As noted in Section 1.5 , the average values of the observables of a quantum system are determined by its wave function Ψ . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C002_equ_0001.tif"/> In this chapter, we introduce some of the mathematical concepts and techniques needed to analyze quantum wave functions. Especially, we will develop the theory of Hermitian eigenvalue problems . Such problems as we will see occur nearly universally in applications of quantum mechanics to chemical problems.
In this chapter, we develop results that will prove to be very important later. First, we show that if two observables A https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C005_equ_0001.tif"/> and B https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C005_equ_0002.tif"/> are compatible , that is, if their Hermitian operator representatives A ^ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C005_equ_0003.tif"/> and B ^ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C005_equ_0004.tif"/> commute, then these representatives have a complete orthonormal set of common eigenfunctions Ψ a n , b m ( r ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C005_equ_0005.tif"/> which are the solutions of the simultaneous eigenvalue problem A ^ Ψ a n , b m ( r ) = a n Ψ a n , b m ( r ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C005_equ_0006.tif"/> and B ^ Ψ a n , b m ( r ) = b n Ψ a n , b m ( r ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C005_equ_0007.tif"/> Consequently, from the discussion at the end of Section 4.5 , when the system is in a quantum state specified by one of the common eigenfunctions Ψ a n , b m ( r ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C005_equ_0008.tif"/> , both A https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C005_equ_0009.tif"/> and B https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C005_equ_0010.tif"/> have definite values, a n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C005_equ_0011.tif"/> for A https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C005_equ_0012.tif"/> and b m https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C005_equ_0013.tif"/> for B . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C005_equ_0014.tif"/> In contrast, we show that if two observables A https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C005_equ_0015.tif"/> and B https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C005_equ_0016.tif"/> are incompatible , that is, if their representatives A ^ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C005_equ_0279.tif"/> and B ^ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C005_equ_0017.tif"/> do not commute, then these representatives have few if any common eigenfunctions. As a consequence rarely if ever can one find quantum states for which two incompatible observables A https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C005_equ_0018.tif"/> and B https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C005_equ_0019.tif"/> both have definite values rather than spreads of values. Therefore, two incompatible observables cannot (with rare exceptions) be simultaneously precisely measured. Rather, in analogy to the Heisenberg uncertainty relation Δ x Δ p x ≥ ℏ 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C005_equ_0020.tif"/> which exists between the incompatible position and momentum observables, a Heisenberg uncertainty relation exists between any pair of incompatible observables A https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C005_equ_0021.tif"/> and B . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C005_equ_0022.tif"/>
Abstract In this chapter, the main features of quantum theory are presented. The chapter begins with a historical account of the invention of quantum mechanics. The meaning of position and momentum in quantum mechanics is discussed and non-commuting operators are introduced. The Schrödinger equation is presented and solved for a free particle and for a harmonic oscillator potential in one dimension. The meaning of the wavefunction is considered and the probabilistic interpretation is presented. The mathematical machinery and language of quantum mechanics are developed, including Hermitian operators, observables and expectation values. The uncertainty principle is discussed and the uncertainty relations are presented. Scattering and tunnelling by potential wells and barriers is considered. The use of variational principles to estimate ground state energies is explained and illustrated with a simple example.
In Section 1.3 , we developed the Bohr theory of the hydrogen-like atoms: H , H e + , L i 2 + https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C010_equ_0001.tif"/> , and so on. Here, we develop the modern quantum theory of these atoms. Again, we deal with a two-particle system comprised of an electron of mass 10.1 m e = 9.10938972 × 10 − 31 k g https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429155741/5d159368-dc20-4763-957b-88b0e101c959/content/C010_equ_0002.tif"/>
This text presents a rigorous mathematical account of the principles of quantum mechanics, in particular as applied to chemistry and chemical physics. Applications are used as illustrations of the basic theory. The first two chapters serve as an introduction to quantum theory, although it is assumed that the reader has been exposed to elementary quantum mechanics as part of an undergraduate physical chemistry or atomic physics course. Following a discussion of wave motion leading to Schrödinger's wave mechanics, the postulates of quantum mechanics are presented along with essential mathematical concepts and techniques. The postulates are rigorously applied to the harmonic oscillator, angular momentum, the hydrogen atom, the variation method, perturbation theory, and nuclear motion. Modern theoretical concepts such as hermitian operators, Hilbert space, Dirac notation, and ladder operators are introduced and used throughout. This text is appropriate for beginning graduate students in chemistry, chemical physics, molecular physics and materials science.
Quantum mechanics (or simply quantum physics) is the branch of physics that applies to particles and systems of particles, such as atoms and molecules. In that realm the laws of classical mechanics and classical electromagnetism break down, and only quantum mechanics provides the concepts and computational methods that explain observed phenomena. Quantum mechanics uses specially developed tools, such as wave functions, that contain all of the relevant information about particles and systems. is approach is needed to understand the true nature of electromagnetic radiation and how that radiation interacts with matter. e mathematics used in quantum mechanics is highly specialized and often complex. In this chapter we’ll address some of the key questions that require quantum mechanics without presenting the full mathematical formalism. e important questions we’ll address all deal with observable physical phenomena. We’ll discuss how quantum mechanics is used to answer these questions but will omit the advanced mathematical details.
The representation of physical observables by Hermitian linear operators was postulated assuming that an expectation value, defined by Eqs.(2.11) and (2.12) , is associated with any observable A measured in a quantum state Ψ of a dynamical system. In view of measurement uncertainties, defined by the dispersion: ( Δ A ) 2 = 〈 A 2 〉 − 〈 A 〉 2 = 〈 ( A − 〈 A 〉 ) 2 〉 = 〈 ( δ A ) 2 〉 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003078265/e77b8ce0-ccb3-48ed-9f6d-a84c8cb529c5/content/eqn3_1_B.tif"/> the expectation value 〈 A 〉 is taken in a state Ψ if and only if (Δ Α ) 2 vanishes. It is convenient to introduce the deviation operator: 3.1 δ  =  − 〈 A 〉 Î https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003078265/e77b8ce0-ccb3-48ed-9f6d-a84c8cb529c5/content/eqn3_2_B.tif"/> such that the condition of a vanishing dispersion for the observable A in a given state Ψ reads: 〈 Ψ |( δ  ) 2 Ψ 〉 = 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003078265/e77b8ce0-ccb3-48ed-9f6d-a84c8cb529c5/content/eqn3_3_B.tif"/> Because δ is a Hermitian operator, we can also write: 〈 δ Â Ψ | δ Â Ψ 〉 = | δ Â Ψ | 2 = 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003078265/e77b8ce0-ccb3-48ed-9f6d-a84c8cb529c5/content/eqn3_4_B.tif"/> which means that the norm of δ Â Ψ is zero and this implies that we have a null function δ Â Ψ = 0. In other words: 3.2 Â Ψ = 〈 A 〉 Ψ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003078265/e77b8ce0-ccb3-48ed-9f6d-a84c8cb529c5/content/eqn3_5_B.tif"/> It follows that the state function Ψ where a definite expectation value 〈 A 〉 of the observable A is observed, should be a solution of Eq.(3.2) rather than an arbitrary 112 function. The action of the operator  on Ψ has to be simply that of rescaling. All the state functions which are solutions of Eq.(3.2) are called eigenfunctions of  , and the constant factor 〈 A 〉 is said to be the corresponding eigenvalue. Hence we have shown that if ((δ  ) 2 ) = 0, the state function should be an eigenfunction of  , which means that a physical observable only takes definite values in states represented by eigenfunctions of the associated quantum operator. Conversely, given a quantum operator  one can systematically determine all its eigenvalues and eigenfunctions, which are solutions of the so-called eigenvalue equation: 3.3 Â Ψ = λ Ψ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003078265/e77b8ce0-ccb3-48ed-9f6d-a84c8cb529c5/content/eqn3_6_B.tif"/> It is visible that the eigenvalue equation fixes the eigenfunctions only up to an overall scale factor, which may always be chosen such that the eigenfunctions be considered normalized. Then, in view of Eq.(2.11) , Eq.(3.3) gives: 〈  〉 = ∫ − ∞ ∞ Ψ * Â Ψ d x = λ ∫ − ∞ ∞ Ψ * Ψ d x = λ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003078265/e77b8ce0-ccb3-48ed-9f6d-a84c8cb529c5/content/eqn3_7_B.tif"/> If we further apply the operator  to Eq.(3.3) , we obtain  2 Ψ = λ 2 Ψ and this leads to: 〈  2 〉 = ∫ − ∞ ∞ Ψ *  2 Ψ d x = λ 2 ∫ − ∞ ∞ Ψ * Ψ d x = λ 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003078265/e77b8ce0-ccb3-48ed-9f6d-a84c8cb529c5/content/eqn3_8_B.tif"/> It follows that: ( Δ A ) 2 = 〈  2 〉 − 〈  〉 2 = 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003078265/e77b8ce0-ccb3-48ed-9f6d-a84c8cb529c5/content/eqn3_9_B.tif"/> and this shows that the dispersion vanishes if a physical observable is measured in states represented by eigenfunctions of the associated operator.