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Review of Time-Independent Quantum Mechanics

While some spectroscopic observations can be understood using purely classical concepts, most molecular spectroscopy experiments probe explicitly quantum mechanical properties. It is assumed that students using this text have already taken a course in basic quantum mechanics, but it is also recognized that there are likely to be some holes in the preparation of most students and that all can benefit from a brief review. As this is not a quantum mechanics textbook, many results in this chapter are given without proof and with minimal explanation. Students seeking a deeper treatment are encouraged to consult the references given at the end of this chapter (Das and Melissinos 1986; Fayer 2001; Levine 2009; McQuarrie 2008; Sakurai 1993).

This chapter, like most introductory quantum chemistry courses, focuses on solutions of the time-independent Schrödinger equation. Because of the importance of time-dependent quantum mechanics in spectroscopy, that topic is discussed in Chapter 5.

1.1 States, Operators, and Representations

A quantum mechanical system consisting of N particles (usually electrons and/or nuclei) is represented most generally by a state function or state vector ?. The state vector contains, in principle, all information about the quantum mechanical system.

In order to be useful, state vectors must be expressed in some basis. In the most commonly used position basis, the state vector is called the wavefunction, written as ?(r 1, r 2, ., r N ) where r i is the position in space of particle i. The position r may be expressed in Cartesian coordinates (x, y, z), spherical polar coordinates (r, ?, f), or some other coordinate system. Wavefunctions may alternatively be expressed in the momentum basis, ?(p 1, p 2, ., p N ) where p i is the momentum of particle i. Some state vectors cannot be expressed as a function of position, such as those representing the spin of an electron. But there's always a state function that describes the system, even if it's not a "function" of ordinary spatial coordinates.

The wavefunction itself, also known as the probability amplitude, is not directly measurable and has no simple physical interpretation. However, the quantity |?(r 1, r 2, ., r N )|2 dr 1dr 2.dr N gives the probability that particles 1, 2, etc. are each in some infinitesimal volume element around r 1, r 2, etc. Integration over a finite volume then gives the probability that the system is found within that volume. A "legal" wavefunction must be single valued, continuous, differentiable, and normalizable.

The scalar product or inner product of two wavefunctions ? and F is given by ??*F, where the asterisk means complex conjugation and the integration is performed over all of the coordinates of all the particles. The inner product is not a function but simply a number, generally a complex number if the wavefunctions are complex. In Dirac notation, this inner product is denoted <??│?F>. The absolute square of the inner product, |<??│?F>|2, gives the probability (a real number) that a system in state ? is also in state F. If <??│?F> = 0, then ? and F are said to be orthogonal. Reversing the order in Dirac notation corresponds to taking the complex conjugate of the inner product: = ???F*? while <??│?F> = ????*F = (?F*?)*.

The inner product of a wavefunction with itself, <??│??> = ????*? = ???|?|2, is always real and positive. Usually single-particle wavefunctions are chosen to be normalized to ??*? = 1. This means that the probability of finding the system somewhere in space is unity.

The quantities we are used to dealing with in classical mechanics are represented in quantum mechanics by operators. Operators act on wavefunctions or state vectors to give other wavefunctions or state vectors. Operator acting on wavefunction ? to give wavefunction F is written as . The action of an operator can be as simple as multiplication, although many (not all) operators involve differentiation.

Quantum mechanical operators are linear, which means that if ? 1 and ? 2 are numbers (not states or operators), then , and . However, it is not true in general that ; the order in which the operators are applied often matters. The quantity is called the commutator of and and is symbolized , and it is zero for some pairs of operators but not for all. Most of what is "interesting" (i.e. nonclassical) about quantum mechanical systems arises from the noncommutation of certain operators.

A representation is a set of basis vectors, which may be discrete (finite or infinite) or continuous. An example of a finite discrete basis is the eigenstates of the z-component of spin for a spin-1/2 particle (two states, usually called a and ß). An example of a discrete infinite basis is the set of eigenstates of a one-dimensional harmonic oscillator {? v } where v must be an integer but can go from 0 to 8. An example of a continuous basis is the one-dimensional position basis {x} where x can take on any real value. To be a representation, a set of basis vectors must obey certain extra conditions. One is orthonormality: <u i |u j > = d ij (the Kronecker delta) for a discrete basis, or (the Dirac delta function) for a continuous basis. The Kronecker delta is defined by d ij = 1 if i = j and d ij = 0 if i???j. The Dirac delta function d(a -?a´) is a hypothetical function of the variable a that is infinitely sharply peaked around a = a´ and has an integrated area of unity. Three useful properties of the Dirac delta function are:

where a is a constant.

A set of vectors in a particular state space is a basis if every state in that space has a unique expansion, such that ? = S i c i u i (discrete basis) or ? = ???da c(a)w a (continuous basis), where the c's are (generally complex) numbers. "In a particular state space" means, e.g. that if we want to describe only the spin state of a system, the basis does not have to include the spatial degrees of freedom. Or, the states of position in one dimension {x} can be a basis for a particle in a one-dimensional box, but not a two-dimensional box, which requires a two-dimensional position basis {(x, y)}. An important property of a representation is closure:

Representations of states and operators in discrete bases are often conveniently written in matrix form (see Appendix C). A state vector is represented in a basis by a column vector of numbers: and its complex conjugate by a row vector: (u 1>u 2>?). The inner product is then obtained by the usual rules for matrix multiplication as

An operator is represented by a square matrix:

For Hermitian operators, A ji * = A ij . It follows that the diagonal elements (A ii ) must be real for Hermitian operators.

The operator expression is represented in the {u i } basis as the matrix equation

1.2 Eigenvalue Problems and the Schrödinger Equation

The state ? is an eigenvector or eigenstate of operator with eigenvalue ? if where ? is a number. That is, operating on ? with just multiplies ? by a constant. The eigenvalue ? is nondegenerate if there is only one eigenstate having that eigenvalue. If more than one distinct state (wavefunctions that differ from each other by more than just an overall multiplicative constant) has the same eigenvalue, then that eigenvalue is degenerate.

To every observable (measurable quantity) in classical mechanics, there corresponds a linear, Hermitian operator in quantum mechanics. Since observables correspond to measurable things, this means all observables have only real eigenvalues. It can be shown from this that eigenfunctions of the same observable having different eigenvalues are necessarily orthogonal (orthonormal if we require that they be normalized).

In Dirac notation using basis {u i }, the eigenvalue equation is . Inserting closure gives , or in a shorter form...