Quantum harmonic oscillator with harmonic perturbation

In summary, the conversation discusses the use of perturbation theory in solving for the ground state energy of a harmonic oscillator with a harmonic perturbation. The exact solution is known, but the goal is to obtain a series for the ground state energy using perturbation theory. The conversation mentions the use of recurrence relations and the ladder operator approach in the calculations. However, the calculation of the nth order of perturbation series is not simple and becomes cumbersome. The use of Kato's theory is also mentioned as a potential aid in this case.
  • #1
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Homework Statement



Let's consider a harmonic oscillator with a harmonic perturbation:
[tex]H = \frac{p^2}{2} + \frac{x^2}{2} + a \frac{x^2}{2}.[/tex]
Exact solution is known, but we want to derive it using perturbation theory. More specifically, suppose we want to obtain a series for the ground state energy [itex]E(a)[/itex]. This series is known to be a convegent one (when [itex]|a| < 1[/itex]) with sum equal to the exact answer ([itex]1/2 \sqrt{1+a}[/itex]). We want to comfirm this fact by a direct calculation.

Homework Equations

and

The Attempt at a Solution



Well-knows recurrence relations of Rayleigh-Schrodinger theory are applicable, and I've manually checked first five orders. The calculations quickly become very cumbersome, and there seems to be no easy way to obtain a general expression for n-th order (known in advance from the exact answer).

What is the most straightforward way to obtain this series? Will the Kato's theory be of any help in this case?

Thank you in advance.
 
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  • #2
Are you using a ladder operator approach ? The calculations should be the simplest.
 
  • #3
Sure, I use them to calculate matrix elements of perturbation. This part of calculation is indeed simple. What seems to be not simple is the calculation of the n-th order of perturbation series and not just 1st and 2nd like most textbooks do.

1st order correction is [itex]A |2\rangle[/itex], 2nd order - [itex]B |2\rangle + C |4\rangle[/itex], 3rd order - [itex]D |2\rangle + F |4\rangle + G |6\rangle[/itex] and so on. The number of terms grows with n.

There should be an elegant way to do the calculation even in the framework of an "old" perturbation theory. I deliberately avoid using diagramms here.
 

1. What is a quantum harmonic oscillator with harmonic perturbation?

A quantum harmonic oscillator with harmonic perturbation is a physical system that follows the principles of quantum mechanics and is described by a Hamiltonian with both a harmonic potential and a perturbation term. The harmonic potential describes the restoring force in a simple harmonic motion, while the perturbation term introduces a small external force that affects the motion of the oscillator.

2. How is the energy spectrum of a quantum harmonic oscillator with harmonic perturbation determined?

The energy spectrum of a quantum harmonic oscillator with harmonic perturbation is determined by solving the Schrödinger equation for the system. This results in a series of discrete energy levels, known as the energy spectrum, which are quantized and spaced evenly apart.

3. What is the significance of the energy levels in a quantum harmonic oscillator with harmonic perturbation?

The energy levels in a quantum harmonic oscillator with harmonic perturbation represent the possible states that the system can occupy. The lowest energy level, or ground state, is the most stable and has the lowest energy, while the higher energy levels represent more excited states of the system.

4. How does the perturbation term affect the energy levels in a quantum harmonic oscillator?

The perturbation term in the Hamiltonian introduces a small external force that affects the motion of the oscillator. This perturbation can cause the energy levels to shift or split, resulting in a more complex energy spectrum compared to a simple harmonic oscillator without a perturbation term.

5. What are some real-world applications of a quantum harmonic oscillator with harmonic perturbation?

A quantum harmonic oscillator with harmonic perturbation has many applications in various fields of science, such as quantum optics, condensed matter physics, and quantum computing. It is also used to model and study the behavior of molecules, atoms, and other quantum systems, providing valuable insights into their properties and interactions.

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