Historical Motivation

1.1 Failures of Classical Physics

By the late 19th century, classical physics could not explain several phenomena:

Blackbody radiation. The Rayleigh-Jeans law predicted infinite energy at short wavelengths (the “ultraviolet catastrophe”). Experiment showed a peak that shifts with temperature.

Photoelectric effect. Classical theory predicted that the kinetic energy of emitted electrons Depends on the intensity of light. Experiment showed on the frequency.

Atomic spectra. Atoms emit light at discrete frequencies, not the continuous spectrum predicted By classical electrodynamics.

Stability of atoms. Classical electrodynamics predicts orbiting electrons radiate energy and Spiral into the nucleus.

1.2 Key Experiments

Planck”s quantisation (1900). Blackbody radiation is explained by assuming energy is emitted in Discrete quanta: $E = h\nu$ where $h = 6.626 \times 10^{-34}$ J$\cdot$S is Planck’s constant.

Einstein’s photon (1905). Light consists of photons, each carrying energy $E = h\nu$ and momentum $p = h/\lambda = h\nu/c$. The photoelectric effect: $E_k = h\nu - \phi$ where $\phi$ is the work Function.

Compton scattering (1923). X-rays scattered off electrons show a wavelength shift:

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

This confirms that photons carry momentum $p = h/\lambda$.

Davisson-Germer experiment (1927). Electrons scattered off a nickel crystal produce a diffraction Pattern, confirming de Broglie’s hypothesis that matter has wave properties: $\lambda = h/p$.

1.3 The Photoelectric Effect: Detailed Derivation

The photoelectric effect provided the first direct evidence for the quantum nature of light. When Monochromatic light of frequency $\nu$ strikes a metal surface, electrons are ejected with a maximum Kinetic energy $K_{\max}$ that depends on $\nu$ but not on the intensity.

Einstein’s quantum hypothesis (1905). Each photon carries energy $E_\gamma = h\nu$. When a photon Strikes the surface, it transfers all its energy to a single electron. By energy conservation:

$h\nu = \phi + K_{\max}$

Where $\phi$ is the work function (minimum energy to remove an electron from the metal).

Key predictions:

1. Threshold frequency. No electrons are emitted if $\nu \lt \nu_0 = \phi/h$Regardless of intensity. This is because each photon must supply at least $\phi$.

2. Linear dependence on frequency. $K_{\max} = h\nu - \phi$ is linear in $\nu$ with slope $h$ (independent of the metal).

3. Intensity affects current, not energy. Higher intensity means more photons per unit time, so more electrons are emitted, but each electron has the same maximum kinetic energy.

4. No time delay. Classically, an electron should accumulate energy gradually; quantum mechanically, a single photon ejects an electron instantaneously.

Proof of the threshold frequency. Setting $K_{\max} = 0$ in the energy balance:

$h\nu_0 = \phi \implies \nu_0 = \frac{\phi}{h}$

For frequencies $\nu \lt \nu_0$The photon energy is insufficient to liberate an electron, and No photoelectric emission occurs regardless of intensity. $\blacksquare$

Millikan’s experimental verification (1916). Robert Millikan, who initially opposed Einstein’s Theory, performed careful experiments measuring $K_{\max}$ versus $\nu$ for various metals. His Results confirmed the linear relation $K_{\max} = h\nu - \phi$ with a universal slope $h$ (Planck’s Constant), providing compelling evidence for the photon concept. Millikan’s measured value of $h$ Agreed with Planck’s value from blackbody radiation to within $0.5\%$.

Example 1.1. Sodium has a work function $\phi = 2.28$ eV. Find the cutoff wavelength.

1.4 Compton Scattering: Derivation

Compton scattering provides direct evidence that photons carry momentum. When an X-ray photon of Wavelength $\lambda$ scatters off a free (or loosely bound) electron at rest, the scattered photon Has a longer wavelength $\lambda'$.

Setup. Incident photon: energy $E = hc/\lambda$Momentum $p = h/\lambda$. Target electron: At rest, energy $m_e c^2$Momentum $0$. After scattering, the photon is deflected by angle $\theta$ And the electron recoils at angle $\phi$.

Energy conservation:

$\frac{hc}{\lambda} + m_e c^2 = \frac{hc}{\lambda'} + E_e$

Momentum conservation (vector equation):

$\frac{h}{\lambda}\hat{n} = \frac{h}{\lambda'}\hat{n}' + \mathbf{p}_e$

Derivation of the wavelength shift. From the relativistic energy-momentum relation for the Electron, $E_e^2 = (p_e c)^2 + (m_e c^2)^2$. Rearranging the energy conservation:

$E_e - m_e c^2 = hc\!\left(\frac{1}{\lambda} - \frac{1}{\lambda'}\right)$

Squaring the momentum equation:

$p_e^2 = \left(\frac{h}{\lambda}\right)^2 + \left(\frac{h}{\lambda'}\right)^2 - \frac{2h^2}{\lambda\lambda'}\cos\theta$

Using $E_e^2 = p_e^2 c^2 + m_e^2 c^4$ and writing $T_e = E_e - m_e c^2$:

$E_e^2 - m_e^2 c^4 = 2m_e c^2 T_e + T_e^2 = p_e^2 c^2$

Substituting $T_e = hc(1/\lambda - 1/\lambda')$ and $p_e^2$ from above, then dividing by $c^2$ and Simplifying:

$2m_e c \cdot \frac{h}{\lambda\lambda'}(1 - \cos\theta) = 2h^2\!\left(\frac{1}{\lambda^2} + \frac{1}{\lambda'^2} - \frac{2\cos\theta}{\lambda\lambda'}\right)$

$\frac{1}{\lambda'} - \frac{1}{\lambda} = \frac{h}{m_e c}(1 - \cos\theta)\cdot\frac{1}{\lambda\lambda'}$

Multiplying through by $\lambda\lambda'$ yields the Compton formula:

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

The quantity $\lambda_C = h/(m_e c) \approx 2.426 \times 10^{-12}$ m is the Compton wavelength of The electron. $\blacksquare$

Classical limit. In the classical limit ($\lambda \gg \lambda_C$), the wavelength shift $\Delta\lambda \to 0$ and the scattering reduces to classical Thomson scattering. The Compton Effect is only significant for X-rays and gamma rays, where $\lambda$ is comparable to $\lambda_C$. For visible light ($\lambda \sim 500$ nm), the Compton shift is negligible compared to the wavelength.

Physical interpretation. The maximum shift $\Delta\lambda = 2\lambda_C \approx 4.85$ pm occurs at $\theta = \pi$ (backscattering). The shift is independent of the material and depends only on the Scattering angle, confirming that the scattering involves individual photons and electrons.

Example 1.2. X-rays of wavelength $0.100$ nm are Compton-scattered at $\theta = 90°$. Find the Wavelength of the scattered photon and the kinetic energy of the recoil electron.

1.5 The Davisson-Germer Experiment

The Davisson-Germer experiment (1927) provided the first direct confirmation of de Broglie’s Hypothesis that particles have wave-like properties.

Experimental setup. A beam of electrons is accelerated through a potential difference $V$ and Directed at a nickel crystal. The scattered electrons are detected at various angles $\phi$.

de Broglie relation. An electron accelerated through potential $V$ has kinetic energy $K = eV$ And momentum:

$p = \sqrt{2m_e eV}$

The de Broglie wavelength is:

$\lambda = \frac{h}{p} = \frac{h}{\sqrt{2m_e eV}}$

Bragg condition. The nickel crystal acts as a diffraction grating with lattice spacing $d$. Constructive interference occurs when:

$n\lambda = 2d\sin\phi$

Where $\phi$ is the angle measured from the crystal surface.

The key observation. At $V = 54$ V, a pronounced peak was observed at $\phi = 50°$. The De Broglie wavelength at this voltage is:

$\lambda = \frac{6.626 \times 10^{-34}}{\sqrt{2(9.109 \times 10^{-31})(1.602 \times 10^{-19})(54)}} = 0.167\;\mathrm{nm}$

The Bragg condition with the nickel lattice spacing gives excellent agreement with this Prediction, confirming that electrons exhibit wave-like diffraction.

Significance. The Davisson-Germer experiment established wave-particle duality for matter. The De Broglie relation $\lambda = h/p$ was subsequently confirmed for neutrons, atoms, and molecules (C60 fullerenes in 1999), establishing it as a universal principle. In 2019, the de Broglie Wavelength of molecules exceeding 25,000 atomic mass units was demonstrated, pushing the boundary Of quantum mechanics to the macroscopic regime.