E

P. W. Atkins

Published Apr 11, 1991• 0 citations

Abstract

Abstract When a mathematical operation (such as multiplication, division, or differentiation) is performed on a function, the result is generally some different function. For instance, differentiation of x2 yields the different function 2x. However, for some combinations of operations and functions the same function is regenerated, but perhaps multiplied by a number (Fig. E.l). Thus differentiation of e2x gives 2e2X, which is the same function multiplied by 2. In such cases, the function is said to be an eigenfunction of the operator (in this case the differential operator d/dx), and the numerical factor (2 in the example) is called the eigenvalue of the operator. (Eigen is the German word meaning ‘own’ or ‘characteristic’.) If an operator Q operates on a function f and produces wf, where w is a number, f is an eigenfunction of Q and w is the corresponding eigenvalue. An expression of the form Qi= wf is called an eigenvalue equation.

DOI: 10.1093/oso/9780198555735.003.0005

Related papers

APPROXIMATION METHODS

Vishnu S. Mathur, Surendra Singh

We have seen that to determine the possible energies En (energy eigenvalues) which a physical system can have and the corresponding eigenfunctions (wave functions) ψn, we have to set up and solve the time-independent Schro¨dinger equation Hψn = Enψn . (8.1.1) In this equation, the wave function ψn, representing the n-th eigenstate of the system in the coordinate representation, is a function of the continuously varying eigenvalues of the position observables of the system, the number of position observables being equal to number of degrees of freedom of the system. The Hamiltonian operator H, in the coordinate representation, is a differential operator involving the continuously varying eigenvalues of the position observables and the derivatives with respect to them. Since we will be dealing with the operators mostly in the coordinate representation, we shall denote them without the caret.

Chapman & Hall/CRC eBooksDec 12, 20081 citation

The Schrödinger Wave Equation

Don Shillady

This chapter is primarily about ‘‘wave mechanics’’ since that is the most convenient way to introduce undergraduates to quantum mechanics using calculus. An equivalent form called ‘‘matrix mechanics’’ will be discussed brieﬂy in a later chapter. Consider again the 1923 paper by De Broglie and the experiments that validated the particle-wave duality in Chapter 10. One might well ask that if there really is some ‘‘wave’’ that describes the behavior of particles, then is there an equation that the wave obeys? Even today it is difﬁcult to say what the ‘‘wave’’ is, but it may help to ﬁnd an equation it obeys. Step back a moment to some basic calculus: d2 dx2 sin (ax) ¼ d dx [a cos (ax)] ¼ (a2) sin (ax): Perhaps you did not notice the pattern before but we can put this into a general form as (Operator)(Eigenfunction) ¼ (Eigenvalue)(Eigenfunction): The word ‘‘eigen’’ in German means ‘‘characteristic, unique, peculiar, special . . . ’’ and only certain functions satisfy this condition called an eigenfunction equation. An analogy that has been successful in explaining this to undergraduates is to consider an apple tree with ripe apples on it. If you hit the branches with a stout stick some apples will fall off the tree but the tree will still be there. The operator is the act or operation of hitting the tree with the stick, the tree is the eigenfunction and the apples are the eigenvalue(s). The eigen word comes from German because this relationship was ﬁrst linked to the De Broglie wave idea by Erwin Schrödinger in 1926 in a series of papers that are among the most important in modern science [1]. Schrödinger (1887-1961) was an Austrian physicist who received the Nobel Prize for his work in 1933 (Figure 11.1).

CRC Press eBooksJul 27, 20110 citations

On a New Treatment of Some Eigenvalue Problems

Léopold Infeld

A new method for treating the most important eigenvalue problems in quantum mechanics is developed. The solutions can be found immediately once the equations are factorized by means of linear operators. These operators acting on a normalized eigenfunction change it into a new normalized eigenfunction and all solutions can be found once the basic eigenfunction is known. This basic eigenfunction is a solution of a simple differential equation of the first order. The underlying theory is explained more fully on a special case (Section 1) and then the rules of procedure are formulated explicitly (Section 2). The rest of the paper contains applications including the Kepler problem treated according to Dirac's theory.

Physical ReviewMay 1, 1941139 citations

Chapter 8. Operators, Eigenfunctions, and Eigenvalues

Dennis Morris

De Gruyter eBooksMay 27, 20160 citations

P. W. Atkins

Abstract According to quantum mechanics, all the dynamical information about a system is contained in its °wave function. The outcome of an observation of a particular property can be predicted by performing the appropriate mathematical operation on the wave function. Each physical observable is therefore represented by an operator, a mathematical action such as multiplication or differentiation. The operator corresponding to the observable Q is often denoted fi, but we shall not make that distinction here. Different choices of operators for observables correspond to the different representations of quantum mechanics.

Apr 11, 19911 citation

MATRIX MECHANICS OF ANGULAR MOMENTUM

Mircea S. Rogalski, Stuart B. Palmer

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.

CRC Press eBooksJan 8, 20140 citations