Lesson: Hilbert Spaces and Linear Operators

Learning Objectives

• Understand the concept of Hilbert spaces and their role in quantum mechanics.
• Explore the properties and applications of linear operators.
• Apply these concepts to solve problems in quantum computing and physics.

Introduction

In quantum mechanics, the state of a quantum system is described by a vector known as a state vector. The set of all possible state vectors for a system forms a vector space called a Hilbert space.

Definition of Hilbert Space

A Hilbert space is an infinite-dimensional, complex vector space that:

• Is complete, meaning every Cauchy sequence converges to a limit.
• Has an inner product, denoted by $<.,.>$, which satisfies the following properties:
  • Linearity
  • Symmetry
  • Positive definiteness

Example: Spin-1/2 System

For a spin-1/2 particle, the Hilbert space is a two-dimensional complex vector space spanned by the states |↑> and |↓>. The inner product is defined as:

<↑|↑> = 1, <↑|↓> = 0, <↓|↑> = 0, <↓|↓> = 1

Linear Operators

Linear operators are mappings from one vector space to another that preserve the vector space structure. In quantum mechanics, linear operators represent observables such as position, momentum, and energy.

Properties of Linear Operators

• Linearity: For any vectors |ψ> and |φ>, and any scalars a and b, the operator satisfies: A(a|ψ> + b|φ>) = aA|ψ> + bA|φ>
• Hermiticity: A linear operator is Hermitian if it is equal to its adjoint: A* = A
• Unitary: A linear operator is unitary if it preserves the inner product: <A|ψ>|A|φ> = <|ψ>|φ>

Applications in Quantum Computing and Physics

• Qubit manipulation: Unitary operators are used to manipulate qubits in quantum computing.
• Schrödinger's equation: The time evolution of a quantum system is described by the Schrödinger equation, which involves linear operators representing the system's dynamics.
• Quantum information theory: Linear operators play a fundamental role in quantifying entanglement and information in quantum systems.

Additional Learning Resources

• Hilbert Spaces in Quantum Mechanics
• Linear Operators in Quantum Mechanics

Assessment Activity

Consider the spin-1/2 system described above.

a) Use the inner product to show that the states |↑> and |↓> are orthogonal. b) Construct a unitary operator that rotates the system's spin by 90 degrees around the y-axis.