Quantum Harmonic Oscillator Study
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Questions and Answers

What is the Hamiltonian operator of the harmonic oscillator in one dimension?

  • Ĥ = p̂2 - mω 2 x̂2 / 2m
  • Ĥ = p̂2 1 + mω 2 x̂2 / 2m (correct)
  • Ĥ = p̂2 - mω 2 x̂2 * 2m
  • Ĥ = p̂2 + mω 2 x̂2 / 2m

    • What does the eigenvalue equation for the harmonic oscillator look like in the {|xi} representation?

  • − ~2 d2ϕ(x)/dx2 + mωxϕ(x) = Eϕ(x) (correct)
  • ~2 d2ϕ(x)/dx2 − mωxϕ(x) = Eϕ(x)
  • ~2 dϕ(x)/dx + mωxϕ(x) = Eϕ(x)
  • ~2 d3ϕ(x)/dx3 + mωxϕ(x) = Eϕ(x)

    • What can be deduced about the eigenvalues of the Hamiltonian from the form of the potential function?

  • The eigenvalues of the Hamiltonian are positive. (correct)
  • The eigenvalues of the Hamiltonian are zero.
  • The eigenvalues of the Hamiltonian are negative.
  • The eigenvalues of the Hamiltonian are complex.

    • What happens to the eigenvalues E of the Hamiltonian if the potential function V (x) is greater than its minimum value Vm?

    <p>The eigenvalues E are greater than Vm.</p> Signup and view all the answers

    What kind of system is described by the Hamiltonian operator for the harmonic oscillator?

    <p>Time-independent (conservative system)</p> Signup and view all the answers

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