PHY1008 Unit 1: Origin of Quantum Theory Study Notes

Introduction to Quantum Mechanics and Classical Limitations

• Foundations of Modern Physics: Quantum mechanics serves as the bedrock for modern physics, chemistry, and biology. It explains phenomena that classical physics cannot, such as blackbody radiation, the photoelectric effect, and Compton scattering.

• Scope of Application: It applies to diverse fields including solid-state, molecular, atomic, nuclear, and particle physics, as well as optics, thermodynamics, and statistical mechanics.

• Classical Characterization:

  • Particles: Described by energy $E$ and momentum $p$, following deterministic trajectories (e.g., a bullet in a straight line).

  • Waves: Described by amplitude and wave vector $\mathbf{k}$; they exhibit interference and diffraction.

• Failure of Classical Physics: Classical theories failed to explain microscopic phenomena, leading to the introduction of wave-particle duality and energy quantization.

Blackbody Radiation

• Definition: A blackbody is an idealized object that absorbs all incident radiation and emits radiation when heated.

• Thermal Radiation: Solid objects emit thermal radiation with a continuous frequency distribution when heated.

• Spectral Energy Density ($u(\nu, T)$): Represents the energy density per unit frequency at temperature $T$. Observations show a pronounced maximum at a specific frequency, which shifts higher as temperature increases.

• Experimental Failures of Classical Theories:

  • Wien’s Law: Given by the formula $u(\nu, T) = A\nu^3 e^{-\beta\nu/T}$. It fits experimental data at high frequencies but fails at low frequencies.

  • Rayleigh-Jeans Law: Given by $u(\nu, T) = \frac{8\pi\nu^2}{c^3} kT$. It fits data at low frequencies but diverges at high frequencies, a phenomenon known as the "ultraviolet catastrophe."

• Planck’s Quantum Hypothesis (1900):

  • Max Planck proposed that energy exchange between matter and radiation occurs in discrete packets called quanta.

  • The energy of a single quantum is $E = h\nu$, where $h$ is Planck’s constant.

  • Planck’s Formula: $u(\nu, T) = \frac{8\pi\nu^2}{c^3} \frac{h\nu}{e^{\frac{h\nu}{kT}} - 1}$.

  • This formula reduces to Rayleigh-Jeans at low frequencies and Wien's Law at high frequencies, ensuring total energy density is finite and marking the birth of quantum mechanics.

The Photoelectric Effect

• Description: The emission of electrons from a material's surface when exposed to light, discovered by Heinrich Hertz (1887) and explained by Albert Einstein (1905).

• Key Experimental Observations:

  • Threshold Frequency ($\nu_0$): No electrons are emitted if the incident light frequency is below a specific threshold, regardless of light intensity.

  • Instantaneous Emission: Electrons are ejected almost immediately, even at very low intensities.

  • Kinetic Energy: The maximum kinetic energy ($K_{max}$) of emitted electrons depends on light frequency, not intensity.

  • Current: The number of emitted electrons (photoelectric current) is proportional to the light intensity.

• Einstein’s Photon Theory:

  • Proposed light consists of discrete packets called photons with energy $E = h\nu$.

  • Photoelectric Equation: $K_{max} = h\nu - W$, where $W$ is the work function (minimum energy required to eject an electron).

  • Stopping Potential ($V_s$): Related to kinetic energy by $eV_s = K_{max}$.

  • Threshold Frequency Relation: $h\nu_0 = W$.

• Calculation Example:

  • Given: Work function $\phi = 2.1\,eV$, Wavelength $\lambda = 400\,nm$.

  • Step 1: Frequency $\nu = \frac{c}{\lambda} = \frac{3 \times 10^8\,m/s}{400 \times 10^{-9}\,m} = 7.5 \times 10^{14}\,Hz$.

  • Step 2: Photon Energy $E = h\nu \approx 4.97 \times 10^{-19}\,J = 3.1\,eV$.

  • Step 3: $K_{max} = 3.1\,eV - 2.1\,eV = 1.0\,eV$.

de-Broglie Hypothesis and Wave-Particle Duality

• Hypothesis (1924): Louis de Broglie suggested that all particles have an associated wave, termed a matter wave or de-Broglie wave.

• De-Broglie Wavelength ($\lambda$):

  • Defined as $\lambda = \frac{h}{p}$, where $p$ is momentum.

  • For non-relativistic particles: $\lambda = \frac{h}{\sqrt{2mK}}$.

  • For thermal particles at temperature $T$: $\lambda = \frac{h}{\sqrt{3mk_B T}}$.

  • For relativistic particles: $\lambda = \frac{h}{\gamma mv}$, where $\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}$.

• Comparison Examples:

  • Electron in an Atom: Velocity $v \approx 2.2 \times 10^6\,m/s$, mass $m = 9.109 \times 10^{-31}\,kg$. Calculated $\lambda \approx 0.33\,nm$. Comparable to atomic size, making wave properties significant.

  • Baseball: Mass $0.15\,kg$, velocity $40\,m/s$. Calculated $\lambda \approx 1.1 \times 10^{-35}\,m$. Negligible for macroscopic objects.

• Experimental Verification:

  • Double Slit Experiment: Demonstrates interference patterns for light and particles.

  • Davisson-Germer Experiment: Confirmed electron diffraction using a nickel crystal and an electron detector.

• Complementarity Principle (Niels Bohr): Particle and wave natures are complementary aspects. The observed nature depends on the experimental setup; for example, observing which slit a particle passes through destroys the wave-like interference pattern.

• Velocities:

  • Phase velocity ($v_p$): Speed of a single wave phase, $v_p = \frac{\omega}{k}$.

  • Group velocity ($v_G$): Speed of the overall wave packet envelope, $v_G = \frac{d\omega}{dk}$.

Heisenberg’s Uncertainty Principle (HUP)

• Core Definition: It is fundamentally impossible to simultaneously determine the exact position and momentum of a particle.

• Mathematical Forms:

  • Position and Momentum: $\Delta x \Delta p \geq \frac{\hbar}{2}$, where $\hbar = \frac{h}{2\pi}$.

  • Energy and Time: $\Delta E \Delta t \geq \frac{\hbar}{2}$.

  • Angular Momentum and Angle: $\Delta L \Delta \theta \geq \frac{\hbar}{2}$.

• Applications of HUP:

  • Spectral Line Width: Because atomic energy levels have a finite lifetime ($\Delta t$), spectral lines possess an inherent width ($\Delta \nu$). For $\Delta t \sim 10^{-8}\,s$, $\Delta \nu \geq 10^6\,Hz$.

  • Electron-Nucleus Constraint: To reside within a nucleus (size $\Delta x \sim 10^{-15}\,m$), an electron would need an energy $E \geq 100\,MeV$. Since beta particles only have $2-3\,MeV$, electrons cannot exist inside the nucleus.

The Wave Function and Schr¨dinger Equation

• Definition: The wave function $\psi(x, t)$ is a complex-valued function describing the quantum state of a particle.

• Schr¨dinger Equation: $i\hbar \frac{\partial \psi}{\partial t} = -\frac{\hbar^2}{2m} \frac{\partial^2 \psi}{\partial x^2} + V(x)\psi$.

• Mandatory Properties:

  • Complex-Valued: Contains a real and imaginary part.

  • Continuous: Both $\psi(x, t)$ and its first derivative must be continuous.

  • Single-Valued: Must have one unique value at every point.

  • Normalizable/Square-Integrable: The total probability must be finite: $\int_{-\infty}^{\infty} |\psi(x, t)|^2 dx = 1$.

• Born’s Probability Interpretation (1926): The quantity $|\psi(x, t)|^2$ represents the probability density of finding the particle at position $x$ at time $t$.

Normalization and Probability Current

• Normalization Procedure:

  1. Integrate the square magnitude: $\int |\psi(x)|^2 dx$.

  2. Multiply $\psi(x)$ by a normalization constant $A = \sqrt{\frac{1}{\int |\psi(x)|^2 dx}}$.

  • Calculated Normalization Weights:

  • For $\psi(x) = Ae^{-\alpha x^2}$, $A = (\frac{2\alpha}{\pi})^{1/4}$.

  • For $\psi(x) = Ae^{-\alpha |x|}$, $A = \sqrt{\alpha}$.

• Probability Current Density ($\mathbf{j}(x, t)$): Describes the flow of probability over time and space, defined as:   $\mathbf{j}(x, t) = \frac{\hbar}{2im} \left( \psi^* \frac{\partial \psi}{\partial x} - \psi \frac{\partial \psi^*}{\partial x} \right)$

• Continuity Equation: Ensures conservation of probability, expressed as:   $\frac{\partial \rho(x, t)}{\partial t} + \frac{\partial \mathbf{j}(x, t)}{\partial x} = 0$, where $\rho(x, t) = |\psi(x, t)|^2$.

• Example Case Study: Free Particle:

  • Wave function: $\psi(x, t) = A e^{i(kx - \omega t)}$.

  • Calculated Gradient: $\frac{\partial \psi}{\partial x} = ik \psi$.

  • Calculated Gradient Conjugate: $\frac{\partial \psi^*}{\partial x} = -ik \psi^*$.

  • Resulting Current: $\mathbf{j}(x, t) = \frac{\hbar k}{m} |A|^2$.

• Example Case Study: Gaussian Wave Packet:

  • Wave function: $\psi(x, t) = (\frac{2}{\pi a^2})^{1/4} e^{i(k_0 x - \omega t)} e^{-\frac{(x - v_g t)^2}{a^2 + \frac{2i\hbar t}{m}}}$.

  • Resulting Current: $\mathbf{j}(x, t) = \frac{\hbar k_0}{m} |\psi|^2$, where the magnitude is proportional to the group velocity $v_g = \frac{\hbar k_0}{m}$.