Topics in quantum mechanics
The book consists of two parts that are rather different in character. The first part is about the basics of quantum mechanics with very detailed calculations. Contrary to the usual habit setting the physical constants in a suitable way to one they are kept all the time. The second part focuses on path integrals and zeta functions and discusses their use in quantum statistical mechanics. The last chapter, contributed by P. Shanahan, gives a short introduction to gauge theory. Chapter 0 introduces the basic physical quantities like length, mass, force, energy etc. as well as their units. The last point is certainly interesting for mathematicians who in most cases ignore physical constants and their units. The units are given in the cgs system. The SI system is mentioned only at the end of the chapter. In Chapter 1 some of the ideas at the beginning of the 20th century which led to quantum mechanics are reviewed. Starting with some remarks on Planck's quantization this chapter presents Bohr's frequency rules and the emission spectra for the hydrogen atom. Then Schrödinger's equation is introduced, followed by a discussion of the de Broglie wavelength and Heisenberg's uncertainty principle where the slit experiment is considered. Finally the particle-wave dualism is noted in the section about Bohr's complementary principle. The next chapter introduces elementarily some concepts of classical mechanics. It begins with Newton's law for the equation of motion to proceed with the explicit calculation of some examples in one dimension. After passing to three dimensions Hamilton's and Lagrange's equations of motion are introduced and again some examples like a ball in constant gravitational field are calculated in detail. This chapter also deals with many particle problems and the transformation into centre of mass coordinates is presented. Chapter 3 deals with the Schrödinger equation, first written in the time dependent way \(i\hbar {\partial \psi \over \partial t} = {\partial^2 \psi \over \partial x^2} + V\psi\) but passes immediately to the time-independent version since the potential does not depend on time. In what follows the classical problems like the harmonic oscillator and the finite potential well are solved. Then an \(n\)-particle system is discussed. As before the relation to the classical problem is mentioned to give an idea on how to pass from the classical problem to the quantum mechanical one. The problems so far only allowed for a discrete set of eigenvalues. The free problem \(\Delta \psi + {8\pi^2 \over h^2}\psi = 0\) on \(R^3\) shows that the free particle is not quantized that is to say the free system has a purely (absolutely) continuous spectrum. Next, one encounters for the first time the spectral zeta function in relation with the Bohr-Wilson-Sommerfeld quantization rule and the high energy asymptotics. The transformation of the Schrödinger equation after passing to centre of mass coordinates follows. Chapter 4 introduces an approach due to \textit{A. E. Nikiforov} and \textit{V. B. Uvarov} [Special functions of mathematical physics, Basel-Boston, Birkhäuser (1988; Zbl 0624.33001)], to solve in a unified way one-dimensional second order differential equations that arise in mathematical physics by separation of variables. These equations are of hypergeometric type and examples are the Legendre, Hermite, Laguerre, Chebyshev, Bessel, Gauss or Euler equation. This method replaces the approach via power series to solve equations that are deduced from systems like the harmonic oscillator or the hydrogen atom. Some of the equations are treated in detail and e.g. the normalized eigenfunctions of the harmonic oscillator are determined. Chapter 5 discusses the hydrogen atom or hydrogen-like atoms. The eigenvalues and eigenfunctions are calculated. Furthermore, the quantization of the angular momentum is considered as well as the average distance of the electron to the nucleus. In Chapter 6 Heisenberg's uncertainty principle, already discussed in Chapter 1, is presented in a precise way. The expression \(\Delta x\Delta p \geq {\hbar \over 2}\) is given a mathematical meaning and it is proved with the help of Heisenberg's inequality. Motivated by the fact that transitions between certain states of the hydrogen atom are forbidden or allowed the representation of groups is presented in Chapter 7 and a lot of examples are exhibited. Having defined the action of topological groups on a topological space, both locally compact, an abstract selection rule is proven and later on applied to transition integrals for the hydrogen atom. In passing the first order time-dependent perturbation theory is introduced needed to formulate spectroscopic selection rules. These allow to justify why only certain transitions occur - in first order perturbation theory. Chapter 8 contains the discussion of the motion of a charged particle in an electromagnetic field both in the classical and in the quantum mechanical setting. Two appendices deal with the electric field due to point charges and introduce the electric dipole moment. The next chapter focuses on the magnetic field and the intrinsic angular momentum or spin of a particle. The Pauli spin matrices are defined and the corresponding Pauli wave functions are introduced. A time-dependent oscillating magnetic field \(B(x,yz,t)=(0,0,B\cos(\omega t))\) serves as an example. The normal Zeeman effect for the hydrogen-like atom in a weak magnetic field is shown and eventually the magnetic dipole moment and its quantization is considered. The last chapter -- No. 10 -- of the first part approaches the problem of multi-electron atoms. Since the problem cannot be solved in a closed form the first order time-independent perturbation theory is developed. This is used to calculate the ground state energy of the Helium atom taking the electron-electron interaction into account perturbationally. The multi-electron atom is also used to illustrate the Pauli exclusion principle. Shell diagrams and electron configurations for the ground state of several atoms are given. The second part of the book starts in Chapter 11 with the calculation of Fresnel integrals. Approximating the paths in the action integral \(S(x(t)):=\int_{t_{a}}^{t_{b}} (m/2) x'(t)^2 - V(x(t)) dt\) by polygons the evaluation of Feynman path integrals for the free case and for the system with the potential \(V(x) = mgx\) is accomplished with the help of the Fresnel integrals. Chapter 12 deals then with the Feynman path integral for the harmonic oscillator \(V(x) = m\omega^2 x^2/2\). With the Wick rotation \(t\to -it\) the Feynman path integral becomes a Euclidean path integral which has a sound mathematical foundation. This integral and its expansion into eigenfunctions is briefly discussed in Chapter 13. Its application in quantum statistical mechanics is the subject of Chapter 14. The relation to the density matrix is shown and several thermodynamic quantities like the partition function \(Z\), the Helmholtz free energy \(F\), entropy \(S\) and the internal energy \(U\) are introduced and explicitly given for the harmonic oscillator. To make sense of \(\log Z\) which appears in \(F=-kT \log Z\), where \(T\) is the temperature and \(k\) the Boltzmann constant, a suitable zeta function \(\zeta(s;T)\) is defined. Given that this zeta function can be meromorphically continued such that \(\zeta\) is holomorphic at \(s=0\) one has \(F(T) = -(kT/2)\zeta'(0;T)\). Again the harmonic oscillator serves as an example. The use of the zeta function in various circumstances is the subject of the following three chapters. Chapter 16 shows formally how to make sense of infinite products that would diverge in the usual sense using a zeta function. This is applied to the free Helmholtz energy for Kaluza-Klein space-times. The material presented is a special case of more general results obtained by the author. Chapter 17 concerns the product of Laplace operators and the corresponding zeta function. A theorem about the meromorphic continuation of the zeta function is stated. The final Chapter 18 gives a short introduction to gauge theory. Starting with the ambiguity of the Schrödinger equation if the magnetic potential \(A\) and the electric potential \(\Phi\) are changed according to \(A \to A+\nabla f\) and \(\Phi\to\Phi + {\partial f \over \partial t}\) one is introduced to differential geometric formulation of the electromagnetic forces. The presentation does not stop there but continues with the weak force and the Yang-Mills equation. The first part of the book being rather modest concerning the mathematical prerequisites is well suited for mathematics students. The second part requires some more advanced knowledge in particular about differential geometry to get a complete understanding of the themes presented.
scientific article; zbMATH DE number 1907573