2.1 Complex numbers and their applications in quantum mechanics

Complex numbers are the backbone of quantum mechanics. They allow us to describe wave functions, superposition, and interference. Without them, we couldn't represent the weird, probabilistic nature of the quantum world.

In this section, we'll explore complex number properties, operations, and their geometric representation. We'll see how they're used in wave functions, quantum states, and observables. Understanding complex numbers is crucial for grasping quantum mechanics.

Complex Numbers: Properties and Arithmetic

Definition and Components

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• Complex numbers take the form a + bi, where a and b represent real numbers and i denotes the imaginary unit satisfying $i^2 = -1$
• Real part (Re) of complex number z = a + bi equals a
• Imaginary part (Im) of complex number z = a + bi equals b
• Complex conjugation defined as z* = a - bi for z = a + bi
  • Property: zz* = |z|²
• Modulus (absolute value) of complex number z = a + bi calculated using $|z| = \sqrt{a^2 + b^2}$

Geometric Representation and Operations

• Argand diagram provides geometric representation of complex numbers
  • Real part plotted on x-axis
  • Imaginary part plotted on y-axis
• Arithmetic operations maintain closure within complex number system
  • Addition: (a + bi) + (c + di) = (a + c) + (b + d)i
  • Subtraction: (a + bi) - (c + di) = (a - c) + (b - d)i
  • Multiplication: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
  • Division: (a + bi) / (c + di) = [(ac + bd) / (c² + d²)] + [(bc - ad) / (c² + d²)]i

Euler's Formula and Applications

• Euler's formula relates complex exponentials to trigonometric functions
  • $e^{i\theta} = \cos(\theta) + i\sin(\theta)$
• Fundamental in quantum mechanics and various mathematical applications
• Enables expression of complex numbers in polar form
  • z = r(cos(θ) + isin(θ)) = re^(iθ), where r = |z| and θ = arg(z)
• Facilitates manipulation of complex expressions involving trigonometric functions

Complex Numbers in Quantum Mechanics

Wave Functions and Probability

• Quantum mechanical wave functions ψ(x,t) generally complex-valued functions of position and time
• Modulus squared |ψ(x,t)|² represents probability density of finding particle at position x at time t
  • Ensures probabilities always real and non-negative
• Complex wave functions enable representation of interference phenomena
  • Example: Double-slit experiment interference patterns
• Time-dependent Schrödinger equation intrinsically involves complex numbers
  • $i\hbar\frac{\partial\psi}{\partial t} = \hat{H}\psi$
  • i ensures equation describes unitary time evolution

Stationary States and Superposition

• Stationary states represented by wave functions of form ψ(x,t) = ψ(x)e^(-iEt/ℏ)
  • E denotes energy eigenvalue
  • Time-independent part ψ(x) can be complex
• Superposition of quantum states involves complex linear combinations
  • General state: |ψ⟩ = c₁|ψ₁⟩ + c₂|ψ₂⟩, where c₁ and c₂ are complex coefficients
• Phase of complex wave function carries information about quantum state's dynamical properties
  • Example: Phase difference between components in superposition determines interference effects

Complex Numbers for Quantum States and Observables

Hilbert Space and Operators

• Quantum states represented by vectors in complex Hilbert space
  • Inner products defined using complex conjugation: ⟨ψ|φ⟩ = ∫ψ*(x)φ(x)dx
• Observables represented by Hermitian operators
  • Real eigenvalues but generally complex eigenvectors
  • Example: Momentum operator $\hat{p} = -i\hbar\frac{d}{dx}$ (Hermitian)
• Expectation value of observable A in state ψ given by ⟨A⟩ = ⟨ψ|Â|ψ⟩
  • Always real despite involving complex numbers in calculation

Uncertainty and Density Matrices

• Uncertainty principle formulated using commutator of observables
  • Can yield imaginary results crucial for determining uncertainty relations
  • Example: [x, p] = iℏ leads to ΔxΔp ≥ ℏ/2
• Density matrix formalism uses complex matrices for mixed quantum states
  • ρ = Σ pᵢ|ψᵢ⟩⟨ψᵢ|, where pᵢ are probabilities and |ψᵢ⟩ are pure states
• Unitary transformations represented by complex unitary matrices
  • Describe evolution of quantum systems and symmetry operations
  • Example: Time evolution operator U(t) = e^(-iHt/ℏ)

Complex Amplitudes in Quantum Mechanics

Probability and Interference

• Complex amplitudes represent magnitude and phase of quantum state's contribution to measurement outcomes
• Born rule states probability of measuring outcome equals squared modulus of corresponding complex amplitude
  • P(outcome) = |⟨outcome|ψ⟩|²
• Relative phases between complex amplitudes in superposition determine interference patterns
  • Example: Electron diffraction through crystal lattice

Quantum Phenomena

• Complex nature of amplitudes allows description of quantum tunneling
  • Particles can penetrate classically forbidden regions
  • Example: Alpha decay in radioactive nuclei
• Complex amplitudes in path integrals formulate sum over histories approach
  • Classical trajectories interfere to produce quantum behavior
• Aharonov-Bohm effect demonstrates observable consequences of complex phases
  • Electron phase shift occurs even in regions with zero electromagnetic field
• Quantum entanglement described using complex amplitudes in multiparticle wave functions
  • Example: Bell states for two-qubit systems (|Φ⁺⟩ = (1/√2)(|00⟩ + |11⟩))