Mathematical Methods for Physics and Engineering - Matematica.NET

19Quantum operatorsAlthough the previous chapter was principally concerned with the use of linearoperators and their eigenfunctions in connection with the solution of givendifferential equations, it is of interest to study the properties of the operatorsthemselves and determine which of them follow purely from the nature of theoperators, without reference to specific forms of eigenfunctions.19.1 Operator formalismThe results we will obtain in this chapter have most of their applications in thefield of quantum mechanics and our descriptions of the methods will reflect this.In particular, when we discuss a function ψ that depends upon variables such asspace coordinates and time, and possibly also on some non-classical variables, ψwill usually be a quantum-mechanical wavefunction that is being used to describethe state of a physical system. For example, the value of |ψ| 2 for a particularset of values of the variables is interpreted in quantum mechanics as being theprobability that the system’s variables have that set of values.To this end, we will be no more specific about the functions involved thanattaching just enough labels to them that a particular function, or a particularset of functions, is identified. A convenient notation for this kind of approachis that already hinted at, but not specifically stated, in subsection 17.1, wherethe definition of an inner product is given. This notation, often called the Diracnotation, denotes a state whose wavefunction is ψ by | ψ〉; sinceψ belongs to avector space of functions, | ψ〉 is known as a ket vector. Ket vectors, or simply kets,must not be thought of as completely analogous to physical vectors. Quantummechanics associates the same physical state with ke iθ | ψ〉 as it does with | ψ〉for all real k and θ and so there is no loss of generality in taking k as 1 and θas 0. On the other hand, the combination c 1 | ψ 1 〉 + c 2 | ψ 2 〉,where| ψ 1 〉 and | ψ 2 〉648