Читать книгу Quantum Mechanical Foundations of Molecular Spectroscopy - Max Diem - Страница 21

Postulate 1: The state of a quantum mechanical system is completely defined by a wavefunction Ψ(x, t). The square of this function, or in the case of complex wavefunction, the product Ψ*(x, t) Ψ(x, t), integrated over a volume element dτ (= dx dy dz in Cartesian coordinates or sin²θ dθ dφ in spherical polar coordinates) gives the probability of finding a system in the volume element dτ. Here, Ψ*(x, t) is the complex conjugate of the function Ψ(x, t). This postulate contains the transition from a deterministic to probabilistic description of a quantum mechanical system. The wavefunctions must be mathematically well behaved, that is, they must be single‐valued, continuous, having a continuous first derivative, and integratable (so they can be normalized).

Postulate 2: The classical linear momentum expression, p = mv, is substituted in quantum mechanics by the differential operator , defined by

(2.2)

operating (or being applied to) the wavefunction Ψ(x, t). In Eq. (2.2), i is the imaginary unit, defined by Equation (2.2) often is considered the central postulate of QM.

The form of Eq. (2.2) can be made plausible from equations of classical wave mechanics, de Broglie's equation (Eq. [1.10]) and Planck's equation (Eq. [1.7]), but cannot be derived axiomatically. It was the genius of E. Schrödinger to realize that the substitution described in Eq. (2.2) yields differential equations that had long been known and had solutions that agreed with experiments. In the Schrödinger equations to be discussed explicitly in the next chapters (for the H atom, the vibrations and rotations of molecules, and molecular electronic energies), the classical kinetic energy T given by

(2.3)

is, therefore, substituted by

(2.4)

which is, of course, obtained by inserting Eq. (2.2) into Eq. (2.3). The total energy of a system is given as the sum of the potential energy V and the kinetic energy T:

(2.5)

Postulate 3: All experimental results are referred to as observables that must be real (not imaginary or complex). An observable is associated with (or is the “eigenvalue” of) a quantum mechanical operator . This can be written as

(2.6)

where a are the eigenvalues and ϕ the corresponding eigenfunctions. The terms “operator,” “eigenvalues,” and “eigenfunctions” are terminology from linear algebra and will be further explained in Section 2.3 where the first real eigenvalue problem, the particle in a box, will be discussed. Notice that the eigenfunctions often are polynomials, and each of these eigenfunctions has its corresponding eigenvalue.

In this book, following generally accepted notations, the total energy operator is generally identified by the symbol and referred to as the Hamilton operator, or the Hamiltonian, of the system. With the definition of the Hamiltonian above, it is customary to write the total energy equation of the system as

(2.7)

Equation (2.7) implies that the energy “eigenvalues” E are obtained by applying the operator on a set of (still unknown) eigenfunctions ψ that are here assumed to be time‐independent and a function of spatial coordinates x only, ψ(x). Solving the differential equations given by Eq. (2.7) yields the eigenfunctions ψ_i and their associated energy eigenvalues E_i.

Postulate 4: The expectation value of an observable a, associated with an operator , for repeated measurements, is given by

(2.8)

If the wavefunctions Ψ(x, t) are normalized, Eq. (2.7) simplifies to

(2.9)

since the denominator in Eq. (2.8) equals 1. This expectation value may be viewed as an expected average of many independent measurements and embodies the probabilistic nature of quantum mechanics.

Postulate 5: The eigenfunctions ϕ_i, which are the solutions of the equation , form a complete orthogonal set of functions or, in other words, define a vector space. This, again, will be demonstrated in Section 2.3 for the particle‐in‐a‐box wavefunctions, which are all orthogonal to each other and therefore may be considered unit vectors in a vector space.

When evaluating the expectation values (Eq. [2.9]), the functions ψ(x) may or may not be eigenfunctions of because the real eigenfunctions ϕ(x) form a complete vector space. Functions that are not eigenfunctions of can be written as linear combinations of the basis functions ϕ(x). Thus, any arbitrary wavefunction ψ of a system can be written in terms of a series expansion of the true eigenfunctions ϕ(x) as follows:

(2.10)

The expansion coefficients a_n indicate how much each wavefunction contributes to, or resembles, the true eigenfunction of the operator. This aspect is particularly important for the approximate methods for solving the Schrödinger equation discussed in Appendix 2.

Postulate 6: Time‐dependent systems are described by the time‐dependent Schrödinger equation

(2.11)

where the time‐dependent wavefunctions are the product of a time‐independent part, ψ(x), and a time evolution part:

(2.12)

We shall encounter the time‐dependent Schrödinger equation mainly in processes where molecular systems are subject to a perturbation by electromagnetic radiation (i.e. in spectroscopy) and shall develop the formalism that predicts whether or not the incident radiation will cause a transition in the molecule between two states with energy difference ΔE = h ν = ħ ω.

Next, a simple operator/eigenvalue example will be presented to illustrate some of the mathematical aspects.

Example 2.1 Operator/eigenvalue problem

Show that the function is an eigenfunction of the operator , that is, show that

Answer:

(E2.1.1)

The function is an eigenfunction of the operator. The eigenvalue c = −1.

Postulate 7: In many‐electron atoms, no two electrons can have identically the same set of quantum numbers. This postulate is known as the Pauli exclusion principle. It is also formulated as follows: the product wavefunction for all electrons in an atom must be antisymmetric with respect to interchange of two electrons. This postulate leads to the formulation of the product wavefunction in the form of Slater determinants (see Section 9.2) in many‐electron systems. The value of a determinant is zero when two rows or two columns are equal; thus, an atomic system where any electrons have exactly the same four quantum numbers would have an undefined product wavefunction. Furthermore, exchange of two rows (or columns) leads to a sign change of the value of the determinant. This last statement implies the antisymmetric property of the product wavefunction that changes its sign upon exchange of two electrons.

Commutation of operators: Although not really a postulate of quantum mechanics (since it follows from well‐defined mathematical principles), a discussion of the effects of operator commutation is included here. In physics, one often wishes to determine several quantities simultaneously, such as the position and momentum of a moving object or the x, y, and z components of the angular momentum. Since Postulate 3 above states that every observable is associated with a quantum mechanical operator, one has to investigate the case of solving for the eigenvalues of two operators simultaneously.

Let and be two operators such that

(2.13)

where a and b are the eigenvalues and φ and ϕ the eigenfunctions of and , respectively. These eigenvalues can be determined simultaneously in the same vector space if and only if the operators commutate, that is, if the order of application of the operators on the eigenfunction is immaterial. This commutator of two operators is written as

(2.14)

or abbreviated as . If the operators commutate and can be determined simultaneously; if the commutator is not zero, then the eigenvalues cannot be determined simultaneously. This case will be demonstrated in Example 2.2.

Example 2.2 Determine the commutator of the momentum operator and the position operator when applied to a function f(x), i.e. determine

(E2.2.1)

(E2.2.2)

Answer:

The derivative of the product needs to be evaluated using the product rule of differentiation. Thus,

(E2.2.3)

(E2.2.4)

Thus, the commutator

(E2.2.5)

which predicts that the position and momentum of a moving particle cannot be determined simultaneously. This was stated earlier in Eq. (2.1) as the Heisenberg uncertainty principle as

(2.1)

To show the equivalency of Eqs. (E2.2.5) and (2.1), one has to determine the standard deviations in momentum and position σ_p and σ_x that can be related to the uncertainties Δp_x and Δx.

Figure 2.1 Potential energy functions and analytical expressions for (a) molecular vibrations and (b) an electron in the field of a nucleus. Here, f is a force constant, k is the Coulombic constant, and e is the electron charge.