Perturbation from a quantum harmonic oscillator potential

In summary, the off-diagonal term in the integration returns 0 due to (p^2+q^2), but (pq+qp) gives a complicated expression due to the quantum harmonic oscillator's wavefunctions. To compute the Hmn terms, using the creation and annihilation operators may be helpful. Another method is to substitute the operators p' and q' with p and q to get the wavefunction in the q space.
  • #1
Mayan Fung
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Homework Statement
Given the hamiltonian ##\hat{H}##:
$$\hat{H} = \frac{\hat{p}^2}{2} + \frac{\hat{q}^2}{2} + \alpha (\hat{p}\hat{q}+\hat{q}\hat{p})$$

Approximate the eigenvectors and eigenvalues numerically using the known eigenvectors with ##\alpha = 0## (That means I am asked to truncate the infinitely large matrix and find the eigenvectors and eigenvalues of the truncated matrix)
Relevant Equations
$$ E = \hbar \omega (n+\frac{1}{2}) $$
For the off-diagonal term, it is obvious that (p^2+q^2) returns 0 in the integration (##<m|p^2+q^2|n> = E<m|n> = 0##). However, (pq+qp) seems to give a complicated expression because of the complicated wavefunctions of a quantum harmonic oscillator. I wonder whether there is a good method to compute the Hmn terms?
 
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  • #2
Try using the creation and annihilation operators.
 
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Likes hutchphd, Abhishek11235 and Mayan Fung
  • #3
vela said:
Try using the creation and annihilation operators.

Thanks! It helps a lot.

I also tried to solve it analytically by substituting:
$$\hat{p'} = \hat{p} + \hat{q}$$
$$\hat{q'} = \hat{p} - \hat{q}$$
I finally got something like ##\hat{H} = a\hat{p'} + b\hat{q'} ##

However, ##p',q'## are composed of both p and q. How I can get the wavefunction in the q space?
 

FAQ: Perturbation from a quantum harmonic oscillator potential

1. What is a quantum harmonic oscillator potential?

A quantum harmonic oscillator potential is a mathematical model used to describe the behavior of a quantum mechanical system. It is a potential energy function that is used to calculate the energy levels and wavefunctions of a particle in a harmonic oscillator system.

2. How does perturbation affect a quantum harmonic oscillator potential?

Perturbation refers to a small disturbance or change in a system. In the case of a quantum harmonic oscillator potential, perturbation can cause a shift in the energy levels and wavefunctions of the system. This is because the perturbation introduces new terms into the potential energy function, which alters the behavior of the system.

3. What are some examples of perturbation in a quantum harmonic oscillator potential?

Some examples of perturbation in a quantum harmonic oscillator potential include external forces acting on the system, changes in the potential energy function due to changes in the environment, and interactions with other particles.

4. How is perturbation from a quantum harmonic oscillator potential calculated?

Perturbation from a quantum harmonic oscillator potential is calculated using perturbation theory, which is a mathematical method for solving problems that involve small changes in a system. This involves expanding the potential energy function in a series of terms and calculating the effects of each term on the energy levels and wavefunctions of the system.

5. What are the applications of studying perturbation from a quantum harmonic oscillator potential?

Studying perturbation from a quantum harmonic oscillator potential has many applications, including understanding the behavior of atoms and molecules, predicting the properties of materials, and developing new technologies such as quantum computers. It is also a fundamental concept in quantum mechanics and is used in many other areas of physics and engineering.

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