Perturbed Harmonic oscil

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In summary, the conversation discusses a problem involving a simple one-dimensional harmonic oscillator subjected to a perturbation. The goal is to calculate the energy shift in the ground state to the lowest non-vanishing order. The solution involves using the matrix representation of the perturbation and considering all terms in the summation. After correcting for errors, the correct expression for the energy shift is determined to be b^2 / (2m \omega ^2).
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Homework Statement



Sakural modern quantum.. ch 5 problem 1

A simple one dimensional harmonic oscillator is subject to a perturbation:

V = bx, where b is a real constant.

Calculate the energy shift in ground state to lowest non vanishing order.

Homework Equations



You may use:

[tex] \langle k \vert x \vert n \rangle = \sqrt{\dfrac{\hbar}{2m\omega}}\left( \sqrt{n+1}\delta_{k,n+1} + \sqrt{n}\delta_{k,n-1} \right) [/tex]

where |n> is eigentkets to unperturbed harm. osc

Energy shift:

[tex]
\Delta _{n} \equiv E_n - E^{(0)}_n = \lambda V_{nn} + \lambda^{2} \sum _{k\neq n} \dfrac{\vert V_{nk}\vert^{2}}{E^{(0)}_n - E^{(0)}_k} + . . . [/tex]

Lamda is order, V_nn is matrix elements.

Energy levels for harm osc

[tex] E_N^{(0)} = \hbar \omega (1/2 + N) [/tex]

The Attempt at a Solution




I first do the matrix representation of V = bx

[tex] V_{nk} \doteq b\sqrt{\hbar / (2m \omega)}\left( \begin{array}{ccccc} 0 & 1 & 0 & 0 & 0 \\1 & 0 & \sqrt{2}& 0 & 0 \\0 & \sqrt{2}& 0 & \sqrt{3} &0\\ 0 & 0 & \sqrt{3}&0&0 \end{array}
[/tex]

Then I choose n = 0, since ground state.

[tex]\Delta _{0} \equiv E_0 - E^{(0)}_0 = \lambda V_{00} + \lambda^{2} \sum _{k\neq 0} \dfrac{\vert V_{0k}\vert^{2}}{E^{(0)}_0 - E^{(0)}_k} + . . . [/tex]

I notice that [tex]V_{00} = 0[/tex] and [tex]V_{0k} [/tex]is zero for all k except 1; so that:

[tex]V_{01} = b\sqrt{\hbar / (2m \omega)} [/tex]

And

[tex]E^{(0)}_0 - E^{(0)}_1} = \hbar \omega [/tex]

So that
[tex] \Delta _{0} = -b^2 / (2m \omega ^2) [/tex]

I have no answer to this problem, does it look right to you?
Thanx!
 
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  • #2




Your approach to solving this problem is on the right track. However, there are a few errors in your calculations. Firstly, the matrix representation of V should have elements V_{nk} = b\sqrt{\hbar / (2m \omega)}\delta_{n,k+1} + b\sqrt{\hbar / (2m \omega)}\delta_{n,k-1}. This can be derived by plugging in the values of n and k in the matrix representation given in the homework equations.

Secondly, when you choose n=0, you have to take into account all the terms in the summation, not just the term with k=1. This is because the summation runs over all k values except n. So, the correct expression for \Delta_{0} would be:

\Delta _{0} = b^2 / (2m \omega ^2) \left( \dfrac{1}{1} + \dfrac{1}{2} + \dfrac{1}{3} +... \right)

This infinite series can be simplified to give \Delta_{0} = b^2 / (2m \omega ^2).

I hope this helps. Keep up the good work!
 

1. What is a perturbed harmonic oscillator?

A perturbed harmonic oscillator is a mathematical model that describes the motion of a particle that is subject to a restoring force (like a spring) and a perturbing force (like friction or external forces). It is a more complex version of the simple harmonic oscillator, which only considers the restoring force.

2. How is a perturbed harmonic oscillator different from a simple harmonic oscillator?

The main difference is that a perturbed harmonic oscillator takes into account additional forces that affect the motion of the particle, while a simple harmonic oscillator only considers the restoring force. This makes the perturbed harmonic oscillator a more accurate model for real-world systems.

3. What are some examples of perturbed harmonic oscillators in real life?

Some examples include a mass attached to a spring that is also subject to friction, a pendulum subject to air resistance, or an electron moving in an electric field. In general, any system that experiences both a restoring force and a perturbing force can be modeled as a perturbed harmonic oscillator.

4. How is the motion of a perturbed harmonic oscillator described mathematically?

The motion of a perturbed harmonic oscillator can be described by a differential equation known as the perturbed harmonic oscillator equation. This equation takes into account the restoring force, the perturbing force, and the mass and position of the particle to determine its motion over time.

5. What is the significance of perturbed harmonic oscillators in science?

Perturbed harmonic oscillators are an important concept in many scientific fields, including physics, chemistry, and engineering. They are used to model and understand the behavior of complex systems and help us make predictions about the motion of particles in various situations. They are also a fundamental concept in the study of vibrations and waves.

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