Time Independant Pertubation Theory - QM

In summary, the conversation discusses using lowest order perturbation theory to determine the shift in the second level of an electron confined to a 1 dimensional infinite well due to a perturbation. The conversation also mentions the use of integrals and the limits of integration in relation to the perturbation. Finally, it is noted that equation [1] only applies in the case of non-degenerate energy eigenstates.
  • #1
knowlewj01
110
0

Homework Statement



An electron is confined to a 1 dimensional infinite well [itex]0 \leq x \leq L[/itex]
Use lowest order pertubation theory to determine the shift in the second level due to a pertubation [itex]V(x) = -V_0 \frac{x}{L}[/itex] where Vo is small (0.1eV).


Homework Equations



[1]
[itex]E_n \approx E_n^{(0)} + V_{nn}[/itex]

[2]
[itex]V_{nn} = \int_{-\infty}^{\infty} \psi_{n}^{(0) *} (x) V(x) \psi_{n}^{(0)} (x) dx[/itex]

the following integral may be useful:

[3]
[itex]\int_{0}^{2\pi}\phi sin^2 \phi d\phi = \pi^2 [/itex]



The Attempt at a Solution



From [1] and the known result for E2 of an infinite well
[itex]E_2 = \frac{4\hbar^2 \pi^2}{2mL^2} - \frac{2V_0}{L^2}\int_{0}^{L} x sin^2\left(\frac{2\pi x}{L}\right) dx[/itex]

I can't see a substitution that will get it into the form in [3], anyone have any ideas?
Also, is equation [1] a general result for the time independant case for first order pertubations?

Thanks
 
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  • #2
If i were to make the substitution:

[itex]\phi = \frac{2\pi x}{L}[/itex]

[itex]\frac{\phi}{x} = \frac{2\pi}{L}[/itex]

does this imply that the limits of integration change from L to 2π ?
 
Last edited:
  • #3
Yes. Simply plug in the limits for x to find the limits for ɸ.
 
  • #4
knowlewj01 said:
[1]
[itex]E_n \approx E_n^{(0)} + V_{nn}[/itex]
Also, is equation [1] a general result for the time independant case for first order pertubations?
No, it isn't. It applies when the energy eigenstates of the unperturbed Hamiltonian states are non-degenerate, like in this problem. You'll have to use a different approach for the degenerate case.
 
  • #5
for any help!
Yes, equation [1] is a general result for first order perturbation theory in the time independent case. To solve this problem, you can use the fact that the integral in [2] can be simplified to [3] since the perturbation potential is linear in x. Therefore, we can rewrite [2] as:

V_{nn} = -\frac{V_0}{L}\int_{0}^{L} x sin^2\left(\frac{2\pi x}{L}\right) dx

Using the substitution u = 2\pi x/L, we can rewrite the integral as:

V_{nn} = -\frac{V_0}{L}\int_{0}^{2\pi} \frac{L}{2\pi} u sin^2(u) du

Using [3], this becomes:

V_{nn} = -\frac{V_0}{2\pi}\left(\frac{L^2}{2}\right) \pi^2 = -\frac{V_0 L^2}{4}

Substituting this into [1], we get the new energy for the second level as:

E_2 = \frac{4\hbar^2 \pi^2}{2mL^2} - \frac{V_0 L^2}{4} = \frac{4\hbar^2 \pi^2}{2mL^2} - \frac{V_0}{4}

Therefore, the shift in the second level due to the perturbation is -V_0/4, which is a small correction to the original energy.
 

Related to Time Independant Pertubation Theory - QM

What is Time Independent Perturbation Theory in Quantum Mechanics?

Time Independent Perturbation Theory is a technique used in quantum mechanics to solve for the energy levels and wavefunctions of a quantum system that has been subjected to a small perturbation (a small change in the system's Hamiltonian). It allows us to approximately solve for the energy levels and wavefunctions of a perturbed system by using the known solutions of the unperturbed system.

How does Time Independent Perturbation Theory work?

Time Independent Perturbation Theory involves treating the perturbation as a small parameter and expanding the perturbed Hamiltonian in a series. The first order term in this series is used to approximate the energy levels and wavefunctions of the perturbed system. This is known as first-order perturbation theory. Higher order terms can also be used to improve the accuracy of the approximation.

When is Time Independent Perturbation Theory applicable?

Time Independent Perturbation Theory is applicable when the perturbation is small compared to the energy differences between the unperturbed energy levels. It is also applicable when the unperturbed system has known solutions, which are usually simple and solvable.

What are the limitations of Time Independent Perturbation Theory?

Time Independent Perturbation Theory has several limitations, including its inability to accurately predict energy changes for large perturbations, its reliance on the unperturbed system having known solutions, and its inability to account for degenerate energy levels (energy levels with the same value). Additionally, higher order terms in the perturbation series may be needed for accurate predictions, making the calculations more complex.

What are some applications of Time Independent Perturbation Theory?

Time Independent Perturbation Theory is widely used in quantum mechanics to study various systems, including atoms, molecules, and solid state materials. It is also used in fields such as quantum chemistry and condensed matter physics to study the effects of external forces on a system. Additionally, perturbation theory is used in other areas of physics, such as classical mechanics and electromagnetism.

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