Atkins & de Paula: Atkins Physical Chemistry 9e Checklist of key ideas. Chapter 7: Quantum Theory: Introduction and Principles

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1 Atkins & de Paula: Atkins Physical Chemistry 9e Checklist of key ideas Chapter 7: Quantum Theory: Introduction and Principles

2 classical mechanics, the laws of motion introduced in the seventeenth century by Isaac Newton. quantum mechanics, the laws of motion introduced in the twentieth century by Heisenberg and Schrödinger. THE ORIGINS OF QUANTUM MECHANICS electromagnetic field, an oscillating electric and magnetic disturbance that spreads as a harmonic wave through space. electric field, a field that acts on charged particles. magnetic field, a field that acts on moving charged particles.

3 wavelength, λ, the peak-to-peak distance of a wave. frequency, v, the number of times per second that a displacement returns to its initial value. wavenumber, vɶ, the reciprocal of the wavelength. electromagnetic spectrum, the range of frequencies exhibited by the electromagnetic field and its classification into regions. 7.1 The failures of classical physics black body, an object capable of emitting and absorbing all frequencies of radiation uniformly.

4 7.1 The failures of classical physics (cont ) Rayleigh Jeans law, de = ρdλ, ρ = 8πkT/λ 4. density of states, ρ, the proportionality constant between the range of wavelengths and the energy density in that range: de = ρdλ. ultraviolet catastrophe, the divergence of the energy density of black-body radiation at high frequencies. quantization of energy, the limitation of energies to discrete values. Planck s constant, h = J s.

5 7.1 The failures of classical physics (cont..) Planck distribution, de= ρdλ, ρ = (8πhc/λ 5 )/(e hc/λkt 1). Dulong and Petit s law: the molar heat capacities of all monatomic solids are the same, and close to 25 J K 1 mol 1. Einstein formula, C V,m = 3Rf, f = (θ E /T) 2 {e θ_e/2t /(e θ_e/t 1)} Einstein temperature, θ E = hv/k. Debye formula, C V,m = 3Rf, f T = θ 3 4 x θ / T x 0 2. D 3 d D x e x ( e 1)

6 7.1 The failures of classical physics (cont..) Debye temperature, θ D = hv D /k. spectrum, the record of intensity of light transmitted, absorbed, or scattered as a function of frequency, wavelength, or wavenumber. spectroscopy, the detection and analysis of a spectrum. spectroscopic transition, a change of state that gives rise to a feature in spectrum. Bohr frequency transition, the relation between the change in energy and the frequency of the radiation emitted or absorbed: E = hv.

7 7.2 Wave particle duality photon, a particle of electromagnetic radiation. photoelectric effect, the ejection of electrons from metals when they are exposed to ultraviolet radiation: ½m e v 2 = hv Φ. work function, Φ, the energy required to remove an electron from the metal to infinity. Davisson Germer experiment, the diffraction of electrons by a crystal. electron diffraction, the diffraction of electrons by an object in their path. de Broglie relation, λ = h/p. wave particle duality, the joint particle and wave character of matter and radiation.

8 THE DYNAMICS OF MICROSCOPIC SYSTEMS wavefunction, ψ, a mathematical function obtained by solving the Schrödinger equation and which contains all the dynamical information about a system. 7.3 The Schrödinger equation time-independent Schrödinger equation, (ħ 2 /2m)(d 2 ψ/dx 2 ) + V(x)ψ = Eψ. 7.4 The Born interpretation of the wavefunction Born interpretation, the value of ψ 2 at a point is proportional to the probability of finding the particle at that point.

9 7.4 The Born interpretation of the wavefunction (cont..) Born interpretation, the value of ψ 2 at a point is proportional to the probability of finding the particle at that point. probability density, the probability of finding a particle in a region divided by the volume of the region. probability amplitude, the square-root of the probability density (the wavefunction itself). normalization constant, N = 1/{ ψ*ψ dx} 1/2. spherical polar coordinates, the radius r, the colatitude θ, and the azimuth φ. The volume element in spherical coordinates is r 2 sin θ drdθdφ. quantization, confinement of a dynamical observable to discrete values.

10 7.4 The Born interpretation of the wavefunction (cont..) constraints on the wavefunction, the conditions a wavefunction must obey (be continuous, have a continuous first derivative, be single-valued, and be square-integrable). QUANTUM MECHANICAL PRINCIPLES 7.5 The information in a wavefunction node, a point where a wavefunction passes through zero. operator, something that carries out a mathematical operation on a function. hamiltonian operator, the operator for the total energy of a system, H ˆ ψ = Eψ.

11 7.5 The information in a wavefunction (cont..) eigenvalue, the constant ω in the eigenvalue equation Ωˆ ψ = ωψ. eigenfunction, the function ψ in the eigenvalue equation Ωˆ ψ = ωψ. eigenvalue equation, an equation of the form Ωˆ ψ = ωψ. observable, measurable properties of a system. position operator, xˆ = x. momentum operator, pˆ x = (ħ/i)d/dx. hermitian operator, an operator for which it is true that ψ Ωˆ ψ dx i j = { } ψ Ωˆ ψ dx j i

12 7.5 The information in a wavefunction (cont..) orthogonal functions, ψ i *ψ j dτ = 0. linear combination of two functions, c 1 f + c 2 g. superposition, a linear combination of wavefunctions. complete set of functions, functions that can be used to express any arbitrary function as a linear combination. expectation value, Ω = ψ Ωˆ ψdτ. 7.6 The uncertainty principle Heisenberg uncertainty principle: it is impossible to specify simultaneously, with arbitrary precision, both the 1 momentum and the position of a particle; p q ħ. 2

13 7.6 The uncertainty principle (cont..) wave packet, a localized wavefunction formed by superimposing a series of wavefunctions. complementary observables, observables corresponding to non-commuting operators. Ω ˆ ˆ. commuting operators, operators for which [, Ω ] 0 ˆ commutator, [ Ω ] 1, Ω2 = Ω1Ω2 Ω2Ω1 ˆ ˆ ˆ ˆ ˆ 1 2 = general form of the Heisenberg uncertainty principle: 1 [ ] Ω 1 Ω ˆ ˆ 2 Ω1, Ω 2 2

14 Atomic Units mass of an electron : charge : 4πε 0ħ length : Bohr radius a 0 = 2 mee vacuum permittivity ε : e ( energy : 27.21eV = kcal/mol 0 m e h ħ = 2π -31 = kg 2-11 = m 1 4πε 1 = SI atomic unit C) J/mol = 1 hartree