7.5 The Quantum Harmonic Oscillator

The vibrational frequency of the hydrogen iodide HI diatomic molecule is $6.69 \times 10^{\textrm{ } 13} \text{Hz}$. (a) What is the force constant of the molecular bond between the hydrogen and the iodine atoms? (b) What is the energy of the emitted photon when this molecule makes a transition between adjacent vibrational energy levels?

Find the expectation value of the position for a particle in the ground state of a harmonic oscillator using symmetry.

Is it possible to measure energy of $0.75 \hbar \omega$ for a quantum harmonic oscillator? Why? Why not? Explain.

If a classical harmonic oscillator can be at rest, why can the quantum harmonic oscillator never be at rest? Does this violate Bohr’s correspondence principle?

Can we simultaneously measure position and energy of a quantum oscillator? Why? Why not?

Show that the two lowest energy states of the simple harmonic oscillator, $\psi_{0} \left(x\right)$ and $\psi_{1} \left(x\right)$ from the equation, satisfy the equation.

When a quantum harmonic oscillator makes a transition from the $\left(n + 1\right)$ state to the n state and emits a 450-nm photon, what is its frequency?

A particle with mass 0.030 kg oscillates back-and-forth on a spring with frequency 4.0 Hz. At the equilibrium position, it has a speed of 0.60 m/s. If the particle is in a state of definite energy, find its energy quantum number.

Determine the expectation value of the potential energy for a quantum harmonic oscillator in the ground state. Use this to calculate the expectation value of the kinetic energy.

Estimate the ground state energy of the quantum harmonic oscillator by Heisenberg’s uncertainty principle. Start by assuming that the product of the uncertainties $\Delta x$ and $\Delta p$ is at its minimum. Write $\Delta p$ in terms of $\Delta x$ and assume that for the ground state $x \approx \Delta x$ and $p \approx \Delta p ,$ then write the ground state energy in terms of x. Finally, find the value of x that minimizes the energy and find the minimum of the energy.