How to Show the Eigenvalue for v=1 in a Harmonic Oscillator?

In summary, the task at hand is to find the v=1 eigenfunction for the harmonic oscillator and then substitute it into the Schrodinger equation to show that the eigenvalue is (3/2)hν. The wavefunction for an oscillator is given by ψv(x) = NvHv(y)e−y2/2, and for V = 1, Hv(y) = 2y. After normalizing the wavefunction, it is necessary to plug it into the Schrodinger equation to complete the task.
  • #1
ahhppull
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Homework Statement



Write down the v=1 eigenfunction for the harmonic oscillator. Substitute this eigenfunction into the Schrodinger equation and show that the eigenvalue is (3/2)hν.

Homework Equations





The Attempt at a Solution



I'm not really sure on how to to this, but here's what I did...

The wavefunction of an oscillator is ψv(x) = NvHv(y)e−y2/2. For V = 1, Hv(y) = 2y.

ψv(x) = Nv(2y)e−y2/2.

Then I normalized it by squaring it and finding out what N is. I tried plugging in the schrodinger's equation but it doesn't work. What am I doing wrong?
 
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  • #2
So far so good, but we can't help you if you don't show the rest of your work.
 

FAQ: How to Show the Eigenvalue for v=1 in a Harmonic Oscillator?

What is a harmonic oscillator eigenvalue?

A harmonic oscillator eigenvalue is a numerical value that represents the energy states of a harmonic oscillator system. It is found by solving the Schrödinger equation for the system, and it corresponds to the allowed energy levels of the oscillator.

2. How is the eigenvalue of a harmonic oscillator calculated?

The eigenvalue of a harmonic oscillator is calculated by solving the Schrödinger equation for the system, which involves finding the eigenfunctions and eigenstates of the system. These eigenstates are then used to determine the energy levels and corresponding eigenvalues of the oscillator.

3. What is the significance of the eigenvalue in a harmonic oscillator?

The eigenvalue of a harmonic oscillator is significant because it represents the allowed energy levels of the system. This allows us to understand the behavior of the oscillator and predict its energy states. It also plays a crucial role in quantum mechanics and the study of wave functions.

4. How does the eigenvalue relate to the frequency of a harmonic oscillator?

The eigenvalue of a harmonic oscillator is directly related to its frequency. The higher the eigenvalue, the higher the frequency of the oscillator. This means that systems with higher energy levels will have a higher frequency of oscillation.

5. Can the eigenvalue of a harmonic oscillator be measured experimentally?

Yes, the eigenvalue of a harmonic oscillator can be measured experimentally through techniques such as spectroscopy. By measuring the energy transitions of the oscillator, the eigenvalues can be determined and compared to theoretical calculations.

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