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

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