Worksheet: Quantum Particle in a Potential Well

In this worksheet, we will practice finding stationary solutions of the Schrödinger wave equation and calculating the energy states of bound particles.

Q1:

An electron confined to a box of width 0.15 nm by infinite potential energy barriers emits a photon when it makes a transition from the first excited state to the ground state. Find the wavelength of the emitted photon.

Q2:

An electron is confined to a box of width 0.250 nm. What is the wavelength of photons emitted when the electron transitions between the second excited state and the ground state?

Q3:

An electron is confined to a box of width 0.250 nm. What is the wavelength of photons emitted when the electron transitions between the third and second excited states?

Q4:

An electron is confined to a box of width 0.250 nm. What is the wavelength of photons emitted when the electron transitions between the fourth and second excited states?

Q5:

An electron in a long, organic molecule used in a dye laser behaves approximately like a quantum particle in a box with width 4.18 nm. Find the wavelength of the emitted photon when the electron makes a transition from the first excited state to the ground state.

Q6:

An electron in a long, organic molecule used in a dye laser behaves approximately like a quantum particle in a box with width 4.18 nm. Find the wavelength of the emitted photon when the electron makes a transition from the second excited state to the first excited state.

Q7:

An electron confined to a box has a ground state energy of 2.5 eV. What is the width of the box?

Q8:

Photons are emitted by an electron that is confined to a box. The longest wavelength of the emitted photons is 500 nm. What is the width of the box?

  • A 5 . 9 6 × 1 0 1 0 m
  • B 5 . 9 2 × 1 0 1 0 m
  • C 5 . 9 9 × 1 0 1 0 m
  • D 6 . 0 3 × 1 0 1 0 m
  • E 6 . 0 6 × 1 0 1 0 m

Q9:

What is the ground state energy of an alpha-particle confined to a one-dimensional box of length 15.0 fm?

Q10:

Assume that a proton in a nucleus can be treated as if it were confined to a one-dimensional box of width 12.0 fm.

What is the energy of the proton when it is in the state corresponding to 𝑛 = 1 ?

  • A 8 . 9 9 × 1 0 1 2 J
  • B 2 . 7 3 × 1 0 2 7 J
  • C 7 . 2 5 × 1 0 1 4 J
  • D 2 . 2 8 × 1 0 1 3 J
  • E 4 . 1 8 × 1 0 1 0 J

What is the energy of the proton when it is in the state corresponding to 𝑛 = 2 ?

  • A 9 . 1 1 × 1 0 1 3 J
  • B 1 . 0 9 × 1 0 2 6 J
  • C 3 . 6 0 × 1 0 1 1 J
  • D 2 . 2 8 × 1 0 1 3 J
  • E 1 . 6 7 × 1 0 9 J

What is the energy of the photon emitted when the proton makes the transition from the first excited state to the ground state?

  • A 8 . 2 0 × 1 0 2 7 J
  • B 1 . 8 2 × 1 0 1 2 J
  • C 6 . 8 4 × 1 0 1 3 J
  • D 2 . 2 8 × 1 0 1 3 J
  • E 2 . 7 0 × 1 0 1 1 J

What is the energy of the photon emitted when the proton makes the transition from the second excited state to the ground state?

  • A 1 . 8 2 × 1 0 1 2 J
  • B 8 . 6 4 × 1 0 2 5 J
  • C 7 . 2 0 × 1 0 1 1 J
  • D 6 . 8 3 × 1 0 1 3 J
  • E 4 . 5 6 × 1 0 1 3 J

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