Abstract
The approach to quantum mechanics in the preceding chapter is based on the description of the time evolution of a state by means of a differential equation. In this chapter, we choose a different approach. We consider (for now) not the Schrödinger equation or another description of the space-time behavior, but instead the emphasis is now on how we can define states (for the moment, time-independent states).
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Notes
- 1.
If this term is not familiar (or forgotten): The basic concepts are summarized in Appendix G, Vol. 1. In addition, we will return to this topic in Chap. 4. For the moment, it is enough to know that e.g. the set of all vectors \(\left( \begin{array}{c} a_{1} \\ a_{2} \end{array} \right) \) with \(a_{i}\in \mathbb {C}\) forms a two-dimensional complex vector space. An important property is that any linear combination of two vectors is itself a valid vector in this space.
- 2.
From the theory of electromagnetism, we know that light is a transverse wave, i.e. that its electric field oscillates perpendicular to its direction of propagation. The polarization describes the orientation of this oscillation.
Polarization is often regarded as an esoteric and specialized topic, possibly because we cannot see directly whether light is polarized. However, it is a ubiquitous phenomenon in our environment—natural light is almost always polarized, at least partially. Many animals, such as bees or other insects, take advantage of this; they can detect and analyze light polarization. In our daily life, polarization is used e.g. in polarizing filters for cameras or some sunglasses. Moreover, the fundamentals of the formal treatment of polarization are also very simple, as we shall see below.
- 3.
We note that a real light wave is only approximately described by a plane wave, since that would have the intensity at all points and all times. However, this approximate description is common for several reasons, and suffices for our purposes here.
- 4.
In this connection also called the light vector.
- 5.
The relative phase could of course be associated with the y component instead of the x component.
- 6.
In physical optics, right and left circular polarization is usually defined the other way around (optics convention).
- 7.
In the following, we want to multiply vectors by matrices. In the usual notation, a matrix acts on a vector from the the left, which therefore—according to the usual rules of matrix multiplication—must be a column vector. See also Appendix F, Vol. 1, on linear algebra.
- 8.
The length of the vector \(\left( \begin{array}{c} 1 \\ i \end{array} \right) \) is given by \(\sqrt{2}\); we explain the reasoning for this in Chap. 4.
- 9.
Different symbols are in use to denote representations; Fließbach writes \(:= \), for example. Apart from that, many authors denote representations not by a special symbol, but by simply writing \(=\).
- 10.
Also, the general mathematical modelling uses concepts that are implemented only approximately in reality. A time-honored example is Euclidean geometry with its points and lines, which strictly speaking do not exist anywhere in our real world. Yet no one doubts that Euclidean geometry is extremely useful for practical calculations. “Although this may be seen as a paradox, all exact science is dominated by the idea of approximation” (Bertrand Russell).
- 11.
Most theoretical results are based on approximations or numerical calculations and are in this sense not strictly precise. This naturally applies a fortiori to experimental results. Even though there are high-precision measurements with small relative errors of less than a part per billion, it has to be noted that each measurement is inaccurate. Nevertheless, one can estimate this inaccuracy in general quite precisely; keyword ‘theory of errors’.
- 12.
If several theories describe the same facts, one should prefer the simplest of them (this is the principle of parsimony in science, also called Occam’s razor: “entia non sunt multiplicanda praeter necessitatem”).
- 13.
The active rotation (rotation of the vector by \(\vartheta \) counterclockwise) is given by \(\left( \begin{array}{cc} \cos \vartheta &{} -\sin \vartheta \\ \sin \vartheta &{} \cos \vartheta \end{array} \right) \); the passive rotation (rotation of the coordinate system) by \(\left( \begin{array}{cc} \cos \vartheta &{} \sin \vartheta \\ -\sin \vartheta &{} \cos \vartheta \end{array} \right) \).
- 14.
Perhaps familiar from school or undergraduate laboratory courses?
- 15.
Single-photon experiments are standard technology these days. In 1952, Schrödinger declared: “We never experiment with just one electron or atom or (small) molecule. In thought-experiments we sometimes assume that we do; this invariably entails ridiculous consequences.” Times have changed: Precision experiments using a single photon or a single atom are the basis of e.g. today’s time standard, and modern quantum-mechanical developments such as the quantum computer rest on those ‘ridiculous consequences’. We recall that photons (as far as we know) have immeasurably small dimensions and are in this sense referred to as point objects (or point particles). Although they represent light of a specific wavelength, they do not have a spatial extension on the order of the wavelength of the light.
- 16.
In a vacuum, photons are indivisible, and that holds also for most interactions with matter. One has to work hard to ‘cut’ photons. This can be achieved for example in the interaction with certain nonlinear crystals, where a single photon breaks up into two photons of lower energy (parametric fluorescence, see Appendix I, Vol. 2). Devices for polarization measurement are of course manufactured in such a way that they leave the photons unsplit.
- 17.
Furthermore, in ‘slow’ conditions—the effects of the theory of relativity are beyond our daily life experience, as well.
- 18.
We point out that this is not an exotic quantum-mechanical procedure—the eyes of every bee, or suitable sunglasses, perform precisely this kind of ‘measurement process’.
- 19.
These cases can be produced by inserting a further analyzer whose orientation is \(\varphi \,+\, 0\) or \(\varphi \,+\,\pi /2\), for example.
- 20.
The systems need not be in the same state, but the preparation process must be the same.
- 21.
Just as the interference pattern in the double slit experiment builds up gradually from scattered spots over time.
- 22.
Another example of an ensemble are electrons which are prepared by a Stern–Gerlach apparatus and a velocity filter so that their spins are pointing upwards and their speeds are confined to a particular interval \( \left( v-\Delta v, v+\Delta v\right) \). A further example is a set of hydrogen atoms in a particular excited state, whereby here the preparation refers to the energy, but not to the angular momentum of the state.
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Pade, J. (2018). Polarization. In: Quantum Mechanics for Pedestrians 1. Undergraduate Lecture Notes in Physics. Springer, Cham. https://doi.org/10.1007/978-3-030-00464-4_2
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DOI: https://doi.org/10.1007/978-3-030-00464-4_2
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