Quantum Mechanics I — formula sheet
Every boxed result from the 20 derivations of this unit on one page, for revision. Each title links to the full derivation, with its assumptions and where the formula breaks down. Constants: physical constants table.
The Time-Dependent Schrödinger Equation
The rate of change of the wavefunction’s phase and amplitude (left) is set by the total-energy operator, the Hamiltonian Ĥ = −(ℏ²/2m)∇² + V (right). The factor i makes evolution unitary rather than diffusive: probability is transported and interferes, not smeared away. Given Ψ now, the equation dictates Ψ at every later instant.
Born Rule and Conservation of Normalisation
The total probability of finding the particle somewhere is conserved. Locally, probability can flow — the density at a point rises or falls — but only by an equal and opposite flux j through neighbouring regions, never by creation or destruction. Normalisation set once holds forever.
Probability Current and the Continuity Equation
Identical in form to charge conservation in electromagnetism and to mass conservation in fluids. Probability does not appear or vanish; it flows. Integrating over all space and using the boundary condition gives d/dt ∫|ψ|2dx = 0 — normalisation, once imposed, is preserved forever.
Separation of Variables and Stationary States
The spatial shape ψ is fixed by the eigenvalue equation and does not move; time enters only through a global complex phase turning at angular frequency ω = E/ℏ (the de Broglie–Planck relation E = ℏω). Because the phase is spatially uniform, |Ψ(r,t)|2 = |ψ(r)|2 and every expectation value of a time-independent operator is constant — hence "stationary".
Hermitian Operators: Real Eigenvalues and Orthogonal Eigenstates
A Hermitian observable has real measurement outcomes λn. Its eigenstates can be chosen orthonormal and, when complete, resolve the identity, so any physical state expands uniquely as a superposition whose coefficients are recovered by a single projection cn = ⟨n|ψ⟩. The squared moduli |cn|² are then the Born probabilities of the outcomes.
Expectation Values and the Operator Dictionary
The measurable average of any observable is the overlap of the state with the operator acting on the state. Position acts by plain multiplication because |ψ|² already lives in position space; momentum acts by differentiation because a spatial derivative is what reads off the local wavenumber k = p/ℏ of the wave. The two operators together generate the whole dictionary A(x,p) → Â(x̂, p̂).
The Canonical Commutator [x,p] = i hbar
Measuring position then momentum is not the same operation as measuring momentum then position; the two orders differ by the fixed amount iℏ, independent of the state. The factor i makes the right-hand side anti-Hermitian in a way consistent with both x̂ and p̂ being Hermitian: [x̂,p̂]† = [p̂,x̂] = −[x̂,p̂], and indeed (iℏ)* = −iℏ. The presence of ℏ sets the scale below which the non-commutativity, and hence quantum behaviour, becomes unavoidable.
The Generalised Uncertainty Relation
Two observables can both be sharp in the same state only if their commutator has vanishing expectation there. Since [x̂,p̂] = iℏ is a non-zero constant, position and momentum can never both be sharp — no state exists with ΔxΔp < ℏ/2. Substituting gives the familiar form.
Ehrenfest's Theorem and the Classical Limit
Read carelessly, this says quantum mechanics contains Newton's laws. Read carefully, it does not. The classical statement would be d⟨p⟩/dt = −V′(⟨x⟩) — the force at the mean position. What the theorem gives is ⟨V′(x)⟩, the mean of the force. These agree only when V′ is linear over the width of the packet: when V is at most quadratic, or the packet is narrow compared with the scale on which the force varies.
Bound States of the Infinite Square Well
The allowed energies form a ladder spaced as the squares of the integers, with a nonzero ground state E1 = π2ℏ2/(2mL2) — the zero-point energy demanded by confinement and the uncertainty principle. The n-th eigenfunction is a standing sine wave fitting exactly n half-wavelengths between the walls, with n − 1 interior nodes; adjacent states are orthogonal.
Finite Square Well: Transcendental Quantisation
With u = ka (interior half-oscillation count) and v = κa (tail decay over a half-width), the allowed levels are the finite set of points where the constraint quarter-circle of radius R = a√(2mV0)/ℏ meets the tangent (even) and cotangent (odd) branches. A larger, deeper, or wider well means larger R and hence more intersections — more bound states. Because κ > 0 for every state, ψ always leaks outside as an evanescent tail e−κ|x|, unlike the infinite well where ψ is strictly confined.
Continuity Conditions at Potential Steps
Wherever the potential is finite, the wavefunction is smooth to first order: no jumps and no kinks. A finite step in V produces only a discontinuity in the second derivative, i.e. a change of curvature, while the value and slope pass through unbroken. Physically, continuity of ψ keeps the probability density single-valued, and continuity of ψ′ keeps the probability current j = (ℏ/m) Im(ψ* ψ′) conserved across the interface.
The Harmonic Oscillator via Ladder Operators
Evenly spaced levels separated by ℏω, with an irreducible ½ℏω that cannot be removed. The even spacing is why an oscillator absorbs and emits at a single frequency, and it is why the classical picture of a vibrating molecule survives at all.
Oscillator Eigenfunctions and Hermite Polynomials
Every oscillator eigenfunction is a fixed Gaussian envelope e⁻ξ²⁄² modulated by a Hermite polynomial Hₙ of degree n. The polynomial supplies exactly n real nodes (zeros), so ψₙ has n nodes and alternates parity: even in x for even n, odd for odd n. The Gaussian sets a universal width scale b = √(ℏ/mω) (the "oscillator length"), while raising n stretches probability outward toward the classical turning points.
Free Gaussian Wave Packet and Its Spreading
Two regimes. For t ≪ τ the width barely changes; for t ≫ τ it grows linearly at rate ħ/2mσ0, which is exactly the velocity spread σp/m implied by the initial momentum uncertainty. The packet spreads because it was made of a range of momenta, and the tighter it was squeezed, the wider that range.
Measurement, Collapse, and Spectral Expansion
Expand the state in the observable's eigenbasis; the modulus-squared of each coefficient is the probability of the associated eigenvalue, and the act of obtaining that eigenvalue discards every other term, leaving the matching eigenstate. The complete distribution {P(an)} carries everything  can reveal about |ψ⟩; its mean is ⟨ψ|Â|ψ⟩.
The Time-Evolution Operator and Energy Basis Dynamics
Time evolution is generated by the Hamiltonian: Û is the exponential of −i/ħ times energy times elapsed time. In the energy basis it is diagonal, so each eigenstate contribution keeps its magnitude |cn| fixed and rotates its phase at angular frequency ωn = En/ħ. A stationary state (single n) has trivial dynamics up to a global phase; all interesting motion comes from relative phases between different energies.
The Photoelectric Effect and Einstein's Light-Quantum Equation
The maximum kinetic energy of photoelectrons is the photon energy \(hf\) minus the work function \(\phi\). If \(hf < \phi\), no electrons are emitted. The stopping potential \(V_s\) is directly proportional to \(f\), with slope \(h/e\) and intercept \(-\phi/e\). The threshold frequency is \(f_0 = \phi/h\), and the threshold wavelength is \(\lambda_0 = hc/\phi\).
The de Broglie Wavelength of a Material Particle
The de Broglie wavelength is inversely proportional to momentum. Heavier particles or faster particles have shorter wavelengths. For electrons, the wavelength depends on the accelerating voltage: higher voltage means higher momentum and shorter wavelength.