An overview of main operators in quantum mechanics is presented. Operators for position, linear momentum, kinetic energy, potential energy, total energy, and angular momentum are introduced. Properties of linear operators are described. Commutative property of operators has profound implications because a pair of non-commutative operators cannot be measured accurately at the same time. Eigenfunctions and eigenvalues of an operator are defined, and operator form of Schrodinger’s equation is given. Most operators in quantum mechanics are of the Hermitian category because they always have real eigenvalues. The condition to test Hermitian nature of operators is specified. Postulates of quantum mechanics are stated to serve as a convenient starting point.
An overview of main operators in quantum mechanics is presented. Operators for position, linear momentum, kinetic energy, potential energy, total energy, and angular momentum are introduced. Properties of linear operators are described. Commutative property of operators has profound implications because a pair of non-commutative operators cannot be measured accurately at the same time. Eigenfunctions and eigenvalues of an operator are defined, and operator form of Schrodinger’s equation is given. Most operators in quantum mechanics are of the Hermitian category because they always have real eigenvalues. The condition to test Hermitian nature of operators is specified. Postulates of quantum mechanics are stated to serve as a convenient starting point.
of theory What you need to know Definitions and properties Operator, linear operator, functions of operators, commuting and noncommuting operators, eigenfunction, eigenvalue, degeneracy, normalised function, orthogonal functions, Hermitian operator. Postulates of quantum mechanics (1) The state of a system with n position variables q 1 , q 2 , … q n is specified by a state (or wave) function ψ( q 1 , q 2 , … q n ). All possible information about the system can be derived from this state function. In general, n is three times the number of particles in the system. So for a single particle n = 3, and q 1 , q 2 , q 3 may be the Cartesian coordinates x, y, z , or the spherical polar coordinates r , θ, φ, or some other set of coordinates. (2) To every observable there corresponds a Hermitian operator given by the following rules: (i) The operator corresponding to the Cartesian position coordinate x is x × similarly for the coordinates y and z . (ii) The operator corresponding to p x , the x component of linear momentum, is (ħ/i)∂/∂ x – similarly for the y and z components. (iii) To obtain the operator corresponding to any other observable, first write down the classical expression for the observable in terms of x, y, z, p x , p y , p z , and then replace each of these quantities by its corresponding operator according to rules (i) and (ii). (3) The only possible result which can be obtained when a measurement is made of an observable whose operator is A is an eigenvalue of A .
According to quantum mechanics, the information with respect to any measurement on a physical system is contained in a mathematical object, the wave function. In this chapter we become familiar with the mathematical objects that represent the measured properties themselves, namely the quantum mechanical operators. We start from a brief introduction into operators and their properties, emphasizing linear operators, and noncommuting operators. Then we introduce the canonical position and momentum operators. Defining functions of operators, we derive different quantum mechanical operators that correspond to different physical observables, including angular momentum, kinetic energy, and the scalar potential energy. Finally, we introduce the quantum mechanical total energy operator (the Hamiltonian) and demonstrate its explicit generic form for nanoscale building blocks such as atoms and molecules.
A basic quantity in quantum mechanics is the wave function ψ. Given a quantum mechanical system we write the Schrödinger equation for it. Its solution is the wave function. When introducing the wave function we stated that the maximum information about the state of a system can be obtained only from ψ. How do we extract observable properties of the system from the wave function? In quantum mechanics for every classical dynamical variable there exists an operator. For example, the operator of momentum p is −i~∇ while the operator of energy variable H = p2/(2m)+V (X) is −(~2/2m)∇2+V (X). Operation of an operator on ψ gives another function. If the new function is λψ, then λ is the eigenvalue of the operator. For example, the energy of a particle in a particular state represented by ψ can be obtained by operating H on ψ. That is, we have Hψ = Eψ where E is the energy eigenvalue. The expectation value of A(X,p) can be obtained from the formula 〈A〉 = ∫ ∞ −∞ ψ∗Aψ dτ , (3.1) where ψ is the solution of the equation Hψ = Eψ. Operators in quantum mechanics which represent observables are of a spe- cial type: They are linear and Hermitian. Such operators not only form computation tools, but also form the most effective language in terms of which the theory can be formulated. The fascinating point is that using linear operators we can understand the quantum mechanics. Therefore, it is important to study the basic theory of linear operators. In this chapter, we present important properties of linear operators and their eigenfunctions. We discuss the features of various operators which are necessary for studying quantum and ematical structure of quantum mechanics is.
Part 1: the need for the quantum mechanics quantization of radiation the special theory of relativity wave-particle duality the nature of matter waves the uncertainty principle the Bohr atom. Part 2 Schrodinger's equation: the postulates of quantum mechanics the wave equation the wave equation in three dimensions the time-independent Schrodinger equation stationary states normalisation probability flux physical constraints on the wave-function wave packets Gaussian wave-packets and the uncertainty principle. Part 3 One-dimensional potential wells: potential wells the one-dimensional infinite square well discussion of infinite square-well results eigenvalue equations the finite square well finite square well eigenfunctions the control of the wave-function by the wave equation numerical integration of Schrodinger's equation continuum solutions of the square-well potential interpretation of unbound-particle wave functions the double potential well the quantum bouncer electric fields and the tilted well the harmonic oscillator - introduction the quantum oscillator discussion of harmonic oscillator eigenvalues and eigenfunctions the classical probability density more potential wells. Part 4 Schrodinger's equation in three dimensions: particle in a three dimensional box degeneracy Schrodingers equation in spherical polar coordinates - the hydrogen atom normalisation. Part 5 Formal development of quantum mechanics: eigenvalues equations, operators and measurements the momentum operator, eigenvalues and eigenfunctions the position operator and its eigenfunctions Hermitian operators orthogonality of eigenfunctions the expansion postulate example - the b-decay of tritium expectation values compatibility of observables - commutators non-commuting operators time-dependence of expectation values - Ehrenfest's theorem abstract operators - the parity operator, matrix mechanics. Part 6 Angular momentum: angular momentum operators angular momentum commutators angular momentum eigenvalues and eigenfunctions interpretation of angular momentum eigenvalues interpretation of angular momentum eigenfunctions the Stern-Gerlach apparatus and electron spin spin operators - the Pauli Spin matrices the Zeeman effect measurements in quantum mechanics. Part 7 Spherically symmetric potential wells: the rigid rotator the radial equation the hydrogen atom and hydrogen like ions the isotropic harmonic oscillator molecular vibrations - the Morse potential the spherical square well general properties of spherically symmetric potential wells. (Part contents)
Abstract Chapter 9 begins laying a foundation for quantum statistical mechanics with a discussion of relevant topics in quantum mechanics. The fundamental postulates of quantum mechanics are reviewed, beginning with the definition of a quantum state and the introduction of the concept of a state vector in a Hilbert space. The representation of observables as Hermitian operators on Hilbert space vectors is described; it is shown that the eigenvalues of Hermitian operators are real and, therefore, constitute possible outcomes of measurements made on these observables. The notions compatible and non-compatible observables and implication for the measurement problem are presented. The time-dependent Schrödinger equation is introduced as generating the time-evolution of the state vector, and it is shown how to cast the Schrödinger equation into a coordinate basis to yield the wave equation of quantum mechanics. Coordinate and momentum bases and their properties are desribed. Several analytically solvable quantum systems are presented as illustrative examples. The chapter concludes with a discussion of spin and the treatment identical particles