Calculating Probability for Harmonic Oscillator States

In summary, to find the probability that a particle is in a particular state, you need to calculate the inner product or overlap integral between your state psi and the states of definite energy. This involves integrating the product of psi(x,0) and u(x) over all space, or from -x0 to x0 if the initial wavefunction is defined within that range. The result of this integral squared gives you the probability of the particle being in that particular state.
  • #1
eku_girl83
89
0
I have the U(x) functions for the ground state and first excited state of the simple harmonic oscillator. I also have the psi (x,0) wave function for this situation. How do I find the probability the particle is in a particular state? Is it simply the integral of psi(x,0) * u(x) dx evaluated from -x0 to x0... and then the square of this value?

Or am I totally off base with this?
Thanks!
 
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  • #2
Hi eku_girl,

You have the right idea except that you have to remember to integrate over all space, not just from -x0 to x0. All you're really doing is calculating the inner product or overlap integral between your state and the states of definite energy. This quantity tells you how much your state psi looks like the ground state or first the excited state or whatever you put in there.

Hope this helps.
 
  • #3
The psi (x,0) is defined from -x0 to x0 in the problem. Do I still integrate over all space?
 
  • #4
I see, your initial wavefunction vanishes outside of -x0 to x0? Sorry for the confusion; yes, you should integrate from -x0 to x0. In fact, you can think about integrating over all space except that the wavefunction is zero outside of -x0 to x0 so it all works out alright.
 

FAQ: Calculating Probability for Harmonic Oscillator States

What is a harmonic oscillator?

A harmonic oscillator is a physical system that exhibits a repetitive, or oscillatory, motion around an equilibrium point. Examples include a mass on a spring and a pendulum.

What is the equation for a harmonic oscillator?

The equation for a harmonic oscillator is given by F = -kx, where F is the restoring force, k is the spring constant, and x is the displacement from equilibrium.

What is the period of a harmonic oscillator?

The period of a harmonic oscillator is the time it takes for one complete oscillation, or cycle, to occur. It is given by T = 2π√(m/k), where m is the mass and k is the spring constant.

What factors affect the frequency of a harmonic oscillator?

The frequency of a harmonic oscillator is affected by the mass, spring constant, and amplitude of the oscillation. Increasing the mass or spring constant will decrease the frequency, while increasing the amplitude will increase the frequency.

What is the difference between a simple harmonic oscillator and a damped harmonic oscillator?

A simple harmonic oscillator has no external forces acting on it, while a damped harmonic oscillator experiences external forces, such as friction, that cause its motion to gradually decrease over time. This results in a decrease in amplitude and frequency over time for a damped oscillator.

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