Semester 2: Equations Flashcards
What is the normalisation condition for a wavefunction in 2D (Cartesian coordinates)?
Give the probability of finding a quantum particle in a 2D region (Born rule)
This is the probability of finding the particle in any eigenstate in that region.
What is the probability of finding a particle in a given eigenstate in 2D (Born rule)?
|Overlap integral|² = Probability to measure a given eigenvalue
Define the 3D position operator
Define the 3D momentum operator
What is the mathematical form of the 3D momentum operator?
Define the del operator (gradient operator)
∇ = del operator
e_ = orthogonal unit vectors
What are the commutation relations for position operators?
What are the commutation relations for momentum operators?
What are the commutation relations for position and momentum operators with the same spatial dependence?
What are the commutation relations for position and momentum operators with different spatial dependence?
Define the Hamiltonian operator in multiple dimension
µ = mass
Momentum operator² = -ħ²∇²
∇² = Laplace operator
What is the time-independent Schrödinger equation in multiple dimensions?
What is the time-dependent Schrödinger equation in multiple dimensions?
What are the solutions of the time-dependent Schrödinger equation in multiple dimensions?
What is the time-independent Schrödinger equation in Dirac notation?
What is the equation for the expectation value of an obervable?
What is the equation for the Bohr magneton?
What are the two spherical harmonic functions (for the quantum angular momentum vector operator)?
Y = spherical harmonic eigenvalues
l = orbital quantum number = 0, 1, 2, …
m = magnetic quantum number = -l, -l + 1, …, l - 1, l
What is the Hamiltonian for a hydrogen atom?
What are the eigenfunctions of the Hamiltonian for the hydrogen atom
R = radial eigenvalues
Y = spherical harmonic eigenvalues
These are the same as the eigenfunctions for angular momentum.
What is the equation for the Hamiltonian of a hydrogen atom in a magnetic field (in the z-direction)?
µ = magnetic moment
B = magnetic field
What are the eigenvalues of the spin vector?
s = 0, 1/2, 1, 3/2, 2, …
What are the values of the spin quantum number?
m = -s, -s + 1, …, s - 1, s
What is the equation that relates the spin of a charged particle to its magnetic moment?
µ = magnetic moment operator
S = spin operator
g = Landé factor ≈ 2
What is the equation that relates the spin of a charged particle to its magnetic moment in terms of the gyromagnetic constant?
γ = gyromagnetic constant
What are the two eigenvalue equations for a spin-1/2 particle in the z-direction (and z basis)?
What is the equation for the complete basis ket set of an arbitrary spin-1/2 quantum state vector?
|ψ⟩ = spin quantum state vector
a, b = expansion coefficients
Write |+⟩ and |-⟩ in vector form
Write an arbitrary spin-1/2 state in vector form
a, b = expansion coefficients
Write the corresponding bra-vector to an arbitrary spin-1/2 state
a, b = complex conjugates of the expansion coefficients
How can the expansion coefficients of a spin state be obtained in Dirac notation (spin-1/2 example)?
⟨ +|+⟩ = orthonormality condition
How can the expansion coefficients of a spin state be obtained in Dirac notation based on vector calculus?
Write the general spin state of a spin-1/2 particle using the orthonormality condition for basis kets
Describe how the projection operator can act on a spin state to find a given eigenstate of the basis
ψ = spin state
uⱼ = eigenstate
What is the equation for the probability of finding a state vector in a given spin-1/2 state?
What are the two eigenvalue equations for a spin-1/2 particle in the x-direction (and z basis)?
What are the two eigenvalue equations for a spin-1/2 particle in the y-direction (and z basis)?
What is the eigenvalue equation for the spin operator in the z basis?
Write |+⟩ and |-⟩ in vector form in the x-direction and z basis
Write |+⟩ and |-⟩ in vector form in the y-direction and z basis
What are the raising and lowering operator equations for a spin-1/2 system?
What are the raising and lowering operators for a spin-1/2 system in vector form?
Write the equation for what the identity operator represents in Dirac notation?
This is the sum of the outer product of basis kets with their adjoint bras.
What are the commutation relations for the spin component operators?
These operators do not commute.
What is the equation for the magnitude operator of a spin vector?
What is the magnitude of a spin-1/2 particle?
What is the equation for a spin operator in an arbitrary direction?
S = spin vector
n = unit vector in an arbitrary direction (specified by polar and azimuthal angles)
What is the matrix representation of a spin component operator in an arbitrary direction?
What are the two eigenvalue equations for a spin-1/2 particle in an arbitrary direction (and z basis)?
What are the three eigenvalue equations for a spin-1 particle in the z-direction (and z basis)?
Write |1⟩, |0⟩, and |-1⟩ in vector form in the z-direction and z basis
What is the Dirac notation representation of the state of an electron based on its 4 quantum numbers?
n = principal
l = azimuthal (orbital)
mₗ = magnetic
mₛ = spin
What is the equation for the expectation value of an observable?
A = operator
ψ = normalised wavefunction
What is the equation for the uncertainty in an observable?
∆A = uncertainty in A
A = operator
Write the uncertainty principle in terms of the uncertainty in an observable and commutator relations
What is the time-dependent Schrödinger equation in Dirac notation?
What is the equation for the Hamiltonian in Dirac notation?
What is the solution to the time-dependent Schrödinger equation for a given eigenstate?
What is the equation for the magnetic moment of a particle with spin?
µ = magnetic moment operator
S = spin operator
γ = gyromagnetic ratio
What is the Hamiltonian operator for a spin in a magnetic field?
H = Hamiltonian operator (total energy of magnetic moment in a magnetic field)
γ = gyromagnetic ratio
B = magnetic field
S = spin operator
What is the matrix representation of the Hamiltonian for the energy of a magnetic moment in a magnetic field?
H = Hamiltonian
γ = gyromagnetic constant
B = magnetic field
What is the equation for the Larmor frequency?
ω₀ = Larmor frequency
γ = gyromagnetic ratio
B = magnetic field
What are the energy eigenvalues for a spin-1/2 particle in a magnetic field in terms of the Larmor frequency (z basis)?
ω₀ = Larmor frequency
What is the equation for the time dependence of an expectation value for an operator?
A = operator
H = Hamiltonian
What is the general equation for the time evolution of a spin operator?
γ = gyromagnetic ratio
S = spin operator
B = magnetic field
What are the equations for the time-evolution of a spin operator in Cartesian coordinates?
What are the general solutions to the equations of motion of the spin vector?
What is the equation for the probability of a spin flip for a static magetic field (Rabi’s formula)?
ω₀ = Larmor frequency in z-direction
ω₁ = Larmor frequency in x-direction
What is the equation for the probability of a spin flip for a rotating magetic field (Rabi’s formula)?
∆ω = change in Larmor frequency in z-direction due to a rotating magnetic field
ω₁ = Larmor frequency in x-direction
