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.

  • A6.11×10 GeV
  • B5.06×10 GeV
  • C6.83×10 GeV
  • D9.11×10 GeV
  • E1.43×10 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?

  • A1.80×10 s
  • B1.60×10 s
  • C1.69×10 s
  • D1.39×10 s
  • E1.51×10 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?

  • A1.86×10
  • B2.58×10
  • C3.14×10
  • D3.81×10
  • E4.49×10

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?

  • A1.89×10 N⋅s
  • B2.11×10 N⋅s
  • C2.88×10 N⋅s
  • D2.38×10 N⋅s
  • E2.62×10 N⋅s

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

  • A5.77×10 m/s
  • B6.31×10 m/s
  • C4.82×10 m/s
  • D6.93×10 m/s
  • E5.21×10 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?

  • A7.25×101
  • B6.56×101
  • C6.23×101
  • D6.89×101
  • E5.94×101

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.

  • A1.3×10 J
  • B7.9×10 J
  • C1.3×10 J
  • D7.9×10 J
  • E1.3×10 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, ±𝜌min.

What is the value of ±𝜌min?

  • A2.8×10 kg⋅m/s
  • B5.5×10 kg⋅m/s
  • C5.5×10 kg⋅m/s
  • D8.8×10 kg⋅m/s
  • E4.4×10 kg⋅m/s

What fraction of the electron’s momentum is ±𝜌min?

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

  • A1.1×10 J
  • B2.4×10 J
  • C1.1×10 J
  • D1.3×10 J
  • E5.8×10 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 ±𝜌min?

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.

  • A5.24×10 s
  • B4.39×10 s
  • C2.76×10 s
  • D1.38×10 s
  • E2.20×10 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?

  • A1.59×10 m
  • B9.98×10 m
  • C5.80×10 m
  • D7.94×10 m
  • E5.23×10 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.

  • A1.93×10 m/s
  • B10.5 m/s
  • C10.5×10 m/s
  • D10.5×10 m/s
  • E1.05×10 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 0.020𝐿 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 0.020𝐿 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 0.020𝐿 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 0.020𝐿 around the given point 𝑥=0.90𝐿?

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

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