Suppose that the potential for a one-dimensional system in x is mw?x²/2. Recall that the (n = 1) eigenfunction is: 1 vi (x) = (") re=*/2 where y = mw/i. What are ALL the eigenvalues of the Harmonic Oscillator Hamiltonian? What is the expectation value of ^p² for the (n = 1)-state wavefunction? What is the expectation value of ^x² for the (n = 1)-state wavefunction?
Suppose that the potential for a one-dimensional system in x is mw?x²/2. Recall that the (n = 1) eigenfunction is: 1 vi (x) = (") re=*/2 where y = mw/i. What are ALL the eigenvalues of the Harmonic Oscillator Hamiltonian? What is the expectation value of ^p² for the (n = 1)-state wavefunction? What is the expectation value of ^x² for the (n = 1)-state wavefunction?
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![Suppose that the potential for a one-dimensional system in \( x \) is \( m\omega^2x^2/2 \).
Recall that the \((n = 1)\) eigenfunction is:
\[
\psi_1(x) = \left(\frac{(4x^3)}{\pi}\right)^{\frac{1}{4}} x e^{-\gamma x^2 / 2}
\]
where \(\gamma = m\omega/\hbar\).
1. What are ALL the eigenvalues of the Harmonic Oscillator Hamiltonian?
2. What is the expectation value of \(\hat{p}^2\) for the \((n = 1)\)-state wavefunction?
3. What is the expectation value of \(\hat{x}^2\) for the \((n = 1)\)-state wavefunction?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe5e96e7c-fbd2-4d06-b86e-1209676a058c%2F7a924048-6336-47ca-813a-405de33507a6%2Fzibh8h7_processed.png&w=3840&q=75)
Transcribed Image Text:Suppose that the potential for a one-dimensional system in \( x \) is \( m\omega^2x^2/2 \).
Recall that the \((n = 1)\) eigenfunction is:
\[
\psi_1(x) = \left(\frac{(4x^3)}{\pi}\right)^{\frac{1}{4}} x e^{-\gamma x^2 / 2}
\]
where \(\gamma = m\omega/\hbar\).
1. What are ALL the eigenvalues of the Harmonic Oscillator Hamiltonian?
2. What is the expectation value of \(\hat{p}^2\) for the \((n = 1)\)-state wavefunction?
3. What is the expectation value of \(\hat{x}^2\) for the \((n = 1)\)-state wavefunction?
Expert Solution
Step 1
The eigenvalues (En) for the nth-state of the quantum harmonic oscillator may be given as follows:
Here, ħ and ω have their usual meanings.
Step 2
The expectation value of the p2-operator in the first state may be evaluated from its one-dimensional operator form as follows:
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