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.
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.
Operators in quantum mechanics which represent observables are of a special 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. This chapter presents important properties of linear operators and their eigenfunctions. It discusses the features of various operators which are necessary for studying quantum mechanical systems. The chapter shows how rich and beautiful the mathematical structure of quantum mechanics is. There exists a class of eigenvalue problems, the so-called parity-time reversal (PT) -symmetric non-Hermitian Hamiltonian problems. They have arisen in recent years in a number of physical contexts such as localization transitions, vortex pinning and population biology. Such non-Hermitian Hamiltonians can be used to describe certain experimental phenomena. PT-symmetry has been noticed in certain optical systems, such as optical couplers, microwave billiard and large-scale temporal lattices.
The basic equation of quantum mechanics is the Schrödinger equation which expresses the wave function Ψ of a quantum system as an eigenfunction of a quantized Hamiltonian operator H : HΨ = λΨ where the (real) eigenvalue λ is the quantum energy of the system in the state Ψ see equations (1.3.2), (1.3.4). Embodied already in this equation is the basic quantum mechanical principle that quantum energies cannot take on arbitrary values but are quantized : they are given by a discrete set of eigenvalues of a suitable second-order differential operator. This mathematical phenomenon of the discreteness of eigenvalues explains, for example, the observed discreteness of absorption and emission atomic spectral lines; compare remarks in Sections 1.2 and 1.3 of Chapter 1. The Schrödinger theory, and the equivalent theory of Heisenberg, Born and Jordan, represents a distinct advancement of the Bohr theory. Some early basic papers on quantum mechanics are compiled in the book [82], which includes a historic introduction by B. van der Waerden. Also see [6, 8, 9, 24, 75, 76]. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Abstract Mathematical formalism of quantum mechanics is closely connected with the theory of linear operators. A key quantum-mechanical principle is that physical quantities (observables) are represented by Hermitian linear operators that act on state vectors that belong to a Hilbert vector space. This chapter analyzes problems that deal with the basic concepts of the theory of linear operators; eigenfunctions, eigenvalues, mean values; the projection operators; quantum-mechanical representations of operators and wave-functions; and unitary operators.
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 .
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)
Ψ=Ψ Iˆ In view of the expansion (29.28), which holds for an arbitrary state function in the form Ψ = Ψ Ψ Ψ Ψ = Ψ Ψ Ψ∑ ∑ we may define the unit or identity operator as i iI ΨΨ=∑ =1 ˆ (33.4) Conversely, for every linear operator we can form a square matrix, in terms of a complete set of functions by multiplying the left-hand side of Eq.(33.3) by and integrating, which gives fˆ ,iΨ ∗Ψi ΨΨ=ΨΨ=ΦΨ=ΦΨ∫ ∗ Iffd iiii ˆˆˆV ΨΨ=ΨΨΨΨ= n ˆˆ Since ii b=ΦΨ this result is identical with that given by Eq.(33.1), provided we set Vˆˆ dfff kikiik ΨΨ=ΨΨ= ∫ ∗ (33.5) which are called the matrix elements of the linear operator in terms of the particular complete set of functions It is a straightforward matter to show that the rules of matrix algebra for addition, multiplication, associativity and distributivity (see Appendix II) apply to the matrix elements (33.5), following from their definition. Two properties of fˆ .iΨ matrix elements that are needed in quantum mechanics can be readily derived from Eq.(33.5). Using Eq.(29.19) we obtain ∗∗ =ΨΨ=ΨΨ= kiikkiik ffff ˆˆ (33.6) which means that if is a Hermitian operator, then its matrix representation in a complete set of functions is Hermitian. Hence, the elements in the principal diagonal of a Hermitian matrix must be real. When all the elements are real, the matrix must be symmetric. From Eq.(29.20) we have fˆ ikiikiik f δλλ =ΨΨ=ΨΨ ˆ (33.7) that is: if the functions are the eigenfunctions of the Hermitian operator the matrix representation is diagonal, and the matrix elements are the eigenvalues of the operator. A quantum mechanical problem can be stated in a differential equation form or in a matrix form. In both cases the solution consists of finding the eigenvalues and the eigenfunctions of an operator associated with a physical observable. In view of a matrix representation in any complete set of functions, the eigenvalue equation (29.20) must be formulated as iΨ ,fˆ (33.8) Ψ=Ψ λfˆ where the arbitrary eigenfunction Ψ can be expanded, as given by Eq.(29.4), in terms of a complete set of functions If we multiply Eq.(33.8) by and integrate, we obtain .iΨ ∗Ψi ˆ ˆ ˆV V ori i if d d f Iλ λ∗ ∗Ψ Ψ = Ψ Ψ Ψ Ψ = Ψ Ψ∫ ∫ i or also, in view of Eq.(33.4) k kki f λ ˆ This can be rewritten as = n 1 λ which reads (33.9) ⎟⎟ ⎟⎟ ⎟ ⎠ ⎞ ⎜⎜ ⎜⎜ ⎜ ⎝ ⎛ = ⎟⎟ ⎟⎟ ⎟ ⎠ ⎞ ⎜⎜ ⎜⎜ ⎜ ⎝ ⎛ ⎟⎟ ⎟⎟ ⎟ ⎠ ⎞ ⎜⎜ ⎜⎜ ⎜ ⎝ ⎛ a a a a a a fff fff fff …… … ……… … … λ Equation (33.6) is called the matrix eigenvalue equation and consists of n linear homogenous equations for the given by ,ka ( ) 0... 1212111 =+++− nnafafaf λ ( ) 0... 2222121 =++−+ nnafafaf λ (33.10) ……… ( ) 0...2211 =−+++ nnnnn afafaf λ This system of simultaneous equations has a nontrivial solution for the if and only if the characteristic determinant vanishes ( ) ( ) ( ) ( ) 01ˆˆdet = ⎟⎟ ⎟⎟ ⎟ ⎠ ⎞ ⎜⎜ ⎜⎜ ⎜ ⎝ ⎛ − − − =− λ λ λ λ nnnn fff fff fff f … ……… … … (33.11) If expanded, the determinant gives a polynominal of nth degree called the characteristic polynominal of fˆ with n roots which are the desired eigenvalues. For each eigenvalue Eqs.(33.10) can be solved for the giving the corresponding eigenfunction that is a column matrix.