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Worksheet: The Schrödinger Equation

Q1:

In the solution of the SchrΓΆdinger equation for the rigid rotor system, what is the degeneracy of an energy level with angular momentum quantum number 𝑙 = 2 ?

Q2:

The SchrΓΆdinger equation for the rigid rotor system must be solved with the spherical coordinate system. What is the correct expression for calculating the average value of 𝐿 π‘₯ , the π‘₯ component of the angular momentum?

  • A ⟨ 𝐿 ⟩ = ο„Έ ο„Έ π‘Œ ( πœƒ , πœ™ ) π‘Œ ( πœƒ , πœ™ ) πœƒ πœ™ π‘₯ πœ‹ 0 2 πœ‹ 0 1 2 βˆ— 1 2 d d
  • B ⟨ 𝐿 ⟩ = ο„Έ ο„Έ π‘Œ ( πœƒ , πœ™ ) Μ‚ 𝐿 π‘Œ ( πœƒ , πœ™ ) πœƒ πœ™ π‘₯ πœ‹ 0 ∞ βˆ’ ∞ 1 2 βˆ— π‘₯ 1 2 d d
  • C ⟨ 𝐿 ⟩ = ο„Έ ο„Έ π‘Œ ( πœƒ , πœ™ ) Μ‚ 𝐿 π‘Œ ( πœƒ , πœ™ ) πœƒ πœ™ π‘₯ ∞ 0 ∞ 0 1 2 βˆ— π‘₯ 1 2 d d
  • D ⟨ 𝐿 ⟩ = ο„Έ ο„Έ π‘Œ ( πœƒ , πœ™ ) Μ‚ 𝐿 π‘Œ ( πœƒ , πœ™ ) πœƒ πœ™ π‘₯ πœ‹ 0 2 πœ‹ 0 1 2 βˆ— π‘₯ 1 2 d d

Q3:

The kinetic energy operator for a one-dimensional system is:

  • A Μ‚ 𝑇 = βˆ’ 𝑖 ℏ π‘₯ 2 2 2 d d
  • B Μ‚ 𝑇 = ℏ 2 π‘š π‘₯ 2 2 2 d d
  • C Μ‚ 𝑇 = βˆ’ 𝑖 ℏ π‘₯ d d
  • D Μ‚ 𝑇 = βˆ’ ℏ 2 π‘š π‘₯ 2 2 2 d d

Q4:

For a one-dimensional system, the potential energy function is described by 𝑉 ( π‘₯ ) = 1 2 π‘˜ π‘₯ 2 . The SchrΓΆdinger equation for the system is:

  • A ο€Ώ ℏ 2 π‘š π‘₯ + 1 2 π‘˜ π‘₯  Ξ¨ = 𝐸 πœ“ 2 2 2 2 d d
  • B βˆ’ ℏ 2 π‘š π‘₯ Ξ¨ + 1 2 π‘˜ π‘₯ = 𝐸 πœ“ 2 2 2 2 d d
  • C ο€Ώ ℏ 2 π‘š π‘₯ βˆ’ 1 2 π‘˜ π‘₯  Ξ¨ = 𝐸 πœ“ 2 2 2 2 d d
  • D ο€Ώ βˆ’ ℏ 2 π‘š π‘₯ + 1 2 π‘˜ π‘₯  Ξ¨ = 𝐸 πœ“ 2 2 2 2 d d

Q5:

The solution of the SchrΓΆdinger equation for the rigid rotor system results in degenerate states for non-zero values of the angular momentum quantum number 𝑙 . For the energy level with 𝑙 = 1 , the degeneracy is:

Q6:

The solution of the SchrΓΆdinger equation for the hydrogen atom results in degenerate states for the principal energy levels characterized by the principal quantum number. For 𝑛 = 3 , what is the degeneracy?

Q7:

For a one-dimensional system, the potential energy function is described by 𝑣 ( π‘₯ ) = π‘˜ π‘₯ . The SchrΓΆdinger equation for the system is:

  • A βˆ’ ℏ 2 π‘š π‘₯ Ξ¨ + π‘˜ π‘₯ = 𝐸 πœ“    d d
  • B ο€Ώ ℏ 2 π‘š π‘₯ + π‘˜ π‘₯  Ξ¨ = 𝐸 πœ“    d d
  • C βˆ’ ℏ 2 π‘š π‘₯ Ξ¨ = 𝐸 πœ“    d d
  • D ο€Ώ βˆ’ ℏ 2 π‘š π‘₯ + π‘˜ π‘₯  Ξ¨ = 𝐸 πœ“    d d

Q8:

Wavefunctions for the hydrogen atom contain nodes. The 4d orbital possesses:

  • Afour angular nodes and two radial nodes.
  • Bthree angular nodes and one radial node.
  • Cfour angular nodes and one radial node.
  • Dtwo angular nodes and one radial node.

Q9:

The SchrΓΆdinger equation for the rigid rotor system must be solved with the spherical coordinate system. For the 𝑦 - c o m p o n e n t of the orbital angular momentum 𝐿 𝑦 , which of the following expressions gives the expectation value?

  • A ⟨ 𝐿 ⟩ = ο„Έ ο„Έ π‘Œ ( πœƒ , πœ™ ) π‘Œ ( πœƒ , πœ™ ) πœƒ πœ™ 𝑦 πœ‹ 0 2 πœ‹ 0 βˆ’ 1 2 βˆ’ 1 2 βˆ— d d
  • B ⟨ 𝐿 ⟩ = ο„Έ ο„Έ Μ‚ 𝐿  π‘Œ ( πœƒ , πœ™ )  πœƒ πœ™ 𝑦 πœ‹ 0 2 πœ‹ 0 𝑦 βˆ’ 1 2 2 d d
  • C ⟨ 𝐿 ⟩ = ο„Έ ο„Έ π‘Œ ( πœƒ , πœ™ ) Μ‚ 𝐿 π‘Œ ( πœƒ , πœ™ ) πœƒ πœ™ 𝑦 2 πœ‹ 0 ∞ βˆ’ ∞ βˆ’ 1 2 𝑦 βˆ’ 1 2 βˆ— d d
  • D ⟨ 𝐿 ⟩ = ο„Έ ο„Έ π‘Œ ( πœƒ , πœ™ ) Μ‚ 𝐿 π‘Œ ( πœƒ , πœ™ ) πœƒ πœ™ 𝑦 πœ‹ 0 2 πœ‹ 0 βˆ’ 1 2 𝑦 βˆ’ 1 2 βˆ— d d

Q10:

For the three-dimensional particle in a box system, the Hamiltonian is separable, i.e., Μ‚ 𝐻 = Μ‚ 𝐻 ( π‘₯ ) + Μ‚ 𝐻 ( 𝑦 ) + Μ‚ 𝐻 ( 𝑧 ) π‘₯ 𝑦 𝑧 . How can the total wavefunction of the system be expressed?

  • A Ξ¨ = Ξ¨ Γ— Ξ¨ Γ— Ξ¨ 2 π‘₯ 2 𝑦 2 𝑧
  • B Ξ¨ =  Ξ¨ + Ξ¨ + Ξ¨ π‘₯ 𝑦 𝑧
  • C Ξ¨ = Ξ¨ + Ξ¨ + Ξ¨ π‘₯ 𝑦 𝑧
  • D Ξ¨ = Ξ¨ Γ— Ξ¨ Γ— Ξ¨ π‘₯ 𝑦 𝑧

Q11:

Which of the following is the correct form of the time-independent SchrΓΆdinger equation?

  • A Μ‚ 𝐻 Ξ¨ = 𝐸 𝑖 πœ“
  • B Μ‚ 𝐻 Ξ¨ = 𝑖 𝐸 πœ“
  • C βˆ’ Ξ¨ Μ‚ 𝐻 = 𝐸 πœ“
  • D Μ‚ 𝐻 Ξ¨ = 𝐸 πœ“

Q12:

Wave functions for the hydrogen atom can contain nodes. The 5p orbital possesses:

  • Afive angular nodes and three radial nodes.
  • Btwo angular nodes and two radial nodes.
  • Cone angular node and five radial nodes.
  • Done angular node and three radial nodes.

Q13:

The time independent form of the SchrΓΆdinger equation is given by:

  • A Μ‚ 𝐻 Ξ¨ = 𝐸 𝑖 πœ“
  • B Μ‚ 𝐻 Ξ¨ = 𝑖 𝐸 πœ“
  • C Ξ¨ Μ‚ 𝐻 = 𝐸 πœ“
  • D Μ‚ 𝐻 Ξ¨ 𝐸 = πœ“