Engineering Physics - BAPHY105 Study Notes

Foundational Principles of Quantum Computing

Core Concepts: Quantum computation is built upon four fundamental principles that distinguish it from classical computing: the quantum state, superposition, measurement, and entanglement. These principles dictate how information is represented, manipulated, and extracted.
Quantum Mechanics as a Tool: Quantum mechanics provides the mathematical language (wavefunctions, probability amplitudes, and linear operators) necessary to understand how qubits behave and why quantum algorithms, such as those by Shor and Grover, provide proven speedups.
Value for Computer Science: Quantum computing extends the foundations of computation by offering a richer model than classical bits and Boolean logic. It broadens complexity theory by revealing new classes of efficiently solvable problems and supports the development of quantum cryptography for secure communication.
Course Structure: The study of Engineering Physics (BAPHY105) is divided into five modules:

Wave–Particle Duality: Historical origins and failures of classical physics.
Mathematical Foundations: Formalism including vector spaces, operators, and probability amplitudes.
Postulates of Quantum Mechanics: The fundamental principles governing quantum systems.
Applications: Physical systems and phenomena where quantum mechanics is central.
Elements of Quantum Computing: Connecting quantum mechanics to qubits, gates, and algorithms.

Transition from Classical to Modern Physics

Successes of Classical Physics: Classical physics comprises Newton’s laws of motion and Maxwell’s equations. Newton’s equations describe the trajectories of macroscopic objects (stones, missiles, rockets, planets) and the motion of molecules in cylinders. Maxwell’s equations describe the behavior of electric and magnetic fields, showing light is an electromagnetic wave.
Failures at Subatomic Scales: Classical physics cannot explain why electrons revolve in specific orbits around a nucleus. Furthermore, Maxwell’s equations fail to explain the blackbody radiation spectrum, the Photoelectric effect, and the Compton effect.
Modern Physics Scope: Modern physics encompasses both the quantum mechanical framework and relativistic physics, addressing dynamics beyond the Newtonian formulation.

The Dual Nature of Radiation: Wave Characteristics

Wave Phenomena: Radiation was long understood as a wave due to interference, diffraction, and polarization.
Young’s Double-Slit Experiment (1801): Designed by Thomas Young to prove light behaves as a wave. It demonstrated two key wave properties:
- Diffraction: The spreading and bending of waves around obstacles or through narrow openings.
- Interference: The overlap of coherent waves producing alternating bright and dark regions through constructive and destructive superposition.
Experimental Setup: A monochromatic light source generates planar waves of wavelength $\lambda$ . Two screens are used; the first has two slits separated by distance $d$ , and the second is at distance $D$ such that $D \gg d$ .
Mathematical Conditions for Interference:
- Path Difference: $\Delta x = d \sin(\theta)$ , where $\theta$ is the angle from the midpoint of the slits.
- Constructive Interference (Bright Fringes): Occurs when the path difference is an integral multiple of the wavelength: $d \sin(\theta) = n\lambda$ for $n = 0, 1, 2, 3, ...$
- Destructive Interference (Dark Fringes): Occurs when path difference is an odd number of half wavelengths: $d \sin(\theta) = (n + \frac{1}{2})\lambda$ for $n = 0, 1, 2, 3, ...$
Fringe Width ( $\beta$ ): Defined as the distance between two successive bright or dark fringes.
- Position of the nth bright fringe: $y_n = \frac{n\lambda D}{d}$
- Formula: $\beta = y_{n+1} - y_n = \frac{\lambda D}{d}$
- The fringe width is independent of the order of the fringe.

The Dual Nature of Radiation: Particle Characteristics

Blackbody Radiation: A blackbody is an idealized object that perfectly absorbs all incident electromagnetic radiation and re-emits thermal radiation based solely on its temperature $T$ .
Key Observed Properties:
1. The wavelength at maximum intensity ( $\lambda_{max}$ ) decreases as temperature increases.
2. The area under the spectral radiance curve (total power emitted) increases with temperature.
Historical Laws and Their Limits:
- Wien’s Displacement Law: $\lambda_{max} = \frac{b}{T}$ , where $b = 2.892 \times 10^{-3}\,m\,K$ .
- Stefan-Boltzmann Law: Total intensity $I = \sigma T^4$ , where $\sigma = 5.67 \times 10^{-8}\,J\,s^{-1}\,m^{-2}\,K^{-4}$ .
- Rayleigh-Jeans Law: $B_{\lambda}(T) = \frac{8\pi k_B T}{\lambda^4}$ . Matches experiments only at long wavelengths (low frequencies).
- Ultraviolet Catastrophe: The failure of the Rayleigh-Jeans law at low wavelengths/high frequencies, where classical theory predicted infinite energy density.
- Wien’s Radiation Law: $B_{\lambda}(T) = \frac{8\pi hc}{\lambda^5} e^{-\frac{hc}{\lambda k_B T}}$ . Matches only at short wavelengths.
Planck’s Resolution (1900): Max Planck proposed that energy is quantized. Sources of radiation are atomic oscillators with discrete vibrational energies.
- Planck Postulate: $E = nh\nu$ , where $n = 1, 2, 3...$ and $h = 6.63 \times 10^{-34}\,Js$ .
- Planck’s Law of Radiation: $B_{\lambda}(T) = \frac{8\pi hc}{\lambda^5} \frac{1}{e^{\frac{hc}{\lambda k_B T}} - 1}$ . This formula perfectly describes the spectrum for all wavelengths.
- Frequency Domain Form: $B_{\nu}(T) = \frac{8\pi h\nu^3}{c^3} \frac{1}{e^{\frac{h\nu}{k_B T}} - 1}$ .
Compton Effect: Discovered by Arthur Holly Compton, showing particle nature through collisions between photons and electrons.
- Compton Shift Formula: $\Delta \lambda = \lambda' - \lambda = \frac{h}{m_e c} (1 - \cos(\theta))$ .
- Compton Wavelength: $\frac{h}{m_e c} = 0.2429 \times 10^{-11}\,m$ .
- The shift is minimum at $\theta = 0^{\circ}$ and maximum at $\theta = 180^{\circ}$ . This effect assumes elastic collisions and conservation of energy and momentum.
Photoelectric Effect: Light striking a metal surface ejects electrons.
- Threshold Frequency: Electrons are only ejected if light frequency exceeds a specific threshold.
- Intensity vs. Frequency: Increasing intensity increases the number of electrons, but not their kinetic energy. Kinetic energy increases solely with light frequency.
- Einstein’s Proposal: Light consists of discrete packets called photons of energy $E = h\nu$ . An electron is ejected only if the photon energy exceeds the material's work function.

Dual Nature of Matter

de Broglie Hypothesis: Louis de Broglie postulated that every moving particle is associated with a wavelength $\lambda = \frac{h}{p}$ .
- Relating Energy: Since kinetic energy $E = \frac{p^2}{2m}$ , then $\lambda = \frac{h}{\sqrt{2mE}}$ .
- Macroscopic vs. Microscopic: For a cricket ball ( $m = 0.15\,kg$ , $v = 44.7\,m/s$ ), $\lambda \approx 1.0 \times 10^{-34}\,m$ , which is unobservable. For an electron moving at $0.1c$ , $\lambda \approx 2.4 \times 10^{-11}\,m$ , which is comparable to atomic dimensions.
- Charged Particles: For a particle of charge $q$ accelerated by potential $V$ , kinetic energy $E = |q|V$ , resulting in $\lambda = \frac{h}{\sqrt{2m|q|V}}$ .
Heisenberg Uncertainty Principle: A consequence of wave-particle duality. A quantum particle cannot have its position ( $\Delta x$ ) and momentum ( $\Delta p_x$ ) measured simultaneously with arbitrary precision.
- Formula: $\Delta x \Delta p_x \ge \frac{h}{4π}$
- Wave Packet Logic: A monochromatic wave has defined momentum but infinite position uncertainty ( $\Delta x \rightarrow \infty$ ). Precise localization ( $\Delta x = 0$ ) requires superimposing many wavelengths, leading to infinite momentum uncertainty ( $\Delta p \rightarrow \infty$ ).
Electron Double-Slit Experiment: Electrons produce interference fringes ( $P_{12}(x) \neq P_1(x) + P_2(x)$ ). This holds even when electrons are sent one at a time, meaning each electron interferes with itself.
Effect of Measurement: Detecting which slit an electron passes through causes the interference pattern to disappear, and the distribution becomes a simple sum: $P_{12}(x) = P_1(x) + P_2(x)$ . Gaining "which-path" information destroys the wave behavior.

Stern-Gerlach Experiment and Spin

Context: Demonstrated in 1921 by Otto Stern and Walter Gerlach. It shows that measurement alters a system even without wave interference.
Discovery of Spin: Electrons possess an intrinsic angular momentum called spin. Along a chosen direction, spin components take only discrete (quantized) values.
Experimental setup: A beam of silver atoms passes through an inhomogeneous (non-uniform) magnetic field gradient $\frac{dB_z}{dz}$ .
Classical Expectation: Classically, magnetic moments ( $\mu_z$ ) should have a continuous range of orientations, leading to a continuous vertical smear on a detection plate.
Quantum Outcome: The beam splits into exactly two sharp spots. For a spin- $\frac{1}{2}$ particle, $\mu_z = \pm \mu_B$ , where $\mu_B = \frac{eh}{4π m}$ is the Bohr magneton.
Force on Atoms: The force is $F_z = \pm \mu_B \frac{dB_z}{dz}$ . This force creates two distinct trajectories corresponding to spin-up and spin-down states.
Silver Atoms vs. Electrons: Silver atoms are used because they are electrically neutral and experience no Lorentz force. Their valence electron is in an s-orbital (orbital magnetic moment is zero), so the splitting is purely due to intrinsic spin.

The Wavefunction and the Schrödinger Equation

Wavefunction ( $\Psi$ ): A mathematical object replacing classical position and momentum as the complete description of a quantum state.
Crucial Properties:
1. Normalization: The total probability of finding the particle must be 1: $\int_{-\infty}^{\infty} |\Psi(x, t)|^2 dx = 1$ .
2. Probability Interpretation: The probability density is $|\Psi(x, t)|^2 = \Psi^*(x, t)\Psi(x, t)$ , which is always real and non-negative.
3. Continuity and Single-Valuedness: $\Psi$ must be continuous to avoid unphysical jumps in probability and single-valued to ensure probability is non-ambiguous at any point.
4. Superposition: If $\Psi_1$ and $\Psi_2$ are solutions, then $\Psi = c_1\Psi_1 + c_2\Psi_2$ is also a valid state.
Schrödinger Wave Equation: The fundamental equation of motion for matter waves.
- Time-Dependent Equation: $i\hbar \frac{\partial \Psi(x, t)}{\partial t} = \hat{H}\Psi$
- Hamiltonian Operator (1D): $\hat{H} = -\frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2} + V(x, t)$
- Complete Form: $i\hbar \frac{\partial \Psi(x, t)}{\partial t} = -\frac{\hbar^2}{2m} \frac{\partial^2 \Psi(x, t)}{\partial x^2} + V(x, t)\Psi(x, t)$
Modern Unified Framework: In modern theory, every quantum state is an abstract vector in a Hilbert space. The position-space wavefunction $\Psi(x, t)$ is one representation of this state vector.

Questions & Discussion

Problem 1 (Double Slit): $\lambda = 500\,nm$ , $n = 1$ , $\theta = 12^{\circ}$ . Calculate separation $d$ . (Use $d = \frac{n\lambda}{\sin(\theta)}$ ).
Problem 2 (Diffraction): $d = 0.25\,mm$ , $\lambda = 650\,nm$ , $n = 2$ . Calculate angle $\theta$ .
Problem 3 (Double Slit): $n = 1$ , $\theta = 8^{\circ}$ , $d = 0.30\,mm$ . Find $\lambda$ .
Problem 4 (Star Radiance): Maximum emission at $480\,nm$ . Find surface temperature $T$. (Use Wien's law: $T = \frac{b}{\lambda_{max}}$ ).
Problem 5 (Power): Find total power per unit area at $T = 1200\,K$ . (Use Stefan-Boltzmann: $I = \sigma T^4$ ).
Problem 6 (Spectral Radiance): Determine $B(\lambda, T)$ for $\lambda = 1.0\,μm$ at $T = 3000\,K$ . (Use Planck's Law).
Problem 7 (Compton): Scattering of X-ray photon ( $\lambda = 0.071\,nm$ ) at $\theta = 60^{\circ}$ . Calculate Compton shift $\Delta \lambda$ , scattered wavelength $\lambda'$ , and electron kinetic energy.
Problem 8 (Electron Acceleration): Calculate $\lambda$ for an electron accelerated through $150\,V$ . (Use $\lambda = \frac{1.227}{\sqrt{V}}\,nm$ for non-relativistic electrons).
Problem 9 (Proton Acceleration): Calculate $\lambda$ for a proton at $5\,kV$ .
Problem 10 (Target Potential): Potential $V$ required for an electron to have $\lambda = 0.10\,nm$ .