Final Exam Review!!!

Question 1

For a free particle, what is the potential energy function?

Answer 1

Zero

Question 2

Critical physical difference between a free particle and a bound particle

Answer 2

There is no confining potential for a free particle

Question 3

For a free particle, is the energy discrete or continuous?

Answer 3

Continuous because there are not enough constraints

Question 4

Consider the time evolution of the energy eigenstates for a free particle. What do you get and what does it look like?

Answer 4

You get a form that represents a wave that retains its shape as it moves, any given point on that shape moves with a speed determined by the parameter v. Has the same form that we know from classical waves

Question 5

Phase velocity (free particle state)

Answer 5

The speed at which points of constant phase move

Question 6

Whenever we see a wave function with spatial dependence exp(p/m kx), the sign of the wave vector in the exponent indicates

Answer 6

The direction of motion

Question 7

what is the momentum eigenvalue for a momentum eigenstate (free particle)

Answer 7

p = h/lambda

Question 8

For the free particle, the eigenstates of the momentum are also eigenstates of the

Answer 8

energy

Question 9

Commuting observables share

Answer 9

Common eigenstates

Question 10

True or false: for a free particle, a given momentum eigenstate has a definite energy

Answer 10

True

Question 11

True or false: a given energy state has a definite momentum

Answer 11

False

Question 12

For a free particle, a general energy eigenstate is a superposition of

Answer 12

Two momentum states with opposite momenta

Question 13

For a free particle, we say that the energy state is degenerate with respect to momentum because

Answer 13

A given energy state corresponds to multiple momentum states

Question 14

In the free particle case, the energy states are

Answer 14

Two-fold degenerate

Question 15

The proper way to use a wave to describe a particle is using the

Answer 15

Group velocity of a wave packet

Question 16

For a free particle, mathematically, we cannot normalize the momentum eigenstates because

Answer 16

The integral of the probability density over all space is infinite

Question 17

How do we get wave functions that are normalizable and localized to finite regions of space for the free particle?

Answer 17

We can construct superpositions of momentum eigenstates to make wave packets

Question 18

What is the continuous analog of the Kronecker delta?

Answer 18

Dirac delta function

Question 19

The wave function and the momentum space wave function are both representations of

Answer 19

The same state, but they are representing that state in different bases

Question 20

For a free particle, the state psi can be written in two different bases. What are they?

Answer 20

The position basis using the wavefunction OR the momentum basis using the momentum space wave function

Question 21

What is the connection between the momentum space wave function and the position space wave function?

Answer 21

The Fourier transform

Question 22

Why is a wave packet called a wave packet?

Answer 22

Because it is localized to a region of space

Question 23

What does a wave packet have in common with a wave? With a particle?

Answer 23

It has a characteristic wavelength determined by the central momentum
it has a limited spatial extent

Question 24

If you have a discrete Fourier sum of momentum eigenstates vs one with a continuum distribution, how are the results different?

Answer 24

For the discrete case, the localization is repeated periodically. For the continuum distribution, only one localized region exists

Question 25

The envelope of the wave packet moves at a different velocity from the carrier and this velocity is known as the

Answer 25

group velocity

Question 26

The group velocity

Answer 26

characterizes the velocity of the group of waves together

Question 27

The Hamiltonian operator corresponds to

Answer 27

The total energy of the system

Question 28

Which operator determines the time evolution of a quantum system?

Answer 28

The Hamiltonian

Question 29

The Schrodinger equation is a

Answer 29

Differential equation that tells you the time evolution of a quantum system

Question 30

What is the energy basis?

Answer 30

The basis of eigenvectors of the Hamiltonian

Question 31

Why do the eigenvectors of the Hamiltonian form a complete basis?

Answer 31

Because the Hamiltonian is an observable and a Hermitian operator

Question 32

How do you find the time dependence of an original state vector?

Answer 32

You multiply each energy eigenstate coefficient by its own phase factor that depends on the energy of that eigenstate

Question 33

What are stationary states?

Answer 33

States that do not evolve with time

Question 34

True or false: if a system begins in an energy eigenstate, then it remains in that state

Answer 34

True

Question 35

Consider an operator A. It commutes with the Hamiltonian. What can you conclude about the probability of measuring any particular eigenvalue of A?

Answer 35

It is time independent

Question 36

If an operator A does not commute with the Hamiltonian, then what can you say in general about the eigenstates of A?

Answer 36

They are superpositions of energy eigenstates

Question 37

Time dependence is determined by

Answer 37

The difference of the energies of the two states involved in the superposition

Question 38

If a state is unbound, then is the energy spectrum discrete or continuous and why?

Answer 38

Continuous because all energies above the energy required to escape the well are allowed

Question 39

If a state is bound, then is the energy spectrum discrete or continuous?

Answer 39

Discrete

Question 40

What does local curvature tell you?

Answer 40

It tells you what the shape of the wavefunction is

Question 41

The second derivative of a wavefunction is proportional to the wavefunction, what does this tell you about the wavefunction?

Answer 41

It’s exponential

Question 42

The second derivative of a wavefunction is proportional to the negative of itself, what does this tell you about the shape of the wavefunction in this region?

Answer 42

It’s oscillatory

Question 43

Suppose you have a Pauli matrix and you multiply it by some value, how are the eigenvectors affected? How are the eigenvalues affected?

Answer 43

The eigenvectors are the same but the eigenvalues are all multiplied by the constant

Question 44

For any spin projection (i.e. something like a superposition of S_z and S_x), what are the eigenvalues?

Answer 44

plus or minus hbar/2

Question 45

If you have something that is a superposition of S_z and S_x but it’s not normalized, then how do you find the eigenvalues?

Answer 45

Recall that the eigenvalues for any spin projection are plus or minus hbar/2, to find the eigenvalues then you just need to find the normalized version and the constant that the normalized version differs from the actual version by

Question 46

What does the quantum number l called and what does it represent

Answer 46

It is the orbital angular momentum quantum number and it gives a measure of the size of the angular momentum vector in that direction.

Question 47

What is the magnitude of the angular momentum vector

Answer 47

sqrt(l(l+1)/hbar)

Question 48

Crucial difference between spin angular momentum and orbital angular momentum (e.g. the allowed quantized values for the spin angular momentum number)

Answer 48

spin case: the allowed quantized values of the spin angular momentum number s are the integers and half integers

orbital case: the quantum number l can only take on integer values

Question 49

for each l, what are the possible values of m?

Answer 49

Integers from -l to l.

Question 50

What is the spherical coordinate representation of the angular momentum operator L_z?

Answer 50

-ihbar(d/d(phi))

Question 51

Does the Hamiltonian commute with the orbital angular momentum operator L^2?

Answer 51

Yes

Question 52

Does the Hamiltonian commute with the orbital angular momentum operator L_z?

Answer 52

Yes

Question 53

What properties does a system have with no radial or polar angle dependence?

Answer 53

A particle confined to move on a ring with constant radius

Question 54

What are the energy eigenstates for the particle on a ring?

Answer 54

The states |m> that satisfy the L_z eigenvalue equation

Question 55

What are the allowed values for a particle on a ring?

Answer 55

E_m = m^2(hbar^2/2I)

Question 56

What is the position representation of the |m> state?

Answer 56

1/sqrt(2pi)exp(im(phi))

Question 57

true or false: |m> is an eigenstate of the ring Hamiltonian

Answer 57

true

Question 58

true or false: |m> is an eigenstate of the z-component of the orbital angular momentum

Answer 58

true