Worksheet: Heisenberg's Uncertainty Principle

In this worksheet, we will practice calculating the energy, duration of existence, momentum, and position of particles using the Heisenberg uncertainty principle.

Q1:

Use the Heisenberg uncertainty principle to calculate the uncertainty in energy for a corresponding time interval of 5 . 3 9 × 1 0 4 4 s.

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

Q2:

An unstable elementary particle has a rest energy of 80.41 GeV and an uncertainty in rest energy of 2.06 GeV. What is the minimum lifetime of this particle?

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

Q3:

Measurements indicate that an atom remains in an excited state for an average time of 50.0 ns before making a transition to the ground state with the simultaneous emission of a 2.10-eV photon.

What is the minimum uncertainty in the frequency of the emitted photon?

What fraction of the emitted photon’s average frequency is the minimum uncertainty in its frequency?

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

Q4:

A gas of helium atoms at a temperature of 273 K is in a cubical container with a side length of 25.000 cm.

What is the minimum uncertainty in the momentum components of helium atoms?

  • A 2 . 3 8 × 1 0 3 4 N⋅s
  • B 1 . 8 9 × 1 0 3 4 N⋅s
  • C 2 . 6 2 × 1 0 3 4 N⋅s
  • D 2 . 1 1 × 1 0 3 4 N⋅s
  • E 2 . 8 8 × 1 0 3 4 N⋅s

What is the minimum uncertainty in the velocity components of the helium atoms?

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

What is the ratio of the minimum uncertainty in the velocity components of helium atoms to the mean speed of the atoms?

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

Q5:

An atom in a metastable state has a lifetime of 4.2 ms. Find the minimum uncertainty in the measurement of the energy of the excited state.

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

Q6:

An electron is confined to a region of length 0.12 nm and its kinetic energy is equal to the ground state energy of the hydrogen atom in Bohr’s model, 13.6 eV. The electron has a minimum uncertainty in momentum, ± 𝜌 min.

What is the value of ± 𝜌 min?

  • A 5 . 5 × 1 0 2 4 kg⋅m/s
  • B 8 . 8 × 1 0 2 5 kg⋅m/s
  • C 2 . 8 × 1 0 2 4 kg⋅m/s
  • D 4 . 4 × 1 0 2 5 kg⋅m/s
  • E 5 . 5 × 1 0 3 3 kg⋅m/s

What fraction of the electron’s momentum is ± 𝜌 min?

What would the uncertainty in kinetic energy of this electron be if its momentum were equal to ± 𝜌 min?

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

What fraction of an electron’s kinetic energy is the minimum uncertainty in its kinetic energy if the momentum of the electron was equal to ± 𝜌 min?

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