Charged harmonic oscillator in an electric field

E \psi + q \epsilon x \psi.In summary, the conversation discusses finding the eigenvalues and eigenstates of energy for a charged harmonic oscillator placed in an external electric field. The attempt at a solution involves completing the square and substituting a new variable, but it does not seem to give the correct results. The correct Hamiltonian for this problem should be \frac{\hbar^2}{2m} \frac{d^2 \psi}{dx^2} + \frac{1
  • #1
bjogae
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Homework Statement



A charged harmonic oscillator is placed in an external electric field [tex]\epsilon[/tex] i.e. its hamiltonian is [tex] H = \frac{p^2}{2m} + \frac{1}{2}m \omega ^2 x^2 - q \epsilon x [/tex] Find the eigenvalues and eigenstates of energy

Homework Equations





The Attempt at a Solution



By completing the square i get
[tex] [-\frac{\hbar^2}{2m}\frac{d^2}{du^2}+\frac{1}{2}m \omega ^2u^2] \phi (u) = (E + \frac{q^2 \epsilon^2}{2m \omega ^2}) \phi (u) [/tex]
where
[tex]u=x-\frac{q^2\epsilon^2}{2m\omega^2}[/tex].

Then usually for Hamiltonians of this kind the energy eigenvalues are
[tex]E_n=\hbar\omega(n+\frac{1}{2})[/tex]
but how do I obtain them in this case? Or is this the right way to go?
Do i call
[tex]E + \frac{q^2 \epsilon^2}{2m \omega ^2}=E'[/tex]
which would give me
[tex]E'_n=\hbar\omega(n+\frac{1}{2})[/tex]
And how do I swich back to x?
 
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  • #2
bjogae said:
By completing the square i get
[tex] [-\frac{\hbar^2}{2m}\frac{d^2}{du^2}+\frac{1}{2}m \omega ^2u^2] \phi (u) = (E + \frac{q^2 \epsilon^2}{2m \omega ^2}) \phi (u) [/tex]
where
[tex]u=x-\frac{q^2\epsilon^2}{2m\omega^2}[/tex].

That doesn't look quite right...
 

What is a charged harmonic oscillator in an electric field?

A charged harmonic oscillator in an electric field is a physical system consisting of a charged particle attached to a spring and placed in an external electric field. The particle oscillates back and forth due to the restoring force of the spring and the force exerted by the electric field.

What is the equation of motion for a charged harmonic oscillator in an electric field?

The equation of motion for a charged harmonic oscillator in an electric field is:

m(d^2x/dt^2) + kx = qE

Where m is the mass of the particle, x is the displacement from equilibrium, k is the spring constant, q is the charge of the particle, and E is the strength of the electric field.

How does the electric field affect the motion of a charged harmonic oscillator?

The electric field exerts a force on the charged particle, causing it to experience an additional acceleration. This results in a change in the amplitude and frequency of the oscillations of the charged harmonic oscillator.

What are the energy levels of a charged harmonic oscillator in an electric field?

The energy levels of a charged harmonic oscillator in an electric field are quantized, meaning they can only take on certain discrete values. The energy levels depend on the strength of the electric field and the properties of the system, such as the mass and charge of the particle and the spring constant.

What is the significance of the charged harmonic oscillator in an electric field in physics?

The charged harmonic oscillator in an electric field is a common model used in physics to study the behavior of charged particles in electric fields. It has applications in various fields such as quantum mechanics, solid state physics, and electromagnetism. It also helps us understand the effects of external forces on oscillating systems and has implications for technologies such as sensors and oscillators.

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