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Key ideas, equations and common mistakes. Open any topic below for the full explanation.

Identify the evidence and the quantity being predicted. A photon energy, a probability density, a momentum spread and an atomic energy gap require different calculations.

Choose the model before the equation

Decisions that prevent common quantum-model errors
Task	Keep this distinction
Light evidence	Threshold frequency supports quantised energy exchange; interference and diffraction support wave behaviour. Frequency changes energy per photon, while intensity at fixed frequency changes photon rate.
Momentum and wavelength	A photon has p = E/c = h/λ despite zero rest mass. A non-relativistic electron uses p = mv and λ = h/p. Do not use E = pc for its kinetic energy.
Position probability	Amplitude can be signed; density \|ψ\|² is nonnegative. An interval probability is density area, not density alone or the square of amplitude area.
Coherent alternatives	Add amplitudes before squaring. Relative signs or phases affect interference. A global sign reversal of one complete real state does not change its density.
Localisation	A narrower position distribution needs a broader momentum distribution. The width relation concerns spread, not a required mean momentum or apparatus error.
Confinement	Boundary conditions select standing shapes and quantised energies. A wavefunction plot is not a particle trajectory, and the box's level sequence is not an atom's exact sequence.
Spectral lines	Use an allowed gap from the actual occupied level. Photon energy is the magnitude of the energy change, not the absolute value of one level.

Photon and matter-wave calculations

Photon: E = hf = hc/λ, p = E/c = h/λ
Threshold: Φ_work = hf₀
Matter wave: λ = h/p
Non-relativistic massive particle: p = mv

For the 6.00 × 10¹⁴ Hz photon, λ = 500 nm, E ≈ 2.49 eV and p ≈ 1.33 × 10^-27 kg m/s. The 1.50 × 10⁶ m/s electron instead has λ ≈ 0.485 nm. Doubling its speed halves wavelength while quadrupling kinetic energy.

Normalisation and area

For a profile that vanishes outside 0 < x < L:

Square profile ψ = C:
C²L = 1, so C = 1/√L

Sine profile ψ = C sin(πx/L):
C²L/2 = 1, so C = √(2/L)

Choose a global sign without changing probabilities. At L = 4.00 nm, the positive coefficients are 0.500 and 0.707107 nm^-1/2. The square profile gives probability 0.375 over 1.00-2.50 nm. The sine profile's central 1.00-3.00 nm probability is about 0.818310; a coarse trapezium estimate need not equal that smooth-model area.

The coordinate and density units must match: nm^-1 times nm gives a dimensionless probability. For coherent real contributions, (a + b)² includes 2ab. Do not replace it with a² + b². Detection-bin frequencies estimate finite-interval probabilities and do not establish a classical path or exact point probability.

Width scales, box states and atomic gaps

Characteristic widths: ΔxΔp ≳ h
Infinite well: ψ_n = √(2/L) sin(nπx/L)
λ_n = 2L/n
E_n = h²n²/(8mL²)

The width relation is an order-of-magnitude convention, not an exact equality. Infinite-well states have n = 1, 2, 3, ... and n - 1 interior nodes. Their first energies are in the ratio 1:4:9. Doubling width quarters energies at fixed n and mass; n = 2 at width 2L has the same energy as n = 1 at width L.

In the supplied atom, levels -6, -3 and -1 eV give permitted gaps of 3, 2 and 5 eV. The emitted wavelengths are about 414, 622 and 249 nm. A ground-state absorber can use the 3 and 5 eV upward gaps; the 2 eV absorption requires the -3 eV state to be occupied. The 249 nm line is ultraviolet.

Quantities and units

Quantum quantities, unit conversions and local symbol meanings
Quantity and symbol	Unit	Meaning
Energy E, E_n	J or eV	1 eV = 1.60 × 10^-19 J.
Work function Φ_work	J	Minimum surface-removal energy, distinct from magnetic flux.
Planck constant h	J s	6.63 × 10^-34 J s in these examples.
Momentum p	kg m/s = N s	Photon and massive-particle relations have different conditions.
Frequency f	Hz	Sets energy per photon.
Vacuum light speed c	m/s	3.00 × 10⁸ m/s.
Wavelength λ; width L; position x	m or nm	1 nm = 10^-9 m. Match density units to the position coordinate.
Mass m, m_e	kg	m_e = 9.11 × 10^-31 kg.
Elementary charge e	C	1.60 × 10^-19 C; eV itself is an energy unit.
One-dimensional amplitude ψ	m^-1/2 or nm^-1/2	Its modulus squared is the corresponding inverse-length density.
Probability; quantum number n	No unit	Probability is between 0 and 1. Here n labels a box state, not amount of substance.
Box potential energy V	J or eV	The local symbol V here does not mean electric potential.

A three-dimensional electron cloud instead uses probability per volume, with SI density unit m^-3. Always identify what the graph's vertical axis represents before reading a height or calculating an area.

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