Semester 1: Equations

Question 1

What is the polar form of a complex number?

Answer 1

z = x + iy
r = |z| = magnitude
φ = phase amplitude

Question 2

What is Euler’s formula (for complex numbers)?

Answer 2

Question 3

Write cos(x) in complex form

Answer 3

Question 4

Write sin(x) in complex form

Answer 4

Question 5

What is the 1D time-dependent Schrödinger equation?

Answer 5

ψ(x, t) = wavefunction

Question 6

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

Answer 6

|ψ 〉= state vector

Question 7

What are the solutions to the 1D time-dependent Schrödinger equation?

Answer 7

u(x) = eigenfunction of the Hamiltonian
E = associated energy eigenvalue

Question 8

What is the 1D time-independent Schrödinger equation?

Answer 8

u(x) = wavefunction (eigenfunction)
E = associated energy eigenvalue

Question 9

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

Answer 9

|n 〉= state vector

Question 10

What is the general solution of the Schrödinger equation?

Answer 10

uₙ(x) = eigenfunctions of the Hamiltonian

Question 11

What is the normalisation condition for a wavefunction?

Answer 11

Question 12

What is the normalisation condition for a state vector (wavefunction) in Dirac notation?

Answer 12

Question 13

Define the Hamiltonian operator

Answer 13

First term = kinetic energy
Second term = potential energy

Question 14

Define the kinetic energy operator

Answer 14

Question 15

Define the position operator

Answer 15

Question 16

Define the momentum operator

Answer 16

Question 17

Define the momentum operator in natural units

Answer 17

Question 18

What is the equation for wavenumber (aka spatial frequency)?

Answer 18

k = wavenumber
λ = wavelength

Question 19

What is the equation for the momentum of a particle (in terms of the de Broglie wavelength)?

Answer 19

p = momentum
h = Planck’s constant
λ = wavelength

Question 20

What is the equation for the Heisenberg uncertainty principle?

Answer 20

∆x = change in position
∆p = change in momentum

Question 21

What is the equation for a plane wave?

Answer 21

ψ(x, t) = wavefunction
k = wavenumber
ω = angular frequency

Question 22

Define the sifting property of the Dirac delta function

Answer 22

Question 23

What is the dispersion relation for a free particle?

Answer 23

E = energy
ω = angular frequency
k = wavenumber
m = mass

Question 24

What is the eigenvalue equation (in terms of operators)?

Answer 24

Ô = operator
uₙ = energy eigenstates (aka energy eigenfunction)
oₙ = observable

Question 25

Define the Born rule for probabilities

Answer 25

|Overlap integral|² = probability of finding system in a given eigenstate uₙ = energy eigenstates (aka energy eigenfunction) ψ(x, t) = wavefunction

Question 26

Define the Born rule for probabilities in Dirac notation

Answer 26

|⟨n|ψ⟩|² = probability to measure n-th eigenvalue n = n-th eigenstate ψ = state vector

Question 27

What is the expectation value integral?

Answer 27

Question 28

What is the expectation value in Dirac notation

Answer 28

Question 29

What is the condition for Hermiticity of operators (in terms of expectation values)?

Answer 29

Question 30

What is the definition of a commutator?

Answer 30

Question 31

What is the equation for group velocity?

Answer 31

Question 32

What is the expectation value for a discrete superposition of eigenstates if the wavefunction is orthonormal?

Answer 32

= expectation value of operator cₙ = coefficients (modulus squared of the coefficients equals the probability of being in a given eigenstate) oₙ = observable

Question 33

What is the expectation value for a discrete superposition of eigenstates if the wavefunction is orthonormal?

Answer 33

= expectation value of operator cₙ = coefficients (modulus squared of the coefficients equals the probability of being in a given eigenstate) oₙ = observable

Question 34

What is the expectation value for a discrete superposition of eigenstates if the wavefunction is orthonormal?

Answer 34

= expectation value of operator cₙ = coefficients (modulus squared of the coefficients equals the probability of being in a given eigenstate) oₙ = observable

Question 35

What is the equation for the energy of a 1D infinite potential well?

Answer 35

a = width