Quantum energy of a particle in a 2 dimensional space

In summary, the conversation discusses solving a problem involving determining the frequency of normal modes of vibration using the potential and kinetic matrix. The natural frequency is found to be 2ω0 and ω0. The student asks for guidance on approaching the problem quantum mechanically and mentions a typographical error. The expert suggests mapping the problem to that of uncoupled harmonic oscillators by performing a rotation due to the symmetry of the problem.
  • #1
Apashanka
429
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Homework Statement

IMG_20181207_104633.jpg
[/B]

Homework Equations


Doing this problem like e.g setting the determinant of potential matrix and the ω2*kinetic matrix equal to 0 ,det(V-ω2T)=0,I got the frequency of the normal modes of vibration to be 2ω0 and ω0 where ω0 is the natural frequency,
But sir how to treat this problem quantum mechanically?
The term z is a typographical error..[/B]

The Attempt at a Solution

 

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  • #2
I imagine that you know the solution to the quantum harmonic oscillator. In that case, you should try to map the problem to that of uncoupled harmonic oscillators. If you notice the symmetry of the problem, that gives you a hint that you can achieve this by performing a certain rotation.
 

Related to Quantum energy of a particle in a 2 dimensional space

1. What is the concept of quantum energy in a 2 dimensional space?

The concept of quantum energy refers to the energy associated with a particle in a 2 dimensional space, which is governed by the principles of quantum mechanics. This energy is described by a wave function, which represents the probability of finding the particle at a certain location in the 2 dimensional space.

2. How is the quantum energy of a particle in a 2 dimensional space calculated?

The quantum energy of a particle in a 2 dimensional space is calculated using the Schrödinger equation, which takes into account the particle's position, velocity, and potential energy. This equation is solved to obtain the energy levels or allowed states for the particle in the 2 dimensional space.

3. What are the implications of the uncertainty principle on the quantum energy of a particle in a 2 dimensional space?

The uncertainty principle states that there is a fundamental limit to how precisely we can know a particle's position and momentum at the same time. Therefore, the quantum energy of a particle in a 2 dimensional space cannot be precisely determined, as the exact position and momentum of the particle cannot be known simultaneously.

4. How does the quantum energy of a particle in a 2 dimensional space differ from that in a 3 dimensional space?

The main difference between the quantum energy of a particle in a 2 dimensional space and a 3 dimensional space is that the former is described by a 2D wave function, whereas the latter is described by a 3D wave function. This results in different energy levels and allowed states for the particle in each respective space.

5. What practical applications does the study of quantum energy in a 2 dimensional space have?

The study of quantum energy in a 2 dimensional space has many practical applications, including the development of quantum computers, quantum sensors, and quantum cryptography. It also helps us better understand the behavior of particles at the nanoscale, which has implications in fields such as nanotechnology and materials science.

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