Practice Midterm

Question 1

A valid wavefunction must be single-valued and not go to infinity

Answer 1

True
(must be single-valued and not go to infinity)

Question 2

A valid wavefunction must be an eigenfunction of the Hamiltonian

Answer 2

False
(A valid wavefunction does not have to be an eigenfunction of the Hamiltonian)

Question 3

The eigenvalues of a Hermitian operator must be real numbers

Answer 3

True
(MUST BE REAL #’S)

Question 4

The eigenfunctions of a nondegenerate Hermitian Operator must be orthogonal

Answer 4

True
(Hermitian = orthogonal)

Question 5

Nondegenerate

Answer 5

quantum states that have unique energy levels, meaning that no two states share the same energy value

Question 6

The most probable location of a particle in a box is always L/2

Answer 6

False
(depends on energy & quantum number)

Question 7

For a particle in a box in a stationary state <p>=0

Answer 7

True
(momentum cancels itself out)

Question 8

If [A,B] = 0, then A and B must have the same eigenfunctions

Answer 8

False.

(If they commute, they can share a common set, but not necessarily having same e-functions)

Question 9

Any valid wavefunction can be constructed using the eigenfunctions of a Hermitian operator.

Answer 9

True.
(Hermitian is good, so yes)

Question 10

A small particle will behave classically when it has low kinetic energy.

Answer 10

False.

(Reverse. remember the amount of oscillations and nodes contributing to a straiter line)

Question 11

The energy of a rigid rotor depends on both quantum numbers l and m.

Answer 11

False
(depends on l more than m)

Question 12

Kinetic Energy equals

Answer 12

(3/2)kB *T

Question 13

Kinetic Energy equals

Answer 13

(1/2)mv^2

Question 14

mass of electron

Answer 14

9.11 x 10^-31

Question 15

If particles travel at the same energy, they must have the same de Broglie wavelength.

Answer 15

False
(need to have the same mass)

Question 16

The expectation value <x> gives the most probable location for the particle in a box</x>

Answer 16

False
(average position not most probable)

Question 17

A valid wavefunction must satisfy integral |Psi(x)|^2 = 1

Answer 17

True
(needs to satisfy the normalization condition, total probability of finding the particle somewhere in space is 1)

Question 18

The Heisenburg representation of a wavefunction is a matrix.

Answer 18

False (they are state vectors)

Question 19

The eigenfunctions of the Hamiltonian are the only possible quantum states of the system.

Answer 19

False
(they represent stationary states, not superpositions)

Question 20

A well-behaved wavefunction cannot go to infinity on the interval where it is defined.

Answer 20

True
(A well-behaved wavefunction must be finite - not go to infinity)

Question 21

A superposition state is also called a nonstationary state.

Answer 21

True.
(unlike a single energy eigenstate, a superposition of multiple eigenstates generally results in time-dependent behavior - leading to a nonstationary state)

Question 22

The nondegenerate eigenstates of a Hermitian operator must be orthogonal (by default)
.

Answer 22

True.
(Follows properties of Hermitian operators, guarantee the eigenstates corresponding to different eigenvalues are orthogonal)

Question 23

The eigenfunctions of the rigid rotor Hamiltonian are non degenerate.

Answer 23

False
(They exhibit degeneracy. E-values are nondegenerate for l, but the E-Functions are degenerate)

Question 24

If two operators commute, there will be an uncertainty relation between their observables.

Answer 24

False
(There is no uncertainty. Heisenburg Uncertainty principle applies to non-commuting observables).

Question 25

If two particles travel at the same velocity, they must have the same deBroglie wavelength.

Answer 25

False

(differing masses could travel at same v)

Question 26

All Hermitian operators are real.

Answer 26

False.

(not necessarily real-valued operators themselves)

Question 27

All eigenvalues of a Hermitian operator is a diagonal matrix

Answer 27

False
(doesn’t have to be, only diagonal if the basis set is eigenfunction of Hermetian operator)

Question 28

An eigenfunction of the Hamiltonian is also called a stationary state.

Answer 28

True
(Eigen function of Hamiltonian is stationary state)

Question 29

A well-behaved wavefunction must have only real values.

Answer 29

False.

(spherical harmonics have an imaginary part)

Question 30

An eigenfunction of the Hamiltonian is also called a stationary state.

Answer 30

True.

Question 31

The non-degenerate eigenstates of a Hermitian operator must be orthogonal.

Answer 31

True.

(Hermitian is usually pretty nice to work with, so you can assume orthogonality works)

Question 32

A valid wavefunction must be normalized.

Answer 32

True.
(Physically, the total probability of the particle existing must equal to 1)

Question 33

If two operators commute, there will be an uncertainty relation between their observables.

Answer 33

False.

(There is no uncertainty relation. Only uncertainty when they don’t commute!)