Quantum mechanics shattered the deterministic paradigm of classical physics, revealing the intrinsic wave-particle duality, probabilistic state evolution, and non-commutative algebraic structures of the microscopic universe.
This cheat sheet organizes 16 foundational physical laws and governing equations in quantum mechanics, categorized into four modules: Wave-Particle Duality, Schrödinger Equation & State Evolution, Uncertainty & Spin, and Atomic & Quantum States, complete with operator formulations, physical interpretations, and modern quantum technology applications.
🔗 Physics Equations Suite: Classical Mechanics · Electromagnetism · Fluid & Thermal · Optics & Acoustics · Nuclear & Particle · Relativity & Cosmology · Astro & Geophysics
1. Wave-Particle Duality (3 Items)
01 · de Broglie Relation
Matter Waves
$$\lambda = \frac{h}{p} = \frac{h}{mv}$$
- ①Physical Essence: All matter exhibits both wave-like and particle-like characteristics. The de Broglie wavelength is inversely proportional to momentum.
- ②Mathematical Notation: $\lambda$ is de Broglie wavelength (m), $h$ is Planck constant ($6.626 \times 10^{-34}\text{ J}\cdot\text{s}$), $p$ is momentum ($\text{kg}\cdot\text{m/s}$).
- ③Theoretical Significance: Bridged corpuscular and wave mechanics, laying the conceptual foundation for wave mechanics and electron microscopy.
- ④Boundary Conditions & Applications: Valid across all subatomic and macroscopic particles. Applied in Transmission Electron Microscopy (TEM), electron diffraction, and neutron scattering.
02 · Planck-Einstein Relation
Energy-Frequency Relation
$$E = h\nu = \hbar\omega$$
- ①Physical Essence: Energy of a quantum of electromagnetic radiation is strictly proportional to its oscillation frequency.
- ②Mathematical Notation: $E$ is photon energy (J), $\nu$ is frequency (Hz), $\hbar = h/(2\pi)$ is reduced Planck constant, $\omega$ is angular frequency.
- ③Theoretical Significance: Birth of quantum theory, resolving ultraviolet catastrophe in blackbody radiation and explaining the photoelectric effect.
- ④Boundary Conditions & Applications: Universal relation for all quantized elementary excitations (photons, phonons, magnons). Applied in semiconductor photodetectors, solar cells, and spectroscopy.
03 · Photon Momentum
Momentum of Radiation
$$p = \frac{E}{c} = \frac{h}{\lambda} = \hbar k$$
- ①Physical Essence: Zero-rest-mass particles (photons) carry real physical momentum proportional to wavenumber.
- ②Mathematical Notation: $p$ is photon momentum, $c$ is speed of light in vacuum ($3 \times 10^8\text{ m/s}$), $k = 2\pi/\lambda$ is angular wavenumber ($\text{rad/m}$).
- ③Theoretical Significance: Demonstrates that electromagnetic radiation exerts radiation pressure and transfers mechanical momentum.
- ④Boundary Conditions & Applications: Strictly applies to massless relativistic gauge bosons in vacuum. Applied in Compton scattering, laser optical tweezers, and solar sail propulsion.
2. Schrödinger Equation & State Evolution (4 Items)
04 · Time-Dependent Schrödinger Equation
Fundamental Equation of Wave Mechanics
$$i\hbar \frac{\partial \Psi(\mathbf{r}, t)}{\partial t} = \hat{H}\Psi(\mathbf{r}, t) = \left[-\frac{\hbar^2}{2m}\nabla^2 + V(\mathbf{r}, t)\right]\Psi(\mathbf{r}, t)$$
- ①Physical Essence: Governs the continuous, unitary, and deterministic temporal evolution of a non-relativistic quantum state wavefunction.
- ②Mathematical Notation: $i$ is imaginary unit, $\Psi(\mathbf{r}, t)$ is complex wavefunction, $\hat{H}$ is Hamiltonian operator, $\nabla^2$ is Laplacian operator, $V$ is potential.
- ③Theoretical Significance: The core dynamical equation of non-relativistic quantum mechanics, analogous to Newton's Second Law.
- ④Boundary Conditions & Applications: Valid for low-speed non-relativistic single particles ($v \ll c$); Dirac equation required for relativistic fermions. Applied in quantum computing simulation, condensed matter, and molecular orbitals.
05 · Time-Independent Schrödinger Equation
Stationary State Eigenvalue Problem
$$\hat{H}\psi(\mathbf{r}) = E\psi(\mathbf{r}) \implies \left[-\frac{\hbar^2}{2m}\nabla^2 + V(\mathbf{r})\right]\psi(\mathbf{r}) = E\psi(\mathbf{r})$$
- ①Physical Essence: When potential energy is time-independent ($V(\mathbf{r})$), the system admits stationary state solutions whose spatial probability distributions are static.
- ②Mathematical Notation: $\hat{H}$ is Hamiltonian operator, $\psi(\mathbf{r})$ is spatial eigenfunction, $E$ is energy eigenvalue.
- ③Theoretical Significance: Explains discrete quantized energy levels, atomic spectral lines, and molecular orbital energy bands.
- ④Boundary Conditions & Applications: Applied in hydrogen atom orbitals, quantum wells, semiconductor heterostructures, and scanning tunneling microscopes (STM).
06 · Born Interpretation
Probability Density of Wavefunction
$$P(\mathbf{r}, t) = |\Psi(\mathbf{r}, t)|^2 = \Psi^*(\mathbf{r}, t)\Psi(\mathbf{r}, t),\qquad \int_{-\infty}^{\infty} |\Psi|^2\,\mathrm{d}^3\mathbf{r} = 1$$
- ①Physical Essence: The complex wavefunction $\Psi$ represents a probability amplitude; its squared modulus yields the probability density of finding the particle.
- ②Mathematical Notation: $P$ is probability density ($\text{m}^{-3}$), $\Psi^*$ is complex conjugate, integral across all space equals 1 (normalization condition).
- ③Theoretical Significance: Established statistical probabilistic interpretation of quantum theory, earning Born the 1954 Nobel Prize.
- ④Boundary Conditions & Applications: Wavefunctions must be square-integrable ($L^2$ Hilbert space). Fundamental to all quantum measurement calculations.
07 · Continuity of Probability Current
Probability Conservation
$$\frac{\partial |\Psi|^2}{\partial t} + \nabla \cdot \mathbf{j} = 0,\qquad \mathbf{j} = \frac{\hbar}{2mi}\left(\Psi^*\nabla\Psi - \Psi\nabla\Psi^*\right)$$
- ①Physical Essence: Guarantees that total probability across all space is strictly conserved over time under unitary Hamiltonian evolution.
- ②Mathematical Notation: $\mathbf{j}$ is probability current density vector ($\text{m}^{-2}\text{s}^{-1}$), $\nabla\cdot\mathbf{j}$ is divergence.
- ③Theoretical Significance: Equivalent to local conservation of charge and matter probability, establishing unitarity of quantum evolution ($U^\dagger U = I$).
- ④Boundary Conditions & Applications: Used in quantum scattering transmission coefficients, reflection probabilities, and tunneling currents.
3. Uncertainty & Spin (5 Items)
08 · Heisenberg Uncertainty Principle
Position-Momentum Complementarity
$$\Delta x \cdot \Delta p_x \ge \frac{\hbar}{2}$$
- ①Physical Essence: Conjugate physical observables cannot simultaneously possess arbitrarily precise definite values. This is an intrinsic wave-mechanical property, not a measurement defect.
- ②Mathematical Notation: $\Delta x$ is position standard deviation, $\Delta p_x$ is momentum standard deviation, $\hbar$ is reduced Planck constant.
- ③Theoretical Significance: Disproves classical trajectory determinism; explains zero-point energy and why electrons do not spiral into atomic nuclei.
- ④Boundary Conditions & Applications: Applies to any non-commuting operator pair. Minimum uncertainty wave packets are Gaussian states. Explains Casimir effect and quantum noise limits in gravitational wave detectors (LIGO).
09 · Energy-Time Uncertainty Relation
State Lifetime & Energy Spread
$$\Delta E \cdot \Delta t \ge \frac{\hbar}{2}$$
- ①Physical Essence: A quantum state with finite lifetime $\Delta t$ necessarily possesses an energy uncertainty $\Delta E$.
- ②Mathematical Notation: $\Delta E$ is energy spread (natural linewidth), $\Delta t$ is characteristic decay lifetime of the excited state.
- ③Theoretical Significance: Dictates the intrinsic natural linewidth of atomic spectral emissions; permits virtual particle pair fluctuations out of vacuum.
- ④Boundary Conditions & Applications: Governs unstable resonance particle decay widths in high-energy colliders, and virtual photon exchange in QED.
10 · Pauli Exclusion Principle
Fermionic Antisymmetry
$$\Psi(\mathbf{x}_1, \mathbf{x}_2) = -\Psi(\mathbf{x}_2, \mathbf{x}_1) \implies \Psi(\mathbf{x}, \mathbf{x}) = 0$$
- ①Physical Essence: Two identical half-integer spin fermions cannot occupy the exact same quantum state simultaneously.
- ②Mathematical Notation: Total multiparticle wavefunction $\Psi$ is completely antisymmetric under particle coordinate and spin exchange.
- ③Theoretical Significance: Explains electron orbital shell filling, the periodic table of elements, and the stability of macroscopic matter.
- ④Boundary Conditions & Applications: Strictly applies to fermions (spin 1/2, 3/2...). Provides electron degeneracy pressure supporting white dwarf stars against gravitational collapse.
11 · Electron Spin Angular Momentum
Intrinsic Quantum Spin
$$S = \sqrt{s(s+1)}\hbar = \frac{\sqrt{3}}{2}\hbar,\qquad S_z = m_s\hbar = \pm\frac{1}{2}\hbar$$
- ①Physical Essence: Particles possess an intrinsic quantum angular momentum (spin) independent of spatial orbital motion.
- ②Mathematical Notation: $s = 1/2$ is spin quantum number for electrons, $S_z$ is projection along quantization axis, $m_s = \pm 1/2$.
- ③Theoretical Significance: First confirmed experimentally by the Stern-Gerlach experiment (1922); naturally derived from Dirac relativistic wave equation.
- ④Boundary Conditions & Applications: Foundation for Nuclear Magnetic Resonance (NMR / MRI), spintronic MRAM storage, and superconducting qubit states.
12 · Canonical Commutation Relation
Quantum Commutator Algebra
$$[\hat{x}_j, \hat{p}_k] = \hat{x}_j\hat{p}_k - \hat{p}_k\hat{x}_j = i\hbar\delta_{jk}\hat{I}$$
- ①Physical Essence: Observables in quantum mechanics are non-commutative linear operators; the order of operations dictates the outcome.
- ②Mathematical Notation: $[\hat{A}, \hat{B}]$ is commutator bracket, $\delta_{jk}$ is Kronecker delta, $\hat{I}$ is identity operator.
- ③Theoretical Significance: Defines canonical quantization, bridging classical Poisson brackets into quantum operator algebras.
- ④Boundary Conditions & Applications: Underlying mathematical origin of all generalized uncertainty relations ($\Delta A \Delta B \ge \frac{1}{2}|\langle[\hat{A}, \hat{B}]\rangle|$).
4. Atomic & Quantum States (4 Items)
13 · Hydrogen Energy Level Formula
Bohr Energy Formula
$$E_n = -\frac{m_e e^4}{32\pi^2\varepsilon_0^2\hbar^2} \frac{1}{n^2} \approx -\frac{13.606\text{ eV}}{n^2}\quad (n = 1, 2, 3\dots)$$
- ①Physical Essence: Bound electrons in hydrogen atoms occupy discrete quantized energy states proportional to $-1/n^2$.
- ②Mathematical Notation: $m_e$ is electron mass, $e$ is elementary charge, $\varepsilon_0$ is vacuum permittivity, $n$ is principal quantum number. Ground state $E_1 \approx -13.6\text{ eV}$.
- ③Theoretical Significance: Analytically verified the Rydberg formula and explained atomic spectral emission series (Lyman, Balmer, Paschen).
- ④Boundary Conditions & Applications: Exact for single-electron hydrogenic ions ($Z^2/n^2$); multi-electron systems require Hartree-Fock or DFT approximations.
14 · Fine-Structure Constant
Electromagnetic Coupling Constant
$$\alpha = \frac{e^2}{4\pi\varepsilon_0\hbar c} \approx \frac{1}{137.035999}$$
- ①Physical Essence: Dimensionless fundamental constant governing the intrinsic strength of the electromagnetic interaction between charged particles.
- ②Mathematical Notation: $\alpha \approx 1/137$, independent of arbitrary human measurement unit systems.
- ③Theoretical Significance: Characterizes relativistic split corrections in atomic spectra; serves as the perturbation expansion parameter in Quantum Electrodynamics (QED).
- ④Boundary Conditions & Applications: Renormalization group shows $\alpha$ runs with energy scale ($\approx 1/128$ at $Z^0$ mass scale). Benchmarks cosmic cosmological constancy.
15 · Quantum Tunneling
Barrier Penetration Probability
$$T \approx e^{-2\gamma L},\qquad \gamma = \frac{\sqrt{2m(V_0 - E)}}{\hbar}\quad (E < V_0)$$
- ①Physical Essence: Microscopic particles have a non-zero probability of penetrating a potential energy barrier higher than their kinetic energy.
- ②Mathematical Notation: $T$ is transmission coefficient, $L$ is barrier thickness, $V_0 - E$ is barrier height deficiency, $m$ is particle mass.
- ③Theoretical Significance: Overcomes classical energy barriers, explaining alpha radioactive decay, nuclear fusion in stars, and flash memory.
- ④Boundary Conditions & Applications: Transmission drops exponentially with thickness and mass. Applied in Scanning Tunneling Microscopes (STM), tunnel diodes, and flash memory NAND cells.
16 · Quantum Entanglement
Bell State & Non-Locality
$$|\Phi^+\rangle = \frac{1}{\sqrt{2}}\left(|00\rangle + |11\rangle\right),\qquad \langle QS\rangle \le 2\xrightarrow{\text{Quantum}} 2\sqrt{2}$$
- ①Physical Essence: Composite quantum states that cannot be factored into product states of individual subsystems; measuring one instantaneously determines the other.
- ②Mathematical Notation: $|\Phi^+\rangle$ is one of four maximally entangled Bell states; Bell-CHSH inequality bound 2 is violated up to Tsirelson bound $2\sqrt{2}$.
- ③Theoretical Significance: Proves quantum mechanics violates local hidden-variable realism (2022 Nobel Prize in Physics).
- ④Boundary Conditions & Applications: No-signaling theorem ensures entanglement cannot transmit classical information faster than light. Core resource for Quantum Key Distribution (QKD) and quantum teleportation.
5. Physical Quantities, Symbols & SI Units
| Physical Quantity | Symbol | Scalar / Vector | SI Base or Derived Unit | Unit Symbol | Dimension ($M, L, T, I, \Theta$) |
|---|---|---|---|---|---|
| Planck Constant | $h$ | Scalar | Joule-second | $\text{J}\cdot\text{s}$ | $M L^2 T^{-1}$ |
| Reduced Planck Constant | $\hbar$ | Scalar | Joule-second | $\text{J}\cdot\text{s}$ | $M L^2 T^{-1}$ |
| Wavefunction (1D) | $\Psi$ | Complex Scalar | Meter to the power minus one-half | $\text{m}^{-1/2}$ | $L^{-1/2}$ |
| Wavefunction (3D) | $\Psi$ | Complex Scalar | Meter to the power minus three-halves | $\text{m}^{-3/2}$ | $L^{-3/2}$ |
| Probability Density | $P, \rho$ | Scalar | Per cubic meter | $\text{m}^{-3}$ | $L^{-3}$ |
| Probability Current Density | $\mathbf{j}$ | Vector | Per square meter second | $\text{m}^{-2}\text{s}^{-1}$ | $L^{-2} T^{-1}$ |
| Hamiltonian Operator | $\hat{H}$ | Linear Operator | Joule | $\text{J}$ | $M L^2 T^{-2}$ |
| Momentum Operator | $\hat{\mathbf{p}}$ | Vector Operator | Kilogram-meter per second | $\text{kg}\cdot\text{m/s}$ | $M L T^{-1}$ |
| Angular Momentum Operator | $\hat{\mathbf{L}}, \hat{\mathbf{S}}$ | Vector Operator | Joule-second | $\text{J}\cdot\text{s}$ | $M L^2 T^{-1}$ |
| Fine-Structure Constant | $\alpha$ | Dimensionless Scalar | Dimensionless | $1$ | Dimensionless |