E.2 Quantum physicsIB Physics HL: Revision notes

When light of high enough frequency falls on a clean metal surface, electrons called photoelectrons are emitted. Key observations:

• There is a threshold frequency $f_0$: below it no electrons are emitted, however intense the light.
• Above it, emission is instantaneous, even at very low intensity.
• The maximum kinetic energy of the photoelectrons depends on frequency, not on intensity.
• Increasing intensity increases the number of photoelectrons per second (the current).

None of these can be explained if light is a continuous wave, so the photoelectric effect is evidence for the particle nature of light.

Light consists of photons, each of energy $E = hf$. One photon gives all its energy to one electron. Some of this energy, at least the work function Φ, is needed to free the electron; the rest becomes kinetic energy. Electrons at the surface need exactly Φ, so they leave with the maximum kinetic energy:

$E_{max} = hf - \Phi$

At threshold $E_{max} = 0$, so $\Phi = hf_0$. Plotting $E_{max}$ against $f$ gives a straight line of gradient h and an intercept on the frequency axis of $f_0$. Remember to convert between eV and J using 1 eV = 1.60 × 10⁻¹⁹ J.

De Broglie proposed that particles have a wavelength too:

$\lambda = \frac{h}{p}$

For a particle accelerated from rest through a potential difference V, $E_k = qV$ and $p = \sqrt{2mE_k}$. At equal kinetic energy a heavier particle has more momentum and a shorter wavelength ($\lambda \propto 1/\sqrt{m}$); at equal speed $\lambda \propto 1/m$. Everyday objects have wavelengths of order 10⁻³⁴ m, far too small to show any wave effects.

Electrons accelerated through a few kV have wavelengths of about 10⁻¹¹ m, comparable with the spacing of atoms in a crystal. Passing through thin graphite they diffract, forming rings on a screen. Diffraction is a wave property, so this is evidence for the wave nature of matter. Raising the accelerating voltage shortens λ, so the rings get smaller.

Wave–particle duality: both light and matter show wave behaviour (diffraction, interference) and particle behaviour (detection at single points, collisions). Which behaviour appears depends on the experiment.

When X-rays or gamma rays scatter off electrons, the scattered radiation has a longer wavelength. A classical wave would be re-emitted at the same wavelength, so the shift is further evidence of the particle nature of light. The photon, of momentum $h/\lambda$, collides with an electron and transfers energy and momentum to it:

$\Delta\lambda = \lambda_f - \lambda_i = \frac{h}{m_e c}(1 - \cos\theta)$

The shift is zero at θ = 0 and largest at θ = 180°, where it equals $2h/(m_e c)$ ≈ 4.85 pm. It does not depend on the incident wavelength, so it is most noticeable for short-wavelength X-rays and gamma rays.