system for larger perturbations. Additionally, perturbation theory may not be applicable to certain non-linear systems or systems with strong interactions between components.

5. How can I determine if perturbation theory is valid for a specific system?

To determine if perturbation theory is a valid approach for a specific system, you can analyze the magnitude of the perturbation in relation to the system's overall behavior. If the perturbation is small and results in only minor changes to the system, perturbation theory is likely a valid method. However, if the perturbation is large and results in significant changes to the system, perturbation theory may not be a suitable approach.

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