Mathematical Methods for Physics and Engineering - Matematica.NET

QUANTUM OPERATORSis to produce a scalar multiple of that ket, i.e.A| ψ〉 = λ| ψ〉, (19.3)then, just as for matrices and differential equations, | ψ〉 is called an eigenket or,more usually, an eigenstate of A, with corresponding eigenvalue λ; tomarkthisspecial property the state will normally be denoted by | λ〉, rather than by themore general | ψ〉. Taking the Hermitian conjugate of this ket vector eigenequationgives a bra vector equation,〈ψ|A † = λ ∗ 〈ψ|. (19.4)It should be noted that the complex conjugate of the eigenvalue appears in thisequation. Should the action of A on |ψ〉 produce an unphysical state (usuallyone whose wavefunction is identically zero, and is therefore unacceptable as aquantum-mechanical wavefunction because of the required probability interpretation)we denote the result either by 0 or by the ket vector |∅〉 according tocontext. Formally, |∅〉 can be considered as an eigenket of any operator, but onefor which the eigenvalue is always zero.If an operator A is Hermitian (A † = A) then its eigenvalues are real andthe eigenstates can be chosen to be orthogonal; this can be shown in the sameway as in chapter 17 (but using a different notation). As indicated there, thereality of their eigenvalues is one reason why Hermitian operators form thebasis of measurement in quantum mechanics; in that formulation of physics, theeigenvalues of an operator are the only possible values that can be obtained whena measurement of the physical quantity corresponding to the operator is made.Actual individual measurements must always result in real values, even if theyare combined in a complex form (x + iy or re iθ ) for final presentation or analysis,and using only Hermitian operators ensures this. The proof of the reality of theeigenvalues using the Dirac notation is given below in a worked example.In the same notation the Hermitian property of an operator A is representedby the double equality〈Aφ|ψ〉 = 〈φ|A| ψ〉 = 〈φ|Aψ〉.It should be remembered that the definition of an Hermitian operator involvesspecifying boundary conditions that the wavefunctions considered must satisfy.Typically, they are that the wavefunctions vanish for large values of the spatialvariables upon which they depend; this deals with most physical systems sincethey are nearly all formally infinite in extent. Some model systems require thewavefunction to be periodic or to vanish at finite values of a spatial variable.Depending on the nature of the physical system, the eigenvalues of a particularlinear operator may be discrete, part of a continuum, or a mixture of both. Forexample, the energy levels of the bound proton–electron system (the hydrogenatom) are discrete, but if the atom is ionised and the electron is free, the energy650