Glossary

1.2 Operators and observables

5 min read•july 30, 2024

Operators and observables are the backbone of quantum mechanics, letting us measure and predict the behavior of particles. They're mathematical tools that transform wave functions, representing physical quantities like position and momentum.

Understanding operators is crucial for grasping how quantum systems evolve and interact. We'll look at their properties, how they relate to measurable quantities, and how they shape our understanding of the quantum world.

Operators in Quantum Mechanics

Definition and Properties of Operators

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  • An is a mathematical entity that acts on a function and produces another function
    • Operators transform functions from one form to another (e.g., differentiation, integration)
    • In quantum mechanics, operators are used to represent physical observables such as position, momentum, and energy
  • Operators can be linear or nonlinear, and they can be represented as matrices in a given basis
    • Linear operators satisfy the properties of linearity: A^(aψ1+bψ2)=aA^ψ1+bA^ψ2\hat{A}(a\psi_1 + b\psi_2) = a\hat{A}\psi_1 + b\hat{A}\psi_2
    • Nonlinear operators do not satisfy these properties
  • The on a wave function is denoted by placing the operator symbol to the left of the wave function
    • For example, A^ψ(x)\hat{A}\psi(x) represents the action of operator A^\hat{A} on the wave function ψ(x)\psi(x)
  • The Hermitian conjugate of an operator is obtained by taking the complex conjugate of the operator and transposing it
    • Denoted as A^†\hat{A}^{\dagger}
    • Hermitian operators satisfy A^=A^†\hat{A} = \hat{A}^{\dagger}, and their eigenvalues are real

Commutators and Simultaneous Measurability

  • The commutator of two operators A^\hat{A} and B^\hat{B} is defined as [A^,B^]=A^B^−B^A^[\hat{A}, \hat{B}] = \hat{A}\hat{B} - \hat{B}\hat{A}
    • If the commutator is zero, the operators commute, and the corresponding observables can be simultaneously measured with arbitrary precision
    • If the commutator is non-zero, the operators do not commute, and there is an inherent uncertainty in simultaneously measuring the corresponding observables (Heisenberg's )
  • The between position (x^\hat{x}) and momentum (p^\hat{p}) operators is [x^,p^]=iℏ[\hat{x}, \hat{p}] = i\hbar
    • This relation leads to the position-momentum uncertainty principle: ΔxΔp≥ℏ2\Delta x \Delta p \geq \frac{\hbar}{2}

Operators and Observables

Hermitian Operators and Physical Observables

  • Physical observables in quantum mechanics are represented by Hermitian operators
    • Hermitian operators have real eigenvalues, which correspond to the possible measurement outcomes of the
    • Examples of physical observables include position, momentum, energy, and angular momentum
  • The of an observable is calculated by taking the inner product of the wave function with the operator acting on the wave function
    • Expectation value of observable A^\hat{A}: ⟨A^⟩=⟨ψ∣A^∣ψ⟩=∫ψ∗(x)A^ψ(x)dx\langle \hat{A} \rangle = \langle \psi | \hat{A} | \psi \rangle = \int \psi^*(x) \hat{A} \psi(x) dx
    • The expectation value represents the average value of the observable over many measurements on identically prepared systems

Uncertainty and Standard Deviation

  • The uncertainty of an observable is quantified by its ΔA\Delta A, given by ΔA=⟨A^2⟩−⟨A^⟩2\Delta A = \sqrt{\langle \hat{A}^2 \rangle - \langle \hat{A} \rangle^2}
    • The standard deviation represents the spread of the measurement outcomes around the expectation value
  • The uncertainty principle states that the product of the standard deviations of two non-commuting observables is always greater than or equal to ℏ2\frac{\hbar}{2}
    • For example, the position-momentum uncertainty principle: ΔxΔp≥ℏ2\Delta x \Delta p \geq \frac{\hbar}{2}
    • This principle limits the precision with which certain pairs of observables can be simultaneously measured

Applying Operators to Wave Functions

Common Operators in Quantum Mechanics

  • The x^\hat{x} is represented by the multiplication of the position variable (x)(x) with the wave function
    • x^ψ(x)=xψ(x)\hat{x}\psi(x) = x\psi(x)
  • The p^\hat{p} is represented by the negative imaginary unit (−iℏ)(-i\hbar) multiplied by the partial derivative with respect to position
    • p^ψ(x)=−iℏ∂∂xψ(x)\hat{p}\psi(x) = -i\hbar \frac{\partial}{\partial x}\psi(x)
  • The , or H^\hat{H}, is the sum of the kinetic and potential energy operators
    • H^=T^+V^=−ℏ22m∇2+V(x)\hat{H} = \hat{T} + \hat{V} = -\frac{\hbar^2}{2m}\nabla^2 + V(x), where mm is the mass of the particle and V(x)V(x) is the potential energy
  • L^x\hat{L}_x, L^y\hat{L}_y, and L^z\hat{L}_z are defined in terms of position and momentum operators
    • L^x=y^p^z−z^p^y\hat{L}_x = \hat{y}\hat{p}_z - \hat{z}\hat{p}_y, L^y=z^p^x−x^p^z\hat{L}_y = \hat{z}\hat{p}_x - \hat{x}\hat{p}_z, L^z=x^p^y−y^p^x\hat{L}_z = \hat{x}\hat{p}_y - \hat{y}\hat{p}_x

Time Evolution and the Schrödinger Equation

  • The time-dependent Schrödinger equation describes the evolution of a wave function under the action of the Hamiltonian operator
    • iℏ∂∂tΨ(x,t)=H^Ψ(x,t)i\hbar \frac{\partial}{\partial t}\Psi(x, t) = \hat{H}\Psi(x, t)
    • The solution to this equation gives the wave function Ψ(x,t)\Psi(x, t) at any time tt, given an initial wave function Ψ(x,0)\Psi(x, 0)
  • The time-independent Schrödinger equation is an for the Hamiltonian operator
    • H^ψ(x)=Eψ(x)\hat{H}\psi(x) = E\psi(x), where EE is the energy
    • Solutions to this equation give the stationary states (energy eigenfunctions) and their corresponding energy eigenvalues

Eigenvalues and Eigenfunctions

Eigenvalues and Eigenfunctions of Operators

  • Eigenvalues are the possible measurement outcomes of an observable, and they are real numbers for Hermitian operators
    • If A^ψn=anψn\hat{A}\psi_n = a_n\psi_n, then ana_n is an eigenvalue of the operator A^\hat{A}, and ψn\psi_n is the corresponding
  • Eigenfunctions are the wave functions that, when acted upon by an operator, result in the same wave function multiplied by the corresponding eigenvalue
    • A^ψn=anψn\hat{A}\psi_n = a_n\psi_n, where ψn\psi_n is an eigenfunction of operator A^\hat{A} with eigenvalue ana_n
  • The set of eigenfunctions for an operator forms a complete orthonormal basis, known as the eigenbasis
    • Orthonormality: ⟨ψm∣ψn⟩=δmn\langle \psi_m | \psi_n \rangle = \delta_{mn}, where δmn\delta_{mn} is the Kronecker delta
    • Completeness: Any wave function can be expressed as a linear combination of the eigenfunctions, ψ(x)=∑ncnψn(x)\psi(x) = \sum_n c_n \psi_n(x)

Probability and Measurement

  • The probability of measuring a particular eigenvalue is given by the square of the inner product between the wave function and the corresponding eigenfunction
    • Probability of measuring eigenvalue ana_n: P(an)=∣⟨ψn∣ψ⟩∣2=∣cn∣2P(a_n) = |\langle \psi_n | \psi \rangle|^2 = |c_n|^2
  • The spectral decomposition theorem states that any wave function can be expressed as a linear combination of the eigenfunctions of an operator
    • ψ(x)=∑ncnψn(x)\psi(x) = \sum_n c_n \psi_n(x), where cn=⟨ψn∣ψ⟩c_n = \langle \psi_n | \psi \rangle are the expansion coefficients
  • Upon measurement, the wave function collapses to one of the eigenfunctions corresponding to the measured eigenvalue
    • The act of measurement changes the state of the system, and subsequent measurements will yield the same eigenvalue until the system is disturbed

Key Terms to Review (26)

Action of an operator: The action of an operator refers to the effect that a mathematical operator has when it is applied to a function or a state vector within quantum mechanics. This concept is crucial for understanding how observables are measured, as operators represent physical quantities, and their action provides the outcomes when these quantities are evaluated in a quantum system.
Angular Momentum Operators: Angular momentum operators are mathematical constructs in quantum mechanics that represent the angular momentum of a quantum system. They are crucial for understanding the behavior of particles in rotational motion and play a key role in determining the allowed states and properties of systems such as atoms and molecules. These operators help quantify the rotational aspects of physical systems and are linked to observable quantities, allowing us to make predictions about measurements.
Canonical Commutation Relation: The canonical commutation relation is a fundamental principle in quantum mechanics that describes how certain pairs of observables, such as position and momentum, are related through their corresponding operators. It states that the product of two operators, specifically position $$ ext{(x)}$$ and momentum $$ ext{(p)}$$, has a specific relationship expressed mathematically as $$[ ext{x}, ext{p}] = ext{x} ext{p} - ext{p} ext{x} = i rac{ ext{h}}{2 ext{π}}$$, where $$ ext{i}$$ is the imaginary unit and $$ ext{h}$$ is Planck's constant. This relationship reflects the inherent uncertainty in quantum mechanics and indicates that not all pairs of measurements can be precisely defined simultaneously.
Collapse of the wave function: The collapse of the wave function 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 probabilistic nature of quantum mechanics, as the wave function provides the probabilities of different outcomes, but only one outcome is observed when a measurement is made. This process connects deeply with the concepts of operators and observables, as measurements correspond to operators acting on the wave function, and it also relates to qubits, which utilize superposition in their quantum state representation.
Commutation Relation: The commutation relation is a mathematical expression that defines how two operators behave when they are applied in succession. It is crucial in quantum mechanics because it helps determine the compatibility of observables, indicating whether they can be simultaneously measured without affecting each other. The most famous example is the position and momentum operators, which satisfy the commutation relation $$[X,P] = XP - PX = i\hbar$$, revealing their fundamental uncertainty relationship.
Eigenfunction: An eigenfunction is a special type of function that remains unchanged, except for a scalar factor, when acted upon by a linear operator. This property makes eigenfunctions crucial in the study of quantum mechanics, as they are directly related to measurable physical quantities or observables. When an observable is measured, the system is found to be in an eigenstate associated with that observable, revealing deep connections between mathematical operators and physical measurements.
Eigenvalue: An eigenvalue is a special scalar associated with a linear transformation represented by an operator, which indicates how much a corresponding eigenvector is stretched or shrunk during that transformation. In quantum mechanics, eigenvalues are particularly significant because they represent measurable quantities, or observables, of a physical system, helping to define the state of the system in a vector space. They are closely related to the concepts of operators and their actions on vectors, as well as fundamental properties like spin angular momentum.
Eigenvalue Equation: The eigenvalue equation is a mathematical expression used in quantum mechanics and linear algebra, represented as $$ ext{A} extbf{v} = ext{λ} extbf{v}$$, where A is an operator, v is a vector (or state), and λ is the eigenvalue corresponding to that vector. This equation illustrates how operators act on quantum states, providing critical insight into observables and their measurable quantities. The eigenvalue corresponds to the measurement outcome associated with the observable represented by the operator.
Energy Operator: The energy operator is an important mathematical tool in quantum mechanics, represented by the Hamiltonian operator, which corresponds to the total energy of a quantum system. This operator plays a crucial role in determining the behavior and evolution of quantum states over time through its relationship with the Schrödinger equation. Understanding the energy operator allows for insights into how quantum systems interact and evolve under various conditions.
Expectation Value: The expectation value is a fundamental concept in quantum mechanics that represents the average value of a physical quantity, measured over many identical experiments or measurements. It connects quantum states with observable outcomes, providing a way to predict the likely result of measuring an observable in a given quantum state. The expectation value is calculated using operators on wave functions, linking it closely to the notion of observables in quantum mechanics and finding applications in atomic and molecular systems.
Hamiltonian: The Hamiltonian is an operator that represents the total energy of a quantum system, encompassing both kinetic and potential energy components. It plays a central role in quantum mechanics, serving as the generator of time evolution in the Schrödinger equation and providing insight into the observable properties of systems such as atoms and molecules. The Hamiltonian is crucial for understanding how systems behave under various interactions, including spin effects and electromagnetic field quantization.
Heisenberg Uncertainty Principle: The Heisenberg Uncertainty Principle states that it is impossible to simultaneously know both the exact position and momentum of a particle with absolute precision. This principle highlights fundamental limitations in measurement and reinforces the concept that at a quantum level, uncertainty is inherent to the properties of particles, making it essential for understanding the behavior of quantum systems.
Hermitian Operator: A Hermitian operator is a type of linear operator in quantum mechanics that is equal to its own adjoint, meaning it satisfies the condition $$A = A^{ ext{†}}$$. This property ensures that the eigenvalues of the operator are real numbers, making them suitable for representing observable physical quantities. As a result, Hermitian operators play a crucial role in defining observables and calculating expectation values, which are essential for understanding measurements in quantum mechanics.
Hilbert Space: Hilbert space is a complete, infinite-dimensional vector space equipped with an inner product that allows for the geometric interpretation of quantum states and their transformations. This mathematical framework is crucial for understanding operators and observables, as well as the behavior of quantum systems in relation to linear operators and time-dependent perturbation theory.
Measurement Postulate: The measurement postulate is a fundamental concept in quantum mechanics that states how the outcome of a measurement relates to the state of a quantum system. When a measurement is performed, the system collapses into one of the eigenstates of the observable being measured, and the result corresponds to the eigenvalue associated with that eigenstate. This idea links measurements, operators, and expectation values, making it essential for understanding phenomena like total angular momentum and how different quantum states can couple together.
Momentum operator: The momentum operator is a fundamental concept in quantum mechanics, defined as the operator corresponding to the momentum observable of a quantum system. It plays a crucial role in measuring momentum and is typically represented as $$ extbf{p} = -i extbf{ abla} ext{hbar}$$ in position space, where $$ extbf{ abla}$$ is the gradient operator. This operator's significance extends to calculating expectation values and understanding how momentum influences the behavior of quantum particles.
Niels Bohr: Niels Bohr was a Danish physicist known for his foundational contributions to understanding atomic structure and quantum mechanics, particularly through the Bohr model of the atom. His work emphasized the quantization of energy levels in atoms and introduced the idea that electrons orbit the nucleus at fixed distances, which directly ties into the concepts of measurement, operators, transition probabilities, and fine structure in quantum systems.
Observable: In quantum mechanics, an observable is a physical quantity that can be measured and is represented mathematically by an operator acting on a state vector in a Hilbert space. Observables are fundamental in linking the mathematical framework of quantum mechanics to physical measurements, allowing us to understand systems in terms of measurable properties such as position, momentum, and angular momentum.
Operator: In quantum mechanics, an operator is a mathematical entity that acts on a quantum state to produce observable outcomes. Operators are essential for describing physical observables, such as position, momentum, and energy, and they help in transitioning between different states within the quantum framework. The interplay between operators and their associated eigenvalues allows for the extraction of measurable quantities from quantum systems.
Position Operator: The position operator is a fundamental mathematical entity in quantum mechanics that represents the observable quantity of position in a quantum system. It is denoted as \( \hat{x} \) and acts on the wave function to extract information about the position of a particle. This operator plays a crucial role in defining measurement processes and calculating expectation values related to spatial coordinates.
Spectral Theorem: The spectral theorem states that every normal operator on a finite-dimensional inner product space can be diagonalized by an orthonormal basis of eigenvectors. This powerful result connects linear operators, their eigenvalues, and the geometry of vector spaces, establishing a deep relationship between algebra and analysis in quantum mechanics.
Standard Deviation: Standard deviation is a statistical measure that quantifies the amount of variation or dispersion in a set of values. In the context of quantum mechanics, it reflects the uncertainty associated with a measurement and helps describe how much the values of a physical observable deviate from their average value. This concept is crucial when dealing with operators and observables, as it connects to the inherent uncertainties present in quantum systems.
State Transformation: State transformation refers to the process by which the quantum state of a system changes due to the application of an operator. This concept is crucial when discussing how measurements affect quantum systems, as operators correspond to physical observables, and their application can alter the state of a system, leading to different outcomes in measurements.
State vector: A state vector is a mathematical representation of the state of a quantum system, typically denoted by a column vector in a complex vector space. It encodes all the information about a system's properties and behaviors, allowing for predictions regarding measurement outcomes when operators act on it. The state vector is essential for understanding how quantum systems evolve over time and how they interact with observables.
Uncertainty Principle: The uncertainty principle is a fundamental concept in quantum mechanics that states it is impossible to precisely measure certain pairs of properties, such as position and momentum, simultaneously. This principle reflects the intrinsic limitations of measurement at the quantum level, emphasizing the wave-particle duality of matter and how measurement affects the state of a system.
Werner Heisenberg: Werner Heisenberg was a German physicist best known for formulating the uncertainty principle, which is a fundamental concept in quantum mechanics. His work has profoundly influenced the understanding of quantum systems, particularly in relation to operators and observables, emphasizing that certain pairs of physical properties cannot both be precisely known at the same time.
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