Glossary

5.2 Observables and Hermitian operators

2 min read•august 9, 2024

Quantum observables are physical quantities we measure in quantum systems, represented by Hermitian operators. These operators have special properties that ensure real eigenvalues and orthonormal eigenvectors, crucial for understanding outcomes and system states.

Expectation values give us the average measurement result for many identical systems. The sets limits on how precisely we can measure certain pairs of observables simultaneously, highlighting the probabilistic nature of quantum mechanics.

Observables and Operators

Fundamental Concepts of Quantum Observables

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  • represents a physical quantity measured in quantum mechanics (position, momentum, energy)
  • Corresponds to a in mathematical formalism
  • Hermitian operator maintains specific mathematical properties ensuring real eigenvalues
  • equivalent to Hermitian operator in finite-dimensional spaces
  • Matrix representation provides a concrete way to express operators in linear algebra

Properties of Hermitian Operators

  • Hermitian operators have real eigenvalues corresponding to possible measurement outcomes
  • Eigenvectors of Hermitian operators form an for the Hilbert space
  • Hermitian operators satisfy the condition A†=AA^† = A, where A†A^† denotes the adjoint of operator AA
  • guarantees the diagonalizability of Hermitian operators
  • Commutation relations between Hermitian operators determine compatibility of measurements

Mathematical Representation of Observables

  • Matrix representation allows expressing operators as square matrices
  • Diagonal elements of Hermitian matrices are real numbers
  • Off-diagonal elements of Hermitian matrices are complex conjugates of each other
  • equation for an observable AA written as A∣ψ⟩=a∣ψ⟩A|ψ⟩ = a|ψ⟩, where aa is the eigenvalue
  • Transformation between different bases utilizes unitary matrices

Expectation Values and Uncertainty

Calculation and Interpretation of Expectation Values

  • Expectation value represents the average outcome of many measurements on identically prepared systems
  • Computed using the formula ⟨A⟩=⟨ψ∣A∣ψ⟩⟨A⟩ = ⟨ψ|A|ψ⟩ for a state ∣ψ⟩|ψ⟩ and observable AA
  • Provides a probabilistic prediction of measurement results in quantum mechanics
  • Can be expressed as a weighted sum of eigenvalues for discrete spectra
  • Continuous spectra require integration over the probability density function

Uncertainty Principle and Measurement Limitations

  • Uncertainty principle sets fundamental limits on simultaneous measurement precision of conjugate variables
  • Quantified by the inequality ΔA⋅ΔB≥12∣⟨[A,B]⟩∣ΔA · ΔB ≥ \frac{1}{2}|⟨[A,B]⟩| for observables AA and BB
  • Standard deviation ΔA=⟨A2⟩−⟨A⟩2ΔA = \sqrt{⟨A^2⟩ - ⟨A⟩^2} measures the spread of measurement outcomes
  • Heisenberg uncertainty principle applies to position and momentum with Δx⋅Δp≥ℏ2Δx · Δp ≥ \frac{ℏ}{2}
  • Generalized uncertainty relations exist for other pairs of non-commuting observables

Linearity and Superposition in Quantum Mechanics

  • Linearity principle states that quantum operators act linearly on state vectors
  • Superposition principle allows quantum states to exist in multiple basis states simultaneously
  • Linear combinations of eigenstates produce new valid quantum states
  • Measurement collapses superposition to a single eigenstate with associated probability
  • Schrödinger equation, governing quantum time evolution, maintains linearity

Key Terms to Review (16)

Bounded operator: A bounded operator is a linear transformation between two normed vector spaces that maps bounded sets to bounded sets. In quantum mechanics, bounded operators are crucial because they often represent observables and are associated with measurable quantities in physical systems. Their significance lies in ensuring that the mathematical operations involved remain well-defined and lead to meaningful physical predictions.
Collapse of the wavefunction: The collapse of the wavefunction refers to the process by which a quantum system transitions from a superposition of states to a single definite state upon measurement. This phenomenon highlights the fundamental nature of quantum mechanics, where the act of observation plays a crucial role in determining the state of a system. It connects deeply with how observables are measured and understood, as well as the challenges faced in building practical quantum computers, where maintaining superpositions without collapse is essential for computation.
Commutation Relation: A commutation relation is a mathematical expression that describes the relationship between two operators in quantum mechanics, specifically indicating whether the operators can be simultaneously measured or not. If two operators commute, their commutation relation equals zero, meaning they share a complete set of eigenstates, allowing for simultaneous measurement. If they do not commute, their commutation relation is non-zero, implying an inherent uncertainty in measuring the corresponding observables at the same time.
Continuous Spectrum: A continuous spectrum refers to a range of wavelengths or frequencies emitted by an object without any gaps or interruptions. In the context of quantum mechanics, it often arises from the presence of a particle that can occupy a continuous range of energy states, indicating that energy levels are not quantized. This concept is essential for understanding how observables are related to physical properties and how measurements can yield a wide array of possible results.
Discrete Spectrum: A discrete spectrum refers to a set of distinct, separate values that describe the possible outcomes of an observable in quantum mechanics. These values are linked to the eigenvalues of Hermitian operators, which represent physical quantities such as position, momentum, or energy. When a system is measured, it can only yield these specific outcomes, illustrating the quantized nature of certain properties in quantum systems.
Eigenvalue: An eigenvalue is a special scalar associated with a linear transformation represented by a matrix or an operator, indicating how much an eigenstate is stretched or shrunk during that transformation. Eigenvalues play a crucial role in quantum mechanics, particularly when dealing with measurements, where they determine the possible outcomes of an observable. They are closely tied to concepts such as operators, wave functions, and the mathematical structure of quantum systems.
Eigenvector: An eigenvector is a non-zero vector that only changes by a scalar factor when a linear transformation is applied to it. This concept is crucial for understanding how certain operators act in quantum mechanics, particularly when discussing measurements and physical observables. The relationship between eigenvectors and their corresponding eigenvalues helps to reveal the properties of the system being analyzed, providing a clear way to understand the structure of vector spaces and the behavior of Hermitian operators.
Hermitian operator: A Hermitian operator is a linear operator on a Hilbert space that is equal to its own adjoint, meaning that the inner product of two vectors remains unchanged when the order of the vectors is swapped. This property makes Hermitian operators crucial in quantum mechanics, as they correspond to observable physical quantities, ensuring real eigenvalues and orthogonal eigenstates that represent possible measurement outcomes.
Linear Operator: A linear operator is a mathematical function that acts on elements of a vector space and satisfies two key properties: additivity and homogeneity. In quantum mechanics, linear operators are essential for representing physical observables, allowing us to calculate measurable quantities while adhering to the principles of superposition. They are closely tied to Hermitian operators, which have real eigenvalues and orthogonal eigenvectors, making them particularly important for the measurement process in quantum systems.
Measurement: Measurement in quantum mechanics refers to the process of obtaining information about a quantum system, which often leads to a change in the state of that system. This concept is crucial because it connects the abstract mathematical framework of quantum mechanics to physical reality, revealing how we can extract observable properties from quantum states. Measurement is closely linked to observables, which are represented by Hermitian operators, and highlights the fundamental role of probability and uncertainty in describing outcomes.
Observable: An observable is a physical quantity in quantum mechanics that can be measured, represented mathematically by a Hermitian operator. This concept connects to the framework of Hilbert space and state vectors, allowing us to describe the possible outcomes of measurements and their probabilities. Observables are crucial for determining expectation values and understanding how a quantum system evolves over time, as highlighted by the Ehrenfest theorem.
Orthonormal Basis: An orthonormal basis is a set of vectors in a vector space that are both orthogonal and normalized, meaning that each vector is perpendicular to the others and has a unit length. This concept is crucial for simplifying calculations and understanding the structure of quantum states in Hilbert spaces. When using an orthonormal basis, any vector in the space can be expressed as a unique linear combination of the basis vectors, which is especially important for representing quantum states and observables.
Quantum State: A quantum state represents the complete information about a quantum system, encapsulating all its properties and potential outcomes when measurements are made. It is a fundamental concept in quantum mechanics that describes how particles behave and interact, relating directly to the probabilistic nature of physical observables and the intrinsic features of the system.
Self-adjoint operator: A self-adjoint operator is a linear operator that is equal to its own adjoint, meaning that for any two vectors in the space, the inner product of the operator applied to one vector with the other vector equals the inner product of the first vector with the operator applied to the second vector. This property ensures that the operator has real eigenvalues and orthogonal eigenvectors, making it essential in quantum mechanics for representing observables, which are physical quantities that can be measured.
Spectral Theorem: The spectral theorem is a fundamental result in linear algebra that states any Hermitian operator can be diagonalized by a unitary transformation. This theorem implies that the eigenvalues of the operator correspond to observable quantities in quantum mechanics, and the eigenvectors represent the states of the system. The spectral theorem is essential for understanding how observables are represented in quantum mechanics, linking mathematical structures with physical interpretations.
Uncertainty Principle: The uncertainty principle is a fundamental concept in quantum mechanics that states that certain pairs of physical properties, like position and momentum, cannot be simultaneously known with arbitrary precision. This inherent limitation arises from the wave-like nature of particles, fundamentally changing our understanding of measurement and observation in quantum systems.
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