Semester 2: Equations

Question 1

What is the normalisation condition for a wavefunction in 2D (Cartesian coordinates)?

Answer 1

Question 2

Give the probability of finding a quantum particle in a 2D region (Born rule)

Answer 2

This is the probability of finding the particle in any eigenstate in that region.

Question 3

What is the probability of finding a particle in a given eigenstate in 2D (Born rule)?

Answer 3

|Overlap integral|² = Probability to measure a given eigenvalue

Question 4

Define the 3D position operator

Answer 4

Question 5

Define the 3D momentum operator

Answer 5

Question 6

What is the mathematical form of the 3D momentum operator?

Answer 6

Question 7

Define the del operator (gradient operator)

Answer 7

∇ = del operator
e_ = orthogonal unit vectors

Question 8

What are the commutation relations for position operators?

Answer 8

Question 9

What are the commutation relations for momentum operators?

Answer 9

Question 10

What are the commutation relations for position and momentum operators with the same spatial dependence?

Answer 10

Question 11

What are the commutation relations for position and momentum operators with different spatial dependence?

Answer 11

Question 12

Define the Hamiltonian operator in multiple dimension

Answer 12

µ = mass
Momentum operator² = -ħ²∇²
∇² = Laplace operator

Question 13

What is the time-independent Schrödinger equation in multiple dimensions?

Answer 13

Question 14

What is the time-dependent Schrödinger equation in multiple dimensions?

Answer 14

Question 15

What are the solutions of the time-dependent Schrödinger equation in multiple dimensions?

Answer 15

Question 16

What is the time-independent Schrödinger equation in Dirac notation?

Answer 16

Question 17

What is the equation for the expectation value of an obervable?

Answer 17

Question 18

What is the equation for the Bohr magneton?

Answer 18

Question 19

What are the two spherical harmonic functions (for the quantum angular momentum vector operator)?

Answer 19

Y = spherical harmonic eigenvalues
l = orbital quantum number = 0, 1, 2, …
m = magnetic quantum number = -l, -l + 1, …, l - 1, l

Question 20

What is the Hamiltonian for a hydrogen atom?

Answer 20

Question 21

What are the eigenfunctions of the Hamiltonian for the hydrogen atom

Answer 21

R = radial eigenvalues
Y = spherical harmonic eigenvalues
These are the same as the eigenfunctions for angular momentum.

Question 22

What is the equation for the Hamiltonian of a hydrogen atom in a magnetic field (in the z-direction)?

Answer 22

µ = magnetic moment
B = magnetic field

Question 23

What are the eigenvalues of the spin vector?

Answer 23

s = 0, 1/2, 1, 3/2, 2, …

Question 24

What are the values of the spin quantum number?

Answer 24

m = -s, -s + 1, …, s - 1, s

Question 25

What is the equation that relates the spin of a charged particle to its magnetic moment?

Answer 25

µ = magnetic moment operator S = spin operator g = Landé factor ≈ 2

Question 26

What is the equation that relates the spin of a charged particle to its magnetic moment in terms of the gyromagnetic constant?

Answer 26

γ = gyromagnetic constant

Question 27

What are the two eigenvalue equations for a spin-1/2 particle in the z-direction (and z basis)?

Answer 27

Question 28

What is the equation for the complete basis ket set of an arbitrary spin-1/2 quantum state vector?

Answer 28

|ψ⟩ = spin quantum state vector a, b = expansion coefficients

Question 29

Write |+⟩ and |-⟩ in vector form

Answer 29

Question 30

Write an arbitrary spin-1/2 state in vector form

Answer 30

a, b = expansion coefficients

Question 31

Write the corresponding bra-vector to an arbitrary spin-1/2 state

Answer 31

a*, b* = complex conjugates of the expansion coefficients

Question 32

How can the expansion coefficients of a spin state be obtained in Dirac notation (spin-1/2 example)?

Answer 32

⟨ +|+⟩ = orthonormality condition

Question 33

How can the expansion coefficients of a spin state be obtained in Dirac notation based on vector calculus?

Answer 33

Question 34

Write the general spin state of a spin-1/2 particle using the orthonormality condition for basis kets

Answer 34

Question 35

Describe how the projection operator can act on a spin state to find a given eigenstate of the basis

Answer 35

ψ = spin state uⱼ = eigenstate

Question 36

What is the equation for the probability of finding a state vector in a given spin-1/2 state?

Answer 36

Question 37

What are the two eigenvalue equations for a spin-1/2 particle in the x-direction (and z basis)?

Answer 37

Question 38

What are the two eigenvalue equations for a spin-1/2 particle in the y-direction (and z basis)?

Answer 38

Question 39

What is the eigenvalue equation for the spin operator in the z basis?

Answer 39

Question 40

Write |+⟩ and |-⟩ in vector form in the x-direction and z basis

Answer 40

Question 41

Write |+⟩ and |-⟩ in vector form in the y-direction and z basis

Answer 41

Question 42

What are the raising and lowering operator equations for a spin-1/2 system?

Answer 42

Question 43

What are the raising and lowering operators for a spin-1/2 system in vector form?

Answer 43

Question 44

Write the equation for what the identity operator represents in Dirac notation?

Answer 44

This is the sum of the outer product of basis kets with their adjoint bras.

Question 45

What are the commutation relations for the spin component operators?

Answer 45

These operators do not commute.

Question 46

What is the equation for the magnitude operator of a spin vector?

Answer 46

Question 47

What is the magnitude of a spin-1/2 particle?

Answer 47

Question 48

What is the equation for a spin operator in an arbitrary direction?

Answer 48

S = spin vector n = unit vector in an arbitrary direction (specified by polar and azimuthal angles)

Question 49

What is the matrix representation of a spin component operator in an arbitrary direction?

Answer 49

Question 50

What are the two eigenvalue equations for a spin-1/2 particle in an arbitrary direction (and z basis)?

Answer 50

Question 51

What are the three eigenvalue equations for a spin-1 particle in the z-direction (and z basis)?

Answer 51

Question 52

Write |1⟩, |0⟩, and |-1⟩ in vector form in the z-direction and z basis

Answer 52

Question 53

What is the Dirac notation representation of the state of an electron based on its 4 quantum numbers?

Answer 53

n = principal l = azimuthal (orbital) mₗ = magnetic mₛ = spin

Question 54

What is the equation for the expectation value of an observable?

Answer 54

A = operator ψ = normalised wavefunction

Question 55

What is the equation for the uncertainty in an observable?

Answer 55

∆A = uncertainty in A A = operator

Question 56

Write the uncertainty principle in terms of the uncertainty in an observable and commutator relations

Answer 56

Question 57

What is the time-dependent Schrödinger equation in Dirac notation?

Answer 57

Question 58

What is the equation for the Hamiltonian in Dirac notation?

Answer 58

Question 59

What is the solution to the time-dependent Schrödinger equation for a given eigenstate?

Answer 59

Question 60

What is the equation for the magnetic moment of a particle with spin?

Answer 60

µ = magnetic moment operator S = spin operator γ = gyromagnetic ratio

Question 61

What is the Hamiltonian operator for a spin in a magnetic field?

Answer 61

H = Hamiltonian operator (total energy of magnetic moment in a magnetic field) γ = gyromagnetic ratio B = magnetic field S = spin operator

Question 62

What is the matrix representation of the Hamiltonian for the energy of a magnetic moment in a magnetic field?

Answer 62

H = Hamiltonian γ = gyromagnetic constant B = magnetic field

Question 63

What is the equation for the Larmor frequency?

Answer 63

ω₀ = Larmor frequency γ = gyromagnetic ratio B = magnetic field

Question 64

What are the energy eigenvalues for a spin-1/2 particle in a magnetic field in terms of the Larmor frequency (z basis)?

Answer 64

ω₀ = Larmor frequency

Question 65

What is the equation for the time dependence of an expectation value for an operator?

Answer 65

A = operator H = Hamiltonian

Question 66

What is the general equation for the time evolution of a spin operator?

Answer 66

γ = gyromagnetic ratio S = spin operator B = magnetic field

Question 67

What are the equations for the time-evolution of a spin operator in Cartesian coordinates?

Answer 67

Question 68

What are the general solutions to the equations of motion of the spin vector?

Answer 68

Question 69

What is the equation for the probability of a spin flip for a static magetic field (Rabi’s formula)?

Answer 69

ω₀ = Larmor frequency in z-direction ω₁ = Larmor frequency in x-direction

Question 70

What is the equation for the probability of a spin flip for a rotating magetic field (Rabi’s formula)?

Answer 70

∆ω = change in Larmor frequency in z-direction due to a rotating magnetic field ω₁ = Larmor frequency in x-direction