1.2 Operators and observables

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 operator 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: $\hat{A}(a\psi_1 + b\psi_2) = a\hat{A}\psi_1 + b\hat{A}\psi_2$
  • Nonlinear operators do not satisfy these properties
• The action of an operator on a wave function is denoted by placing the operator symbol to the left of the wave function
  • For example, $\hat{A}\psi(x)$ represents the action of operator $\hat{A}$ on the wave function $\psi(x)$
• The Hermitian conjugate of an operator is obtained by taking the complex conjugate of the operator and transposing it
  • Denoted as $\hat{A}^{\dagger}$
  • Hermitian operators satisfy $\hat{A} = \hat{A}^{\dagger}$, and their eigenvalues are real

Commutators and Simultaneous Measurability

• The commutator of two operators $\hat{A}$ and $\hat{B}$ is defined as $[\hat{A}, \hat{B}] = \hat{A}\hat{B} - \hat{B}\hat{A}$
  • 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 uncertainty principle)
• The canonical commutation relation between position ($\hat{x}$) and momentum ($\hat{p}$) operators is $[\hat{x}, \hat{p}] = i\hbar$
  • This relation leads to the position-momentum uncertainty principle: $\Delta x \Delta p \geq \frac{\hbar}{2}$

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 observable
  • Examples of physical observables include position, momentum, energy, and angular momentum
• The expectation value 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 $\hat{A}$: $\langle \hat{A} \rangle = \langle \psi | \hat{A} | \psi \rangle = \int \psi^*(x) \hat{A} \psi(x) dx$
  • 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 standard deviation $\Delta A$, given by $\Delta A = \sqrt{\langle \hat{A}^2 \rangle - \langle \hat{A} \rangle^2}$
  • 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 $\frac{\hbar}{2}$
  • For example, the position-momentum uncertainty principle: $\Delta x \Delta p \geq \frac{\hbar}{2}$
  • 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 position operator $\hat{x}$ is represented by the multiplication of the position variable $(x)$ with the wave function
  • $\hat{x}\psi(x) = x\psi(x)$
• The momentum operator $\hat{p}$ is represented by the negative imaginary unit $(-i\hbar)$ multiplied by the partial derivative with respect to position
  • $\hat{p}\psi(x) = -i\hbar \frac{\partial}{\partial x}\psi(x)$
• The energy operator, or Hamiltonian $\hat{H}$, is the sum of the kinetic and potential energy operators
  • $\hat{H} = \hat{T} + \hat{V} = -\frac{\hbar^2}{2m}\nabla^2 + V(x)$, where $m$ is the mass of the particle and $V(x)$ is the potential energy
• Angular momentum operators $\hat{L}_x$, $\hat{L}_y$, and $\hat{L}_z$ are defined in terms of position and momentum operators
  • $\hat{L}_x = \hat{y}\hat{p}_z - \hat{z}\hat{p}_y$, $\hat{L}_y = \hat{z}\hat{p}_x - \hat{x}\hat{p}_z$, $\hat{L}_z = \hat{x}\hat{p}_y - \hat{y}\hat{p}_x$

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
  • $i\hbar \frac{\partial}{\partial t}\Psi(x, t) = \hat{H}\Psi(x, t)$
  • The solution to this equation gives the wave function $\Psi(x, t)$ at any time $t$, given an initial wave function $\Psi(x, 0)$
• The time-independent Schrödinger equation is an eigenvalue equation for the Hamiltonian operator
  • $\hat{H}\psi(x) = E\psi(x)$, where $E$ is the energy eigenvalue
  • 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 $\hat{A}\psi_n = a_n\psi_n$, then $a_n$ is an eigenvalue of the operator $\hat{A}$, and $\psi_n$ is the corresponding eigenfunction
• Eigenfunctions are the wave functions that, when acted upon by an operator, result in the same wave function multiplied by the corresponding eigenvalue
  • $\hat{A}\psi_n = a_n\psi_n$, where $\psi_n$ is an eigenfunction of operator $\hat{A}$ with eigenvalue $a_n$
• The set of eigenfunctions for an operator forms a complete orthonormal basis, known as the eigenbasis
  • Orthonormality: $\langle \psi_m | \psi_n \rangle = \delta_{mn}$, where $\delta_{mn}$ is the Kronecker delta
  • Completeness: Any wave function can be expressed as a linear combination of the eigenfunctions, $\psi(x) = \sum_n c_n \psi_n(x)$

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 $a_n$: $P(a_n) = |\langle \psi_n | \psi \rangle|^2 = |c_n|^2$
• The spectral decomposition theorem states that any wave function can be expressed as a linear combination of the eigenfunctions of an operator
  • $\psi(x) = \sum_n c_n \psi_n(x)$, where $c_n = \langle \psi_n | \psi \rangle$ 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