Particle in harmonic oscillator potential

In summary, the particle with mass m moving in 3-dimensions with potential V(x,y,z)=\frac{1}{2}m\omega^{2}x^{2} has allowed energy eigenvalues that can be found by solving the Schrodinger equation and assuming a solution of the form \psi(x,y,z)=X(x)Y(y)Z(z). The total energy is quantized in the x direction, but not in the y and z directions. The eigenvalues for y and z are continuous because the particle is not bound in those directions. The solution for X(x), Y(y), and Z(z) should all be the same, leading to the conclusion that the total energy can be expressed as the sum
  • #1
gumpyworm
3
0

Homework Statement


A particle with mass m moves in 3-dimensions in the potential [tex]V(x,y,z)=\frac{1}{2}m\omega^{2}x^{2}[/tex]. What are the allowed energy eigenvalues?

Homework Equations


The Attempt at a Solution


The Hamiltonian is given by [tex]H=\frac{P^{2}}{2m}+\frac{1}{2}m\omega^{2}X^{2}[/tex] where P is the momentum operator in three dimensions. Projecting this into the coordinate basis, we have [tex]\left( -\frac{\hbar^{2}}{2m}\frac{d^{2}}{dx^{2}}+\frac{1}{2}m\omega^{2}x^{2}\right)\psi = \left(E + \frac{\hbar^{2}}{2m}\left( \frac{d^{2}}{dy^{2}}+\frac{d^{2}}{dz^{2}}\right) \right)\psi [/tex]
The left side is now in the form of a simple harmonic oscillator. However, I am stuck after this point.
 
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  • #2
Try assuming a solution of the form [itex]\psi(x,y,z)=X(x)Y(y)Z(z)[/itex].
 
  • #3
Hmm so if we assume that, then we find that [itex]E=E_{x}+E_{y}+E_{z}[/itex]. But [itex]E_{y}[/itex] and [itex]E_{z}[/itex] correspond to the eigenvalues of a free particle, since the particle is free in the y and z directions. So does this just mean that the eigenvalues are continuous, since the energy is only quantized along one direction?
 
  • #4
Why do you say it's only quantized in one direction? And no, the energy is not continuous in this case.
 
  • #5
Don't we just have [tex]E=E_{x}+E_{y}+E_{z}=(n+\frac{1}{2})\hbar\omega+\frac{p_{y}^{2}}{2m}+\frac{p_{z}^{2}}{2m}[/tex]Then, since p_y and p_z are continuous, wouldn't we just have that the ground state energy is [itex]\frac{\hbar\omega}{2}[/itex] and that above that, the eigenvalues are continuous?
 
  • #6
gumpyworm said:
Hmm so if we assume that, then we find that [itex]E=E_{x}+E_{y}+E_{z}[/itex]. But [itex]E_{y}[/itex] and [itex]E_{z}[/itex] correspond to the eigenvalues of a free particle, since the particle is free in the y and z directions. So does this just mean that the eigenvalues are continuous, since the energy is only quantized along one direction?
Yes.
cbetanco said:
Why do you say it's only quantized in one direction? And no, the energy is not continuous in this case.
Because the potential only depends on x. The particle isn't bound when moving in the y and z directions, so Ey and Ez can take on any value. Ex is quantized, but the total energy isn't.
 
  • #7
Why do you treat y and z differently from x? your solution should be the same for X(x), Y(y) and Z(z). I would say [itex]n=n_x+n_y+n_z[/itex]
 
  • #8
vela said:
Because the potential only depends on x.

OMG, I'm sorry. I didn't read the OP carefully enough. Yes, then you are right. I was thinking of [itex]\frac{1}{2}m\omega r^2[/itex] where [itex]r^2=x^2+y^2+z^2[/itex]
 

1. What is a particle in a harmonic oscillator potential?

A particle in a harmonic oscillator potential is a theoretical model used in quantum mechanics to describe the behavior of a particle moving in a potential energy field that resembles a spring. The particle oscillates back and forth within the potential well, similar to a mass attached to a spring.

2. What is the equation for a particle in a harmonic oscillator potential?

The equation for a particle in a harmonic oscillator potential is given by the Schrödinger equation, which describes the wave function of the particle. It can be written as HΨ = EΨ, where H is the Hamiltonian operator, Ψ is the wave function, and E is the energy of the particle.

3. What are the energy levels of a particle in a harmonic oscillator potential?

The energy levels of a particle in a harmonic oscillator potential are quantized, meaning they can only take on certain discrete values. The energy levels are given by En = (n + 1/2)ħω, where n is a positive integer and ω is the angular frequency of the particle's oscillations.

4. What is the ground state of a particle in a harmonic oscillator potential?

The ground state of a particle in a harmonic oscillator potential is the lowest energy level that the particle can occupy. It corresponds to n = 0 in the energy equation, and has an energy of E0 = 1/2 ħω. The ground state is also the most stable state of the particle.

5. How does the potential energy affect the behavior of a particle in a harmonic oscillator potential?

The potential energy in a harmonic oscillator potential affects the behavior of the particle by determining the shape of the potential well. A steeper potential well will result in higher energy levels and shorter oscillation periods, while a shallower potential well will result in lower energy levels and longer oscillation periods. The potential energy also determines the probability of finding the particle at different positions within the potential well.

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