Introduction: Welcome to Year 3
You have arrived at Quantum Mechanics. Before we can write down the equations that govern atoms, we need a new mathematical language: Linear Algebra. The central object is the Vector Space—an abstract arena where quantum states "live."
What is a Vector Space?
A Vector Space \(V\) over a field \(\mathbb{F}\) (usually real numbers \(\mathbb{R}\) or complex numbers \(\mathbb{C}\)) is a set of objects called vectors that obey specific rules for addition and scalar multiplication.
The key axioms are:
- Closure under Addition: If \(\vec{u}, \vec{v} \in V\), then \(\vec{u} + \vec{v} \in V\).
- Closure under Scalar Multiplication: If \(\vec{v} \in V\) and \(c \in \mathbb{F}\), then \(c\vec{v} \in V\).
- Associativity of Addition: \((\vec{u} + \vec{v}) + \vec{w} = \vec{u} + (\vec{v} + \vec{w})\).
- Commutativity of Addition: \(\vec{u} + \vec{v} = \vec{v} + \vec{u}\).
- Additive Identity: There exists \(\vec{0}\) such that \(\vec{v} + \vec{0} = \vec{v}\).
- Additive Inverse: For every \(\vec{v}\), there exists \(-\vec{v}\) such that \(\vec{v} + (-\vec{v}) = \vec{0}\).
- Distributive Laws: \(c(\vec{u} + \vec{v}) = c\vec{u} + c\vec{v}\) and \((c + d)\vec{v} = c\vec{v} + d\vec{v}\).
- Scalar Multiplication Identity: \(1 \cdot \vec{v} = \vec{v}\).
Worked Examples
Example 1: \(\mathbb{R}^2\) is a Vector Space
The set of all pairs \((x, y)\) where \(x, y \in \mathbb{R}\) forms a vector space under standard addition and scalar multiplication:
- \((1, 2) + (3, 4) = (4, 6)\) ✓ (closure)
- \(2 \cdot (1, 2) = (2, 4)\) ✓ (scalar closure)
- The zero vector is \((0, 0)\)
Example 2: The Space of Polynomials
All polynomials of degree ≤ 2 form a vector space. If \(p(x) = 2x^2 + x\) and \(q(x) = x^2 - 1\):
- \(p(x) + q(x) = 3x^2 + x - 1\) (still degree ≤ 2)
- \(3 \cdot p(x) = 6x^2 + 3x\) (still degree ≤ 2)
Example 3: Functions as Vectors
The set of all continuous functions on \([0, 1]\) is a vector space. If \(f(x) = \sin(x)\) and \(g(x) = e^x\), then \(f + g\) and \(cf\) are also continuous functions on \([0, 1]\).
The Quantum Connection
In quantum mechanics, the state of a particle is represented by a vector in a vector space called a Hilbert Space. The wavefunction \(\psi(x)\) is just one possible "coordinate representation" of this abstract state vector. The rules of vector spaces guarantee that we can superpose quantum states: if \(|\psi\rangle\) and \(|\phi\rangle\) are valid states, so is \(\alpha|\psi\rangle + \beta|\phi\rangle\). This is the mathematical foundation of quantum superposition.