Momentum perturbation to harmonic oscillator

In summary, the conversation discusses a problem and a possible solution attached as a PDF, with the person expressing difficulty in understanding it. They mention the use of a space translation operator and their own solution using perturbation theory. The other person suggests looking at problems involving translation operators in a book by W. Greiner and B. Müller.
  • #1
m1rohit
22
0

Homework Statement



the problem and a possible solution(obtained from a book) is attached as a pdf to the post.However Iam unable to understand it.Please download the attachment.

Homework Equations


equation no (2) in the pdf.Is there any use of space translation operator in here.

The Attempt at a Solution


I have solved this problem using perturbation theory.however Iam intrigued by the method used here.
 

Attachments

  • PERTURBATION PROBLEM.pdf
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  • #2
It can be understood using the properties of the Fourier transform. Since [itex]x[/itex] and [itex]p[/itex] are canonically conjugate, transformation from [itex]\psi(x)[/itex] to [itex]\psi(p)[/itex] is done by a Fourier transform. And from their properties, it is known that shifting the zero in one variable will be equivalent to a complex phase shift in the other.
 
  • #3
Thank you very much for your reply.can you refer me any link or book where i could get problems on translation operators.
 
  • #4
I don't if it's the right level for you, but you can take a look at W. Greiner & B. Müller, Quantum Mechanics: Symmetries (Springer).
 
  • #5


Thank you for sharing this problem and possible solution from a book. After reviewing the attached PDF, I can see that the method used is known as the "momentum perturbation method" or "time-dependent perturbation theory." This method is commonly used in quantum mechanics to solve for the time evolution of a system when a perturbation is applied. In this case, the perturbation is a change in the momentum of the harmonic oscillator.

To answer your question about the use of the space translation operator, it is not explicitly used in this method. However, the Hamiltonian operator (equation (2) in the PDF) contains both the position and momentum operators, which are related to the space translation operator through the Heisenberg uncertainty principle. This is why the position operator appears in the time-dependent perturbation theory expression for the time evolution of the system.

Overall, this method is a powerful tool for solving problems involving perturbations to a system and can be applied to a wide range of physical systems, not just the harmonic oscillator. I hope this helps to clarify the approach used in the attached solution.
 

1. What is momentum perturbation to harmonic oscillator?

Momentum perturbation to harmonic oscillator is a concept in physics that involves changing the momentum of a harmonic oscillator by applying an external force or disturbance. This perturbation causes the oscillator to deviate from its usual motion and can result in interesting phenomena.

2. How does momentum perturbation affect the motion of a harmonic oscillator?

Momentum perturbation can change the amplitude, frequency, and phase of a harmonic oscillator's motion. It can also introduce new frequencies and cause the oscillator to undergo transient or chaotic behavior.

3. What is the mathematical representation of momentum perturbation to harmonic oscillator?

The mathematical representation of momentum perturbation to harmonic oscillator is a differential equation known as the driven harmonic oscillator equation. It describes the motion of the oscillator under the influence of an external force or perturbation.

4. What are some real-world applications of momentum perturbation to harmonic oscillator?

Momentum perturbation to harmonic oscillator has applications in various fields such as optics, electronics, and quantum mechanics. It is used to study the behavior of systems with periodic motion, such as pendulums, springs, and atoms in a crystal lattice.

5. How is momentum perturbation different from position perturbation in a harmonic oscillator?

Momentum perturbation affects the momentum of a harmonic oscillator, while position perturbation affects the position or displacement of the oscillator. Momentum perturbation causes the oscillator to undergo changes in motion, while position perturbation causes changes in the equilibrium position of the oscillator.

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