Quantum equations + more

Question 1

What is the probability of measuring an eigenvalue w? (born rule)

Answer 1

The scalar product of the w eigenstate with the wavefunction, squared.

Question 2

What is the is the average measurement of a value w?

Answer 2

The expectation value of the operator corresponding to a measurement of w.

The expectation value is the scalar product of the wavefunction with the wavefunction acted on by the operator.

Question 3

Why must operators corresponding to observables be hermitian operators?

Answer 3

They give real eigenvalues. Another important property of hermitian operators is that their eigenvectors will form an orthogonal set, so a state can be expressed in the basis of these eigenvectors.

Question 4

What is the potential for the quantum harmonic oscillator? What about the energy?

Answer 4

In ND:

V = the sum of: (1/2 m omega^2 x^2) for all x_i

E = hbar omega * (n_1 + n_2 + … + N/2)
where n_i take values 0, 1, 2, 3…

Question 5

What is the energy of an ND infinite potential well?

Answer 5

The ND energy is simply the linear sum of N independent 1D potential wells.

For 1D, the energy for a well of width L is:
(hbar k)^2 / 2m
where k = n pi / L
n can take values 1, 2, 3, 4…

Question 6

What is the potential for the hydrogen atom (coulomb potential)?
What are the energy levels?
How could we convert this to apply to a non-hydrogen atom with a single electron?

Answer 6

V = - e^2 / 4pi e0 r = - (alpha hbar c)/r
E = - E_r / n^2
where E_r is the Rydberg energy and n takes values 1, 2, 3, 4…
For a non hydrogen atom with Z protons, transform alpha to Z * alpha.
**it is important to note that the Rydberg energy contains a factor alpha^2

Question 7

What is the momentum operator in the x-basis?

What is the x-operator in the momentum-basis?

Answer 7

-i hbar grad (w.r.t. x)

i hbar grad (w.r.t. p)

Question 8

What is the hamiltonian operator?

Answer 8

(momentum operator)^2 / 2m + potential operator

in the x-basis, the kinetic energy term is - (hbar^2 / 2m) laplacian (w.r.t. x)

Question 9

What is the angular momentum operator?
Give the z-component.
Give the operator in the x-basis.

Answer 9

r-operator cross p-operator

L_z = x p_y - y p_x (where those are all operators)

in the x-basis, L -> -i hbar (r-vector cross grad)

Question 10

What is the scalar product of an x-eigenstate and a p-eigenstate?
What about in 3D (i.e. the scalar product of an r-eigenstate with a p-eigenstate)

Answer 10

1/sqrt(2 pi hbar) * e^(i / hbar * p x)

in 3D:
[1/sqrt(2 pi hbar)]^3 * e^(i / hbar * p dot r)

Question 11

What is the scalar product of two different x-eigenstates, x and x’?

Answer 11

Basically zero unless x=x'
so delta(x-x')

Question 12

What is the form of the propagator?

What about in the basis of energy eigenstates?

Answer 12

e^ (-i / hbar * hamiltonian operator * (t - t_0))

In the energy-basis:
the sum over states labelled by n of
e^ (- i / hbar * E_n * (t - t_0)) * the outer product of n-eigenvectors

Question 13

What operation does the propagator perform?

Answer 13

The wavefunction at time t = The propagator(t=t, t_0 = 0) acting on the wavefunction at time 0.

Question 14

What is the magnetic field energy operator?

What about in general?

Answer 14

mu_b / hbar * (L + 2S) dot B

where L and S are the orbital angular momentum and spin angular momentum operators, and mu_b is the bohr magneton

Question 15

How can the spin operator (for spin 1/2) be represented in terms of the pauli matrices?

Answer 15

S = hbar / 2 * the pauli matrices

Question 16

How can an operator A be represented in matrix form in the basis of orthonormal vectors e_i?

Answer 16

A_ij =

where i denotes the rows and j the columns

Question 17

What is the hermitian conjugate of an operator in a matrix representation?

Answer 17

The conjugate of the transpose.

Question 18

What are the special properties of the eigensolutions to a hermitian operator?

Answer 18

Hermitian operators have real-valued eigenvalues.
Hermitian operators have orthogonal eigenvectors, such that an orthonormal basis can be formed of the eigenvectors, and as such the operator can be diagonalised.

Question 19

What is Ehrenfest’s theorem?

Answer 19

-

Question 20

How many basis states do we need to span a product space of subsystems of dimension M and N?

Answer 20

MN states

Question 21

What is the eigenvalue of a J^2 operator acting on an eigensate |j, m_j> ?

Answer 21

hbar^2 * j(j+1)

Question 22

What is the eigenvalue of a J_z operator acting its own eigenstate |j, m_j> ?

Answer 22

hbar m_j

Question 23

For a system with two independent angular momenta, why do we have a choice of two bases?
What are these bases?

Answer 23

L^2 and S^2 commute with J_i
L_i and S_i obviously commute
J^2 commutes with L^2 and S^2 but does not commute with L_i or S_i

i.e: we can work in the |l m_l, s m_s> basis
or the |l, s, j m_j> basis.

m_j = m_l + m_s

Question 24

If the total J operator is the addition of the L and S operators (acting on distinct vector spaces), what is an expression for the J^2 operator?

Answer 24

L^2 + S^2 + 2 L dot S

Question 25

How many states are there for total angular momentum J = L + S?

Answer 25

j_max = l + s
j_min = | l-s |
for each j there are 2j+1 states labelled by m_j

Another way of calculating the total number of states is by realising that the J space is the product space of L and S spaces, which have 2l+1 and 2s+1 states respectively, so there are:
(2l+1)(2s+1) states for total angular momentum.

Question 26

What are the Clebsch-Gordan coefficients?

What is the phase convention for the coefficients?

Answer 26

Coefficients for the expansion of |j m_j> states in terms of |l m_l, s m_s> states and vice-versa.
All coefficients are be real valued.

Question 27

What is the gyromagnetic ratio for spin?

Answer 27

~2

This is the 2 that features in the magnetic hamiltonian.

Question 28

What is the use of the variational method?

Answer 28

To give an upper bound on the ground state energy.

Question 29

How is the variational method implemented?

Answer 29

Make an educated guess for the ground state wavefunction. The trial wavefunction must be continuous and satisfy all relevant boundary conditions.
Take the expectation value of the Hamiltonian using this wavefunction, normalise it.

This expectation value gives an upper bound on the ground-state energy.

Choose a wavefunction that has an adjustable parameter, the expectation value can then be minimized w.r.t. this parameter.

Question 30

In what special case can we use the variational method to get an upper bound on the energy of an excited state?

Answer 30

In the case where a trial state can be constructed in such a way that it is orthogonal to the (known) ground state.

Question 31

What are the conditions required for the WKB approximation to be appropriate?

Answer 31

1D system (position-space) -> the TISE approximates SHM with constant k.
k is dependent on position, but it must not vary too much for the WKB approximation to be valid.

More accurately, the potential must not vary too much on the scale of the wavelength 2pi / k, especially around the “turning points”. This means that the WKB approximation can be invalid in general but valid for large k (small wavelength).

Question 32

What is the form of the constant of SHM k in the WKB approximation?

Answer 32

the second derivative w.r.t. x of the wavefunction is equal to - k^2 * the wavefunction (~SHM)

Where k = 2 pi / lambda = sqrt[(2m/hbar^2) * (E - V)]

Question 33

What is the dependence of the WKB approximate wavefunction on the relation between E and V?

Answer 33

If E > V, k is real and the wavefunction describes an oscillatory bound state.

If E ~ V, k goes to zero and the WKB wavefunction diverges.

If E < V, k is imaginary and the wavefunction describes exponential decay (tunnelling solution).

The two valid cases must be “stitched together” across the V ~ E gap using an airy function.

Question 34

What is the “net phase” and what are the constraints on it for different WKB bound states?

Answer 34

The integral of k dx between the two “turning points” where E = V.

For two hard walls (infinite well) = (n + 1) pi
for one soft wall = (n + 3/4) pi
for two soft walls = (n + 1/2) pi

n = 0, 1, 2, 3…

Question 35

In what case can we use perturbation theory?

Answer 35

In the case where the hamiltonian can be split into a known hamiltonian and a small perturbing hamiltonian.

The solutions are expanded as a series in this small perturbation, so the series must converge quickly.

Question 36

In what case do we need to use degenerate perturbation theory?
How is this done?

Answer 36

If working in a basis where the known hamiltonian gives degenerate states, the expansion diverges.

We must find a new basis in the subspace of degenerate states such that the perturbing hamiltonian is diagonal (in this subspace), and so cancels out the divergent terms in the expansion.

This must be done even if only considering the first order shift!

Question 37

What different interactions combine to give the fine structure of the hydrogen atom?

Answer 37

The relativistic perturbation (extra term in the energy for expansion) O(p^4)

The spin-orbit interaction

The Darwin term (effectively gives the s.o. perturbation for the l=0 state).

Question 38

What is the form of the fine structure energy perturbation for hydrogen?
What does this tell us about the degeneracy of the fine structure?

Answer 38

-
Dependent only on j
-> states of different l, m_j are degenerate (mixed).

Question 39

What is the difference in the application of perturbation theory to the weak field and strong field zeeman effects?

Answer 39

For weak field (B«1T), we use the |j, m_j> basis for perturbation theory as the states of different j are not degenerate as the spin-orbit perturbation is dominant. Also, states of different l, m_j are not mixed by the perturbation, so we can use non-degenerate perturbation theory.

For strong field (B»1T), the magnetic perturbation is the dominant perturbation, and states of definite j become ~degenerate. Use the |l m_l, s m_s> basis as it diagonalizes the magnetic perturbation.

Question 40

What is the perturbation energy for the weak field zeeman effect?

Answer 40

mu_b * B * m_j * the Lande g factor

where mu_b is the bohr magneton

Question 41

What is the perturbation energy for the strong field zeeman effect?

Answer 41

mu_b * B * (m_l + 2m_s)

Question 42

What are the constraints on the quantum number l for the electron in a hydrogen atom?

Answer 42

l must be greater than or equal to 0, and less than or equal to n-1

Question 43

What is the form of the hamiltonian for 1D QHO in terms of the raising and lowering operators?

Answer 43

hbar omega / 2 * (raising lowering + lowering raising)

Question 44

What is the key commutation relation?

Answer 44

[AB, C] = A[B, C] + [A, C]B

[A, BC] = B[A, C] + [A, B]C

Question 45

What is the result of the stark effect on the ground state of hydrogen?

Answer 45

The ground state has no intrinsic dipole moment (symmetric state) so there is no first-order shift. There is a second-order shift due to induced dipole moment by the field {epsilon zhat}.

The second order shift is - 9/4 (e epsilon)^2 * a_0^3 / alpha hbar c

Question 46

What is the result of the stark effect on the n=2 states of hydrogen?

Answer 46

There is a shift of 3 e epsilon a_0 for the two states that have intrinsic dipole moment.
These states are a linear combination of the l=1 m_l=0 and the l=0 m_l=0 states.

Question 47

How can we make a measurement of spin in an arbitrary direction for a spin-1/2 system?

Answer 47

Use the operator n dot S where n is the unit vector in the direction of measurement, and S is the general vector spin operator.

To use spherical polars:
n = sin(theta)cos(phi) [x] + sin(theta)sin(phi) [y] + cos(theta) [z]

Question 48

What is the equation for the uncertainty in an observable?

Answer 48

uncertainty squared = expectation of square observable - expectation of observable, squared

Question 49

What is the number operator? (QHO)

Answer 49

raising operator lowering operator

Question 50

What is the commutation relation for angular momentum operators?

Answer 50

-