Finding Eigenstates of J_z and the Harmonic Oscillator Operators

In summary, the conversation is discussing the process of finding expressions for operators x, p_x, y, and p_y in terms of creation/annihilation operators and using them to express the angular momentum operator. The individual has successfully found the expressions for the operators but is having trouble using them to find J_z. They suggest formulating an eigenequation involving J_z and H to find the eigenstates of J_z.
  • #1
ausdreamer
23
0

Homework Statement



http://img191.imageshack.us/i/questionyw.png/

Homework Equations



Given in problem

The Attempt at a Solution



a) I've been able to find expressions of operators x, p_x, y and p_y in terms of the creation/annihilation operators and hence been able to express the angular momentum operator as:

http://img232.imageshack.us/i/solutionq.png/

However, I'm having trouble just writing the operators x, p_x, y and p_y in terms of the "new" creation/annihilation operators and so I can proceed to find J_z in terms of the new operators.

b) I'm guessing I need to formulate an eigenequation involving J_z and H to find the eigenstates of J_z? I'm pretty stumped on this question to be honest.
 
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  • #2
anyone?
 

FAQ: Finding Eigenstates of J_z and the Harmonic Oscillator Operators

What is a 2D harmonic oscillator?

A 2D harmonic oscillator is a physical system that moves back and forth in two dimensions with a restoring force that is proportional to its displacement from equilibrium. It can be described by a mathematical equation known as the 2D harmonic oscillator equation.

What is the difference between a 2D and 3D harmonic oscillator?

The main difference between a 2D and 3D harmonic oscillator is the number of dimensions in which the system can move. A 2D harmonic oscillator can only move in two dimensions, while a 3D harmonic oscillator can move in three dimensions. This affects the mathematical equations used to describe their motion.

How is the energy of a 2D harmonic oscillator related to its motion?

The energy of a 2D harmonic oscillator is directly related to its motion. As the oscillator moves back and forth, its kinetic energy and potential energy continuously interchange. At the equilibrium point, the energy is completely potential, while at the extremes of motion, it is completely kinetic.

What are the applications of the 2D harmonic oscillator?

The 2D harmonic oscillator has many applications in physics and engineering. It is used to model the motion of pendulums, atoms, molecules, and other physical systems. It also has applications in optics, electronics, and acoustics.

How do you solve the 2D harmonic oscillator equation?

The 2D harmonic oscillator equation can be solved using various mathematical techniques, such as separation of variables and the method of undetermined coefficients. The solution involves finding the general solution for the position and velocity of the oscillator, which can then be used to determine its motion given initial conditions.

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