# Perturbed Hamiltonian Matrix for Quantum Harmonic Oscillator

• Luke1121
In summary, to calculate the matrix elements of the quantum harmonic oscillator Hamiltonian with perturbation, you need to calculate the matrix elements of the perturbation term -2cos(\pi x) and add it to the matrix of the normal Hamiltonian using the formula H=H_o +H'. This can be done by applying perturbation theory or directly calculating the matrix elements as shown in the attempt at a solution.
Luke1121
Homework Statement

How to calculate the matrix elements of the quantum harmonic oscillator Hamiltonian with perturbation to potential of $-2cos(\pi x)$
The attempt at a solution
$H=H_o +H'$ so $H=\frac{p^2}{2m}+\frac{1}{2} m \omega x^2-2cos(\pi x)$

I know how to find the matrix of the normal Hamiltonian as $H \psi_j =E_j \psi_j$ then $H_{ij}=<i｜H｜j>=E_j\delta_{ij}=(j+1/2)\hbar \omega \delta_{ij}$ therefore we get 1/2, 3/2,5/2 etc on the diagonal. However i am not sure how to apply this to this situation. Can I obtain the matrix just from here or do I need to do perturbation theory first?

Thanks

You need to calculate ##\langle i | \hat{H}' | j \rangle = -2 \langle i | \cos(\pi x) | j \rangle##.

## 1. What is a perturbed Hamiltonian matrix?

A perturbed Hamiltonian matrix refers to a matrix representation of the Hamiltonian operator for a quantum system that has been modified or "perturbed" by an external influence or force. This perturbation can affect the energy levels and dynamics of the system, leading to a more complex behavior compared to the unperturbed system.

## 2. What is a quantum harmonic oscillator?

A quantum harmonic oscillator is a system in quantum mechanics that models the behavior of a particle in a harmonic potential. It is characterized by a discrete spectrum of energy levels, where the energy increases in a regular pattern with increasing quantum numbers. This system is widely studied as it provides insight into the properties of more complex quantum systems.

## 3. How does the perturbed Hamiltonian matrix affect the quantum harmonic oscillator?

The perturbed Hamiltonian matrix introduces modifications to the energy levels and dynamics of the quantum harmonic oscillator. This can lead to changes in the frequencies or amplitudes of the oscillations, as well as the addition of new energy states. The specific effects depend on the type and strength of the perturbation.

## 4. What are some examples of perturbations that can affect the quantum harmonic oscillator?

Some common perturbations that can affect the quantum harmonic oscillator include external forces such as an applied electric or magnetic field, changes in the shape or size of the potential well, and interactions with other particles. These perturbations can arise from various physical phenomena, such as electromagnetic radiation or collisions with other particles.

## 5. How is the perturbed Hamiltonian matrix calculated for a quantum harmonic oscillator?

The perturbed Hamiltonian matrix is typically calculated using mathematical techniques such as perturbation theory or numerical methods. These methods involve expanding the Hamiltonian operator into a series of terms, with the first term representing the unperturbed system and subsequent terms representing the perturbation. The resulting matrix can then be used to solve for the modified energy levels and wavefunctions of the perturbed system.

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