MODULE 2(!) Integrals over Operators In quantum chemistry we seek to make contact between calculations done using operators and the actual outcome of experiments. This usually requires us to evaluate certain integrals, all of which have the form In Dirac notation the integral would be symbolized as or the RHS has the appearance of an element of a matrix placed at the intersection of row f and column g. For this reason the integral is referred to as a "matrix element". A closed bracket
MODULE 2 In many cases, the operator is simply multiplication by 1 and then the matrix element has a special symbol The parameter S is given the name "overlap integral" and is a measure of the similarity of a pair of functions. S can take values from 0 to 1. We have seen the limiting values of S in the orthonormality condition.
MODULE 4 MORE ABOUT POSTULATES We need to establish a firm link between what we calculate and what we measure in the laboratory. operator eigenket (function) The operator represents some dynamical observable (energy, momentum, dipole moment, ...) and has an eigenvalue w It is a common practice to name the ket with its eigenvalue.
MODULE 4 Postulate 4 “The only possible values that can result from a series of measurements of a dynamical variable are the eigenvalues of the operator that corresponds to the dynamical variable.” Thus if a system is in the state defined by the ket every measurement of the property represented by will yield as the result (within the error of the experiment).
MODULE 4 For example, the 1s, 2s, 2px,y,z, 3s, etc states of the hydrogen atom are the designations for a series of eigenstates of the hamiltonian operator (see later). If we prepare a large number of H atoms in the 2s state and measure their energies, every one will yield the same result, since the 2s state is an eigenfunction (eigenket) of the hamiltonian operator. Recalling our first equation, with the ket normalized Multiplying from the left by the (normalized) bra closed bracket
MODULE 4 when the ket (and the bra) are eigenvectors (eigenfunctions) of the operator, the closed bracket is the eigenvalue, a real number. However, the system might not be in an eigenstate of the operator we are interested in how to deal with such a situation? To determine the result of experiments of the observable represented by the operator on the state represented by the ket that is not an eigenket of , we must evaluate the expression
MODULE 4 a general function can be expressed as a linear combination of a complete set of eigenfunctions (e.g., a straight line from sine functions or exponentials) of an operator. This is known as the Superposition Principle, and it is centrally important in Quantum Theory. Thus: When the eigenkets form a discrete series, then
MODULE 4 Considering only two states applying the operator to a ket that is not one of its own eigenkets gives a linear combination of its eigenkets multiplied by their eigenvalues.
MODULE 4 Now multiply the last equation from the left by the bra and realizing that Combining these and applying the condition of orthonormality The closed bracket on the LHS is a real number because the square moduli are real The wi are eigenvalues of a Hermitian operator (always real).
MODULE 4 the algebra for the general case is too boring, so we simply state the result Thus where the system is not in an eigenstate of the operator of the required observable the result of a large number of measurements of the observable is seen to be a weighted sum of the eigenvalues of the operator. Each individual eigenvalue contributes to the sum according to the square modulus of the coefficient that governs the contribution of its corresponding eigenket in the LC. New postulate
MODULE 4 POSTULATE 5 "If the system is in an eigenstate of the operator that corresponds to the observable in question, determination of the observable (on a large number of identical preparations) always yields a single, unique result that is the eigenvalue of the operator. If the system is not in an eigenstate of the operator in question, it can be expressed as a superposition of the eigenstates of the operator. Then a single measurement of the observable yields a result that corresponds to one of the eigenvalues of the set of eigenfunctions of the operator. Many determinations of the observable (on a large number of identical preparations) will lead to a distribution of the eigenvalues of the operator. The probability that a particular eigenvalue is measured is equal to where ci is the coefficient of in the superposition."
MODULE 4 A single measurement can provide only one result. A pointer can indicate only one value per determination. Repeated measurements on a large number of identical systems will generate a mean value. This mean is either a unique number (the eigenvalue case). Or is a distribution of numbers about an average value (the not-an-eigenvalue case).
MODULE 4 The RHS is an average value. Thus the matrix element on the LHS is also an average value It is called the expectation value of the operator (a QM average value). In real laboratory experiments it is very unusual for a large number of identical systems to be in a single eigenstate. Measurements will generate the expectation value of the operator.
MODULE 4 In the case of un-normalized wavefunctions the expectation value is given by For particle in a one-dimensional well with infinite walls Suppose we make measurements of the kinetic energy (E) for a series of identical wells in a particular eigenstate because E is an eigenvalue of the KE operator (V = 0) every measurement of E will be identical to every other one (within experimental error.
MODULE 4 So much for the energy, now what about measurements of momentum? For the classical particle we can write p2 =2mE Since E is an eigenvalue of KE operator we might expect p2 and therefore p to be eigenvalues of their corresponding operators Operate on the wavefunction with the momentum operator Thus yn is not an eigenfunction of the momentum operator (sine has become cosine on differentiation)
MODULE 4 according to our postulate, a series of measurements of momentum will yield a distribution of results. Let’s obtain the expectation value of the momentum operator for one of the set of wavefunctions, the kth, for a normalized wavefunction Thus we find that the expectation value of the momentum operator is zero thereforethe average value of a large number of measurements of the momentum will be zero!!!!
MODULE 4 To resolve this apparent paradox we examine the square of the momentum in the x direction. The appropriate operator for this is Thus we see that yk is an eigenfunction of the operator for the square of the momentum a set of measurements of px2 on identical systems will always provide the same result, namely the eigenvalue.
MODULE 4 For the n = 1 energy level we have Now we see the resolution of our paradox. The quantity p2 (an eigenvalue) is the same for all determinations, but p can either be +(2mE)1/2 or – (2mE)1/2. Both of these quantities is an eigenvalue of the momentum operator, but the wavefunction is not an eigenfunction thereof.
MODULE 4 A single measurement of p will always yield one of these values if we make a large number of such measurements we shall find an equal number of the positive and negative values, from which the average value will be zero. the momentum solutions appear with equal weights. We never know in advance whether a single determination will yield the positive or the negative value; but we know that we shall find one or the other and that eventually we shall find an equal number of each. In terms of the motion of the particle we can imagine that the two momentum values correspond to motions in the +x and -x directions.