Worksheet: The Quantum Harmonic Oscillator

In this worksheet, we will practice modeling a quantum harmonic oscillator with wave functions analogous to vibrational energy states of classical oscillators.


A particle with mass 0.0300 kg oscillates back and forth on a spring with frequency 4.00 Hz. At the spring’s equilibrium position, the particle has a speed of 0.600 m/s. If the particle is in a state of definite energy, find its energy quantum number.

  • A2.32×10
  • B2.17×10
  • C1.89×10
  • D2.04×10
  • E2.45×10


A quantum mechanical oscillator vibrates at a frequency of 250.0 THz. What is the least energetic radiation that it can emit?


A quantum harmonic oscillator makes a transition from the (𝑛+1) state to the 𝑛 state and emits a photon that has a wavelength of 0.450 μm. What is the oscillator’s frequency?

  • A6.17×10 Hz
  • B6.30×10 Hz
  • C6.82×10 Hz
  • D6.66×10 Hz
  • E6.43×10 Hz


Atoms in a crystal lattice vibrate in simple harmonic motion. What is the force constant of the lattice if a lattice atom makes a transition from the ground state to the first excited state when it absorbs a 525 μm photon? Use a value of 9.4×10 kg for the mass a lattice atom.


A billiard ball of mass 0.200 kg bounces back and forth between the cushions of a 1.50 m long table. No energy is lost during the ball’s motion or during its collisions with the cushions.

If the ball is in its ground state, how many years does it take to move from one cushion to the other?

  • A4.31×10years
  • B1.48×10 years
  • C1.21×10 years
  • D1.92×10 years
  • E1.74×10 years

How much energy is required to excite the ball from its ground state to its first excited state?

  • A2.05×10 J
  • B3.66×10 J
  • C2.41×10 J
  • D1.84×10 J
  • E2.79×10 J


If the ground state energy of a simple harmonic oscillator is 1.20 eV, what is the frequency of its motion?

  • A2.90×10 Hz
  • B1.14×10 Hz
  • C1.82×10 Hz
  • D0.64×10 Hz
  • E5.80×10 Hz


Exciting an electron in a one-dimensional box from its second excited state to its third excited state requires 15.0 eV of energy. What is the width of the box?

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