 # Quantum Mechanics in a Nutshell

Gerald D. Mahan
Series: In a Nutshell
https://doi.org/10.2307/j.ctt7s8nw
Pages: 414
https://www.jstor.org/stable/j.ctt7s8nw

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1. Front Matter
(pp. i-iv)
(pp. v-x)
3. Preface
(pp. xi-xiv)
4. 1 Introduction
(pp. 1-13)

Quantum mechanics is a mathematical description of how elementary particles move and interact in nature. It is based on the wave–particle dual description formulated by Bohr, Einstein, Heisenberg, Schrödinger, and others. The basic units of nature are indeed particles, but the description of their motion involves wave mechanics.

The important parameter in quantum mechanics is Planck’s constant$h = 6.626 \times {10^{ - 34}}{\rm{J s}}$. It is common to divide it by$2\pi$, and to put a slash through the symbol:$\hbar = 1.054 \times {10^{ - 34}}{\rm{J s}}$. Classical physics treated electromagnetic radiation as waves. It is particles, calledphotons, whose quantum of energy is$\hbar \omega$where$\omega$is the classical angular...

5. 2 One Dimension
(pp. 14-61)

It is useful to study one-dimensional problems, since there are many more exact solutions in one dimension compared to two and three dimensions. The study of exact solutions is probably the best way to understand wave functions and eigenvalues. This chapter gives some examples of exact solutions. There are many more exactly solvable examples than the few cases given here.

A person with a knowledge of quantum mechanics should, if presented with some potential$v(x)$, be able to give a freehand sketch of the wave function$\psi (x)$. This sketch need not be precise, but should oscillate in the right regions, and...

6. 3 Approximate Methods
(pp. 62-86)

The previous chapter has exact solutions to Schrödinger’s equation in one dimension for some simple potentials. There are only a few potentials V(x) for which Schrödinger’s equation is exactly solvable. Many other potentials need to be solved besides those with exact solutions. This chapter presents two approximate methods for solving Schrödinger’s equation: WKBJ and variational. They were both important in the precomputer days of quantum mechanics, when many problems could be solved only approximately. Now numerical methods usually permit accurate solutions for most problems. However, the present methods are often useful even in the computer age. Many computer solutions are...

7. 4 Spin and Angular Momentum
(pp. 87-107)

Angular momentum is an important entity in quantum mechanics. The two major contributors to the angular momentum are the spin$(\vec s)$of the particle and the orbital angular momentum$(\vec \ell )$from the rotational motion. In classical mechanics, the total angular momentum$\vec j = \vec \ell + \vec s$obtained by the vector addition of the two component vectors. In quantum mechanics, all three forms of angular momentum$(\vec s,\vec \ell ,{\rm{ }}\vec j)$are individually quantized and only selected values are allowed. Then the problem of adding angular momentum$\vec j = \vec \ell + \vec s$is more complicated, since both constituents and final value are quantized.

In systems of more than one particle, the problem of...

8. 5 Two and Three Dimensions
(pp. 108-156)

Most quantum mechanical problems are three dimensional. Nuclei, atoms, solids, and stars are systems that must be solved in three dimensions. Two-dimensional solutions are required when a particle is restricted to move on a surface. Examples are when electrons move on the surface of liquid helium or in a semiconductor quantum well. Many of the techniques discussed in prior chapters are now applied to two and three dimensions. Some exact solutions are provided, as well as approximate methods such as WKBJ and variational. Spin and angular momentum are both utilized.

The most important solution to Schrödinger’s equation is where the...

9. 6 Matrix Methods and Perturbation Theory
(pp. 157-212)

When solving any Hamiltonian, the first step is to try to solve it exactly. Finding an exact solution is always a bit of luck. Usually an exact solution is impossible, and then one is faced with a variety of choices. WKBJ, variational, and numerical methods are all available as possible approaches. However, probably the most important approximation is perturbation theory.

The discussion of perturbation theory is divided into two categories. The first is the effect of static perturbations: those that are independent of time. They are treated in this chapter. The second category is perturbations that depend on time, which...

10. 7 Time-Dependent Perturbations
(pp. 213-243)

Many perturbations depend on time. This dependence is expressed as a perturbation$V({\bf{r}},\;t)$that has explicit time dependence. This chapter is devoted to solving Hamiltonians that can be written in the form

$H = {H_0} + V({\bf{r}},\;t)$(7.1)

$i\hbar \frac{\partial }{{\partial t}}\psi ({\bf{r}},\;t)\; = \;H\psi ({\bf{r}}{\rm{,}}\;t)$(7.2)

Of course, this approach is actually nonsense. From a rigorous viewpoint, Hamiltonians cannot depend on time.

As an example, consider the problem of a very fast alpha-particle that whizzes by atom. The proper Hamiltoniian for this problem would contain the complete Hamiltonian for the atom, the kinetic energy of the alpha-particle, and the Coulomb interaction between the alpha-particle, the nucleus, and the electrons in...

(pp. 244-287)

Electromagnetic radiation is composed of elementary particles called photons. They are massless, chargeless, and have spin -1. They have a momentum${\bf{p}}\; = \;\hbar k$, energycp, and a frequency$\omega = cp/\hbar = ck$. The light has two choices of polarization, which are given the label$\lambda (\lambda = 1,2)$. The frequency is the same one that is associated with an oscillating electric field. The correspondence between the classical electric field and the quantum picture of photons is that the intensity of light is proportional to the number of photons${h_{{\rm{k}}\lambda }}$. For an electric field of classical amplitude${E_0}$, wave vectork, and frequency$\omega = ck$, the energy flux is...

12. 9 Many-Particle Systems
(pp. 288-319)

Consider quantum mechanical systems with many identical particles. A typical system is the many electrons in an atom. Similar statistics are found for many nucleons in a nuclei. Later we shall consider systems of many bosons such as superfluid 4 He. The nonrelativistic Hamiltonian for electrons in an atom contains three types of terms—(kinetic energy, (2) potential energy with the nucleus, and (3) electron–electron Coulomb interactions:

$H = \sum\limits_i {\left[ {\frac{{p_i^2}} {{2m}} - \frac{{Z{e^2}}} {{{r_i}}}} \right]} + \sum\limits_{i > j} {\frac{{{e^2}}} {{{r_{ij}}}}}$(9.1)

We work in the Coulomb gauge and neglect the role of photons. Since the electrons nucleus are close together, the long-range forces from photons are not important. It is assumed...

13. 10 Scattering Theory
(pp. 320-351)

The previous chapters have discussed scattering theory for electrons and photons. Usually the matrix elements were calculated in the first Born approximation, which is the Fourier transform of the potential energy. This chapter discusses scattering theory with more rigor. The exact matrix element will be obtained for a variety of systems.

The interaction of photons with charged particles is described well by perturbation theory. Perturbation theory for photons has a smallness parameter that is the finestructure constant${\alpha _f} \approx 1/137$. Each order of perturbation theory is smaller by this parameter. So the calculations in chapter 8 for photons are fine. In the...

14. 11 Relativistic Quantum Mechanics
(pp. 352-396)

The previous chapters have been concerned with nonrelativistic quantum mechanics. The basic equation is Schrödinger’s equation. It does not apply when the particles have kinetic energies of the same size as their rest energy$m{c^2}$. Then a full relativistic equation is required. In this chapter we discuss two different relativistic equations: Klein-Gordon and Dirac. The K-G (Klein-Gordon) equation applies to spinless particles such as mesons. The Dirac equation applies to spin-1/2 fermions, such as electrons, nucleons, or neutrinos.

In special relativity, the energy of a free particle is given by

$E(p) = \sqrt {{m^2}{c^4} + {c^2}{p^2}}$(11.1)

wheremis the mass, p is the...

15. Index
(pp. 397-399)