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 5.39×10 s.

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

Q4:

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

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

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

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

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

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

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

Q7:

The mass of a 𝜌-meson is measured to be 770 MeV/c2 with an uncertainty of 150 MeV/c2. Calculate the lifetime of this meson.

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

Q8:

A velocity measurement of an 𝛼 particle has been performed with a precision of 0.0100 mm/s. What is the minimum uncertainty in its position?

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

Q9:

The uncertainty in the 𝑦-component of a proton’s position is 3.00 pm.

Find the minimum uncertainty in the simultaneous measurement of the proton’s 𝑦-component of velocity.

  • A 1 . 9 3 × 1 0 m/s
  • B10.5 m/s
  • C 1 0 . 5 × 1 0 m/s
  • D 1 0 . 5 × 1 0 m/s
  • E 1 . 0 5 × 1 0 m/s

Find the minimum uncertainty in the simultaneous measurement of the proton’s 𝑥-component of velocity given that uncertainty about its 𝑥-position is infinity.

  • AIt cannot be determined.
  • BThe minimum uncertainty in the simultaneous measurement of the proton’s 𝑥-component of velocity is 0.
  • CThe minimum uncertainty in the simultaneous measurement of the proton’s 𝑥-component of velocity is infinity.
  • DThe minimum uncertainty in the simultaneous measurement of the proton’s 𝑥-component of velocity is 2.
  • EThe minimum uncertainty in the simultaneous measurement of the proton’s 𝑥-component of velocity is 1.

Q10:

A particle of mass 𝑚 is confined to a box of width 𝐿. The particle is in the first excited state.

What is the probability of finding the particle in a region of width 0.020 𝐿 around the given point 𝑥=0.25𝐿?

  • A4.0%
  • B5.1%
  • C6.2%
  • D1.1%
  • E2.7%

What is the probability of finding the particle in a region of width 0.020 𝐿 around the given point 𝑥=0.40𝐿?

  • A2.3%
  • B1.9%
  • C1.4%
  • D3.1%
  • E2.7%

What is the probability of finding the particle in a region of width 0.020 𝐿 around the given point 𝑥=0.75𝐿?

  • A4.8%
  • B4.0%
  • C3.1%
  • D6.2%
  • E5.4%

What is the probability of finding the particle in a region of width 0.020 𝐿 around the given point 𝑥=0.90𝐿?

  • A1.9%
  • B1.1%
  • C2.5%
  • D3.1%
  • E1.4%

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.