Sections#

Review & Summary#

The Photon#

Light is quantized in photons. The energy and momentum of a photon of frequency \(f\) and wavelength \(\lambda\) are

(345)#\[ E = hf = \frac{hc}{\lambda}, \quad p = \frac{h}{\lambda} = \frac{E}{c} \]

where \(h = 6.63 \times 10^{-34}\) J·s is Planck’s constant.

The Photoelectric Effect#

When light strikes a metal surface, electrons can be ejected. Einstein’s equation:

(346)#\[ K_{\max} = hf - \Phi \]

where \(K_{\max}\) is the maximum kinetic energy of ejected electrons and \(\Phi\) is the work function of the metal. The threshold frequency below which no electrons are ejected is \(f_0 = \Phi/h\).

Compton Scattering#

When an x-ray photon scatters from an electron, the wavelength shift (Compton shift) is

(347)#\[ \Delta\lambda = \frac{h}{m_e c}(1 - \cos\phi) \]

where \(\phi\) is the scattering angle and \(h/(m_e c)\) is the Compton wavelength of the electron.

Electrons and Matter Waves#

de Broglie wavelength: A particle with momentum \(p\) has an associated wavelength \(\lambda = h/p = h/(mv)\). Electrons and other particles exhibit wave-like behavior (diffraction, interference).

Schrödinger’s Equation#

The wave function \(\Psi\) obeys the Schrödinger equation, which governs the time evolution of quantum systems. For a free particle, \(\Psi \propto e^{i(kx - \omega t)}\) with \(E = \hbar\omega\) and \(p = \hbar k\), where \(\hbar = h/(2\pi)\).

Heisenberg’s Uncertainty Principle#

Position and momentum cannot both be known with arbitrary precision:

(348)#\[ \Delta x \, \Delta p \geq \frac{\hbar}{2} \]

Similarly, \(\Delta E \, \Delta t \geq \hbar/2\) for energy and time.

Reflection and Tunneling#

A matter wave can reflect from a potential step even when the particle’s energy exceeds the step height. Quantum tunneling: a particle can penetrate a potential barrier of height \(U > E\) with a probability that decreases exponentially with barrier width.