1
Space

Introduction

The goal of this chapter is to understand motion in free space, i.e. in the absence of any potential. For quantum particles, this already turns out to be rather complex. In order to follow the thought process, it can be useful to think of the concepts in terms of pairs. To understand the motion, we initially look at vectors and then extend it to higher order vectors. There are two different types of motion: translation and rotation. Translation changes the length of a vector, but not its direction. Rotation changes the direction of a vector, but not its length. We can construct spaces with translational or rotational symmetry. This leads to conserved quantities, momentum, and angular momentum, respectively, that form a vector space by themselves. This is known as a dual space. Vectors are ideal in tracking rigid objects or point particles, where one only tracks the position in space. Vectors consist of two parts: basis vectors and coefficients. Motion can be achieved in two ways: by manipulation of the basis vectors or by performing an operation on the coefficients. Whichever method you use, the end result should be the same. Quantum particles are not rigid objects. The shape of the particle adjusts itself depending on the geometry (e.g. the potential landscape) of the problem. We therefore need to consider the entire particle, which is described by a function. The goal is to obtain functions that correspond to the conserved quantities (momentum or angular momentum). In free space, this can be done by taking the vectors and raising them to an arbitrary positive integer power. This leads to higher order vectors, which again are expressed in terms of basis vectors and coefficients. The coefficients are now called functions, and all the coefficients together form a function space. Motion can still be obtained by either manipulating the basis vectors or performing an operation on the functions.

Along the way, try to follow what part of the following pairs we are trying to describe: translation or rotation; vectors or higher order vectors (function spaces); obtaining motion by manipulating the basis vectors or by performing an operation on the coefficients/functions; and are we looking at the problem in real space or in its dual space (i.e. the space of the conserved quantities, momentum and angular momentum). The approach followed here is rather different from most quantum mechanics textbooks, where generally the Schrödinger equation and other concepts are postulated, and function spaces are produced for a wide variety of problems. Here, we start very simple and discover that many of the concepts that are often considered quintessential quantum mechanics simply arise from the manipulation of vectors or our description of extended nonrigid objects in space.

Since the approach is different from most textbooks on quantum mechanics, we start with a brief outline of the sections and their goals.

1. 1.1 The exponential function is crucial for the understanding of rotation and translation. This section discusses the exponential function in terms of the repeated application of an infinitesimally small action.
2. 1.2 In order to better understand how to manipulate vectors, we start with the relatively simple example of the rotation of a unit vector in two dimensions. Since a vector consists of unit vectors and coefficients, it is important to realize that this operation can be achieved by either manipulating the unit vectors or by performing an operation on the coefficients. In this section, the former is achieved using anticommuting unit vectors. This is known as Clifford or geometric algebra.
3. 1.3 This section looks again at the rotation of a unit vector, but now by performing an operation on the coefficients. This approach directly leads to differential calculus. The operator approach is generally more emphasized in introductory quantum mechanics courses that focus on wave functions. The geometric approach becomes more prevalent when symmetry and geometry dominate the physics. Despite only looking at the rotation of a vector in two dimensions, an operator similar to the quantum angular momentum operator is obtained.
4. 1.4 Although a vector can indicate a position in space, it does not allow the description of each point in space independently. This is necessary if we want to describe functions or fields in space. For the unit circle, a function space can be obtained by raising the unit vector to all integer powers. This function space allows us to describe any function on the unit circle.
5. 1.5 In this section, the anticommuting algebra in two dimensions is extended to three dimensions. It is shown that this allows the multiplication of vectors without the need to split it into an inner and an outer product.
6. 1.6 In physics, the machinery of anticommuting algebra is often incorporated using the properties of matrices. Contrary to what one might expect, three‐dimensional space is described in and not matrices. This reduction is possible since imaginary units give us an additional degree of freedom. This section introduces spinors, which form the basis of the matrices. Spinors are conceptually complicated since they are effectively the square root of the rotation axis.
7. 1.7 In this section, the Pauli matrices are derived. The three matrices effectively form a set of unit vectors, with the same properties as the algebraic anticommuting unit vectors.
8. 1.8 The Pauli matrices can be directly related to imaginary units or bivectors that allow us to rotate vectors in three dimensions. The eigenvectors of the rotation matrices are related to the spinors described earlier.
9. 1.9 The relationship between spinors and vectors is described in terms of the Pauli matrices. Additionally, it is shown how Pauli matrices can be described in terms of spinors.
10. 1.10 The function space for rotation in Section 1.4 is adapted to describe one‐dimensional translational systems and is then extended to three dimensions. Rotation was treated before translation, since translation is effectively treated like a rotation, which leads to some conceptual complexities that do not occur for rotation.
11. 1.11 The function space from the preceding section allows the description of any arbitrary function in space. For complex exponentials, this procedure is generally called a Fourier transform. Since the function space consists of eigenfunctions of the momentum operator, it allows for the translation of functions in space.
12. 1.12 The concepts for a homogeneous space are modified for a distorted and discrete space. An example of such a space is the periodic lattice of a solid. Here, we encounter a conceptual problem with momentum space. In free space, the translation in the ‐, ‐, or ‐direction is directly related to the momentum or wavevector in the same direction. However, in a distorted space, this intuitive concept falls apart, and we have to rethink what constitutes an orthogonal space. This problem did not occur for rotation, where the rotation occurs in a plane, and the conserved quantity, angular momentum, is perpendicular to this plane, i.e. parallel to the axis of rotation.
13. 1.13 The connection between obtaining the reciprocal/momentum space and linear algebra techniques such as matrix inversion is demonstrated. In many ways, these operations are related to vector division.
14. 1.14 This section returns to rotations. Although the rotation of vectors is relatively straightforward, the function space for rotations is rather complex. The function space for the unit circle was obtained by taking integer powers of the unit vector. This is extended to three dimension by studying the unit vector in spherical polar coordinates and expressing it in spinors. To create higher order unit vectors, the spinors are taken as indistinguishable quantities.
15. 1.15 The function space for rotations in three dimensions is derived by raising the unit vector to an integer arbitrary power. This leads to the function space of spherical harmonics and the associated unit vectors.
16. 1.16 In order to perform operations on higher order unit vectors, the concepts of the Pauli matrices are extended to arbitrary rank. It is shown that the transformation properties of the angular momentum operators are equivalent to those of the unit vectors.
17. 1.17 This section demonstrates that the use of operators on the functions leads to results equivalent to the manipulation of the unit vectors. This establishes the intimate connection between the operator and the transformation techniques.

1.1 The Exponential Function

Introduction. – A recapitulation of the exponential function might seem like a peculiar start of a quantum mechanics textbook. Although most readers will be familiar with the exponential function, it is important to understand the physical meaning of the exponential function as the repeated application of an infinitesimally small action on an object, particle, or any other kind of quantity.

The multiple applications of a particular action were first considered by Jacob Bernoulli in 1683 when studying compound interest. Given an interest rate , the capital increases as , whenever the interest is paid (let us say, once a year). However, if the same interest rate is divided over periods, the capital increases as

Suppose the interest rate is 6% or . If the interest is paid once a year, then . If the interest is paid monthly (), then for an interest rate of 0.5% per month. If interest is paid daily, then . So,...