Mathematical Methods for Physics and Engineering - Matematica.NET

19.2 PHYSICAL EXAMPLES OF OPERATORSquantum-mechanical operators are those corresponding to position r and momentump. One prescription for making the transition from classical to quantummechanics is to express classical quantities in terms of these two variables inCartesian coordinates and then make the component by component substitutionsr → multiplicative operator r and p → differential operator − i∇.(19.22)This generates the quantum operators corresponding to the classical quantities.For the sake of completeness, we should add that if the classical quantity containsa product of factors whose corresponding operators A and B do not commute,then the operator 1 2(AB + BA) is to be substituted for the product.The substitutions (19.22) invoke operators that are closely connected with thetwo that we considered at the start of the previous subsection, namely x and∂/∂x. One,x, corresponds exactly to the x-component of the prescribed quantumposition operator; the other, however, has been multiplied by the imaginaryconstant −i, where is the Planck constant divided by 2π. This has the (subtle)effect of converting the differential operator into the x-component of an Hermitianoperator; this is easily verified using integration by parts to show that it satisfiesequation (17.16). Without the extra imaginary factor (which changes sign undercomplex conjugation) the two sides of the equation differ by a minus sign.Making the differential operator Hermitian does not change in any essentialway its commutation properties, and the commutation relation of the two basicquantum operators reads[[ p x ,x] = −i ∂ ]∂x ,x = −i. (19.23)Corresponding results hold when x is replaced, in both operators, by y or z.However, it should be noted that if different Cartesian coordinates appear in thetwo operators then the operators commute, i.e.[ p x ,y] = [ p x ,z] = [ p y ,x ] = [ p y ,z ] = [ p z ,x] = [ p z ,y] =0.(19.24)As an illustration of the substitution rules, we now construct the Hamiltonian(the quantum-mechanical energy operator) H for a particle of mass m movingin a potential V (x, y, z) when it has one of its allowed energy values, i.e itsenergy is E n ,whereH|ψ n 〉 = E n |ψ n 〉. This latter equation when expressed in aparticular coordinate system is the Schrödinger equation for the particle. In termsof position and momentum, the total classical energy of the particle is given byE = p22m + V (x, y, z) =p2 x + p 2 y + p 2 z+ V (x, y, z).2mSubstituting −i∂/∂x for p x (and similarly for p y and p z ) in the first term on the657