Semester 1: Equations Flashcards
What is the polar form of a complex number?
z = x + iy
r = |z| = magnitude
φ = phase amplitude
What is Euler’s formula (for complex numbers)?
Write cos(x) in complex form
Write sin(x) in complex form
What is the 1D time-dependent Schrödinger equation?
ψ(x, t) = wavefunction
What is the time-dependent Schrödinger equation in Dirac notation?
|ψ 〉= state vector
What are the solutions to the 1D time-dependent Schrödinger equation?
u(x) = eigenfunction of the Hamiltonian
E = associated energy eigenvalue
What is the 1D time-independent Schrödinger equation?
u(x) = wavefunction (eigenfunction)
E = associated energy eigenvalue
What is the time-independent Schrödinger equation in Dirac notation?
|n 〉= state vector
What is the general solution of the Schrödinger equation?
uₙ(x) = eigenfunctions of the Hamiltonian
What is the normalisation condition for a wavefunction?
What is the normalisation condition for a state vector (wavefunction) in Dirac notation?
Define the Hamiltonian operator
First term = kinetic energy
Second term = potential energy
Define the kinetic energy operator
Define the position operator
Define the momentum operator
Define the momentum operator in natural units
What is the equation for wavenumber (aka spatial frequency)?
k = wavenumber
λ = wavelength
What is the equation for the momentum of a particle (in terms of the de Broglie wavelength)?
p = momentum
h = Planck’s constant
λ = wavelength
What is the equation for the Heisenberg uncertainty principle?
∆x = change in position
∆p = change in momentum
What is the equation for a plane wave?
ψ(x, t) = wavefunction
k = wavenumber
ω = angular frequency
Define the sifting property of the Dirac delta function
What is the dispersion relation for a free particle?
E = energy
ω = angular frequency
k = wavenumber
m = mass
What is the eigenvalue equation (in terms of operators)?
Ô = operator
uₙ = energy eigenstates (aka energy eigenfunction)
oₙ = observable
Define the Born rule for probabilities
|Overlap integral|² = probability of finding system in a given eigenstate
uₙ = energy eigenstates (aka energy eigenfunction)
ψ(x, t) = wavefunction
Define the Born rule for probabilities in Dirac notation
|⟨n|ψ⟩|² = probability to measure n-th eigenvalue
n = n-th eigenstate
ψ = state vector
What is the expectation value integral?
What is the expectation value in Dirac notation
What is the condition for Hermiticity of operators (in terms of expectation values)?
What is the definition of a commutator?
What is the equation for group velocity?
What is the expectation value for a discrete superposition of eigenstates if the wavefunction is orthonormal?
<O> = expectation value of operator
cₙ = coefficients (modulus squared of the coefficients equals the probability of being in a given eigenstate)
oₙ = observable

</O>
What is the expectation value for a discrete superposition of eigenstates if the wavefunction is orthonormal?
<O> = expectation value of operator
cₙ = coefficients (modulus squared of the coefficients equals the probability of being in a given eigenstate)
oₙ = observable

</O>
What is the expectation value for a discrete superposition of eigenstates if the wavefunction is orthonormal?
<O> = expectation value of operator
cₙ = coefficients (modulus squared of the coefficients equals the probability of being in a given eigenstate)
oₙ = observable

</O>
What is the equation for the energy of a 1D infinite potential well?
a = width