Quantum Mechanics Course

Total: 450+ pages covering mathematical foundations through advanced quantum mechanics

Part I, Chapter 1 | Page 1 of 8

Hilbert Spaces

The mathematical arena where quantum mechanics lives

1.1 Vector Spaces Over ℂ

A vector space V over the complex numbers ℂ is a set equipped with two operations: vector addition and scalar multiplication, satisfying certain axioms.

Definition: Vector Space

A set V is a vector space over ℂ if for any vectors $|\psi\rangle, |\phi\rangle, |\chi\rangle \in V$and scalars $\alpha, \beta \in \mathbb{C}$, the following hold:

  • 1. Closure under addition: $|\psi\rangle + |\phi\rangle \in V$
  • 2. Commutativity: $|\psi\rangle + |\phi\rangle = |\phi\rangle + |\psi\rangle$
  • 3. Associativity: $(|\psi\rangle + |\phi\rangle) + |\chi\rangle = |\psi\rangle + (|\phi\rangle + |\chi\rangle)$
  • 4. Zero vector: ∃ $|0\rangle \in V$ such that $|\psi\rangle + |0\rangle = |\psi\rangle$
  • 5. Additive inverse: ∀ $|\psi\rangle$ ∃ $-|\psi\rangle$ such that $|\psi\rangle + (-|\psi\rangle) = |0\rangle$
  • 6. Closure under scalar multiplication: $\alpha|\psi\rangle \in V$
  • 7. Distributivity (vectors): $\alpha(|\psi\rangle + |\phi\rangle) = \alpha|\psi\rangle + \alpha|\phi\rangle$
  • 8. Distributivity (scalars): $(\alpha + \beta)|\psi\rangle = \alpha|\psi\rangle + \beta|\psi\rangle$
  • 9. Associativity (scalar): $(\alpha\beta)|\psi\rangle = \alpha(\beta|\psi\rangle)$
  • 10. Identity: $1|\psi\rangle = |\psi\rangle$

Examples of Vector Spaces

Example 1: $\mathbb{C}^n$

The space of n-tuples of complex numbers:

$$|\psi\rangle = \begin{pmatrix} \psi_1 \\ \psi_2 \\ \vdots \\ \psi_n \end{pmatrix}, \quad \psi_i \in \mathbb{C}$$

Example 2: $L^2(\mathbb{R})$

Square-integrable functions on the real line:

$$L^2(\mathbb{R}) = \left\{\psi(x) : \int_{-\infty}^{\infty} |\psi(x)|^2 dx < \infty\right\}$$

This is the space of wave functions in position representation!

Example 3: Fock Space

The space of quantum field states with variable particle number (important in QFT).

1.2 Inner Product

An inner product is a map $\langle \cdot | \cdot \rangle : V \times V \to \mathbb{C}$ that assigns a complex number to each pair of vectors, generalizing the dot product.

Definition: Inner Product

An inner product on V must satisfy:

  1. Conjugate Linearity in first argument:
    $$\langle \alpha\psi + \beta\phi | \chi \rangle = \alpha^* \langle \psi | \chi \rangle + \beta^* \langle \phi | \chi \rangle$$
  2. Linearity in second argument:
    $$\langle \psi | \alpha\phi + \beta\chi \rangle = \alpha \langle \psi | \phi \rangle + \beta \langle \psi | \chi \rangle$$
  3. Hermitian symmetry:
    $$\langle \psi | \phi \rangle = \langle \phi | \psi \rangle^*$$
  4. Positive definiteness:
    $$\langle \psi | \psi \rangle \geq 0, \quad \text{and} \quad \langle \psi | \psi \rangle = 0 \iff |\psi\rangle = |0\rangle$$

Examples of Inner Products

In $\mathbb{C}^n$:

$$\langle \psi | \phi \rangle = \sum_{i=1}^n \psi_i^* \phi_i$$

In $L^2(\mathbb{R})$:

$$\langle \psi | \phi \rangle = \int_{-\infty}^{\infty} \psi^*(x) \phi(x) \, dx$$

This is the inner product we use for wave functions!

The Norm

The inner product induces a norm (length) on vectors:

$$\||\psi\rangle\| = \sqrt{\langle \psi | \psi \rangle}$$

A vector is normalized if $\||\psi\rangle\| = 1$, i.e., $\langle \psi | \psi \rangle = 1$. This is crucial in quantum mechanics: probability conservation requires normalized states!

Physical Interpretation:

In QM, the inner product $\langle \phi | \psi \rangle$ gives the probability amplitude for a system in state $|\psi\rangle$ to be found in state $|\phi\rangle$. The probability is$|\langle \phi | \psi \rangle|^2$.

1.3 Cauchy-Schwarz Inequality

One of the most important inequalities in mathematics and quantum mechanics:

$$|\langle \psi | \phi \rangle|^2 \leq \langle \psi | \psi \rangle \langle \phi | \phi \rangle$$

Or in norm notation:

$$|\langle \psi | \phi \rangle| \leq \||\psi\rangle\| \cdot \||\phi\rangle\|$$

Equality holds if and only if $|\psi\rangle$ and $|\phi\rangle$ are linearly dependent, i.e., $|\psi\rangle = \alpha|\phi\rangle$ for some $\alpha \in \mathbb{C}$.

Proof Sketch

Consider the non-negative quantity (for normalized $|\phi\rangle$):

$$\left\| |\psi\rangle - \langle \phi | \psi \rangle |\phi\rangle \right\|^2 \geq 0$$

Expanding this gives:

$$\langle \psi | \psi \rangle - |\langle \phi | \psi \rangle|^2 \geq 0$$

Rearranging yields Cauchy-Schwarz! □

Physical Consequence:

For normalized states, $|\langle \phi | \psi \rangle|^2 \leq 1$, meaning probabilities (probability amplitudes squared) cannot exceed 1. This ensures consistency of the probabilistic interpretation!

Key Concepts (Page 1)

  • • Quantum states live in complex vector spaces over ℂ
  • • Inner product $\langle \psi | \phi \rangle$ gives probability amplitudes
  • • Normalized states satisfy $\langle \psi | \psi \rangle = 1$
  • • Cauchy-Schwarz ensures probabilities ≤ 1
  • • $L^2(\mathbb{R})$ is the space of square-integrable wave functions
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Chapter 1: Hilbert Spaces
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