Quantum Physics: Chapter 2

• Chapter 2 explores Quantum Physics

Objectives

• To learn experimental results explained by the particle theory of electromagnetic waves
• To learn the particle properties of waves and the wave properties of particles
• To understand the uncertainty principle

Blackbody Radiation and Planck's Hypothesis

• Electromagnetic radiation emitted by a black body is called black-body radiation

Basic Laws of Radiation

• All objects emit radiant energy
• Hotter objects emit more energy per unit area than colder ones.
• Stefan's Law: P = σAeT⁴, where P is power, σ is the Stefan-Boltzmann constant, A is the surface area, e is the emissivity, and T is temperature
• The peak of the wavelength distribution shifts to shorter wavelengths as black body temperature increases
• Wien's Displacement Law: λmT = constant
• Rayleigh-Jeans Law states the intensity or power per unit area I(λ,T)dλ emitted in the wavelength interval λ to λ+dλ from a blackbody: I(λ,T) = (2πckBT) / λ⁴
• Rayleigh-Jeans Law agrees with experimental measurements only for long wavelengths
• Rayleigh-Jeans Law predicts an energy output that diverges towards infinity as wavelengths become smaller, known as the ultraviolet catastrophe.
• Planck's Law states the intensity or power per unit area I(λ,T)dλ emitted in the wavelength interval λ to λ+dλ from a blackbody: I(λ,T) = (2πhc²/λ⁵) * (1 / (e^(hc/λkT) - 1) )
• Assumptions of Planck's Law include energy of an oscillator in cavity walls being given by En = nhf
• Emission/absorption of energy will be integral multiples of hf, where h is Planck's constant and f is frequency
• Planck's Law resolves the ultraviolet catastrophe because the denominator tends to infinity faster than the numerator (λ⁻⁵), thus aligning with experimental observations: I(λ, T) → 0 as λ → 0
• For very large λ, I(λ, T) → 0 as λ → ∞, which can be approximated by the equation exp(hc/λkT) - 1 ≈ hc/λkT, leading to I(λ, T) → 2πcλ⁻⁴kT
• Planck's constant was derived from a fit between Planck's law and experimental data: h = 6.626 × 10⁻³⁴ J·s

Photoelectric Effect

• Photoelectric Effect is the ejection of electrons from the surface of certain metals when irradiated by electromagnetic radiation of suitable frequency
• Classical predictions for the photoelectric effect: electron ejection should be frequency independent, KE of electrons should increase with light intensity, measurable time interval between incidence of light and ejection of photoelectrons, and KMAX should not depend on the incident light frequency.
• Experimental observations of the photoelectric effect: no photoemission below threshold frequency, KMAX is independent of light intensity, effect is instantaneous, KE of the most energetic photoelectrons is KMAX = eΔVs and increases with increasing frequency.

Einstein's Interpretation

• Classical predictions contradict actual experimental results
• Electromagnetic waves carry discrete energy packets called photons
• The energy E per packet depends on frequency f: E = hf
• More intense light corresponds to more photons, not higher energy photons
• Each photon moves in vacuum at the speed of light (c = 3 × 10⁸ m/s) and carries a momentum p = E/c
• Einstein's photoelectric equation: Kmax = hf − ϕ, where ϕ is the work function.

Compton Effect

• When X-rays are scattered by free/nearly free electrons, they suffer a change in their wavelength which depends on the scattering angle
• Classical predictions regarding the effect of oscillating electromagnetic waves on electrons include: oscillations in electrons in all directions, radiation pressure causing electrons to accelerate in the direction of propagation, different electrons moving at different speeds after the interaction, and the scattered wave frequency showing Doppler-shifted values.
• Photon is treated as a particle having energy E = hf = hc/λ₀ and zero rest energy
• In the scattering process, the total energy and total linear momentum of system is conserved
• λ₀ = wavelength of the incident photon
• p₀ = h/λ₀ = momentum of the incident photon
• E₀ = hc/λ₀ = energy of the incident photon
• λ' = wavelength of the scattered photon
• p' = h/λ' = momentum of the scattered photon
• E' = hc/λ' = energy of the scattered photon

Compton Shift

• Conservation of energy in Compton scattering: E₀ = E' + K
• Conservation of momentum: p₀ = p'cosθ + pcosΦ (x-component) and 0 = p'sinθ − psinΦ (y-component)
• Relativistic equations: v = speed of electron, m = mass of electron, p = γmv = momentum of the electron
• γ = 1 / √(1 - v²/c²), E = √(p²c² + m²c⁴) = total relativistic energy of the electron
• K = E − mc² = kinetic energy of the electron
• The Compton shift formula: λ' − λ₀ = h/mc (1 − cosθ)

Photons and Electromagnetic Waves

• Light exhibits diffraction and interference phenomena explicable in terms of wave properties
• The Photoelectric effect and Compton Effect can only be explained taking light as photons/particles
• Light's true nature is best described by considering both wave and particle nature together, which complement each other.

de Broglie Hypothesis

• Wavelength associated with a particle of mass m moving with velocity v: λ = h/p = h/mv
• The momentum (p) of an electron accelerated through a potential difference of ΔV: p = mv = √(2meΔV)
• Frequency of the matter wave associated with the particle: f = E/h, where E is the total relativistic energy of the particle.

Davisson and Germer Experiment

• Davisson and Germer Experiment verified the de Broglie hypothesis
• Central assumption: electron as a wave
• Idea: determine the wavelength of the electron using Bragg's diffraction law and compare it with the de Broglie's wavelength
• A beam of electrons is produced by a heated filament and accelerated by a potential V (around 54V)
• The electron beam is then scattered by a nickel crystal
• Intensities of the scattered electrons are measured as a function of the angle $
• The angle $ is between the incident beam and the scattered beam
• Bragg's diffraction law: d sin φ = nλ
• d is the inter-atomic spacing in nickel which is roughly 0.215 nm
• For the first diffraction maximum, n equals 1
• Substituting experimental numbers, the electron wavelength is approximately λ = 0.165 nm
• Conservation of energy is used to calculate the electron wavelength according to de Broglie's hypothesis: ½mv² = eV
• The momentum of the electron is defined p = mv = √(2meV)
• Wavelength is defined as λ= h/p, which expands to h/(mv) further expands to h/√(2meV)
• Substituting, for V=54 V, the result is λ = 0.167 nm
• The derived wavelength agrees with that from the Bragg's diffraction law

Quantum Particle

• Adding a large number of waves with constructive interference in a small localized region of space forms a wavepacket, representing a quantum particle
• Mathematical representation of a wave packet formed by two waves:
• y₁ = A cos(k₁x − ω₁t) and y₂ = A cos(k₂x − ω₂t), where k = 2π/λ and ω = 2πf
• The resultant wave is given by y = y₁ + y₂
• y = 2A[cos((Δk/2)x − (Δω/2)t) cos(((k₁+k₂)/2)x − ((ω₁+ω₂)/2)t)], where Δk = k₁ − k₂ and Δω = ω₁ − ω₂
• Phase speed: the speed with which wave crest of individual wave moves, (vp = fλ) or (vp = ω/k)
• Group speed: the speed of the wave packet, (vg= Δω/Δk)
• Relation between group speed (vg) and phase speed (vp)is vp = ω/k = fλ, where ω = kvp
• vg = dw/dk = d(kvp)/dk = k (dvp/dk) + vp
• By substituting for k, vg = vp − λ( dvp/dλ)
• The relation between group speed (vg) and particle speed (u): ω = 2πf = (2π/h)E, and k = (2π/λ) = p/(h/π) = 2πp/h
• vg = dw/dk = (2π/h) dE/dp
• For a classical particle moving with speed u, the kinetic energy E is given by E = (½)mu² = p²/2m and dE = (2p dp)/2m, or dE/dp = p/m = u
• Group velocity is defined as vg = dw/dk= dE/dp= u

Double-Slit Experiment Revisited

• d sin θ = mλ, where m is the order number and λ is the electron wavelength
• Electrons are detected at localized spots on the detector screen
• The probability of arrival at the spot is determined by finding the intensity of two interfering waves

Uncertainty Principle

• Heisenberg uncertainty states it is fundamentally impossible to make simultaneous measurements of a particle's position and momentum with infinite accuracy, (Δx)(Δpx) ≥ h/4π
• A relation stating the uncertainty principle related to energy and time is (ΔE)(Δt) ≥ h/4π.