Advanced Mathematical Physics and Quantum Electron Theory of Solids

Complex Numbers

• Fundamental Definition: A complex number $z$ is represented in rectangular form as $z = x + iy$, where $x$ is the real part and $y$ is the imaginary part.

• Argand Diagram: A geometric representation of complex numbers where the horizontal axis represents the real part and the vertical axis represents the imaginary part. A point $P$ can be identified by coordinates $(x, iy)$ or polar coordinates $(r, \theta)$.

• Polar Form: $z = r(\text{cos}(\theta) + i\text{sin}(\theta))$. Here, $r$ is the modulus and $\theta$ is the argument.

• Modulus of $z$: Denoted as $|z|$ or $r$, it is defined as |z| = \text{modulus of } z = \text{\sqrt{x^2 + y^2}}.

• Argument of $z$: Denoted as $\theta$ or $\text{arg}(z)$, defined as $\theta = \text{tan}^{-1}(\frac{y}{x})$. The standard range is typically -\pi < \theta \le \pi.

• Example Representation: To represent $5 - 3i$ and $5 + 3i$ on an Argand diagram, $5 + 3i$ is in the first quadrant, while its conjugate $5 - 3i$ is in the fourth quadrant.

• Complex Conjugate ($z^*$): If $z = x + iy$, then the complex conjugate is $z^* = x - iy$.

Properties and Fundamental Laws of Complex Algebra

• Equality: If two complex numbers are equal, then their real parts must be equal and their imaginary parts must be equal ($x_1 = x_2$ and $y_1 = y_2$).

• Addition: $(x_1 + iy_1) + (x_2 + iy_2) = (x_1 + x_2) + i(y_1 + y_2)$.

• Commutative Law: $z_1 + z_2 = z_2 + z_1$ and $z_1 z_2 = z_2 z_1$.

• Associative Law: $z_1 + (z_2 + z_3) = (z_1 + z_2) + z_3$ and $z_1 (z_2 z_3) = (z_1 z_2) z_3$.

• Additive Identity: The complex number $0 = 0 + i0$ such that $z + 0 = z$.

• Additive Inverse: For $z = x + iy$, the inverse is $-z = -x - iy$ such that $z + (-z) = 0$.

• Multiplicative Inverse: For $z' = x' + iy'$, if $zz' = 1$, then $z' = \frac{1}{x + iy} = \frac{x - iy}{x^2 + y^2}$. Thus, $x' = \frac{x}{x^2 + y^2}$ and $y' = \frac{-y}{x^2 + y^2}$.

• Distributive Law: $(z_1 + z_2)z_3 = z_1z_3 + z_2z_3$.

• Identity of Multiplication: $z \times 1 = z$.

• Sum of Conjugates: The complex conjugate of a sum is the sum of the conjugates: \text{\overline{z_1 + z_2}} = z_1^* + z_2^*.

• Product of Conjugates: $z \cdot z^* = (x + iy)(x - iy) = x^2 + y^2 = |z|^2$.

• Real and Imaginary Parts via Conjugates:

  • $z + z^* = 2x = 2\text{Re}(z)$.

  • $z - z^* = 2iy = 2i\text{Im}(z)$.

Vector Spaces

• Definition: A vector space $V$ is a collection of vectors in $n$-dimensional space over a scalar field $F$ that satisfies:

  • Closure Property: Under addition ($u + v \in V$) and scalar multiplication ($au \in V$).

  • Commutative Law: Under addition ($u + v = v + u$).

  • Associative Law: Under addition ($u + (v + w) = (u + v) + w$).

  • Existence of Identity: Additive identity $0$ exists.

  • Additive Inverse: For every $u$, there exists $-u$ such that $u + (-u) = 0$.

  • Distributive Properties: $a(u + v) = au + av$ and $(a + b)u = au + bu$.

  • Associative Law (Scalars): $a(b \cdot u) = (ab)u$.

  • Identity of Scalar Multiplication: $v \cdot 1 = v$.

• Linear Independence: A subset $B$ is linearly independent if for distinct vectors $u_1, u_2, \dots, u_m$ and scalars $c_1, c_2, \dots, c_m$, the equation $\sum c_i u_i = 0$ implies all $c_i = 0$.

• Properties of Sets:

  • Any set containing a linearly dependent subset is itself linearly dependent.

  • Any subset of a linearly independent set is linearly independent.

  • Any set containing the zero vector ($0$) is always linearly dependent.

• Basis and Dimension: A basis is a linearly independent set of vectors that spans the vector space $V$. The dimension is the number of elements in the basis.

• Examples of Vector Spaces:

  • Matrices: e.g., a $2 \times 3$ matrix set where addition and scalar multiplication are defined.

  • Polynomials: Sets of polynomials such as expressions of the form $a_0 + a_1x + a_2x^2 + \dots + a_nx^n$.

Hermite Polynomials (Orthogonal Polynomials)

• Context: Used in signal processing, Brownian motion probability, numerical analysis, quantum mechanics (harmonic oscillator), and system theory.

• Definitions (Rodrigues' Formula):

  • Probabilists' version: $H_n(x) = (-1)^n e^{\frac{x^2}{2}} \frac{d^n}{dx^n}(e^{-\frac{x^2}{2}})$.

  • Physicists' version (More commonly used): $H_n(x) = (-1)^n e^{x^2} \frac{d^n}{dx^n}(e^{-x^2})$.

• Generated Polynomials:

  • $H_0(x) = 1$

  • $H_1(x) = 2x$

  • $H_2(x) = 4x^2 - 2$

  • $H_3(x) = 8x^3 - 12x$

  • $H_4(x) = 16x^4 - 48x^2 + 12$

  • $H_5(x) = 32x^5 - 160x^3 + 120x$

  • $H_6(x) = 64x^6 - 480x^4 + 720x^2 - 120$

• Properties:

  • Parity: For even $n$, the function is symmetric (even parity). For odd $n$, the function is anti-symmetric (odd parity).

  • Orthogonality: $\int_{-\infty}^{\infty} H_m(x) H_n(x) e^{-x^2} dx = \sqrt{\pi} 2^n n! \delta_{mn}$. If $m \neq n$, the integral is $0$.

  • Recursive Relation: $H_n(x) = (2x - \frac{d}{dx}) H_{n-1}(x)$.

Legendre Polynomials

• Definition: A system of complete and orthogonal polynomials used in gravitational and Coulomb potential problems.

• Standardization: Defined over the interval $[-1, 1]$ with weight function $w(x) = 1$. Normalized such that $P_n(1) = 1$.

• Values:

  • $P_0(x) = 1$

  • $P_1(x) = x$

  • $P_2(x) = \frac{1}{2}(3x^2 - 1)$

• Orthogonality Condition: $\int_{-1}^{1} P_m(x) P_n(x) dx = \frac{2}{2n+1} \delta_{mn}$. For $m \neq n$, the integral vanishes.

• Generating Function: $(1 - 2xz + z^2)^{-\frac{1}{2}} = \sum_{n=0}^{\infty} P_n(x) z^n$, where |z| < 1.

• Differential Equation: $(1 - x^2) P_n''(x) - 2x P_n'(x) + n(n + 1) P_n(x) = 0$.

Quantum Mechanical Observables and Operators

• Classical vs. Quantum Measurement: In classical mechanics, measurement does not disturb the system. In quantum mechanics, measurement often disturbs the system, and accuracy is limited for conjugate variables.

• Heisenberg Uncertainty Principle: Simultaneous measurement of conjugate variables (e.g., position/momentum, time/energy) is impossible. Error is defined by $\Delta x \Delta p \ge \frac{\hbar}{2}$.

• Operators: A mathematical rule performed on a function $f$ that produces a function $g$. Order of operations matters ($[\hat{A}, \hat{B}] \neq 0$).

  • Position Operator: $\hat{x} = x$

  • Momentum Operator: $\hat{p} = -i\hbar \frac{d}{dx}$, where $\hbar = \frac{h}{2\pi}$.

  • Energy/Hamiltonian Operator: $\hat{H} = -\frac{\hbar^2}{2m} \nabla^2 + V$.

  • Laplacian Operator: $\nabla^2 = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}$.

Spherical Harmonics

• Context: Specific functions defined on the surface of a sphere, used for solving partial differential equations like the Schrodinger equation for a hydrogen atom.

• Spherical Coordinates: $(r, \theta, \phi)$ where 0 \le r < \infty, $0 \le \theta \le \pi$, and $0 \le \phi \le 2\pi$.

• Wavefunction Decomposition: $\psi(r, \theta, \phi) = R(r) \Theta(\theta) \Phi(\phi) = R(r) Y_l^m(\theta, \phi)$.

• Associated Legendre Polynomials: $P_l^m(\text{cos}(\theta))$ is used to define the angular part.

• Angular Momentum: $L = \sqrt{l(l + 1)} \hbar$. The $z$-component is $L_z = m\hbar$.

• Physical Significance: Spherical harmonics are the eigenfunction of the orbital angular momentum operator.

The Drude Model and Free Electron Gas

• Characteristics of Metals: High electrical and thermal conductivity, obedience to Ohm's Law ($J = \sigma E$), and positive coefficient of resistance.

• Wiedemann-Franz Law: The ratio of thermal conductivity ($K$) to electrical conductivity ($\sigma$) is proportional to temperature ($T$). $K/(\sigma T) = L$ (Lorentz number).

• Drude Assumptions (1900):

  • Metals consist of positive ion cores and valence electrons moving freely.

  • Repulsion between electrons is neglected.

  • Potential energy is assumed constant (zero), so total energy = kinetic energy.

  • Electron behavior follows Maxwell-Boltzmann statistics (distinguishable particles).

  • Electrons collide with ion cores, losing velocity after short periods.

• Electrical Conductivity Equation: $\sigma = \frac{ne^2 \tau}{m}$, where $n$ is charge density and $\tau$ is relaxation time (mean time between collisions).

• Thermal Conductivity: $K = \frac{1}{3} C_v v \lambda$. Using classical gas theory, $K = \frac{3 n k_B^2 T \tau}{2m}$.

• Limitations of Classical Free Electron Theory:

  • Cannot explain why only some crystals are metallic.

  • Cannot explain specific structures (e.g., cubic shape of iron).

  • Observed heat capacity is only $1\%$ of the calculated value.

  • Fails to explain paramagnetic behavior and temperature dependence of conductivity correctly.

Quantum Free Electron Gas (Sommerfeld Model)

• Approach: Problem treated quantum mechanically using Fermi-Dirac distribution and Pauli Exclusion Principle.

• 1-D Fermi Gas: Particles of mass $m$ confined in a box of length $L$ with infinite potential barriers.

  • Energy Levels: $E_n = \frac{\hbar^2}{2m} (\frac{n\pi}{L})^2$.

  • Fermi Level: The highest energy level occupied at absolute zero temperature ($0\,K$).

• 3-D Fermi Gas: Represented in $K$-space (reciprocal space).

  • Fermi Wavevector ($k_F$): $k_F = (3\pi^2 n)^{\frac{1}{3}}$, where $n = N/V$.

  • Fermi Velocity ($v_F$): $v_F = \frac{\hbar k_F}{m}$.

  • Fermi Energy ($E_F$): $E_F = \frac{\hbar^2 k_F^2}{2m}$.

• Density of States (DOS): The number of available energy states per unit energy range.

  • 3D Case: $D(E) = \frac{V}{2\pi^2} (\frac{2m}{\hbar^2})^{\frac{3}{2}} E^{\frac{1}{2}}$.

  • Quantum Confinement impacts DOS:

  • $3D$ (Bulk): Continuous curve proportional to $\sqrt{E}$.

  • $2D$ (Thin film/Quantum Well): Step-like function.

  • $1D$ (Nanowire/CNT): Discrete spikes.

  • $0D$ (Quantum Dot): Delta functions (discrete states).

• Fermi-Dirac Distribution at Temperature: $f(E) = \frac{1}{e^{(E - E_F)/k_B T} + 1}$.

  • At $T = 0\,K$: $f(E) = 1$ for E < E_F and $f(E) = 0$ for E > E_F.

  • At finite $T$: At $E = E_F$, $f(E) = 0.5$.