Quantum Mechanical Formalism and Operators

Vocabulary flashcards covering operator algebra, Hermitian properties, Schmidt orthogonalisation, and the basic postulates of quantum mechanics.

Operator

A symbol for a rule for transforming a given mathematical function into another function, which possesses no physical meaning when written alone.

Commutative Operators

Two operators whose successive applications yield the identical result regardless of the order of operation, satisfying $\hat{A}\hat{B}f(x) = \hat{B}\hat{A}f(x)$.

Linear Operator

An operator $\hat{A}$ whose application satisfies $\hat{A}[f(x) + g(x)] = \hat{A}f(x) + \hat{A}g(x)$ and $\hat{A}[Cf(x)] = C\hat{A}f(x)$, where $C$ is a constant.

Commutator Operator

The difference between the successive operations of two operators in reversed orders, denoted by $[\hat{A}, \hat{B}] = \hat{A}\hat{B} - \hat{B}\hat{A}$; it equals the zero operator if the operators commute.

Del (Vector) Operator ($\nabla$)

A differential vector operator defined as $\nabla = \mathbf{i}\frac{\partial}{\partial x} + \mathbf{j}\frac{\partial}{\partial y} + \mathbf{k}\frac{\partial}{\partial z}$, which transforms a scalar function into its gradient vector.

Laplacian Operator ($\nabla^2$)

The square of the del operator, defined in Cartesian coordinates as $\nabla^2 = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}$.

Eigenvalue Equation

An equation of the form $\hat{A}f = \lambda f$, in which an operator operates on a well-behaved function to produce the identical function multiplied by a constant.

Eigenvalue

The constant multiplier ($\lambda$ or $a$) obtained when an operator operates on an eigenfunction in an eigenvalue equation.

Eigenfunction

A well-behaved (finite, continuous, and single-valued) function that is transformed by a given operator into itself multiplied by a constant eigenvalue.

Hermitian Operator

An operator $\hat{A}$ that satisfies the integral condition $\int \psi^* (\hat{A}\phi)\,d\tau = \int (\hat{A}\psi)^* \phi\,d\tau$, where $\psi^*$ is the complex conjugate of $\psi$.

Orthogonality Condition

The property that two eigenfunctions $\psi_1$ and $\psi_2$ of a Hermitian operator corresponding to non-degenerate eigenvalues satisfy $\int \psi_1^* \psi_2\,d\tau = 0$.

Schmidt Orthogonalisation Process

A mathematical procedure used to construct an orthogonal pair of functions from degenerate, normalized, but non-orthogonal eigenfunctions using suitable linear combinations, yielding $\phi_2 = (1-S^2)^{-1/2}(\psi_2 - S\psi_1)$.

Complete Set of Functions

A series of functions $\psi_1, \psi_2, \dots$ subject to specific boundary conditions such that any arbitrary function $F$ with the same restrictions can be expressed as their linear combination ($F = \sum_i a_i \psi_i$).

Postulate I of Quantum Mechanics

The postulate stating that the state of a microsystem is described in terms of a wave function $\Psi$ of position coordinates and time, which contains all obtainable information about the system.

Postulate II of Quantum Mechanics

The postulate stating that to every experimentally measurable physical quantity (observable), there corresponds a quantum mechanical operator.

Momentum Operator ($\hat{p}_x$)

The quantum mechanical operator corresponding to momentum along the $x$-coordinate, defined as $\hat{p}_x = -\frac{h}{2\pi i} \frac{\partial}{\partial x}$.

Hamiltonian Operator ($\hat{H}$)

The quantum mechanical operator representing total energy (the sum of kinetic and potential energies), expressed for a single particle as $\hat{H} = -\frac{h^2}{8\pi^2 m}\nabla^2 + V$.

Angular Momentum Operator ($\hat{\mathbf{L}}$) Component $\hat{L}_z$

The operator for the $z$-component of angular momentum, given by $\hat{L}_z = -\frac{h}{2\pi i}\left(x\frac{\partial}{\partial y} - y\frac{\partial}{\partial x}\right)$.

Postulate III of Quantum Mechanics

The postulate stating that the possible values of any physical quantity of a system are given by the eigenvalues $a$ in the operator equation $\hat{A}\psi = a\psi$.

Postulate IV of Quantum Mechanics

The postulate stating that the expected average (expectation) value of an observable $M$ for a system described by state function $\Psi$ is given by $\langle M \rangle = \frac{\int \Psi^* \hat{M}\Psi\,d\tau}{\int \Psi^* \Psi\,d\tau}$.

Postulate V of Quantum Mechanics

The postulate stating that every system satisfies the time-dependent Schrödinger equation, $\hat{H}\Psi = -\frac{ih}{2\pi} \frac{\partial \Psi}{\partial t}$.

Stationary State

A quantum state described by a wave function for which the probability density $|\Psi|^2$ and physical properties (such as energy) are completely independent of time.