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 m
  • B 5 . 9 9 × 1 0 m
  • C 6 . 0 6 × 1 0 m
  • D 6 . 7 4 × 1 0 m
  • E 5 . 9 2 × 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 J
  • B 4 . 1 8 × 1 0 J
  • C 7 . 2 5 × 1 0 J
  • D 2 . 7 3 × 1 0 J
  • E 2 . 2 8 × 1 0 J

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

  • A 3 . 6 0 × 1 0 J
  • B 2 . 2 8 × 1 0 J
  • C 1 . 0 9 × 1 0 J
  • D 9 . 1 1 × 1 0 J
  • E 1 . 6 7 × 1 0 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 1 . 8 2 × 1 0 J
  • B 2 . 2 8 × 1 0 J
  • C 6 . 8 4 × 1 0 J
  • D 8 . 2 0 × 1 0 J
  • E 2 . 7 0 × 1 0 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 6 . 8 3 × 1 0 J
  • B 1 . 8 2 × 1 0 J
  • C 4 . 5 6 × 1 0 J
  • D 7 . 2 0 × 1 0 J
  • E 8 . 6 4 × 1 0 J

Q11:

Find the energy in electron-volts of the first excited state of a proton confined to a one-dimensional box that has a radius of 15.0 fm, approximately the length of a uranium nucleus.

Q12:

An electron in a box is in the ground state with energy 1.50 eV.

Find the width of the box.

How much energy is needed to excite the electron to its first excited state?

If the electron makes a transition from an excited state to the ground state with the simultaneous emission of 20.0 eV photon, find the quantum number of the excited state.

Q13:

The energy of the ground state of an electron in a box is 20.0 eV. What is the width of the box?

Q14:

Calculate the lattice spacing in a one-dimensional crystal where an electron is in the first excited state and has a wavelength of 0.900 nm.

  • A 8 . 8 9 × 1 0 m
  • B 1 9 . 8 × 1 0 m
  • C 8 . 9 8 × 1 0 m
  • D 1 6 . 4 × 1 0 m
  • E 4 . 5 0 × 1 0 m

Q15:

An electron is confined to a metal cube of side length 0.60 cm. The density of states for the electron in the cube varies with the electron’s energy.

Find the density of states for 𝐸=0.90eV.

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

Find the density of states for 𝐸=2.1eV.

  • A 2 . 6 × 1 0 m−3⋅J−1
  • B 7 . 7 × 1 0 m−3⋅J−1
  • C 6 . 6 × 1 0 m−3⋅J−1
  • D 1 . 3 × 1 0 m−3⋅J−1
  • E 3 . 3 × 1 0 m−3⋅J−1

Find the density of states for 𝐸=5.2eV.

  • A 4 . 2 × 1 0 m−3⋅J−1
  • B 1 . 0 × 1 0 m−3⋅J−1
  • C 2 . 1 × 1 0 m−3⋅J−1
  • D 1 . 0 × 1 0 m−3⋅J−1
  • E 5 . 2 × 1 0 m−3⋅J−1

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