Quantum and nuclear physics (A Level)

22. Quantum Physics

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22. Quantum Physics — Cambridge AS & A Level Physics 9702 — CAMBRIDGE AS & A LEVEL PHYSICS 9702 — COMPLETE NOTES notes: Use the photon model, the photoelectric equation, de Broglie wavelength and atomic energy levels to explain light–matter interactions.Evidence — electromagnetic…

Photon energy
Photon momentum
Photon rate
Radiation pressure
Electronvolt (eV)
Accelerated charge

Section 1 of 10The photon: energy and momentum

Evidence

Evidence electromagnetic radiation shows diffraction and interference (wave behaviour),
- while the photoelectric effect and line spectra need a particle model.

Photon

Photon a massless packet (quantum) of electromagnetic energy, transferring its energy all at once rather than continuously.

Photon energy

E = hf, or E = hc/λ using c = fλ, where h = 6.63 × 10⁻³⁴ J s is Planck's constant.
Higher frequency means a more energetic photon.
Energy is inversely proportional to wavelength, so long-wavelength photons have less energy.

Photon momentum

Photon momentum a photon has momentum even though it has no mass, p = E/c = hf/c = h/λ.

Photon rate

Photon rate photons per second = power ÷ energy of one photon.

Radiation pressure

Radiation pressure force = photons per second × momentum of each photon, and pressure = force ÷ area.
- (if all are absorbed)

Section 2 of 10The electronvolt

Electronvolt (eV)

Electronvolt (eV) the energy gained by an electron moving through a potential difference of one volt, 1 eV = 1.60 × 10⁻¹⁹ J.

eV to joules: multiply by 1.60 × 10⁻¹⁹; joules to eV: divide by 1.60 × 10⁻¹⁹.
It is convenient because quantum energies are far smaller than a joule.

Accelerated charge

Accelerated charge a charge accelerated from rest through V gains eV = ½mv², so v = √(2eV/m).

Section 3 of 10The photoelectric effect

The photoelectric effect

Photoelectric effect

Photoelectric effect electrons (photoelectrons) are emitted from a metal surface when radiation of sufficient frequency is absorbed.

It shows that light is quantised, since each electron absorbs a single photon.
The electrons were already in the metal and are not created by the light.

Threshold frequency f0

Threshold frequency f0 the minimum frequency of radiation that can release a photoelectron from the surface.

Threshold wavelength λ0

Threshold wavelength λ0 the longest wavelength that can release a photoelectron.

Work function φ

Work function φ the minimum energy needed to release an electron from the surface, φ = hf₀.

A high work function means electrons are bound more tightly.
Alkali metals (sodium, potassium) have thresholds in the visible region because surface electrons are weakly attracted.
Transition metals such as iron have thresholds in the ultraviolet because binding is stronger.

Analogy

Analogy many low-frequency quanta cannot dislodge a coconut, but one high-frequency quantum (a bullet) can.
- (table-tennis balls)

Metal	f₀ / Hz	λ₀ / nm
Sodium	4.40 × 10¹⁴	682
Potassium	5.56 × 10¹⁴	540
Zinc	1.02 × 10¹⁵	294
Iron	1.04 × 10¹⁵	289
Copper	1.13 × 10¹⁵	266
Gold	1.23 × 10¹⁵	244

Section 4 of 10The photoelectric equation and stopping potential

The photoelectric equation and stopping potential

Energy conservation

Energy conservation hf = φ + E_{k max}, where E_{k max} = ½mv_max².
If hf is less than φ, no electrons are emitted.
The maximum kinetic energy depends only on frequency and work function, not on intensity.
Most photoelectrons have less than the maximum, because deeper electrons lose energy reaching the surface.

Graph of Ek max against f

Graph of Ek max against f a straight line E_{k max} = hf − φ.
The x-intercept is the threshold frequency f₀.
The gradient is Planck's constant h.
The magnitude of the y-intercept is the work function φ.

Stopping potential Vs

Stopping potential Vs the potential difference that just stops photoelectrons crossing from the emitter to the collector.

A variable supply makes the emitter positive so electrons are attracted back, and V is raised until the photocurrent is zero.
At V_s, eV_s = E_{k max}, and only the magnitude of V_s matters.

Section 5 of 10Intensity, frequency and the failure of wave theory

Intensity

Intensity it is the energy per unit time per unit area, proportional to the number of photons hitting the surface per second.

Photocurrent

Photocurrent it is proportional to the intensity, because each photoelectron comes from one photon.

Kinetic energy

Kinetic energy it is unaffected by intensity, so the stopping potential is the same for any intensity.

Higher frequency

Higher frequency at constant intensity fewer photons arrive per second,
so the photocurrent falls; at constant photon number the current stays constant but the kinetic energy rises.

Wave theory predicts	Observation
Any frequency gives emission if the exposure is long enough	No emission below the threshold frequency, however long or intense the light
Energy builds up gradually in each electron	Emission is instantaneous above the threshold
Kinetic energy rises with intensity	Kinetic energy depends only on frequency; greater intensity gives more electrons per second

Wave–particle duality
- light travels as a wave but interacts with matter as particles, and the photon model explains the observations.

Section 6 of 10Electron diffraction and the de Broglie wavelength

De Broglie's idea

De Broglie's idea matter such as electrons also shows wave behaviour.

Evidence

Evidence a beam of electrons through a thin graphite film produces concentric rings on a fluorescent screen.
The gaps between neighbouring planes of carbon atoms act as slits of a size similar to the electron wavelength.
Particles would be spread uniformly across the screen instead of forming rings.

Electron gun

Electron gun electrons are accelerated through a p.d.
V to gain E_k = ½mv² = eV.
A higher voltage gives faster electrons, a shorter wavelength and smaller rings.

De Broglie wavelength

De Broglie wavelength the wavelength associated with a moving particle, λ = h/p = h/mv.

What the symbols mean: λ = de Broglie wavelength (m) · h = 6.63 × 10⁻³⁴ J s · p = momentum (kg m s⁻¹) · m = mass (kg) · v = speed (m s⁻¹)

In terms of kinetic energy p = √(2mE), so λ = h/√(2mE).
For the same kinetic energy λ ∝ 1/√m, so heavier particles have shorter wavelengths.

Size

Size for everyday objects the wavelength is far too small to notice, and quantum effects appear only for samples of about that size.

Section 7 of 10Atomic energy levels and line spectra

Discrete levels

Discrete levels electrons in atoms can only have certain energies, drawn as horizontal lines.

Ground state

Ground state the lowest level that an electron normally occupies.

Excitation

Excitation an electron moves up by absorbing a photon, through collisions or by heating.

Ionisation

Ionisation the electron gains enough energy to leave the atom.

De-excitation

De-excitation an electron falling to a lower level emits a photon.

Emission spectrum

Emission spectrum coloured lines on a dark background, each from a particular downward transition.

Absorption spectrum

Absorption spectrum a continuous spectrum with dark lines where photons were absorbed by atoms of a cool, low-pressure gas.
The re-emitted photons go in all directions, so those wavelengths seem missing from the original beam.
The dark lines match the bright lines in the emission spectrum of the same element.

Unique

Unique each element has its own set of lines, so a spectrum identifies the element.

Evidence

Evidence discrete lines show that electrons occupy discrete energy levels.

Section 8 of 10Calculating transition wavelengths

Photon energy

Photon energy hf = E₂ − E₁, so λ = hc/(E₂ − E₁).

Energy gap

Energy gap a larger gap gives a shorter wavelength and a smaller gap a longer wavelength.

Units

Units convert eV to J (× 1.60 × 10⁻¹⁹) before using h and c.

Section 9 of 10Key terms

Term	Meaning
Photon	A packet (quantum) of electromagnetic energy, E = hf.
Electronvolt	The energy gained by an electron accelerated through 1 V, 1.60 × 10⁻¹⁹ J.
Photoelectric effect	Emission of electrons from a metal surface when radiation above the threshold frequency falls on it.
Work function	The minimum energy needed to release an electron from a metal surface.
Threshold frequency	The minimum frequency of radiation that causes photoelectric emission.
Stopping potential	The p.d. that just prevents photoelectrons crossing the gap, eV_s = E_{k max}.
De Broglie wavelength	The wavelength associated with a moving particle, λ = h/p.
Line spectrum	A set of discrete bright (emission) or dark (absorption) lines characteristic of an element.
Ground state	The lowest energy level of an atom.

Section 10 of 10Exam technique

Applying E = hf to particles other than photons — it is only the photon energy equation.

Saying increasing intensity increases the kinetic energy of photoelectrons — it increases their number only.

Forgetting to convert eV to joules before using h and c — convert first.

Thinking the light creates the electrons — they are already in the metal.

Mixing up threshold frequency (minimum) and threshold wavelength (maximum) — a longer wavelength than λ₀ cannot cause emission.