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Quantum Physics overview

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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
TaskKeep this distinction
Light evidenceThreshold 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 wavelengthA 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 probabilityAmplitude can be signed; density |ψ|2 is nonnegative. An interval probability is density area, not density alone or the square of amplitude area.
Coherent alternativesAdd amplitudes before squaring. Relative signs or phases affect interference. A global sign reversal of one complete real state does not change its density.
LocalisationA narrower position distribution needs a broader momentum distribution. The width relation concerns spread, not a required mean momentum or apparatus error.
ConfinementBoundary 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 linesUse 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: Φwork = 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: ψn = √(2/L) sin(nπx/L)
λn = 2L/n
En = 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 symbolUnitMeaning
Energy E, EnJ or eV1 eV = 1.60 × 10-19 J.
Work function ΦworkJMinimum surface-removal energy, distinct from magnetic flux.
Planck constant hJ s6.63 × 10-34 J s in these examples.
Momentum pkg m/s = N sPhoton and massive-particle relations have different conditions.
Frequency fHzSets energy per photon.
Vacuum light speed cm/s3.00 × 108 m/s.
Wavelength λ; width L; position xm or nm1 nm = 10-9 m. Match density units to the position coordinate.
Mass m, mekgme = 9.11 × 10-31 kg.
Elementary charge eC1.60 × 10-19 C; eV itself is an energy unit.
One-dimensional amplitude ψm-1/2 or nm-1/2Its modulus squared is the corresponding inverse-length density.
Probability; quantum number nNo unitProbability is between 0 and 1. Here n labels a box state, not amount of substance.
Box potential energy VJ or eVThe 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.

Practise exam questions

Write your answer first, then tick the mark points it makes.

Exam-style question

Define the work function of a metal and the threshold frequency. [2]

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Exam-style question

Ultraviolet light of wavelength 250 nm falls on a metal with a work function of 2.30 eV. Calculate the maximum kinetic energy of the emitted electrons, in eV. Take h = 6.63 x 10-34 J s, c = 3.00 x 108 m s-1 and e = 1.60 x 10-19 C. [4]

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Exam-style question

Explain how the photoelectric effect gives evidence that light consists of photons. [3]

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Exam-style question

A graph of maximum kinetic energy of photoelectrons (in eV) against frequency of the light is a straight line. It crosses the frequency axis at 4.8 x 1014 Hz and passes through (8.0 x 1014 Hz, 1.32 eV). Use the graph to find the Planck constant and the work function in eV. Take e = 1.60 x 10-19 C. [3]

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Exam-style question

Data-based question: Measuring the Planck constant with a photocell [20]

Light of a single wavelength shines on a clean metal surface inside an evacuated photocell. A reverse potential difference is increased until the photocurrent just falls to zero; this is the stopping potential Vs. The experiment is repeated for five wavelengths. Take c = 3.00 x 108 m s-1 and e = 1.60 x 10-19 C.

Table 1. Stopping potential at different wavelengths
wavelength / nm400450500550600
Vs / V1.110.760.490.260.07
  1. (a) Explain what the stopping potential tells you about the emitted electrons. [2]
  2. (b) Write down the photoelectric equation and show that eVs = hf - Φ, where Φ is the work function. [2]
  3. (c) Calculate the frequency of the light for each wavelength. [1]
  4. (d) Use the data to find the gradient of a graph of Vs against f, and hence a value for the Planck constant h. [3]
  5. (e) Find the threshold frequency, and the work function of the metal in eV. [3]
  6. (f) A student says: "Red light of wavelength 700 nm will release electrons from this metal if it is bright enough." Evaluate this claim. [2]
  7. (g) The intensity of the 400 nm light is doubled. State and explain the effect on the stopping potential and on the photocurrent before the reverse p.d. is applied. [3]
  8. (h) The 450 nm beam delivers 2.0 mW to the surface. Calculate the number of photons arriving each second and the photocurrent if 5.0% of them each release an electron. [3]
  9. (i) State one assumption you made in (h). [1]

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Quick recall

Card 1 of 8. Answer in your head, then check.

A surface has threshold frequency 5.00 x 10^14 Hz. In the ordinary single-photon model, what happens if the intensity at 4.00 x 10^14 Hz increases while frequency and other conditions stay fixed?

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