Energy expectation values of harmonic oscillator

In summary, the conversation discusses finding the expectation values of energy at t=0 for a particle with mass m. The function describing the particle is A.SUM[(1/sqrt2)^n].\varphi_n, with A = 1/sqrt2 and energy eigenstates \varphi_n with eigenvalue E_n=(n + 1/2)hw. The speaker mentions encountering a diverging sum, but it can be split into two series and solved using the geometric series formula and the series for [1/(1-x)]'.
  • #1
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I'm looking at a question...

The last part is this: find the expectation values of energy at t=0

The function that describes the particle of mass m is

A.SUM[(1/sqrt2)^n].[tex]\varphi[/tex]_n

where I've found A to be 1/sqrt2. The energy eigenstates are [tex]\varphi[/tex]_n with eigenvalue E_n=(n + 1/2)hw

I tried the usual expectation value way but I run into a horrible sum which seems to diverge I think. How shouldf I go about this??

Cheers guys!
 
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  • #2
The sum shouldn't diverge because of the [itex](1/\sqrt{2})^n[/itex] factor. You can split it into two series. One will be geometric, so it's easy to sum. The other one may require slightly more work to sum, but it's pretty straightforward. Hint: Consider the series for [1/(1-x)]'.
 

What is a harmonic oscillator?

A harmonic oscillator is a physical system that exhibits periodic motion around an equilibrium point. It follows Hooke's Law, which states that the force exerted by a spring is directly proportional to the displacement from its equilibrium position.

What is the energy expectation value of a harmonic oscillator?

The energy expectation value of a harmonic oscillator is the average energy that the oscillator possesses over a large number of measurements. It takes into account both the kinetic energy, associated with the oscillator's motion, and the potential energy, associated with its position.

How is the energy expectation value of a harmonic oscillator calculated?

The energy expectation value of a harmonic oscillator is calculated using the Schrödinger equation, which describes the time evolution of a quantum system. It involves solving the time-dependent Schrödinger equation and taking the average of the resulting energy values over time.

What is the significance of the energy expectation value in quantum mechanics?

In quantum mechanics, the energy expectation value serves as a measure of the average energy of a quantum system. It is related to the probability of finding the system in a particular energy state, and it can be used to predict the behavior of the system over time.

Can the energy expectation value of a harmonic oscillator be measured experimentally?

Yes, the energy expectation value of a harmonic oscillator can be measured experimentally using various methods, such as spectroscopy or other types of energy measurements. However, due to the uncertainty principle in quantum mechanics, the exact energy value of a system cannot be measured simultaneously with its position or momentum.

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