# 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 . What is the SchrΓΆdinger equation for the system?

• A
• B
• C
• D

Q2:

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

• A
• B
• C
• D

Q3:

For the three-dimensional particle in a box system, the Hamiltonian is separable; that is, . How can the total wave function of the system be expressed?

• A
• B
• C
• D

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 , the component of the angular momentum?

• A
• B
• C
• 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 . For the energy level with , what is the degeneracy?

Q6:

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

• A
• B
• C
• 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 ?

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 , 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
• B
• C
• D

Q12:

For a one-dimensional system, the potential energy function is described by . What is the SchrΓΆdinger equation for the system?

• A
• B
• C
• D

Q13:

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

• A
• B
• C
• D

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
• Dincreases as the quantum number becomes larger

Q15:

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

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

Q16:

What is the expectation value for a physical observable represented by a Hermitian operator where is the wave function of the system observed?

• A
• B
• C
• D

Q17:

What is the expectation value for the linear momentum at a given quantum mechanical state ?

• A
• B
• C
• 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 . What is the probability density for finding the particle?

• A
• B
• C
• D
• E