Lecture Notes on Wavefunctions and Quantum Mechanics

Introduction to Wavefunctions

• The wavefunction is represented by the Greek letter psi (Ψ).
• Previous lecture focused on applying operators to wavefunctions; this lecture will delve deeper into the wavefunction itself.
• Understanding wavefunctions requires knowledge of linear algebra, particularly vector spaces.

Hilbert Space

• A Hilbert space is a specific type of vector space where wavefunctions reside.
• Wavefunctions can be visualized as vectors within a potentially infinite-dimensional vector space.
• In a standard three-dimensional space, basis vectors (x, y, z) are orthogonal and linearly independent.
• Quantum systems can be described by multiple wavefunctions, all orthogonal and linearly independent.

Bra-Ket Notation

• Introduced bra-ket notation (|ψ⟩ for kets and ⟨ψ| for bras).
• Kets represent wavefunctions in vector form.
• Bras are the complex conjugate of kets.
• The inner product in a Hilbert space is defined using this notation:
  • ⟨ψ₁|ψ₂⟩ = ∫ψ₁*ψ₂ dx.
• Properties of inner products:
  • ⟨ψ|ψ⟩ = 1 (normalization)
  • ⟨ψ₁|ψ₂⟩ = 0 for orthogonal wavefunctions.*

Probability Density Function

• Probability density function (PDF) is defined as:
  • P(x, t) = |Ψ(x,t)|² = Ψ*Ψ.
• Provides probability of finding a particle in a given position.
• PDF must integrate to 1 over all space for normalization.
• Quantum probability densities are more diffuse compared to classical particles.*

Normalization of Wavefunctions

• Normalization condition ensures PDFs integrate to 1.
• Eigenfunctions of observables must be normalized.
• Non-normalized wavefunctions can be adjusted by a scaling factor.
• If |φ₁⟩ has an inner product C with itself, scale it to satisfy normalization:
  • |φ₂⟩ = A|φ₁⟩, where A = 1/√C.

Superposition Principle

• Quantum systems can be described as superpositions of wavefunctions.
• Example: Double-slit experiment can be described mathematically as:
  • |Ψ⟩ = |φₗ⟩ + |φᵣ⟩, where φₗ and φᵣ represent wavefunctions at each slit.
• The superposition illustrates that particles can exist in multiple states simultaneously.

Expectation Values

• Expectation values indicate the most probable measurement outcomes for observables.
• Defined mathematically using operators and the wavefunction:
  • ⟨x⟩ = ∫ Ψ* (x operator) Ψ dx.
• The expectation value is a weighted average of all possible measurement outcomes.
• Example: Kinetic energy (T) in quantum mechanics involves replacing classical momentum with the momentum operator:
  • T = p²/(2m).
• Expectation value for kinetic energy:
  • ⟨T⟩ = ∫ Ψ* (T operator) Ψ dx.

Summary of Key Concepts

• Wavefunctions are vectors in Hilbert space, normalized for physical relevance.
• Inner products serve similar functions to dot products in vector spaces.
• Probability density functions provide insights into the likelihood of particle locations.
• Expectation values are crucial for predicting measurement outcomes.
• Next lecture will explore the Schrodinger equation and related concepts further.