16.3 Operators and observables

Quantum mechanics uses operators to represent physical quantities like position and momentum. These mathematical tools act on wave functions, extracting information about the system's state. Operators are crucial for calculating probabilities and expectation values of observables.

Eigenvalues and eigenfunctions of operators reveal possible measurement outcomes and corresponding quantum states. This concept connects to the broader Schrödinger equation, showing how operators help describe a system's behavior and energy levels in quantum mechanics.

Operators in Quantum Mechanics

Definition and Role of Operators

• Operators are mathematical objects that act on functions to produce other functions in quantum mechanics
• They represent physical quantities (position, momentum, energy) and extract information about the state of a quantum system
• The action of an operator on a quantum state is defined by its matrix elements, determined by the specific form of the operator
• Operators can be linear or nonlinear and may be Hermitian (self-adjoint) or non-Hermitian

Eigenvalues and Eigenfunctions

• The eigenvalues and eigenfunctions of an operator provide important information about the possible outcomes of measurements and the corresponding probabilities
• Eigenvalues represent the possible outcomes of a measurement of an observable associated with an operator
• Eigenfunctions are the quantum states that, when acted upon by the operator, yield a specific eigenvalue multiplied by the original eigenfunction
• The eigenvalues and eigenfunctions of an operator form a complete set, meaning any arbitrary quantum state can be expressed as a linear combination of the eigenfunctions
• The eigenvalues of the Hamiltonian operator correspond to the allowed energy levels of the quantum system, while the eigenfunctions represent the stationary states of the system

Operators vs Observables

Relationship between Operators and Observables

• Observables are physical quantities that can be measured in a quantum system (position, momentum, energy)
• Each observable is associated with a corresponding operator that acts on the quantum state to extract information about the observable
• The eigenvalues of an operator represent the possible outcomes of a measurement of the associated observable
• The eigenfunctions represent the quantum states that yield those specific measurement outcomes

Calculating Expectation Values

• The expectation value of an observable is calculated by taking the inner product of the quantum state with the product of the associated operator and the quantum state
• Expectation values provide a way to characterize the average value of an observable for a given quantum state
• For example, the expectation value of the position operator $\hat{x}$ for a quantum state $\psi(x)$ is given by $\langle x \rangle = \int \psi^*(x) \hat{x} \psi(x) dx$
• Similarly, the expectation value of the momentum operator $\hat{p}$ is calculated as $\langle p \rangle = \int \psi^*(x) \hat{p} \psi(x) dx$

Applying Quantum Postulates

Postulates of Quantum Mechanics

• The first postulate states that the state of a quantum system is described by a wave function or state vector in a complex Hilbert space
• The second postulate states that observables are represented by Hermitian operators acting on the Hilbert space, and the possible outcomes of a measurement are the eigenvalues of the operator
• The third postulate states that the probability of measuring a particular eigenvalue is given by the squared modulus of the inner product between the quantum state and the corresponding eigenfunction
• The fourth postulate describes the time evolution of a quantum state under the influence of a Hamiltonian operator, governed by the Schrödinger equation

Solving Problems with Operators and Observables

• Applying these postulates allows for the calculation of probabilities, expectation values, and the time evolution of quantum systems in terms of operators and observables
• For example, to find the probability of measuring a specific eigenvalue $\lambda_i$ of an operator $\hat{A}$, one calculates $P(\lambda_i) = |\langle \psi | \phi_i \rangle|^2$, where $|\psi\rangle$ is the quantum state and $|\phi_i\rangle$ is the eigenfunction corresponding to $\lambda_i$
• To determine the time evolution of a quantum state $|\psi(t)\rangle$, one solves the time-dependent Schrödinger equation $i\hbar \frac{\partial}{\partial t}|\psi(t)\rangle = \hat{H}|\psi(t)\rangle$, where $\hat{H}$ is the Hamiltonian operator

Eigenvalues and Eigenfunctions

Physical Meaning of Eigenvalues and Eigenfunctions

• Eigenvalues and eigenfunctions provide a way to characterize the physical properties and behavior of quantum systems
• The eigenvalues of an operator represent the possible outcomes of a measurement of the associated observable
• The eigenfunctions represent the quantum states that yield those specific measurement outcomes
• For example, the eigenvalues of the Hamiltonian operator correspond to the allowed energy levels of the quantum system, while the eigenfunctions represent the stationary states of the system

Completeness and Expansion of Quantum States

• The eigenvalues and eigenfunctions of an operator form a complete set, meaning any arbitrary quantum state can be expressed as a linear combination of the eigenfunctions
• This property allows for the expansion of a quantum state in terms of the eigenfunctions of an operator
• For example, a quantum state $|\psi\rangle$ can be written as $|\psi\rangle = \sum_i c_i |\phi_i\rangle$, where $|\phi_i\rangle$ are the eigenfunctions of an operator and $c_i$ are the expansion coefficients
• The expansion coefficients $c_i$ are given by the inner product between the quantum state and the corresponding eigenfunction, $c_i = \langle \phi_i | \psi \rangle$

Commutation Relations

Definition and Properties of Commutation Relations

• Commutation relations describe the behavior of operators when their order of application is exchanged
• Two operators $\hat{A}$ and $\hat{B}$ are said to commute if their commutator, defined as $[\hat{A}, \hat{B}] = \hat{A}\hat{B} - \hat{B}\hat{A}$, is equal to zero
• If two operators commute, they can be simultaneously diagonalized, meaning they share a common set of eigenfunctions
• Non-commuting operators cannot be simultaneously measured with arbitrary precision, as described by the uncertainty principle

Implications of Commutation Relations

• The commutation relation between the position and momentum operators, $[\hat{x}, \hat{p}] = i\hbar$, is a fundamental result in quantum mechanics and leads to the Heisenberg uncertainty principle
• The uncertainty principle states that the product of the uncertainties in the measurement of position and momentum is always greater than or equal to $\hbar/2$, i.e., $\Delta x \Delta p \geq \hbar/2$
• Commutation relations play a crucial role in determining the compatibility of observables and the limitations on simultaneous measurements in quantum systems
• For example, the angular momentum operators $\hat{L}_x$, $\hat{L}_y$, and $\hat{L}_z$ do not commute with each other, implying that it is not possible to simultaneously measure all three components of angular momentum with arbitrary precision