How Does a Shifted Harmonic Oscillator Decompose into Eigenfunctions?

In summary, the conversation discusses a wave function that is the ground state of a harmonic oscillator, but shifted by a constant along the position axis. The question is raised about how this shift affects the decomposition into eigenfunctions and the time development of average position and momentum. The response suggests that the shift will affect the average position and explains that the shifted wavefunction must be composed of eigenfunctions centered at zero, each with their own energies and time development. An example is given using an applet.
  • #1
prairiedogj
3
0
I have a wave function which is the ground state of a harmonic oscillator (potential centered at x=0)... but shifted by a constant along the position axis (ie. (x-b) instead of x in the exponential).

How does this decompose into eigenfunctions?? I know it's an infinite sum... but I can't nail the coefficients.

...and...
without knowing the decomposition, how can I get the time development of the average position and momentum?

Help.

Thanks.
 
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  • #2
What does [itex] \exp\left[-(x-a)^{2}\right] [/itex] instead of [itex] \exp\left(-x^{2}\right) [/itex] have to do with time dependence of [itex] \langle \hat{x}\rangle [/itex]...?


Daniel.
 
  • #3
Well, most intuitively, I would assume that shifting the initial wavefunction along the x-axis would affect average position.

Less intuitively, a shifted initial wavefunction in a zero-centered harmonic potential MUST be composed of eigenfunctions... ones centered at zero. Each of these have their own energies and own time development.

As an example... go to THIS applet http://groups.physics.umn.edu/demo/applets/qm1d/index.html
set it to harmonic oscillator, choose the ground state eigenfunction, and then adjust the offset.
 

FAQ: How Does a Shifted Harmonic Oscillator Decompose into Eigenfunctions?

What is a Shifted Harmonic Oscillator?

A Shifted Harmonic Oscillator is a type of oscillator that follows a sinusoidal motion, but with a constant shift in its equilibrium position. This shift can be caused by an external force or change in the system's parameters.

How is a Shifted Harmonic Oscillator different from a regular Harmonic Oscillator?

A regular Harmonic Oscillator has an equilibrium position at the origin, while a Shifted Harmonic Oscillator has an equilibrium position at a non-zero point. This means that the restoring force in a Shifted Harmonic Oscillator is not zero at the equilibrium point, unlike in a regular Harmonic Oscillator.

What factors affect the behavior of a Shifted Harmonic Oscillator?

The behavior of a Shifted Harmonic Oscillator is affected by the amplitude and frequency of the driving force, as well as the magnitude and direction of the shift in equilibrium position. Additionally, the mass and stiffness of the oscillator also play a role in its behavior.

What are some real-world examples of a Shifted Harmonic Oscillator?

One example is a pendulum that is experiencing a horizontal displacement due to an external force. Another example is a mass-spring system with an added weight that causes a shift in the equilibrium position.

How is a Shifted Harmonic Oscillator used in scientific research?

A Shifted Harmonic Oscillator is commonly used in research related to vibration and oscillation, as well as in the study of non-linear systems. It can also be used to model various physical systems, such as molecular vibrations and electronic circuits.

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