Premed · Premed · Physics 2
Lecture 21: Quantum Mechanics: Photoelectric Effect and Wave-Particle Duality
Physics II — Electromagnetism, Optics & Modern Physics
Learning Objectives
By the end of this lecture, students will be able to:
- Describe blackbody radiation and explain Planck's quantum hypothesis
- Explain the photoelectric effect and why classical physics fails to account for it
- Apply Einstein's photon model to solve photoelectric effect problems
- Describe the de Broglie hypothesis and calculate the wavelength of matter waves
- Explain the wave-particle duality and the Heisenberg uncertainty principle
Lecture Content
I. Blackbody Radiation and Planck's Quantum Hypothesis
A blackbody is an idealized object that absorbs all incident radiation and re-emits it in a characteristic spectrum that depends only on its temperature. Classical physics, through the Rayleigh-Jeans law, predicted that the intensity of blackbody radiation would increase without limit at short wavelengths, a disastrous prediction known as the ultraviolet catastrophe that implied infinite total radiated energy.
Two empirical laws describe blackbody radiation successfully. Wien's displacement law states that the peak wavelength shifts to shorter wavelengths at higher temperatures: lambda_max T = 2.898 x 10^-3 m K. For example, the sun (T approximately 5800 K) has lambda_max approximately 500 nm (visible light), while the human body (T approximately 310 K) has lambda_max approximately 9.4 um (infrared). The Stefan-Boltzmann law gives the total power radiated per unit area as P/A = sigma T^4, where sigma = 5.67 x 10^-8 W/(m^2 K^4).
In 1900, Planck resolved the ultraviolet catastrophe with a revolutionary hypothesis: the energy of oscillators in the blackbody is quantized, meaning E = nhf, where n is an integer, h = 6.626 x 10^-34 J s is Planck's constant, and f is the frequency. Energy comes in discrete packets called quanta rather than in continuous values. This assumption fit the observed spectrum perfectly and marked the birth of quantum physics.
II. The Photoelectric Effect
When light shines on a metal surface, electrons are ejected, a phenomenon called the photoelectric effect. Classical wave theory made several predictions: any frequency should work if the intensity is high enough, higher intensity should give electrons more kinetic energy, and there should be a time delay at low intensity while energy accumulates. Experiments flatly contradicted every one of these predictions.
The experimental results revealed that below a threshold frequency (f_0), no electrons are emitted regardless of intensity. Above f_0, electrons are emitted immediately with no time delay, even at very low intensity. The maximum kinetic energy of the ejected electrons depends on the frequency of the light, not on its intensity. Higher intensity increases only the number of ejected electrons (the current) but not their individual energy.
III. Einstein's Photon Explanation (1905)
Einstein proposed that light consists of discrete energy packets called photons, each carrying energy E = hf = hc/lambda. Each photon interacts with a single electron. The photon's energy is used first to overcome the work function (phi) of the metal, which is the minimum energy needed to free an electron from the surface, with any remaining energy going into the electron's kinetic energy. This gives the photoelectric equation: KE_max = hf - phi.
If hf < phi, no electrons can be emitted, which explains the threshold. The threshold frequency is f_0 = phi / h. The stopping potential V_0, the voltage needed to halt the most energetic electrons, satisfies eV_0 = KE_max = hf - phi. Increasing the intensity means more photons striking the surface, so more electrons are ejected and the current increases, but each photon still carries the same energy, so the maximum kinetic energy remains unchanged. Increasing the frequency means each photon carries more energy, so KE_max increases. Einstein received the Nobel Prize in 1921 for this explanation.
<image>Panel A: A diagram of the photoelectric effect setup. UV light illuminates a metal plate (cathode) in a vacuum tube. Ejected electrons travel to the anode, creating a measurable current. A variable voltage source can apply a retarding potential. Panel B: A graph of KE_max (or stopping potential V_0) versus frequency f of the incident light. The graph shows a straight line with slope h (Planck's constant) and x-intercept at the threshold frequency f_0. Below f_0, KE_max = 0 (no emission). The y-intercept gives -phi (work function). Three different metals are shown with different work functions but the same slope h. Panel C: A graph of photocurrent versus applied voltage for two different light intensities (same frequency), showing that higher intensity gives higher saturation current but the same stopping potential.</image>
IV. The Photon: Properties and Evidence
A photon has energy E = hf, momentum p = h/lambda = hf/c = E/c, zero rest mass, and spin 1 (making it a boson). It always travels at speed c in all reference frames.
Compton scattering (1923) provided further compelling evidence for the photon picture. When X-ray photons scatter off electrons, the scattered photon has a longer wavelength than the incident one. The wavelength shift is Delta lambda = (h/m_e c)(1 - cos theta), where h/(m_e c) = 2.43 x 10^-12 m is the Compton wavelength of the electron. This result is explained perfectly by treating the photon as a particle with momentum p = h/lambda, colliding with an electron in a manner consistent with conservation of energy and momentum. Classical wave theory could not account for the wavelength shift.
V. Wave-Particle Duality and de Broglie Waves
In 1924, de Broglie proposed a striking symmetry: if light (traditionally a wave) can behave as a particle, then particles should exhibit wave behavior. The de Broglie wavelength of any particle is lambda = h/p = h/(mv) for a non-relativistic particle. This applies to all matter: electrons, protons, atoms, and even macroscopic objects. For macroscopic objects, lambda is immeasurably small (for example, a baseball has lambda of roughly 10^-34 m), but for electrons, lambda is on the order of atomic dimensions (approximately 0.1 nm), making wave effects significant.
Experimental confirmation came in 1927 with the Davisson-Germer experiment, which demonstrated electron diffraction from a nickel crystal. The diffraction pattern was consistent with lambda = h/p. Since then, diffraction has been observed with neutrons, atoms, and even large molecules such as C60 buckyballs.
Wave-particle duality is the recognition that all quantum objects exhibit both wave and particle properties. The wave nature is most prominent when lambda is comparable to the size of the structures encountered, while the particle nature is most evident when individual detection events are observed.
VI. The Heisenberg Uncertainty Principle
The Heisenberg uncertainty principle (1927) states that there are fundamental limits to how precisely certain pairs of physical quantities can be simultaneously known. For position and momentum: Delta x Delta p >= h/(4 pi) = hbar/2. For energy and time: Delta E Delta t >= hbar/2, where hbar = h/(2 pi) = 1.055 x 10^-34 J s.
This is not a statement about measurement limitations or instrumental imprecision. It is a fundamental property of nature: a particle simply does not simultaneously possess a precise position and a precise momentum.
The implications are far-reaching. Electrons cannot have well-defined orbits in atoms, leading to the orbital (probability cloud) model. The concept of zero-point energy arises because a confined particle cannot have zero kinetic energy. The smaller the confinement region (Delta x), the larger the momentum uncertainty (Delta p), and hence the larger the minimum kinetic energy. The uncertainty principle also explains why atoms are stable: if the electron spiraled into the nucleus, Delta x would become very small, making Delta p and the kinetic energy very large, which would drive the electron back out.
<image>Panel A: A visual representation of the de Broglie wavelength. An electron with momentum p = mv is shown alongside its associated matter wave with wavelength lambda = h/p. A table compares de Broglie wavelengths for different objects: electron at 100 eV (lambda ≈ 0.12 nm), proton at 100 eV (lambda ≈ 0.003 nm), baseball at 40 m/s (lambda ≈ 10^-34 m). Panel B: Illustration of the uncertainty principle — a narrow slit (small Delta x) causes a wide diffraction pattern (large Delta p_x), while a wide slit (large Delta x) produces a narrow pattern (small Delta p_x). This demonstrates the tradeoff between position and momentum certainty.</image>
