Worksheet: The Bohr Hydrogen Atom

In this worksheet, we will practice calculating the solutions of the wave function of particle bound states that correspond to the electrons in a hydrogen atom.

Q1:

What is the number of possible combinations of quantum states of an electron in the 𝑛 = 5 shell?

Q2:

Find the probability that the 1s electron of a hydrogen atom is found outside the Bohr radius, 5 . 2 9 2 × 1 0 m from the atom’s nucleus. Give your answer to two significant figure precision.

Q3:

A 3 𝑑 electron is in an external magnetic field of 2 . 5 0 × 1 0 T.

Find the current associated with the orbital angular momentum of the electron.

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

Find the maximum torque on the electron.

  • A 5 . 5 9 × 1 0 N⋅m
  • B 5 . 6 7 × 1 0 N⋅m
  • C 5 . 5 3 × 1 0 N⋅m
  • D 5 . 7 0 × 1 0 N⋅m
  • E 5 . 6 3 × 1 0 N⋅m

Q4:

What is the longest wavelength that light can have if it can ionize the hydrogen atom in its ground state?

Q5:

Consider hydrogen in the ground state, where 𝑛 = 1 and all other quantum numbers equal 0. Use a value of 5 . 2 9 1 7 7 × 1 0 for 𝑎 , the Bohr radius.

Use the derivative of the electron’s wave function to determine the radial position for which the probability density, 𝑃 ( 𝑟 ) = 4 𝑎 𝑟 𝑒 2 𝑟 𝑎 , is a maximum. Give your answer to 3 significant figures.

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

Use the integral of the electron’s wave function to determine the electron’s average radial position, which is known as the expectation value of the electron’s radial position. Give your answer to 3 significant figures.

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

Q6:

What is the probability that the 1s electron of a hydrogen atom is found between 𝑟 = 0 and 𝑟 = ?

  • A0
  • B0.5
  • C0.8
  • D
  • E1

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