Glossary

3.1 Operators and their properties

3 min read•july 31, 2024

Quantum mechanical operators are the mathematical tools that help us understand the weird world of quantum physics. They represent physical properties like position and momentum, acting on wave functions to give us useful information about quantum systems.

These operators have special properties that make them crucial for quantum mechanics. They're linear, follow specific rules when combined, and can be Hermitian or non-Hermitian. Understanding how they work is key to grasping quantum theory.

Quantum Mechanical Operators

Definition and Role in Quantum Mechanics

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  • Mathematical entities representing physical observables in quantum mechanics (position, momentum, energy)
  • Act on wave functions to extract information about physical properties of quantum systems
  • Relate to eigenvalues and eigenfunctions through equation A^∣ψ⟩=a∣ψ⟩Â|ψ⟩ = a|ψ⟩
  • Represented as matrices or differential operators depending on and representation
  • Provide expectation values ⟨A^⟩=⟨ψ∣A^∣ψ⟩⟨Â⟩ = ⟨ψ|Â|ψ⟩ giving average observable values in quantum states
  • Determine simultaneous measurability of observables through [A^,B^]=A^B^−B^A^[Â,B̂] = ÂB̂ - B̂Â

Examples and Applications

  • x^\hat{x} multiplies wave function by position variable
  • Momentum operator p^=−iℏ∇\hat{p} = -i\hbar\nabla yields momentum distribution information
  • (Hamiltonian) H^=−ℏ22m∇2+V(r)\hat{H} = -\frac{\hbar^2}{2m}\nabla^2 + V(r) returns energy eigenvalues
  • L^=r^×p^\hat{L} = \hat{r} \times \hat{p} applied to spherical harmonics gives angular momentum information
  • e−iH^t/ℏe^{-i\hat{H}t/\hbar} describes how quantum states change over time

Properties of Linear Operators

Linearity and Superposition

  • Satisfy superposition principle: A^(c1ψ1+c2ψ2)=c1A^ψ1+c2A^ψ2\hat{A}(c_1\psi_1 + c_2\psi_2) = c_1\hat{A}\psi_1 + c_2\hat{A}\psi_2
  • Preserve principle of superposition in quantum mechanics
  • Obey distributive property: (A^+B^)ψ=A^ψ+B^ψ(\hat{A} + \hat{B})\psi = \hat{A}\psi + \hat{B}\psi
  • Enable representation of quantum states as linear combinations of basis states

Operator Algebra and Products

  • Product rule states (A^B^)ψ=A^(B^ψ)(\hat{A}\hat{B})\psi = \hat{A}(\hat{B}\psi), allowing sequential application to wave functions
  • Order of application matters for non-commuting operators (momentum and position)
  • Adjoint (Hermitian conjugate) A^†\hat{A}^\dagger defined by ⟨ϕ∣A^ψ⟩=⟨A^†ϕ∣ψ⟩\langle\phi|\hat{A}\psi\rangle = \langle\hat{A}^\dagger\phi|\psi\rangle for any state vectors ∣ϕ⟩|\phi\rangle and ∣ψ⟩|\psi\rangle
  • Commutators [A^,B^]=A^B^−B^A^[\hat{A},\hat{B}] = \hat{A}\hat{B} - \hat{B}\hat{A} quantify non-commutativity of operators

Applying Operators to Wave Functions

Transformation and Interpretation

  • Transform wave functions into new functions or return scalar values
  • Yield information about system's physical properties (energy levels, momentum distributions)
  • Used to calculate probabilities, expectation values, and uncertainties of observables
  • Results interpreted within probabilistic framework of quantum mechanics

Specific Operator Applications

  • Momentum operator p^=−iℏ∇\hat{p} = -i\hbar\nabla applied to plane wave eikxe^{ikx} yields ℏk\hbar k
  • Energy operator H^\hat{H} applied to stationary state ψn\psi_n returns energy eigenvalue EnψnE_n\psi_n
  • Position operator x^\hat{x} multiplying wave function ψ(x)\psi(x) gives xψ(x)x\psi(x)
  • Angular momentum operator L^z=−iℏ∂∂ϕ\hat{L}_z = -i\hbar\frac{\partial}{\partial\phi} applied to eimϕe^{im\phi} yields mℏm\hbar

Hermitian vs Non-Hermitian Operators

Hermitian Operators

  • Self-adjoint operators satisfying A^=A^†\hat{A} = \hat{A}^\dagger
  • Represent physical observables in quantum mechanics
  • Have real eigenvalues ensuring measured values are real numbers
  • Eigenfunctions form complete orthonormal sets for state expansion
  • Examples include position, momentum, energy, and angular momentum operators

Non-Hermitian Operators

  • Do not represent physical observables directly
  • May have complex eigenvalues
  • Include unitary operators like time evolution operator e−iH^t/ℏe^{-i\hat{H}t/\hbar}
  • Useful mathematical tools (lowering and raising operators in harmonic oscillator)
  • Commutator of two Hermitian operators [A^,B^][\hat{A},\hat{B}] is anti-Hermitian
  • Non-Hermitian Hamiltonians describe open quantum systems with gain or loss

Key Terms to Review (19)

Angular Momentum Operator: The angular momentum operator is a fundamental concept in quantum mechanics, represented by the vector operator \\mathbf{L} = -i\\hbar(\\mathbf{r} \\times \\mathbf{p}) where \\hbar is the reduced Planck's constant, \\mathbf{r} is the position vector, and \\mathbf{p} is the momentum operator. This operator is essential for understanding the angular momentum of quantum systems and relates directly to the properties of operators and their eigenvalues and eigenfunctions.
Bra-ket notation: Bra-ket notation is a mathematical notation used in quantum mechanics to describe quantum states. It provides a compact way to represent vectors and dual vectors in a Hilbert space, where 'bra' vectors denote linear functionals and 'ket' vectors denote state vectors. This notation simplifies operations like inner products and the application of operators, making it essential for working with operators and their properties in quantum physics.
Commutation relations: Commutation relations are mathematical expressions that describe the relationship between two operators in quantum mechanics, specifically whether they commute or not. If two operators commute, their commutation relation is zero, indicating that measurements of the associated observables can be made simultaneously without uncertainty. On the other hand, non-commuting operators reflect fundamental limits on the precision of measurements, connecting deeply to the properties of observables and the nature of quantum states.
Composition of operators: Composition of operators refers to the process of combining two or more linear operators to create a new operator. This concept is essential in understanding how operators can be applied sequentially in mathematical and physical contexts, revealing the interplay between different transformations in a system. When composing operators, their order matters, as the result can vary depending on how they are combined, leading to properties such as associativity but not necessarily commutativity.
Dirac Notation: Dirac notation is a mathematical notation used in quantum mechanics to represent quantum states and their properties. It provides a concise and powerful way to work with vectors in a Hilbert space, making it easier to manipulate complex quantum states and operators. The notation simplifies calculations involving inner products, outer products, and operators acting on states, thereby enhancing our understanding of quantum systems.
Eigenvalue: An eigenvalue is a special scalar associated with a linear transformation represented by an operator, where a corresponding eigenvector is not changed in direction by that transformation, only scaled. This concept is crucial in quantum mechanics, particularly in understanding the outcomes of measurements and the behavior of systems described by the Schrödinger equations. Eigenvalues help determine quantized states and observable properties of physical systems.
Eigenvector: An eigenvector is a non-zero vector that only changes by a scalar factor when a linear transformation is applied to it, represented mathematically as $$A\mathbf{v} = \lambda\mathbf{v}$$, where $$A$$ is the operator, $$\mathbf{v}$$ is the eigenvector, and $$\lambda$$ is the corresponding eigenvalue. Eigenvectors reveal important properties about the operators that transform them, indicating the directions along which these transformations act simply by scaling rather than changing direction. Understanding eigenvectors is crucial for analyzing systems described by linear operators, especially in physics and engineering contexts.
Energy operator: The energy operator, often denoted as \( \hat{H} \), represents the total energy of a quantum system in the framework of quantum mechanics. It acts on wave functions to yield the energy eigenvalues, which correspond to the possible measured values of energy for a given quantum state. This operator is crucial because it connects physical observables to their mathematical representations, allowing for the analysis of how energy influences the behavior of particles at a quantum level.
Hermitian operator: A Hermitian operator is a linear operator in quantum mechanics that is equal to its own adjoint or conjugate transpose, meaning it satisfies the property $$A = A^\dagger$$. This characteristic ensures that the eigenvalues of the operator are real, which is essential for physical observables, as measurements yield real values. Additionally, Hermitian operators play a crucial role in the context of commutation relations and compatible observables, influencing the measurement outcomes and uncertainty in quantum systems.
Linear Operator: A linear operator is a mathematical function that maps one vector space to another while preserving the operations of vector addition and scalar multiplication. This means if you apply the operator to a sum of vectors or a scalar multiplied by a vector, the outcome is the same as applying the operator separately and then combining the results. Understanding linear operators is essential for analyzing systems, particularly in physics, where they relate to transformations, eigenvalues, and observables.
Non-Hermitian Operator: A non-Hermitian operator is a linear operator that does not equal its own adjoint, meaning that for an operator \( A \), the condition \( A \neq A^{\dagger} \) holds. These operators can lead to complex eigenvalues and are essential in quantum mechanics, particularly in systems that are open or driven by external forces, thus connecting them to the properties of various physical systems.
Non-linear operator: A non-linear operator is a mathematical function that does not satisfy the principles of superposition, meaning that the output of the operator is not directly proportional to its input. This concept is critical in many areas of physics, especially in systems where interactions are complex, leading to behaviors that cannot be captured by linear approximations. Non-linear operators can create rich and diverse phenomena, including chaotic behavior, which can be observed in various physical systems.
Observable: An observable is a physical quantity in quantum mechanics that can be measured and is represented by an operator. Observables are fundamental to understanding the behavior of quantum systems, as they help relate mathematical descriptions to real-world measurements. The properties of these observables and the operators that represent them are crucial for predicting outcomes and understanding the relationships between different measurements.
Operator Product: An operator product is a mathematical expression formed by the combination of two or more operators in quantum mechanics. It reflects the way in which operators interact and can be essential in understanding physical quantities such as observables and their measurements. The properties of operator products help elucidate the structure of quantum theory, especially when analyzing systems with multiple degrees of freedom or when applying transformation rules.
Position Operator: The position operator is a fundamental concept in quantum mechanics representing the measurement of a particle's position within a defined space. It acts on the wave function of a quantum system and is typically denoted as \( \hat{x} \). The position operator's properties are essential for understanding how measurements affect quantum states and are key in the formulation of quantum mechanics.
Self-Adjoint Operator: A self-adjoint operator is a linear operator that is equal to its own adjoint, meaning it satisfies the condition \( A = A^\dagger \). This property ensures that the operator has real eigenvalues and orthogonal eigenvectors, making it crucial for quantum mechanics and other areas of physics. Self-adjoint operators are important because they guarantee that observable quantities in quantum systems are measurable and correspond to real values.
Spectral Theorem: The spectral theorem states that any Hermitian operator can be diagonalized by a complete set of orthonormal eigenvectors. This theorem is fundamental in quantum mechanics, as it connects observable physical quantities to their corresponding operators and eigenvalues, providing insight into how measurements affect quantum systems.
Stone's Theorem: Stone's Theorem refers to a result in functional analysis that connects the properties of certain types of operators, specifically bounded linear operators on a Hilbert space, to their spectral characteristics. This theorem is crucial because it establishes the conditions under which a self-adjoint operator can be expressed in terms of its spectral decomposition, allowing for an understanding of how operators behave and interact in a mathematical framework.
Time Evolution Operator: The time evolution operator is a mathematical construct that describes how a quantum state changes over time under the influence of a Hamiltonian. This operator is crucial in quantum mechanics as it encapsulates the dynamics of quantum systems, enabling the calculation of the state of a system at any future time based on its initial condition.
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