Quantum eqs

λ_max T = 2.898 × 10⁻³ m·K

Wien's Displacement Law: Relates the temperature of a blackbody to the wavelength at which its emission is maximum. λ_max is the peak wavelength; T is the absolute temperature in kelvin.

I = σT⁴

Stefan-Boltzmann Law: Gives the total power radiated per unit area by a blackbody. I is intensity (power per area); σ is the Stefan-Boltzmann constant; T is absolute temperature.

u(ν)dν = (8πhν³ / c³) · [dν / (e^{hν/kT} − 1)]

Planck's Law: Describes the spectral energy density of blackbody radiation. u(ν) is energy density per frequency; ν is frequency; h is Planck's constant; c is speed of light; k is Boltzmann's constant; T is temperature.

E = hν

Photon Energy: Energy carried by a single photon. E is energy; h is Planck's constant; ν is frequency.

eV = hν − W

Photoelectric Effect Equation: Relates the maximum kinetic energy of emitted electrons to photon energy. eV is electron kinetic energy; V is stopping potential; W is the work function of the metal.

|ψ⟩ = a|↑⟩ + b|↓⟩

General Spin Superposition: Any spin-½ state can be written as a linear combination of spin-up and spin-down states along the z-axis. a and b are complex probability amplitudes.

|↑ₓ⟩ = (1/√2)|↑⟩ + (1/√2)|↓⟩

Spin Eigenstate Along x-Axis (Up): State giving +ℏ/2 when Sₓ is measured. Written in the z-basis.

|↓ₓ⟩ = (1/√2)|↑⟩ − (1/√2)|↓⟩

Spin Eigenstate Along x-Axis (Down): State giving −ℏ/2 when Sₓ is measured.

|↑ᵧ⟩ = (1/√2)|↑⟩ + (i/√2)|↓⟩

Spin Eigenstate Along y-Axis (Up): Eigenstate of Sᵧ with eigenvalue +ℏ/2. Imaginary coefficient encodes phase.

|↓ᵧ⟩ = (1/√2)|↑⟩ − (i/√2)|↓⟩

Spin Eigenstate Along y-Axis (Down): Eigenstate of Sᵧ with eigenvalue −ℏ/2.

|↑ₙ⟩ = cos(θ/2)|↑⟩ + e^{iφ} sin(θ/2)|↓⟩

General Spin-Up State: Spin-up along an arbitrary direction n. θ is the polar angle from z; φ is the azimuthal angle in the x-y plane.

|↓ₙ⟩ = sin(θ/2)|↑⟩ − e^{iφ} cos(θ/2)|↓⟩

General Spin-Down State: Spin-down along an arbitrary direction n.

1/λ_mn = R∞(1/m² − 1/n²)

Rydberg Formula: Calculates the wavelength of light emitted when an electron transitions from level n to m in hydrogen. R∞ is the Rydberg constant.

L = nℏ

Bohr Quantization Condition: Angular momentum of an electron in a hydrogen atom is quantized. n is a positive integer; ℏ is reduced Planck's constant.

r_n = (4πϵ₀ / Ze²)(n²ℏ² / m)

Bohr Radius Formula: Radius of the nth orbit in a hydrogen-like atom. Z is atomic number; m is electron mass.

λ = h/p

De Broglie Wavelength: Associates a wavelength with a particle of momentum p, demonstrating wave-particle duality.

Ŝ_z|↑⟩ = +ℏ/2 |↑⟩

Spin Operator Eigenvalue Equation: Measuring S_z on a spin-up state yields +ℏ/2.

Ŝ_z|↓⟩ = −ℏ/2 |↓⟩

Spin Operator Eigenvalue Equation: Measuring S_z on a spin-down state yields −ℏ/2.

p̂ = −iℏ ∂/∂x

Momentum Operator: Quantum mechanical operator for momentum in the position representation.

Ĥ = iℏ ∂/∂t

Hamiltonian Operator: Governs time evolution of a quantum state and corresponds to total energy.

ΔA = √(⟨²⟩ − ⟨Â⟩²)

Definition of Uncertainty: Standard deviation of observable A, measuring spread of outcomes.

ΔAΔB ≥ ½ |⟨[Â, B̂]⟩|

Generalized Uncertainty Principle: Non-commuting operators cannot be simultaneously known with arbitrary precision.

E_n = n²π²ℏ² / (2mL²)

Particle in a Box Energy Levels: Allowed energies for a particle confined to an infinite square well of width L.

N = N₀ e^{−λt}

Radioactive Decay Law: Describes exponential decay of unstable nuclei. λ is the decay constant; t is time.

Locality

the principle of locality states that an object is influenced directly only by its immediate surroundings.

Realism

The stance that properties have definite values independent of measurement is realism.

completeness

any physical state can be fully described as a superposition (linear combination) of basis states (like energy or position eigenstates)

Copenhagen interpretation

states a quantum particle exists in a superposition of all its possible states until a measurement is made