Perturbation theory infinite well

In summary, the conversation discusses the calculation of total energy in an infinite well with a small potential. The speaker used the time independent Schrodinger equation and the slope equation to calculate the total energy, and found that the addition of the small potential did not affect the total energy of the system. However, the other speaker suspects that the integral was done incorrectly and explains that a perturbation can still have an effect on the energy to a higher order.
  • #1
Dammes
15
0
in the infinite well with small potential shown in the attachment.
I calculated the total energy by using the time independent Schrodinger equation and adding the correction energy to the equation of the slope k=(Vo/L)x.

E=h^2/8mL^2 +∫ ψkψ dx

ψ=√(2/L) sin⁡(∏/L x)

when integrating ∫ ψkψ dx between 0 and L
I got Zero, ∫ ψkψ dx=0
∴ total energy=h^2/8mL^2 +0

so what i don't understand is when adding a small potential it doesn't affect the total energy of the system? that is what it shows when i integrated it.
 
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  • #2
I suspect you've done the integral wrong. That said, it's perfectly possible for a perturbation to have no effect on the energy of any given state, especially if you only compute the change in energy to first order in perturbation theory.
 
  • #3
With a symmetric density and an antisymmetric additional potential, the first order correction will be 0. The second order includes that the wave function changes (based on the modified potential), and that will give a non-zero correction.
 

1. What is perturbation theory in the context of an infinite well?

Perturbation theory is a mathematical method used to approximate solutions to a problem that cannot be solved exactly. In the case of an infinite well, it is used to calculate the energy levels and wavefunctions of a particle confined within the well, taking into account small perturbations or disruptions to the system.

2. How does perturbation theory work in an infinite well?

In perturbation theory, the Hamiltonian (energy operator) of the system is split into two parts: the unperturbed Hamiltonian, which describes the system without any perturbations, and the perturbation Hamiltonian, which accounts for the small disruptions. The solutions to the unperturbed Hamiltonian are used as a basis to calculate the perturbation corrections to the energy levels and wavefunctions.

3. What are some common perturbations in an infinite well?

Some common perturbations in an infinite well include changes in the shape or size of the well, the addition of a potential barrier, or the presence of an external electric or magnetic field.

4. What are the limitations of perturbation theory in an infinite well?

Perturbation theory is most accurate for small perturbations, and the accuracy decreases as the perturbation becomes larger. Additionally, it may not work well for systems with degenerate energy levels, where multiple unperturbed states have the same energy. In these cases, other methods such as variational techniques may be more appropriate.

5. How is perturbation theory used in other areas of science?

Perturbation theory is a widely used mathematical tool in various fields of science, including physics, chemistry, and engineering. It is used to approximate solutions in systems that are too complex to solve exactly, such as in quantum mechanics, celestial mechanics, and fluid dynamics. It is also commonly used in data analysis and signal processing to estimate the effects of noise or other disturbances on a system.

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