2.4 Operators and Observables in Quantum Mechanics
4 min read•august 14, 2024
Quantum mechanics relies heavily on operators and observables to describe physical systems. These mathematical tools allow us to extract information about particles and waves, connecting the abstract world of wave functions to measurable quantities.
Operators act on wave functions, while observables represent physical quantities we can measure. Understanding their relationship is crucial for grasping how quantum mechanics predicts experimental outcomes and describes the behavior of particles at the atomic scale.
Operators in Quantum Mechanics
Definition and Properties
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Operators are mathematical entities that act on wave functions to extract information or transform them in quantum mechanics
Linear operators satisfy the linearity property: A^(αψ1+βψ2)=αA^ψ1+βA^ψ2, where α and β are complex numbers, and ψ1 and ψ2 are wave functions
Nonlinear operators do not satisfy this property
have real eigenvalues and orthogonal eigenfunctions
They satisfy the condition ⟨ψ1∣A^ψ2⟩=⟨A^ψ1∣ψ2⟩∗, where ∗ denotes the complex conjugate
Unitary operators preserve the inner product between wave functions
They satisfy the condition U^U^†=U^†U^=I^, where U^† is the adjoint of U^, and I^ is the identity operator
Expectation Values
The expectation value of an operator A^ for a given state ψ is calculated as ⟨A^⟩=⟨ψ∣A^ψ⟩
Represents the average value of the observable associated with the operator
For example, the expectation value of the x^ gives the average position of a particle in a given state
Expectation values are crucial for making predictions and comparing theoretical results with experimental measurements in quantum mechanics
Operators and Observables
Relationship between Operators and Observables
Observables are physical quantities that can be measured in quantum mechanics
Examples include position, momentum, energy, and angular momentum
Each observable is associated with a Hermitian operator that acts on the wave function to extract the observable's value
The eigenvalues of an operator represent the possible outcomes of a measurement of the corresponding observable
For instance, the eigenvalues of the energy operator (Hamiltonian) give the possible energy levels of a quantum system
The eigenfunctions of an operator form a complete set of basis states for the Hilbert space
Any state can be expressed as a linear combination of these eigenfunctions
Uncertainty Principle
The states that certain pairs of observables cannot be simultaneously measured with arbitrary precision
This is related to the non-commutative nature of their corresponding operators
The most well-known example is the Heisenberg uncertainty principle for position and momentum: ΔxΔp≥ħ/2
Δx is the uncertainty in position, Δp is the uncertainty in momentum, and ħ is the reduced Planck's constant
The uncertainty principle imposes fundamental limitations on the precision of simultaneous measurements of incompatible observables
Commutation Relations of Operators
Definition and Properties
The commutator of two operators A^ and B^ is defined as [A^,B^]=A^B^−B^A^
If the commutator is zero, the operators are said to commute
are essential for determining the compatibility of observables and the uncertainty relations between them
Examples of Commutation Relations
The position and momentum operators in one dimension satisfy the canonical commutation relation: [x^,p^]=iħ
This commutation relation leads to the Heisenberg uncertainty principle for position and momentum
The angular momentum operators L^x, L^γ, and L^x satisfy the cyclic commutation relations:
[L^x,L^γ]=iħL^x
[L^γ,L^x]=iħL^x
[L^x,L^x]=iħL^γ
These commutation relations are crucial for understanding the properties of angular momentum in quantum mechanics, such as quantization and conservation laws
Eigenfunctions and Eigenvalues of Operators
Eigenvalue Equation
Eigenfunctions ψn of an operator A^ satisfy the : A^ψn=anψn
an is the corresponding eigenvalue
To find the eigenfunctions and eigenvalues, one solves the eigenvalue equation by applying the operator to a general wave function and solving the resulting differential or algebraic equation
Significance of Eigenfunctions and Eigenvalues
The eigenvalues of an operator represent the possible outcomes of a measurement of the associated observable
The eigenfunctions of an operator form a complete set of basis states for the Hilbert space
Any state can be expressed as a linear combination of these eigenfunctions
Eigenfunctions and eigenvalues play a central role in determining the energy levels, angular momentum states, and other properties of quantum systems
Examples of Operators and Their Eigenfunctions
The position operator x^ has eigenfunctions δ(x−x0) with eigenvalues x0
These eigenfunctions represent states with a definite position x0
The p^ has eigenfunctions exp(ipx/ħ) with eigenvalues p
These eigenfunctions represent states with a definite momentum p
The Hamiltonian operator H^ has eigenfunctions ψn with eigenvalues En
The eigenfunctions represent the stationary states of the system, and the eigenvalues give the corresponding energy levels