physica.
47 formulas

Quantum

Wave-particle duality. Every formula below opens into a live, hands-on simulation.

wave particle duality
λ=hp\lambda = \frac{h}{p}

de Broglie Wavelength

Momentum and wavelength are inversely related through Planck's constant — big things have unmeasurably tiny wavelengths.

quantization
E=hfE = hf

Photon Energy (Planck Relation)

Light energy is quantized: each photon carries a fixed packet of energy set by its frequency.

quantization
Kmax=hf−ϕK_{max} = hf - \phi

Photoelectric Effect

A photon gives all its energy to one electron; the work function is the minimum escape cost.

foundations
Δx Δp≥ℏ2\Delta x \, \Delta p \geq \frac{\hbar}{2}

Heisenberg Uncertainty Principle

Position and momentum are conjugate — pinning one down spreads the other.

atomic physics
En=−13.6 eVn2E_n = -\frac{13.6 \text{ eV}}{n^2}

Bohr Hydrogen Energy Levels

Bound electrons can only sit on a quantized energy ladder; jumping down emits a photon.

foundations
iℏ∂Ψ∂t=H^Ψi\hbar \frac{\partial \Psi}{\partial t} = \hat{H}\Psi

Time-Dependent Schrödinger Equation

The Hamiltonian generates time evolution of the wavefunction in complex Hilbert space.

bound states
En=n2π2ℏ22mL2E_n = \frac{n^2 \pi^2 \hbar^2}{2mL^2}

Particle in a 1D Box

Standing waves must fit inside the box; only integer half-wavelengths are allowed.

relativistic qm
(iγμ∂μ−mc/ℏ)ψ=0(i\gamma^\mu \partial_\mu - mc/\hbar)\psi = 0

Dirac Equation

A first-order relativistic wave equation whose solutions naturally carry spin and antiparticles.

wave mechanics
H^ψ=Eψ\hat{H}\psi = E\psi

Time-Independent Schrödinger Equation

A wavefunction is allowed only if applying the Hamiltonian gives back the same wavefunction scaled by a number — that number is the energy.

wave mechanics
En=ℏω(n+12)E_n = \hbar\omega\left(n + \tfrac{1}{2}\right)

Quantum Harmonic Oscillator

Energy comes in equal steps of hbar*omega, with a built-in floor of hbar*omega/2 — the zero-point motion required by Heisenberg.

atomic spectra
1λ=RH(1n12−1n22)\frac{1}{\lambda} = R_H\left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right)

Rydberg Formula

Light emerges when an electron drops between two energy rungs — the wavelength is set by the difference of two inverse squares.

many body
Ψ(r1,r2)=−Ψ(r2,r1)\Psi(\mathbf{r}_1,\mathbf{r}_2) = -\Psi(\mathbf{r}_2,\mathbf{r}_1)

Pauli Exclusion Principle

Swap two identical fermions and the wavefunction flips sign — so the wavefunction vanishes if they share the same state.

spin
S^i=ℏ2σi\hat{S}_i = \tfrac{\hbar}{2}\sigma_i

Spin-1/2 Operators

Three 2×2 matrices encode every spin-1/2 measurement — they don't commute, which is why spin in x and y can't be known simultaneously.

wave mechanics
T≈exp⁡ ⁣(−2ℏ∫ab2m(V(x)−E) dx)T \approx \exp\!\left(-\frac{2}{\hbar}\int_a^b\sqrt{2m\bigl(V(x)-E\bigr)}\,dx\right)

Quantum Tunneling Probability

The wavefunction decays exponentially inside a barrier — make the barrier thinner or shorter and a measurable tail emerges on the far side.

spin
Fz=μz ∂Bz∂zF_z = \mu_z\,\frac{\partial B_z}{\partial z}

Stern–Gerlach Deflection

A magnetic dipole in a field gradient feels a force along the gradient; quantum spin offers only two values of mu_z, so the beam splits in two.

open systems
ρ^=∑ipi ∣ψi⟩⟨ψi∣\hat{\rho} = \sum_i p_i\,|\psi_i\rangle\langle\psi_i|

Density Matrix

Replace one wavefunction with a weighted bookkeeper of many — diagonal entries are probabilities, off-diagonal entries are coherences that decoherence destroys.

foundations
P(x) dx=∣ψ(x)∣2 dxP(x)\,dx = |\psi(x)|^2\,dx

Born Rule

The squared magnitude of the wavefunction is the probability map for measurement outcomes.

operator algebra
[x^,p^]=iℏ[\hat{x}, \hat{p}] = i\hbar

Canonical Commutation Relation

Order matters: measuring x then p differs from p then x by exactly iℏ — the seed of uncertainty.

angular momentum
∣L∣=l(l+1) ℏ,Lz=mℏ|\mathbf{L}| = \sqrt{l(l+1)}\,\hbar, \qquad L_z = m\hbar

Quantization of Angular Momentum

Angular momentum comes in rungs of ℏ: its length is √(l(l+1))ℏ and its z-shadow is mℏ.

spin dynamics
ωL=γB\omega_L = \gamma B

Larmor Precession

A spin in a magnetic field precesses like a gyroscope, at a rate exactly proportional to the field.

dynamics
d⟨x^⟩dt=⟨p^⟩m,d⟨p^⟩dt=−⟨∂V∂x⟩\frac{d\langle \hat{x}\rangle}{dt} = \frac{\langle \hat{p}\rangle}{m}, \qquad \frac{d\langle \hat{p}\rangle}{dt} = -\left\langle \frac{\partial V}{\partial x}\right\rangle

Ehrenfest Theorem

Quantum averages obey Newton-like equations — the wavepacket's center moves classically.

two level systems
Pe(t)=Ω2Ω2+Δ2 sin⁡2 ⁣(12Ω2+Δ2  t)P_e(t) = \frac{\Omega^2}{\Omega^2 + \Delta^2}\,\sin^2\!\left(\tfrac{1}{2}\sqrt{\Omega^2+\Delta^2}\;t\right)

Rabi Oscillations

A driven two-level system cycles between ground and excited state; detuning caps the swing.

entanglement
S=E(a,b)−E(a,b′)+E(a′,b)+E(a′,b′),∣S∣≤2S = E(a,b) - E(a,b') + E(a',b) + E(a',b'), \qquad |S| \le 2

Bell–CHSH Inequality

Local hidden variables cap a four-correlation sum at 2; entangled particles push it to 2√2.

particle physics
Lint=−e ψˉγμψAμ\mathcal{L}_{\text{int}} = -e\,\bar{\psi}\gamma^{\mu}\psi A_{\mu}

Standard Model Interactions

Every particle reaction in nature is built from a handful of vertices — an electron emitting a photon, a quark emitting a W boson, a gluon splitting. Glue these elementary moves together and you get beta decay, annihilation, Compton scattering, pair production. The bookkeeping rules are absolute: electric charge, lepton number, and baryon number in must equal out. If a reaction conserves them all, somewhere in the universe it happens.

foundations
iℏ ∂∂t Ψ(r,t)=H^ Ψ(r,t)i\hbar\,\frac{\partial}{\partial t}\,\Psi(\mathbf{r},t) = \hat{H}\,\Psi(\mathbf{r},t)

The Schrödinger Equation

Newton's law tells a particle where to go next given a force; the Schrödinger equation tells a *wavefunction* how to evolve given an energy operator. The whole state of a quantum system lives in Ψ — a complex amplitude over every possible configuration — and the Hamiltonian Ĥ acts like a clock, rotating that amplitude forward in time. Everything you can ever measure is hidden inside |Ψ|². It is the F = ma of the quantum world: a single deterministic rule for an object that is anything but.

quantum optics
g(ν)=σ(ν) (N2−N1)g(\nu) = \sigma(\nu)\,(N_2 - N_1)

How a Laser Achieves Coherence

Normally a photon hitting an atom is more likely to be absorbed than to trigger emission, because the ground state is more populated — light dies out. Einstein showed that an excited atom struck by a photon can be stimulated to emit a *clone* photon: same frequency, phase and direction. To make light grow instead of fade you need more atoms up than down — a 'population inversion' — which can't happen at thermal equilibrium, so you pump energy in. Put that gain medium between two mirrors and the cloned photons bounce, stimulate more clones, and a coherent beam explodes out of the noise.

condensed matter
ψnk(r)=eik⋅r unk(r),unk(r+R)=unk(r)\psi_{n\mathbf{k}}(\mathbf{r}) = e^{i\mathbf{k}\cdot\mathbf{r}}\,u_{n\mathbf{k}}(\mathbf{r}),\qquad u_{n\mathbf{k}}(\mathbf{r}+\mathbf{R}) = u_{n\mathbf{k}}(\mathbf{r})

Why Metals Conduct (Band Theory)

Bring two atoms together and the Pauli principle splits each shared level into two. Bring 10²³ atoms together and each level splits into 10²³ sub-levels so closely spaced they form a continuous 'band'. Electrons fill these bands from the bottom up. If the topmost occupied band is only partly full, electrons sit right next to empty states and an electric field nudges them freely — a metal. If a band is completely full and the next is far above across a wide gap, no nearby empty states exist, electrons can't move, and you have an insulator. A small gap gives a semiconductor.

foundations
Δφ=qℏ∮A⋅dℓ=q ΦBℏ\Delta\varphi = \frac{q}{\hbar}\oint \mathbf{A}\cdot d\boldsymbol{\ell} = \frac{q\,\Phi_B}{\hbar}

Aharonov-Bohm Effect

Classically a charged particle feels only forces, and where the magnetic field B is zero there is no force — so nothing should happen. The Aharonov–Bohm effect says otherwise. Send electrons around both sides of a thin, perfectly shielded solenoid. Outside the solenoid B = 0 everywhere, yet the two paths enclose magnetic flux, and the vector potential A is not zero there. Each path picks up a quantum phase from A, and when the beams recombine their interference fringes *shift* — controlled entirely by flux the electrons never touched. It proved the potentials A and φ, long thought to be mere mathematical bookkeeping, are physically real in quantum mechanics.

foundations
∣Ψ−⟩=12(∣↑⟩A∣↓⟩B−∣↓⟩A∣↑⟩B)|\Psi^-\rangle = \frac{1}{\sqrt{2}}\big(|\uparrow\rangle_A|\downarrow\rangle_B - |\downarrow\rangle_A|\uparrow\rangle_B\big)

The EPR Paradox

Prepare two particles in a single entangled state — say total spin zero — and send them far apart. The pair has no definite individual spins; only the *correlation* is fixed. Measure particle A along any axis and you instantly know B will give the opposite, no matter the distance. EPR said: either the answer was secretly predetermined (a 'hidden variable' quantum theory left out — so it's incomplete), or measuring A really does reach across space to set B (which violates locality). Einstein bet on hidden variables. Bell later showed the two options make *different* numerical predictions — and experiments side with quantum mechanics, not Einstein.

foundations
K(b,a)=∫D[x(t)]  e iℏS[x(t)]K(b,a) = \int \mathcal{D}[x(t)]\; e^{\,\frac{i}{\hbar} S[x(t)]}

Path-Integral Formulation

Ask how a particle gets from A to B and quantum mechanics answers: it takes *every* path at once. Each conceivable trajectory contributes an arrow (a complex phase) of equal length but angle set by that path's classical action S divided by ℏ. Add up all the arrows. Wild, jagged paths have wildly varying phases that cancel by destructive interference; only paths near the one that makes S stationary add up in step. That surviving bundle *is* the classical trajectory — so Newton's principle of least action emerges as the place where quantum phases stop cancelling. It reformulates all of quantum mechanics as a single 'sum over histories.'

Foundations
[x^,p^]=x^p^−p^x^=iℏ[\hat{x},\hat{p}]=\hat{x}\hat{p}-\hat{p}\hat{x}=i\hbar

Canonical Commutation [x, p]

In classical physics, measuring position then momentum gives the same answer as momentum then position — order doesn't matter. In quantum mechanics it does: swap the order of x̂ and p̂ and you don't get zero, you get iℏ. That tiny leftover is the mathematical seed of the entire uncertainty principle. Because position and momentum operators refuse to commute, no state can have both sharply defined. The whole strangeness of quantum mechanics is encoded in this one non-zero bracket.

Foundations
⟨A^⟩=∫ψ∗ A^ ψ dx=⟨ψ∣A^∣ψ⟩\langle \hat{A}\rangle=\int \psi^*\,\hat{A}\,\psi\,dx=\langle\psi|\hat{A}|\psi\rangle

Expectation Value of an Observable

A quantum measurement gives random results, but repeat it on many identically prepared systems and the results have a definite average. The expectation value is that average — sandwich the operator between the wavefunction and its conjugate and integrate. For position it's literally the center of mass of |ψ|². It is not the 'expected' single outcome (you may never measure it) but the mean of a huge ensemble of measurements.

Foundations
∫−∞∞∣ψ(x)∣2 dx=1\int_{-\infty}^{\infty}|\psi(x)|^2\,dx=1

Wavefunction Normalization

The particle is somewhere, so the total probability of finding it anywhere must be exactly 1. Since |ψ(x)|² is the probability density, adding it up over all space has to equal one. This condition fixes the otherwise-free overall scale of the wavefunction. If your ψ integrates to some number N, just divide by √N and it's normalized. Any physically meaningful state must be square-integrable — plane waves that don't die off need special (delta-function) treatment.

Foundations
A^=A^†  ⇒  a∈R,  ⟨ϕm∣ϕn⟩=δmn\hat{A}=\hat{A}^{\dagger}\;\Rightarrow\; a\in\mathbb{R},\;\langle\phi_m|\phi_n\rangle=\delta_{mn}

Hermitian Operators & Real Eigenvalues

Measured quantities are always real numbers — you never read '3+2i volts' off a meter. Quantum mechanics guarantees this by demanding every observable be a Hermitian (self-adjoint) operator. Hermiticity forces the eigenvalues (the possible measurement outcomes) to be real and the eigenstates to be orthogonal, so different outcomes are perfectly distinguishable. It's the algebraic condition that makes operators safe to call 'observables.'

Foundations
⟨ϕ∣ψ⟩=∫ϕ∗(x)ψ(x) dx\langle\phi|\psi\rangle=\int\phi^*(x)\psi(x)\,dx

Dirac Bra-Ket Notation

Dirac split the bracket ⟨φ|ψ⟩ into a 'bra' ⟨φ| and a 'ket' |ψ⟩. A ket is a state vector; a bra is its dual, waiting to be paired with a ket to produce a number — their overlap, which measures how similar two states are. This notation hides the messy integrals and matrices: it works identically whether the state lives in an infinite-dimensional function space or a 2-level qubit. It's the universal shorthand of quantum mechanics.

Exactly Solvable Systems
a^=mω2ℏ(x^+imωp^),    En=ℏω(n+12)\hat{a}=\sqrt{\tfrac{m\omega}{2\hbar}}\Big(\hat{x}+\tfrac{i}{m\omega}\hat{p}\Big),\;\;E_n=\hbar\omega\left(n+\tfrac12\right)

Ladder (Creation/Annihilation) Operators

Instead of solving the harmonic oscillator's differential equation, Dirac found a shortcut: two operators that step you up and down the energy ladder. â lowers the energy by one quantum ℏω (destroys a quantum), ↠raises it (creates one). Apply â to the lowest state and you get zero — that floor is why there's a ½ℏω zero-point energy. The whole spectrum falls out from algebra alone, and the same trick builds all of quantum field theory's particles.

Angular Momentum & Spin
L^2∣l,m⟩=ℏ2l(l+1)∣l,m⟩,    L^z∣l,m⟩=ℏm∣l,m⟩\hat{L}^2|l,m\rangle=\hbar^2 l(l+1)|l,m\rangle,\;\;\hat{L}_z|l,m\rangle=\hbar m|l,m\rangle

Angular Momentum Eigenvalues

Angular momentum in quantum mechanics can't point just anywhere or take any size — it's quantized. The total magnitude squared comes in lumps ℏ²l(l+1), and its projection on any axis is a whole (or half) multiple of ℏ. Curiously, the magnitude √(l(l+1)) always exceeds the maximum projection l, so the vector can never fully align with an axis — it must always tilt, a direct fingerprint of the uncertainty principle for the angular-momentum components.

Uncertainty & Wave-Particle
ΔE Δt≳ℏ2\Delta E\,\Delta t\gtrsim\frac{\hbar}{2}

Energy-Time Uncertainty

A state that lasts only a short time cannot have a sharply defined energy. The shorter its lifetime Δt, the fuzzier its energy ΔE. This is why short-lived particles and excited atomic states have a 'width' — their energy is spread out. It also lets the vacuum briefly borrow energy to create virtual particles, as long as they vanish fast enough. Unlike position-momentum, time isn't an operator here; Δt is a characteristic timescale of change.

Foundations
lim⁡n→∞(quantum)=(classical)\lim_{n\to\infty}\text{(quantum)}=\text{(classical)}

Correspondence Principle

Quantum mechanics can't just contradict the classical physics we see every day — it must reproduce it in the right limit. Bohr's principle says: for large quantum numbers (highly excited states) or when actions are huge compared to ℏ, quantum predictions blur into classical ones. A hydrogen electron in orbit n=1000 radiates almost exactly like a classical charge going around a loop. It's the guardrail that kept early quantum theory tethered to known physics.

Exactly Solvable Systems
ktan⁡(ka)=κ  (even),k=2mE/ℏ,  κ=2m(V0−E)/ℏk\tan(ka)=\kappa\;(\text{even}),\quad k=\sqrt{2mE}/\hbar,\;\kappa=\sqrt{2m(V_0-E)}/\hbar

Finite Square Well

Trap a particle in a box with walls of finite height V₀ and two things change versus the infinite box: there are only finitely many bound states, and the wavefunction leaks past the walls, decaying exponentially into the 'forbidden' region. Matching the wiggly inside solution smoothly to the leaking outside solution only works at special energies — the transcendental condition k tan(ka)=κ. That leakage is the seed of quantum tunneling.

foundations
A^ ∣an⟩=an ∣an⟩\hat{A}\,|a_n\rangle = a_n\,|a_n\rangle

Operators and Observables

Every measurable quantity is a Hermitian operator; its eigenvalues are the only possible readings.

foundations
∣ψ⟩=c1∣ψ1⟩+c2∣ψ2⟩,∣c1∣2+∣c2∣2=1|\psi\rangle = c_1|\psi_1\rangle + c_2|\psi_2\rangle,\quad |c_1|^2 + |c_2|^2 = 1

Quantum Superposition

Add valid states with complex weights and you get another valid state that can interfere with itself.

foundations
j=ℏm Im ⁣(ψ∗∂ψ∂x)j = \frac{\hbar}{m}\,\mathrm{Im}\!\left(\psi^{*}\frac{\partial \psi}{\partial x}\right)

Probability Current Density

The imaginary part of psi-star times its gradient is the flow of probability, obeying a continuity law.

angular momentum
[L^x,L^y]=iℏ L^z[\hat{L}_x, \hat{L}_y] = i\hbar\,\hat{L}_z

Angular Momentum Operators

The three angular-momentum components don't commute, so only its magnitude and one axis are sharp together.

angular momentum
S^i=ℏ2 σi\hat{S}_i = \frac{\hbar}{2}\,\sigma_i

Pauli Spin Matrices

Three 2x2 matrices generate every spin-1/2 observable and every single-qubit rotation.

angular momentum
Ylm(θ,ϕ)=Nlm Plm(cos⁡θ) eimϕY_l^m(\theta,\phi) = N_l^m\,P_l^m(\cos\theta)\,e^{im\phi}

Spherical Harmonics

The eigenfunctions of angular momentum on a sphere set the lobed shapes of atomic orbitals.

hydrogen atoms
n=1,2,3,… ;  l=0,…,n−1;  ml=−l,…,+l;  ms=±12n=1,2,3,\dots;\ \ l=0,\dots,n-1;\ \ m_l=-l,\dots,+l;\ \ m_s=\pm\tfrac12

Atomic Quantum Numbers

Four integers (n, l, m_l, m_s) label every electron and, with Pauli's rule, build the periodic table.