Quantum Mechanics Flashcards

Flashcards for Quantum Mechanics Exam Review

What is Stefan-Boltzmann Law?

j = σT⁴, where σ = 5.67 × 10⁻⁸ W m⁻²K⁻⁴

What is Wien's Displacement Law?

λmaxT = b, where b = 2.898 × 10⁻³ m K

What is Planck's Law?

E(λ, T) = (8πhc) / (λ⁵(e^(hc/(λkT)) - 1))

What is the de Broglie Wavelength?

λ = h/p = h/(mv)

What is the Heisenberg Uncertainty Principle?

Δx Δp ≥ ħ/2 = h/(4π)

What is the maximum kinetic energy of electrons in the photoelectric effect?

Kmax = hf - Φ = hf - hf₀

What is the threshold frequency (f₀) in the photoelectric effect?

f₀ = Φ/h

What is the wavelength shift in the Compton Effect?

Δλ = λ' - λ = (h/(mec))(1 - cos θ)

What are the energy levels for a particle in an infinite potential well (1D)?

En = (n²h²) / (8mL²), n = 1, 2, 3,…

What is the normalized wavefunction for a particle in an infinite potential well (1D)?

Ψn(x) = √(2/L) sin(nπx/L)

What type of solutions are found inside and outside the barrier of a finite potential well?

Oscillatory solutions inside, exponential decay outside (tunnelling)

What is the wavefunction of a free particle?

Ψ(x, t) = Aei(kx - ωt)

What is the energy continuum for a free particle?

E = ħ²k² / (2m)

What is the time-dependent Schrödinger equation (1D)?

iħ(∂Ψ/∂t) = (-ħ² / (2m))(∂²Ψ/∂x²) + V(x)Ψ

What is the time-independent Schrödinger equation (3D)?

(-ħ² / (2m))∇²Ψ + VΨ = EΨ

Give an example of the commutation relation between position and momentum?

[x, px] = iħ

What is the approximate quantum tunnelling probability?

T ≈ e^(-2κa), where κ= √(2m(V₀ - E))/ħ

What is the probability current (flux)?

J = (ħ / (2im)) (Ψ(∂Ψ/∂x) - Ψ(∂Ψ/∂x))

What is the tunnelling current in a Scanning Tunnelling Microscope (STM)?

I ∝ e^(-2ks)

What is the ground-state expectation value of position for a quantum harmonic oscillator?

⟨x⟩ = 0

What is the ground-state energy of a quantum harmonic oscillator?

E₀ = (1/2)ħω

What is the Schrodinger equation for the Hydrogen atom (3D Cartesian)?

(-ħ² / (2m))∇²Ψ - (e² / (4πε₀r))Ψ = EΨ

What are the energy levels of the Hydrogen atom?

En = -13.6 eV / n²

What is the de Broglie wavelength in the first orbit of the hydrogen atom?

λ = 2πa₀

What is the reduced mass?

μ = (m₁m₂) / (m₁ + m₂)

What is the vibrational frequency?

ν = (1 / (2π))√(k / μ)

What are the vibrational energy levels?

En = (n + 1/2)hν

What is the zero-point energy?

E₀ = (1/2)hν

What is the difference between a microstate and a macrostate?

Microstate describes a specific configuration, macrostate describes overall properties

What are the three main statistical ensembles and what parameters are fixed in each?

Microcanonical (E fixed), canonical (T fixed), grand-canonical (T, μ fixed)

What is the Maxwell-Boltzmann distribution?

nᵢ ∝ e^(-Eᵢ/kT)

What is the Bose-Einstein distribution?

f(E) = 1 / (e^((E-μ)/kT) - 1)

What is the Fermi-Dirac distribution?

f(E) = 1 / (e^((E-μ)/kT) + 1)

What is the probability of a distribution (MB)?

W = N! / (Πᵢ nᵢ!)

What type of particles behave as bosons?

Photons behave as bosons

The Bose-Einstein distribution leads directly to which formula?

Planck formula

What is the photon number density?

n(λ, T) = 1 / (λ⁴(e^(hc/(λkT)) - 1))

What is the equipartition law?

⟨E⟩ = (1/2)kT

What are some examples of degrees of freedom?

Translational (3), rotational (2 or 3), vibrational (2 per vibration)