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?

  • Aβˆ’β„2π‘šπ‘₯Ξ¨+π‘˜π‘₯=πΈπœ“οŠ¨οŠ¨οŠ¨dd
  • Bβˆ’β„2π‘šπ‘₯Ξ¨=πΈπœ“οŠ¨οŠ¨οŠ¨dd
  • Cο€Ώβˆ’β„2π‘šπ‘₯+π‘˜π‘₯Ψ=πΈπœ“οŠ¨οŠ¨οŠ¨dd
  • Dℏ2π‘šπ‘₯+π‘˜π‘₯Ψ=πΈπœ“οŠ¨οŠ¨οŠ¨dd

Q2:

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

  • Â𝑇=βˆ’π‘–β„π‘₯dd
  • B̂𝑇=ℏ2π‘šπ‘₯dd
  • Ĉ𝑇=βˆ’β„2π‘šπ‘₯dd
  • D̂𝑇=βˆ’π‘–β„π‘₯dd

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?

  • AΞ¨=Ξ¨+Ξ¨+Ξ¨ο—ο˜ο™
  • BΞ¨=Ψ+Ξ¨+Ξ¨ο—ο˜ο™
  • CΞ¨=Ξ¨Γ—Ξ¨Γ—Ξ¨οŠ¨ο—οŠ¨ο˜οŠ¨ο™
  • DΞ¨=Ξ¨Γ—Ξ¨Γ—Ξ¨ο—ο˜ο™

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„Έο„Έπ‘Œ(πœƒ,πœ™)Μ‚πΏπ‘Œ(πœƒ,πœ™)πœƒπœ™ο—βˆžοŠ¦βˆžοŠ¦οŠ§οŠ¨βˆ—ο—οŠ§οŠ¨dd
  • B⟨𝐿⟩=ο„Έο„Έπ‘Œ(πœƒ,πœ™)Μ‚πΏπ‘Œ(πœƒ,πœ™)πœƒπœ™ο—οŽ„οŠ¦βˆžοŠ±βˆžοŠ§οŠ¨βˆ—ο—οŠ§οŠ¨dd
  • C⟨𝐿⟩=ο„Έο„Έπ‘Œ(πœƒ,πœ™)π‘Œ(πœƒ,πœ™)πœƒπœ™ο—οŽ„οŠ¦οŠ¨οŽ„οŠ¦οŠ§οŠ¨βˆ—οŠ§οŠ¨dd
  • D⟨𝐿⟩=ο„Έο„Έπ‘Œ(πœƒ,πœ™)Μ‚πΏπ‘Œ(πœƒ,πœ™)πœƒπœ™ο—οŽ„οŠ¦οŠ¨οŽ„οŠ¦οŠ§οŠ¨βˆ—ο—οŠ§οŠ¨dd

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„Έο„ΈΜ‚πΏο‘π‘Œ(πœƒ,πœ™)οπœƒπœ™ο˜οŽ„οŠ¦οŠ¨οŽ„οŠ¦ο˜οŠ±οŠ§οŠ¨οŠ¨dd
  • B⟨𝐿⟩=ο„Έο„Έπ‘Œ(πœƒ,πœ™)Μ‚πΏπ‘Œ(πœƒ,πœ™)πœƒπœ™ο˜οŽ„οŠ¦οŠ¨οŽ„οŠ¦οŠ±οŠ§οŠ¨ο˜οŠ±οŠ§οŠ¨βˆ—dd
  • C⟨𝐿⟩=ο„Έο„Έπ‘Œ(πœƒ,πœ™)Μ‚πΏπ‘Œ(πœƒ,πœ™)πœƒπœ™ο˜οŠ¨οŽ„οŠ¦βˆžοŠ±βˆžοŠ±οŠ§οŠ¨ο˜οŠ±οŠ§οŠ¨βˆ—dd
  • D⟨𝐿⟩=ο„Έο„Έπ‘Œ(πœƒ,πœ™)π‘Œ(πœƒ,πœ™)πœƒπœ™ο˜οŽ„οŠ¦οŠ¨οŽ„οŠ¦οŠ±οŠ§οŠ¨οŠ±οŠ§οŠ¨βˆ—dd

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 .

  • Â𝐻Ψ𝐸=πœ“
  • B̂𝐻Ψ=π‘–πΈπœ“
  • CΨ̂𝐻=πΈπœ“
  • D̂𝐻Ψ=πΈπ‘–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ℏ2π‘šπ‘₯+12π‘˜π‘₯Ψ=πΈπœ“οŠ¨οŠ¨οŠ¨οŠ¨dd
  • Bℏ2π‘šπ‘₯βˆ’12π‘˜π‘₯Ψ=πΈπœ“οŠ¨οŠ¨οŠ¨οŠ¨dd
  • Cο€Ώβˆ’β„2π‘šπ‘₯+12π‘˜π‘₯Ψ=πΈπœ“οŠ¨οŠ¨οŠ¨οŠ¨dd
  • Dβˆ’β„2π‘šπ‘₯Ξ¨+12π‘˜π‘₯=πΈπœ“οŠ¨οŠ¨οŠ¨οŠ¨dd

Q13:

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

  • Â𝐻Ψ=πΈπœ“
  • B̂𝐻Ψ=π‘–πΈπœ“
  • Ĉ𝐻Ψ=πΈπ‘–πœ“
  • Dβˆ’Ξ¨Μ‚π»=πΈπ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?

  • Aο„Έβˆ’Μ‚π‘‚Ξ¨(π‘₯)π‘₯d
  • Bο„ΈΞ¨(π‘₯)̂𝑂Ψ(π‘₯)π‘₯βˆ—οŠοŠd
  • Cο„ΈΞ¨(π‘₯)Ξ¨(π‘₯)π‘₯βˆ—οŠοŠd
  • D̂𝑂Ψ(π‘₯)π‘₯d

Q17:

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

  • Aο„ΈΞ¨(π‘₯)ℏπ‘₯Ψ(π‘₯)π‘₯βˆ—οŠοŠddd
  • Bο„ΈΞ¨(π‘₯)Ξ¨(π‘₯)π‘₯βˆ—οŠοŠd
  • Cο„Έβˆ’π‘–β„π‘₯Ξ¨(π‘₯)π‘₯ddd
  • Dο„ΈΞ¨(π‘₯)ο€½βˆ’π‘–β„π‘₯Ψ(π‘₯)π‘₯βˆ—οŠοŠddd

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?

  • A𝐴
  • B𝐴π‘₯d
  • C𝐴π‘₯d
  • Dβˆ’π΄
  • E𝐴

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