2.2 Quantum states and measurements

Quantum states are the foundation of quantum mechanics, describing a system's properties and behavior. Using Dirac notation, these states are represented mathematically in a complex Hilbert space, allowing for precise calculations and analysis of quantum systems.

Observables and measurements are key concepts in quantum mechanics. Observables represent measurable quantities, while the measurement process is inherently probabilistic, collapsing quantum states to eigenstates. Understanding these concepts is crucial for interpreting quantum phenomena and performing calculations.

Quantum States and Their Representation

Quantum states and ket notation

• Quantum states fully describe the state of a quantum system at a specific point in time
  • Encapsulate all the information about the system's properties and behavior
• Represented mathematically using Dirac notation, specifically ket notation
  • Ket notation denotes a quantum state as $|\psi\rangle$, where $\psi$ is a label identifying the state (e.g., $|0\rangle$, $|1\rangle$)
  • Bra notation $\langle\psi|$ represents the conjugate transpose of the ket $|\psi\rangle$
• State vectors reside in a complex Hilbert space, a complete inner product space
  • Hilbert space provides a mathematical framework for describing quantum systems
• Quantum states can be expressed as linear combinations of basis states
  • Basis states form a set of orthonormal vectors that span the Hilbert space (e.g., $|0\rangle$ and $|1\rangle$ for a qubit)
  • Example: A general qubit state is written as $|\psi\rangle = \alpha|0\rangle + \beta|1\rangle$
    • $\alpha$ and $\beta$ are complex amplitudes representing the probability amplitudes of the basis states
    • The normalization condition $|\alpha|^2 + |\beta|^2 = 1$ ensures the total probability sums to 1

Observables and Measurements

Observables and quantum states

• Observables represent measurable physical quantities in quantum systems
  • Examples include position, momentum, energy, and spin
• Mathematically described by Hermitian operators, denoted by $\hat{A}$
  • Hermitian operators satisfy $\hat{A}^\dagger = \hat{A}$, where $\hat{A}^\dagger$ is the adjoint of $\hat{A}$
• Eigenstates of an observable are quantum states for which the observable has a definite value
  • The eigenvalue equation $\hat{A}|\psi\rangle = a|\psi\rangle$ relates the eigenstate $|\psi\rangle$ to its corresponding eigenvalue $a$
• The expectation value of an observable, given by $\langle\hat{A}\rangle = \langle\psi|\hat{A}|\psi\rangle$, represents the average value obtained from multiple measurements on identical quantum states

Quantum measurement process

• Quantum measurements are inherently probabilistic, collapsing the quantum state to an eigenstate of the measured observable
• The probability of measuring an eigenvalue $a_i$ is given by $P(a_i) = |\langle\psi|a_i\rangle|^2$
  • $|a_i\rangle$ represents the eigenstate corresponding to the eigenvalue $a_i$
• Repeated measurements on identical quantum states produce a distribution of eigenvalues
• Individual measurement outcomes are fundamentally random and cannot be predicted with certainty

Probability calculations in quantum mechanics

• To calculate the probabilities of different measurement outcomes for a given quantum state $|\psi\rangle$ and an observable $\hat{A}$:
  1. Expand the state in the eigenbasis of the observable: $|\psi\rangle = \sum_i c_i|a_i\rangle$
     • $c_i = \langle a_i|\psi\rangle$ are the expansion coefficients
  2. Calculate the probability of measuring eigenvalue $a_i$ using $P(a_i) = |c_i|^2 = |\langle a_i|\psi\rangle|^2$
• Example: For a qubit state $|\psi\rangle = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)$
  • The probability of measuring $|0\rangle$ is $P(0) = |\langle0|\psi\rangle|^2 = \frac{1}{2}$
  • The probability of measuring $|1\rangle$ is $P(1) = |\langle1|\psi\rangle|^2 = \frac{1}{2}$