Worksheet: The Schrödinger Equation

In this worksheet, we will practice explaining the meaning of the terms in the Schrödinger equation and the general features of molecular wave functions.

Q1:

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

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

Q2:

What is the kinetic energy operator for a one-dimensional system?

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

Q3:

For the three-dimensional particle in a box system, the Hamiltonian is separable; that is, ̂𝐻=̂𝐻(π‘₯)+̂𝐻(𝑦)+̂𝐻(𝑧)ο—ο˜ο™. How can the total wave function of the system be expressed?

  • A Ξ¨ = Ξ¨ + Ξ¨ + Ξ¨   
  • B Ξ¨ =  Ξ¨ + Ξ¨ + Ξ¨   
  • C Ξ¨ = Ξ¨ Γ— Ξ¨ Γ— Ξ¨      
  • D Ξ¨ = Ξ¨ Γ— Ξ¨ Γ— Ξ¨   

Q4:

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, the π‘₯ component of the angular momentum?

  • A ⟨ 𝐿 ⟩ = ο„Έ ο„Έ π‘Œ ( πœƒ , πœ™ ) Μ‚ 𝐿 π‘Œ ( πœƒ , πœ™ ) πœƒ πœ™  ∞  ∞    βˆ—    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™  οŽ„  ∞  ∞   βˆ—    d d
  • C ⟨ 𝐿 ⟩ = ο„Έ ο„Έ π‘Œ ( πœƒ , πœ™ ) π‘Œ ( πœƒ , πœ™ ) πœƒ πœ™  οŽ„   οŽ„    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™  οŽ„   οŽ„    βˆ—    d d

Q5:

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

Q6:

The SchrΓΆdinger equation for the rigid rotor system must be solved with the spherical coordinate system. For the 𝑦-component of the orbital angular momentum 𝐿, which of the following expressions gives the expectation value?

  • A ⟨ 𝐿 ⟩ = ο„Έ ο„Έ Μ‚ 𝐿  π‘Œ ( πœƒ , πœ™ )  πœƒ πœ™  οŽ„   οŽ„       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™  οŽ„   οŽ„         βˆ— 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™   οŽ„  ∞  ∞        βˆ— d d
  • D ⟨ 𝐿 ⟩ = ο„Έ ο„Έ π‘Œ ( πœƒ , πœ™ ) π‘Œ ( πœƒ , πœ™ ) πœƒ πœ™  οŽ„   οŽ„        βˆ— d d

Q7:

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?

Q8:

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?

Q9:

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

  • Aone angular node and three radial nodes
  • Bone angular node and five radial nodes
  • Cfive angular nodes and three radial nodes
  • Dtwo angular nodes and two radial nodes

Q10:

Fill in the blank: Wave functions for the hydrogen atom contain nodes. The 4d orbital possesses .

  • Atwo angular nodes and one radial node
  • Bthree angular nodes and one radial node
  • Cfour angular nodes and one radial node
  • Dfour angular nodes and two radial nodes

Q11:

Fill in the blank: The time-independent form of the SchrΓΆdinger equation is given by .

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

Q12:

For a one-dimensional system, the potential energy function is described by 𝑉(π‘₯)=12π‘˜π‘₯. What is the SchrΓΆdinger equation for the system?

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

Q13:

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

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

Q14:

Complete the following sentence: The energy gap between two adjacent quantum states for the harmonic oscillator system .

  • Adecreases as the quantum number becomes larger
  • Bis a constant
  • Cequals 12ℏ𝑀
  • Dincreases as the quantum number becomes larger

Q15:

What is the energy operator in quantum mechanics, ̂𝐻, called?

  • AThe momentum
  • BThe Hamiltonian
  • CThe Laplacian
  • DThe Hermitian

Q16:

What is the expectation value for a physical observable represented by a Hermitian operator ̂𝑂 where Ξ¨(π‘₯) is the wave function of the system observed?

  • A ο„Έ βˆ’ Μ‚ 𝑂 Ξ¨ ( π‘₯ ) π‘₯   d
  • B ο„Έ Ξ¨ ( π‘₯ ) Μ‚ 𝑂 Ξ¨ ( π‘₯ ) π‘₯ βˆ—   d
  • C ο„Έ Ξ¨ ( π‘₯ ) Ξ¨ ( π‘₯ ) π‘₯ βˆ—   d
  • D ο„Έ Μ‚ 𝑂 Ξ¨ ( π‘₯ ) π‘₯   d

Q17:

What is the expectation value for the linear momentum at a given quantum mechanical state Ξ¨(π‘₯)?

  • A ο„Έ Ξ¨ ( π‘₯ ) ο€½ ℏ π‘₯  Ξ¨ ( π‘₯ ) π‘₯ βˆ—   d d d
  • B ο„Έ Ξ¨ ( π‘₯ ) Ξ¨ ( π‘₯ ) π‘₯ βˆ—   d
  • C ο„Έ βˆ’ 𝑖 ℏ π‘₯ Ξ¨ ( π‘₯ ) π‘₯ d d d  
  • D ο„Έ Ξ¨ ( π‘₯ ) ο€½ βˆ’ 𝑖 ℏ π‘₯  Ξ¨ ( π‘₯ ) π‘₯ βˆ—   d d d

Q18:

The phenomenon that wave functions can extend into the classically forbidden region, that is, the region where a classical particle would have negative kinetic energy, is termed .

  • Azero-point energy
  • Bquantum mechanical tunneling
  • Cdiscretization
  • Dnormalization
  • Equantization

Q19:

The wave function for a particle is given by Ξ¨=π΄π‘’οŠ±οƒο‡ο—. What is the probability density for finding the particle?

  • A 𝐴
  • B 𝐴 π‘₯  d
  • C 𝐴 π‘₯ d
  • D βˆ’ 𝐴
  • E 𝐴 

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