Quantum Mechanics: Introduction to Mathematical Formulation (essentials) 3658326441, 9783658326449

Table of contents :
What You Can Find in This essential
Preface
Acknowledgment
Contents
1 Introduction
2 Quantum Mechanical Phenomena
2.1 Atomic Models According to Bohr and Sommerfeld
2.2 Particle in a Box
2.2.1 Quantum Mechanical States
2.2.2 Discrete Energy Levels and Position Probability
2.2.3 Correspondence Principle and Hamiltonian
2.2.4 Preparation and Superposition of States
3 Mathematical formulation
3.1 Heisenberg and Schrödinger Representation
3.2 Postulates of Quantum Mechanics
3.2.1 Postulate 1: States in the Hilbert Space
3.2.2 Postulate 2: Measured Values, Operators and Eigenvalues
3.2.3 Postulate 3: Probabilities
3.3 Solution Algorithm
4 Examples of Use
4.1 A Thought Experiment with Toy Blocks
4.2 Schrödinger’s Cat
4.3 The Stern-Gerlach Experiment
What You Learned From This essential
Literature

Citation preview

Martin Pieper

Quantum Mechanics Introduction to Mathematical Formulation

essentials

Springer essentials

Springer essentials provide up-to-date knowledge in a concentrated form. They aim to deliver the essence of what counts as "state-of-the-art" in the current academic discussion or in practice. With their quick, uncomplicated and comprehensible information, essentials provide: • an introduction to a current issue within your field of expertis • an introduction to a new topic of interest • an insight, in order to be able to join in the discussion on a particular topic Available in electronic and printed format, the books present expert knowledge from Springer specialist authors in a compact form. They are particularly suitable for use as eBooks on tablet PCs, eBook readers and smartphones. Springer essentials form modules of knowledge from the areas economics, social sciences and humanities, technology and natural sciences, as well as from medicine, psychology and health professions, written by renowned Springer-authors across many disciplines.

More information about this subseries at http://www.springer.com/series/16761

Martin Pieper

Quantum Mechanics Introduction to Mathematical Formulation

Martin Pieper FH Aachen, University of Applied Sciences Jülich, Germany

ISSN 2197-6708 ISSN 2197-6716 (electronic) essentials ISSN 2731-3107 ISSN 2731-3115 (electronic) Springer essentials ISBN 978-3-658-32644-9 ISBN 978-3-658-32645-6 (eBook) https://doi.org/10.1007/978-3-658-32645-6 © Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2021 The translation was done with the help of artificial intelligence (machine translation by the service DeepL.com). A subsequent human revision was done primarily in terms of content. This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Planung/Lektorat: Iris Ruhmann This Springer imprint is published by the registered company Springer Fachmedien Wiesbaden GmbH part of Springer Nature. The registered company address is: Abraham-Lincoln-Str. 46, 65189 Wiesbaden, Germany

For Elli … the cat’s alive!

What You Can Find in This essential

• • • • • •

Postulates (principles) of quantum mechanics Quantum states in Hilbert space Operators as observables Eigenvalues as measured values Born’s rule Discrete energy levels with degeneration

vii

Preface

Nils Bohr, Nobel Prize winner and co-founder of quantum mechanics, quotes: “If quantum mechanics hasn’t profoundly shocked you, you haven’t understood it yet.” This is certainly irritating at first. What does this mean? The results of quantum mechanics represent a radical break with classical physics. For example, it is no longer possible to make statements about concrete particle trajectories. Instead, only probabilities are given. Albert Einstein commented this rather skeptically with “At any rate, I am convinced that He (God) does not throw dice.” Nevertheless, modern physics, especially quantum mechanics, is currently more popular than ever. This is especially true for a broad audience outside the physics community. This is probably due to numerous popular science books on the subject, but not least to the success of the television series “Big Bang Theory.” Thus, many people know the fate of Schrödinger’s cat and one or the other has surely wondered which hieroglyphics are on Sheldon’s board and what they mean. These are exactly the questions that the present text wants to answer—at least partially. The essential therefore is aimed at an interested readership with a certain basic education in mathematics, such as that imparted in natural science and engineering courses. In detail, the education should include linear algebra (vector calculus) in particular. Previous knowledge of physics, especially quantum mechanics, is not necessary. Basic knowledge in the context of school physics is sufficient here, which includes terms such as energy and momentum. In this text, we can only give an introduction to the subject, focusing in particular on mathematical formalism. Therefore, while we will go beyond the content of popular science books, we will of course not cover all the details completely. For this, we refer to the numerous textbooks. Thus, we try the balancing act of using as few formulas and abstract terms as possible on the one hand,

ix

x

Preface

while providing enough information to understand the mathematical formulation of quantum mechanics on the other. To this end, we motivate mathematics by means of well-known examples from linear algebra, that is, vector algebra.

Acknowledgment

The concept for the present text was developed in the context of different electives in the course of studies of physical engineering at the FH Aachen. I would therefore like to thank all students who played “guinea pigs” and gave me valuable feedback. For the critical review of the text and numerous hints, I would like to thank especially my “unwavering first readers” Stephanie Kahmann, Elisabeth Nierle, Darius Mottaghy, Philipp Weyer and Nadja Hansen, the latter especially for the support with the graphics. Further thanks are due to Springer Verlag for giving me the opportunity to write this essential, in particular Mrs Ruhmann and Mrs Schulz, who accompanied the project. In July 2019 Martin Pieper

xi

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

2 Quantum Mechanical Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Atomic Models According to Bohr and Sommerfeld . . . . . . . . . . . . 2.2 Particle in a Box . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Quantum Mechanical States . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Discrete Energy Levels and Position Probability . . . . . . . . . 2.2.3 Correspondence Principle and Hamiltonian . . . . . . . . . . . . . 2.2.4 Preparation and Superposition of States . . . . . . . . . . . . . . . .

3 3 5 5 8 9 10

3 Mathematical formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Heisenberg and Schrödinger Representation . . . . . . . . . . . . . . . . . . . 3.2 Postulates of Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Postulate 1: States in the Hilbert Space . . . . . . . . . . . . . . . . . 3.2.2 Postulate 2: Measured Values, Operators and Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Postulate 3: Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Solution Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13 13 14 14

4 Examples of Use . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 A Thought Experiment with Toy Blocks . . . . . . . . . . . . . . . . . . . . . . 4.2 Schrödinger’s Cat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 The Stern-Gerlach Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25 25 27 27

Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33

18 22 24

xiii

1

Introduction

Quantum mechanics is a subarea of theoretical physics and belongs to modern physics, which emerged in the twentieth century. The main task of theoretical physics is to describe natural phenomena by mathematical models. This allows to explain experiments and predict the results. It investigates which measured variables are relevant for the description, analyses the universal relationship between them and relates them to each other by mathematical equations. It is always attempted to use as few equations as possible and to find a uniform formulation that can describe as many phenomena as possible in general terms. This is where theoretical physics differs from engineering sciences, for example. In these, the aim is usually always a direct application of technology, for which specially adapted models and equations are applied. Before the twentieth century, the movement of bodies under the influence of forces, such as celestial mechanics or ball throwing, was described by classical mechanics. The basis for this are, for example, Newton’s laws, which provide the trajectories. However, numerous experiments at the end of the nineteenth century showed that classical mechanics fails in the description of atomic systems (see e.g. [1]). Many phenomena, such as the photoelectric effect or the spectral lines of the hydrogen atom, can no longer be explained in classical terms. Therefore, a new mechanics for atomic systems had to be developed. The result was quantum mechanics. The main task of quantum mechanics is therefore to predict the results of physical measurements on atomic systems (elementary particles, atoms and molecules). This includes, for example, the movement of atomic particles in electromagnetic fields, the scattering of atomic particles among themselves or the structure of atoms and molecules. In order to be able to describe all these different tasks with only one uniform theory, that is, few equations and assumptions, a radical break with classical © Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2021 M. Pieper, Quantum Mechanics, Springer essentials, https://doi.org/10.1007/978-3-658-32645-6_1

1

2

1

Introduction

mechanics was made. Numerous classical concepts such as trajectories or deterministic measurement results had to be abandoned. This required a completely new mathematics, namely functional analysis. It was developed at the beginning of the twentieth century, almost simultaneously with quantum mechanics, mainly by David Hilbert in Göttingen. Exactly this uniform mathematical description is the subject of the essential. We want to bring the theoretical and mathematical fundamentals of quantum mechanics closer to a broad readership and thus create an interest in quantum mechanics. The essential begins in Chap. 2 with two examples, the atomic models according to Nils Bohr and Arnold Sommerfeld and the description of a particle in a box. The goal is to introduce typical quantum mechanical phenomena like discrete, degenerate energy states and superpositions of states and to discuss them briefly from a mathematical point of view. The knowledge gained from this will be used in the following main chapter to motivate the uniform mathematical formulation and make it plausible. Chapter 3 forms the core of the essential. We give the basic mathematical structure of quantum mechanics in the form of three postulates (principles of theory). In particular, the quantum mechanical states in Hilbert spaces are the fundament. Furthermore, we deal with Hermitian operators and their eigenvalues, which stand for possible measured values. We also explain how the associated probabilities are calculated. In the last chapter, we will finally discuss three small examples. These are intended to demonstrate how the rather abstract and theoretical postulates are applied in practice. We start with a thought experiment, on the one hand, to deepen the notation according to Paul Dirac with bra and ket vectors and on the other hand to explain and treat degenerate states, as observed in Sect. 2.1 with the hydrogen atom. The second example takes up Schrödinger’s cat. We will discuss the superposition of states and their interpretation. The Stern-Gerlach experiment concludes. Based on experimental observations, we apply the postulates to describe the system. As a result, we find that the previously unknown property “particle spin” is the reason for the measurement results in the experiment.

2

Quantum Mechanical Phenomena

2.1

Atomic Models According to Bohr and Sommerfeld

Bohr’s atomic model (1913) assumes a heavy, positively charged nucleus, which is orbited by negatively charged electrons. However, not all classically possible orbits are allowed, but only certain orbits to which discrete energy values E n are assigned. Here the energy level is characterized by the principal quantum number n ∈ N. If an electron jumps or falls from one orbit to another, electromagnetic radiation is emitted or absorbed with the frequency f = E/h, whereby E is called the energy difference of the orbits and h ≈ 6,62 · 10−34 Js is Planck’s constant (see Fig. 2.1). We consider the hydrogen atom as an example. Here we can apply the atomic model to explain, for example, the experimentally determined spectral lines. The formula according to Nils Bohr corresponds exactly with the empirically found formula of Johann Jakob Balmer and Johannes Rydberg. Further confirmation of the model is provided by the Franck-Hertz experiment, in which discrete energy levels in atoms are also observed. However, Bohr’s simple model fails if a magnetic field is applied. In this case, the spectral lines split (Zeeman effect). The reason is that the energy levels E n are degenerate. This can be explained by the Bohr-Sommerfeld atomic model. In 1916, Arnold Sommerfeld extended Bohr’s model by also allowing elliptical orbits on which the electrons orbit the nucleus. A total of three parameters are required to describe an ellipse, so the quantum numbern is no longer sufficient. Sommerfeld also introduced the secondary quantum numbers  and m. For the exact characterization of the quantum mechanical state, the principal quantum number n and the two secondary quantum numbers  and m are necessary.

© Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2021 M. Pieper, Quantum Mechanics, Springer essentials, https://doi.org/10.1007/978-3-658-32645-6_2

3

4

2

orbit with E orbit with E

2 3

E

E

photon 3 E

photon 3

E=E

2 E

2

= photon =h

3

-E

2

E f

energy levels

orbit with E

E

1

Quantum Mechanical Phenomena

E

1

Fig. 2.1 Emitted photon at the energy jump from E 3 to E 2

Without external fields, the discrete energy levels E n still depend only on the principal quantum number n and not on  and m. However, the energy state E n is called degenerate because several possible orbits (states) with energy level E n exist (see Fig. 2.2). A measurement of the energy value E n therefore does not

Fig. 2.2 Three possible orbits for the energy state E 3

2.2 Particle in a Box

5

uniquely determine the state, as we do not know on which orbit the electron is located. If an external magnetic field is present, the discrete energy values also depend on the magnetic quantum number m (hence the name). In this case, the energy level E n splits into 2 + 1 equidistant Zeeman levels. Thus, part of the energy degeneration is cancelled and the state is more precisely defined. The idea of orbits on which the electrons orbit the atomic nucleus is now obsolete. In modern quantum mechanics, the specification of a particle trajectory no longer makes sense. Instead, statements are made about the position probability of the particle. Nevertheless, the observations on energy degeneration are still valid.

2.2

Particle in a Box

In this section, we want to understand the occurrence of discrete energy levels. As an example, we consider a free particle trapped in a one-dimensional box with infinitely high walls (see Fig. 2.3). Since no forces act on the particle in the box of length L, the potential energy disappears. According to the laws of classical mechanics, the particle oscillates in the box between the walls. The motion is clearly defined by the state, that is, the indication of the position x(t) and the momentum p(t) at time t. The particle can assume any energy value E ≥ 0.

2.2.1

Quantum Mechanical States

In microscopic systems described by quantum mechanics, we can no longer specify the classical state of the particle exactly. The reason is the Heisenberg uncertainty principle (1927). It states that position and momentum cannot be measured sharply at the same time. More precisely, the following inequality holds for the standard deviation of position σx and the standard deviation of momentum σ p σx · σ p ≥

 , 2

where  ≈ 1,05·10−34 Js denotes the reduced Planck’s constant. So if we measure the position of the particle very precisely, σx is small. But this means that σ p must be at least so large that the inequality is satisfied. In particular, σ p cannot

6

2

Quantum Mechanical Phenomena

Fig. 2.3 u 2n (x) for E 1 to E 3 ; position probability for x corresponds to the area

also become arbitrarily small at the same time. Similarly, a very precise measurement of the momentum will result in an uncertainty in the position, corresponding to the inequality. In this context, it is important to note that this is a fundamental law of nature, which does not result from any measuring errors or inaccurate measuring equipment. For a better understanding, we give a semiclassical explanation for the uncertainty principle, which goes back to Werner Heisenberg himself (see [2] for further explanations). We need a measuring device, for example, a microscope, to measure the position of a particle. The accuracy is determined by the wavelength of the light used. A smaller wavelength gives a more accurate result. During the measurement, a photon hits our particle, is reflected or diffracted and registered in the microscope. At the moment of the position measurement, when the photon hits the particle, the momentum of our particle changes unsteadily. This change depends on the energy of the photon. The smaller the wavelength of the light, that is, the more accurate the spatial measurement, the greater the change. Therefore, at the moment we know the position, the momentum of the particle becomes uncertain.

2.2 Particle in a Box

7

We have found that we cannot determine the classical state in atomic systems. Instead, we introduce a new quantum mechanical state. According to Erwin Schrödinger, this state is described by a wave function ψ(t, x) that depends on the position x and the time t. However, it has no direct physical meaning and serves rather as an “auxiliary quantity” in the calculation. Physical statements are only possible in connection with Born’s rule: For the subinterval [a, b] ⊂ [0, L], the probability of the particle being present at time t in the interval [a, b] can be determined by the integral using the square of the absolute value |ψ(t, x)|2 of the wave function: b W (t, [a, b]) =

|ψ(t, x)|2 d x.

(2.1)

a

As already mentioned, we note that in quantum mechanics, we can only make statements about possible measurement results and the probabilities with which they occur. The question of concrete particle trajectories makes no sense. In this section, we are not interested in the time-evolution of the state, so we only consider the position-dependent part u(x) of the wave functions ψ(t, x) = v(t) · u(x). This is determined by the time-independent Schrödinger equation −

2 d 2 u(x) = Eu(x) 2m d x 2

(2.2)

Here m denotes the particle mass. On the left side is the second derivative of u(x) with respect to x. In addition, E is a constant, which will turn out to be an energy quantity. The particle can only stay in the box. So it is enough to look at the equation in the interval [0, L]. Outside the box and at the walls, the position probability is zero, that is, the wave function u(x) disappears here. This fact is called boundary condition. The solution of Eq. 2.2 is discussed in the next section. Before that we will take a closer look at the constant E. For this purpose, we reformulate Eq. 2.2 and compare the units:

We recognize that the constant E is an energy quantity, with the unit Joule.

8

2

2.2.2

Quantum Mechanical Phenomena

Discrete Energy Levels and Position Probability

Next, we are interested in the possible energy values for the particle. For this, we have to solve Eq. 2.2, that is, we are looking for functions that are derived twice to give a multiple of themselves. The trigonometric functions sin(λx) and cos(λx), with a constant λ to be determined, are suitable. Because of the boundary condition, u must disappear at the boundary. For the cosine, however, cos(0) = 1 always holds. As possible solution, only the sine remains. We now determine λ in such a way that on the one hand the sine fits in our box, that is, that we have zeros at the walls, and on the other hand that sin(λx) satisfies the differential Eq. 2.2. We obtain two conditions: λ · L = n · π and λ2 =

2m E . 2

The discrete energy values are obtained by combining the two expressions: 2m E n2π 2 = 2 L2 

⇒

E=

2 π 2 2 n . 2m L 2

The energy values E are usually marked with an index n: E n . Analogous to the atomic model, n is called a quantum number and runs through the natural numbers (without zero). The corresponding solutions also receive an index: u n (x). According to Born’s rule, the probability of the particle’s position is given by the square of the wave function u 2n (x). But first we have to normalize: The integral of u 2n (x) over the entire box must equal 1 for the particle to be safely in the box with probability of 1. We finally obtain:  u n (x) =

 nπ  2 sin x L L

⇒

u 2n (x) =

 nπ  2 sin2 x . L L

Figure 2.3 shows u 2n (x) for the first three energy levels. The area under the curve indicates the position probability of the particle (see Eq. 2.1). In contrast to classical mechanics, there are preferred areas (e.g. peaks) where the particle is located and areas where the position probability is low (e.g. zeros). In addition, the particle cannot assume any arbitrary energy value E ≥ 0, but only the discrete energy levels E n .

2.2 Particle in a Box

2.2.3

9

Correspondence Principle and Hamiltonian

We now want to understand Eq. 2.2, that is, especially the mathematical structure. Because of the constant E, we first consider the total energy of the particle. In our problem, the energy consists only of the kinetic energy, as the potential energy disappears in the box according to preconditions: E tot = E kin =

1 2 p2 mv = , 2 2m

(2.3)

We denote with v the particle velocity and with p = mv the corresponding momentum. In quantum mechanics, the observables of classical mechanics become operators through correspondence rules (see [3], Chap. 3). From a mathematical point of view, operators are calculation rules that are applied to states. For identification purposes, we use a hat above the corresponding letters. As an example, we consider the position operator xˆ and the momentum operator p: ˆ x[u(x)] ˆ = x · u(x) and p[u(x)] ˆ = −i

d u(x). dx

We see that the operators are applied to the wave function u(x). The position operator is a multiplication operator that multiplies the wave function by the position x. The momentum operator, on the other hand, is a differential operator, in which the wave function is derived with respect to x. The correspondence principle now means that we replace the observable position and momentum in the classical expressions by their corresponding operators. In the case of total energy (see Eq. 2.3), we obtain E tot =

p2 2m

⇒

2 d 2 Hˆ = − , 2m d x 2

by using the differential operator for the momentum p. The second derivative occurs because p 2 means that the derivative is applied twice. In classical mechanics, the Hamiltonian function is closely related to the total energy. For this reason, we speak of the Hamiltonian, which is referred to as Hˆ . A comparison with Eq. 2.2 gives:

10

2

−

2 d 2 u(x) = Eu(x) 2m d x 2

⇔

Quantum Mechanical Phenomena

Hˆ u(x) = Eu(x).

The stationary Schrödinger equation can therefore be seen as an eigenvalue problem for the Hamiltonian: We are looking for eigenfunctions u(x), which are mapped to a multiple when using the Hamiltonian. The multiple is the energy eigenvalue E. This eigenvalue problem is completely analogous to the search for eigenvalues and eigenvectors of matrices in linear algebra (see Sect. 3.2.2).

2.2.4

Preparation and Superposition of States

From a mathematical point of view, any linear combination u(x) =

∞ 

cn u n (x)

(2.4)

n=1

of the functions u n (x) satisfies the Schrödinger equation and the boundary conditions and therefore is a possible state. Here cn ∈ C are series coefficients. But how can we interpret such linear combinations physically? u(x) is a superposition of different states u n (x), similar to how different waves can be superimposed and a new resulting wave is produced. In this case, we combine the eigenstates u n (x) of the Hamiltonian. Such superpositions are needed, for example, to describe the experimental initial conditions that are determined by the preparation of the particle source. Mathematically, we again use a wave function u(x), which describes this initial condition at t = 0 as a quantum mechanical state. In linear algebra, we represent arbitrary vectors by linear combinations of basis vectors. Analogously, we can expand certain functions in Fourier series, that is, write them as linear combinations of sine and cosine functions. This also applies to the initial state u(x), which we expand into a series with respect to the eigenfunctions u n (x) of the Hamiltonian: u(x) =

∞  n=1

cn u n (x) =

∞  n=1

 cn

 nπ  2 sin x . L L

We obtain a linear combination, similar to Eq. 2.4. The coefficients cn are to be determined in accordance with the function u, analogous to Fourier series.

2.2 Particle in a Box

11

We already understand the meaning of the wave functions u n (x). But how can we interpret the series coefficients cn physically? To do this, we calculate the expectation for the energy described by the Hamiltonian (see [4], Sect. 5.4): Hˆ =

L

u(x)∗ Hˆ u(x)d x =

∞ 

|cn |2 E n .

(2.5)

n=1

0

The brackets Hˆ indicate the expectation of the energy. The asterisk stands for the complex conjugate, as wave functions can generally assume complex values. For a complex number z = a + bi, the complex conjugate is defined by z ∗ = a − bi. We compare the expression in Eq. 2.5 with the standard formula from probability theory for the expectation E(X ) =



pn x n .

n

Here pn is the probability for the random variable X to take the value x n . By comparing coefficients, we find that the squares of the magnitudes |cn |2 of the series coefficients cn can be interpreted as probabilities. They indicate the probability that the particle is in the state u n (x) and takes on the energy value E n during a measurement, after preparation by u(x).

3

Mathematical formulation

3.1

Heisenberg and Schrödinger Representation

In Sect. 2.2, we have considered a particle that is trapped in a box. The particle was described by the quantum mechanical state using a wave function u(x). This approach was developed by Erwin Schrödinger (1926). He conceives particles as waves to which wave functions are assigned. At the same time, matrix mechanics was developed by Werner Heisenberg, Max Born and Pascual Jordan (1925–1926). They describe observables by matrices with an infinite number of entries. The states are correspondingly vectors with infinitely many components. Which is the right representation? The answer is: “both,” because they provide the same physical predictions, that is, they are equivalent. Schrödinger was the first to show equivalence in 1926. Additional proofs followed. John von Neumann in particular put quantum mechanics on a strict mathematical basis (see [5]): The wave functions and the state vectors are only different forms of representations of a general mathematical structure, the Hilbert space, which is the basis of quantum mechanics. We want to understand this point better and consider a 2π periodic function f (t), which we develop into a Fourier series: f (t) =



cn eint .

(3.1)

n∈Z

It does not matter whether we work with the function f (t) or whether we use cn the Fourier series coefficients. Both representations have the same information. We have the choice between the function and an infinitely long vector formed from the expansion coefficients: © Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2021 M. Pieper, Quantum Mechanics, Springer essentials, https://doi.org/10.1007/978-3-658-32645-6_3

13

14

3

f (t)

⇔

Mathematical formulation

(. . . , c−2 , c−1 , c0 , c1 , c2 , . . .).

(3.2)

Analogously, in quantum mechanics, we have the freedom to choose a concrete form of representation for the states and observables. What is decisive is the basic structure of the description, which will turn out to be a separable Hilbert space on which linear, Hermitian operators are defined.

3.2

Postulates of Quantum Mechanics

In the following sections, we summarize the general description of quantum mechanics in three basic postulates. We derive these from physically and mathematically reasonable demands and assumptions. As special cases, the postulates include wave mechanics and matrix mechanics. In particular, we motivate the abstract terms by well-known examples from linear algebra. We refrain from formal mathematical definitions and proofs and instead refer to the numerous literature (e.g., [3, 4] or [5]). In addition, we restrict ourselves to the case of discrete eigenvalue spectra, which is sufficient for a basic understanding.

3.2.1

Postulate 1: States in the Hilbert Space

In Sect. 2.2, we used wave functions to describe the particle in a box. These were interpreted as a quantum mechanical state. In physics, states generally form the basis of any system description. Therefore, we first discuss the necessary properties of general quantum mechanical states. For motivation, we use the wave functions. To develop his wave mechanics, Schrödinger started from de Broglie’s matter waves. He tried to describe mathematically that particles behave like waves in certain experiments (see [6], Chap. 2). One of the important wave phenomena is the so-called superposition principle: different waves superimpose to form a new resulting wave. In the same way, we can also superimpose wave functions and obtain general states (see Sect. 2.2.4 and 4.2). The superposition is mathematically a linear combination. In linear algebra, we encounter linear combinations when, for example, we express arbitrary vectors through the standard basis vectors:

3.2 Postulates of Quantum Mechanics

15

      1 0 2 2· +6· = . 0 1 6

(3.3)

Through regular use of vectors, we know that we can add two vectors or multiply them by a number. As a result, we get a vector again. The same applies to our quantum mechanical states. Here, a linear combination yields a state, too. For vectors, we have certain rules of calculation, which also apply to states. For example, the order of addition is arbitrary: a + b = b + a . In mathematics, all objects that behave like vectors a , that is, obey the same rules of calculation as vectors, are grouped together in a so-called vector space or linear space. Therefore, our states are also abstract vectors in a vector space. Unfortunately, this is not enough for us. We need a scalar product as an add-on to calculate expectations. For two-dimensional vectors, the following holds:  a · b = (a1 , a2 ) · T

 b1 b2

= a 1 b1 + a 2 b2 ,

(3.4)

where a T is the transpose of a , that is, a row vector, if we generally consider a as column vector. But where have we used a scalar product in wave mechanics so far? To answer this question, we first note that the expression  u |v =

u(x)∗ · v(x)d x

(3.5)

defines a scalar product for the wave functions u(x) and v(x). We now show that we can express the expectation from Eq. 2.5 using this scalar product. For this purpose, we replace v by Hˆ u and obtain u | Hˆ u =



u(x)∗ · Hˆ u(x)d x =  Hˆ .

(3.6)

The expressions in Eqs. 3.4 and 3.5 fulfill certain calculation rules, which define a general scalar product in mathematics. However, not every vector space has a scalar product. Therefore, we call vector spaces with scalar product preHilbert spaces. Before we expand the pre-Hilbert space into a Hilbert space, we introduce the bra-ket notation according to Paul Dirac: Generally speaking, states are described by ket vectors |u . Here the notation in the brackets is arbitrary. Often the naming

16

3

Mathematical formulation

of states is chosen analogous to the measured values, as for example, |a with the measured value a. Also common is the use of quantum numbers, which describes the state |n , which, for example, belongs to the energy value E n of the n-th Bohr’s orbit. Dirac’s notation has the advantage that a bra vector a| can be introduced for each ket vector |a . This is to be seen analogous to the transpose a T . Put together, the bra and ket vectors form the usual bracket that characterizes the general scalar product: a T · b

↔

a|b,

where in the middle, there is only one stroke and no double stroke. In linear algebra, two vectors are orthogonal if the scalar product disappears: a T · b = 0. This can be transferred analogously to general states. If we have a|b = 0, then the two states |a and |b are orthogonal. Unfortunately, this property cannot be illustrated as easily for states as for vectors that are perpendicular to each other. But in the following sections, we will see that it is nevertheless very useful. Let us return to the superposition of states and consider Eq. 2.4, now in bra-ket notation: |u =

∞ 

cn |u n  .

(3.7)

n=1

In contrast to the linear combination of vectors in Eq. 3.3, we sum infinitely many state vectors. This is because the vector space of the wave functions is infinite, but the vectors a are only two-dimensional. We illustrate this fact with the function f (t) from paragraph 3.1: We need an infinite number of Fourier coefficients (see Eqs. 3.1 and 3.2). For the description of a only two vector components are sufficient. In principle, we can calculate in infinite-dimensional vector spaces in the same way as in finite-dimensional ones; there are only a few differences that we have to take into account. The branch of mathematics that deals with infinite-dimensional vector spaces is functional analysis (see e.g. [7]). It thus forms the mathematical basis of quantum mechanics. We would be finished at this point if we would only consider states from finite-dimensional vector spaces. The previous properties (vector space with scalar product) are sufficient for description. In quantum mechanics, we are usually

3.2 Postulates of Quantum Mechanics

17

dealing with infinite-dimensional vector spaces, so we must make two additional demands: When evaluating the infinite sum in Eq. 3.7, we calculate a limit from the mathematical point of view. However, it can happen that the limit no longer belongs to our vector space. Then it is not a state and thus has no physical meaning. As an example, we consider the following sequence (see Heron’s method) x0 = 1, xn+1 =

1 xn + , n = 0, 1, 2, . . . , 2 xn

which is defined by rational numbers (fractions) and we calculate the first four elements: 3 17 577 = 1.5, x2 = = 1.4166 . . . , x3 = = 1.4142 . . . 2 12 408 √ We suspect the limit value 2 = 1.4142 . . ., which is not a fraction, that is, does √ not belong to the rational numbers. We must add the irrational numbers (such as 2) to the rational numbers so that our sequence has a limit. In mathematics, the real numbers are called complete. Here the limits of all convergent sequences are again real numbers. The rational numbers are not complete, as the example shows. Thus, we must demand that the vector space is complete. Only in this way can we ensure that infinite sums, as in Eq. 3.7, have a limit in our vector space, that is, provide a quantum mechanical state. A pre-Hilbert space that is complete is called Hilbert space. Let us come to the second additional demand: For motivation, we consider the linear combination in Eq. 3.3. Here we have used the standard basis vectors e1 = (1, 0) T and e2 = (0, 1) T . These form a possible basis of the two-dimensional vector space. We can write any vector uniquely as a linear combination of this basis vectors. We would like to have something similar for the general quantum mechanical states. To construct this basis, we go back to the procedure in Sect. 2.2, where we solved the eigenvalue problem for the Hamiltonian and found the eigenstates |u n  . Then, in Eq. 2.4, we formed the arbitrary states by linear combination of the |u n  . We would now like the eigenstates |u n  to be a basis of our Hilbert space, that is, we would like to be able to represent all possible states as linear combinations of the |u n  . For this, we have to make sure that we only have countably infinite x0 = 1, x1 =

18

3

Mathematical formulation

basis vectors, so that we can form the infinite sum. If we had an uncountable number of vectors (a “greater” infinity than the countable one), this would not be possible. A Hilbert space with a countable basis is called separable. Such Hilbert spaces are in a certain way not “boundlessly large” and thus “manageable.” We summarize the properties for general quantum mechanical states in the first postulate: Postulate 1 (Pure states) For every quantum mechanical system, there exists a separate Hilbert space H, so that every pure state of the system can be represented by a vector |a ∈ H of the Hilbert space, respectively that every vector |a ∈ H from the Hilbert space corresponds to a possible physical state.

3.2.2

Postulate 2: Measured Values, Operators and Eigenvalues

The examples in Chap. 2 show that quantum mechanics can only make statements about possible measurement results and their probability of occurrence. The second postulate therefore deals with observables and their measured values. We generalize the observations from the box example: All quantum mechanical observables are described as operators applied to Hilbert space vectors. The eigenvalues correspond to the possible measured values and the eigenvectors are the associated eigenstates. In the following, we will only specify which properties the operators must have in order for us to interpret them in a physically meaningful way. Due to the linear structure of quantum mechanics, only linear operators are used. The motivation is as follows: We can write states as linear combinations (see Eq. 3.7). An obvious requirement is then that an operator Aˆ should act individually on the summands: ˆ A|u =

∞ 

ˆ n . cn A|u

n=1

But this property characterizes linear operators. In the example above, Aˆ is applied to the states |u and all |u n  . For the two-dimensional vectors, the linear operators correspond to the 2 × 2 matrices, which we will use more often for clarification in the following.

3.2 Postulates of Quantum Mechanics

19

Before we demand another property from the operators, we turn to the measured values, that is, the eigenvalues of the operators. Eigenvalues are complex numbers with a specific property: For each eigenvalue a, there exists an eigenvector |a  = |0 that is not equal to zero. If we apply the operator Aˆ to this eigenvector |a , we obtain a multiple of |a . The multiple corresponds to the ˆ eigenvalue a. We can express this more briefly by mathematical formulas: A|a = a|a . For eigenvectors, the mapping is very simple; it consists only of a multiplication with the eigenvalue. It should be noted that eigenvalues are uniquely determined, but for eigenvectors, we have certain freedoms, for example, with respect to the “length.” For clarification, we look at the matrix  A=

 12 . 21

The eigenvalues of A are a1 = −1 and a2 = 3 and possible eigenvectors are, for example 

       12 −1 1 −1 · = = (−1) · 21 1 −1 1         12 1 3 1 · = = 3· , 21 1 3 1

(3.8)

but every multiple of the vectors would also be an eigenvector. Eigenvalues can also be complex numbers, as the following example shows. The matrix   12 B= . −2 1 has the complex eigenvalues b1,2 = 1 ± 2i. In quantum mechanics, the eigenvalues correspond to the measured values. In experiments, however, we do not measure complex numbers, so only real eigenvalues are allowed. We need an additional property of the operators to ensure this requirement: All operators must be Hermitian operators. We explain this property for 2 × 2 matrices. A comparison of the two matrices A and B in the example above shows that A is symmetric: A coincides with the

20

3

Mathematical formulation

transpose A T (A = A T ). The transpose is formed by mirroring the entries on the diagonal. For complex-valued matrices A, we have to transpose and additionally form the complex conjugate of the matrix entries. The resulting matrix A+ is called the adjoint matrix. We call a matrix self-adjoint or Hermitian, if A = A+ . Similarly, the definition is extended to general operators by calculating the matrix entries using a scalar product. Hermitian operators have a number of useful properties. For us, the following two are particularly important: All eigenvalues and all expectations calculated according to Eq. 3.6 are real. We know almost all properties to describe observables and their measured values. However, one property is still missing. This concerns the eigenvectors. For this, we look again at our example for the matrix A. In Eq. 3.8, we have found that A has the eigenvectors a1 = (−1, 1) T and a2 = (1, 1) T . The scalar product   1 =0 a1T · a2 = (−1, 1) · 1 of the eigenvectors is zero, so they are orthogonal. This is not a coincidence, but applies generally to Hermitian operators. In addition, we can normalize all eigenvectors, that is, bring them to length one. Such a system is then called orthonormalized. In addition, we would like the orthonormal system of eigenvectors to be complete, that is, to form a basis of the corresponding Hilbert space. Then all states can be written as linear combinations of the system. We demonstrate this fact with the vector (2, 6) T , which we write as a linear combination of the eigenvectors of the matrix A (see also Eq. 3.3):  2·

−1 1



    1 2 +4· = . 1 6

Unfortunately, this property is not fulfilled by all Hermitian operators, so we have to demand it explicitly. Fortunately, it is possible to prove it for all operators that describe physical observables. We summarize our observations in the second postulate:

3.2 Postulates of Quantum Mechanics

21

Postulate 2 (Observables and measured values) All physical observables, except time, are represented by linear, Hermitian operators. These have a complete system of orthonormalizable eigenstates. The possible measured values are the eigenvalues of the operators. Before we derive the third postulate in the next section, we want to look at degenerate states, as we have observed in Bohr’s atomic model. In quantum mechanics, degeneration is always referred to when there are several different linearly independent eigenstates for a measured value (eigenvalue). For clarification, we consider an observable, which is represented by the matrix 

 30 03

C=

and has the eigenvalue c = 3 with multiplicity two. Two possible eigenvectors are, the standard basis vectors e1 and e2 . So we find two different linear independent eigenstates, which belong to the same measured value c = 3. Therefore, the states are degenerate. By measuring c = 3, we cannot determine whether the system is in the state e1 or e2 . Mathematically the degeneration is related to the eigenvalue. We have found an eigenvalue with multiplicity two. In general, eigenvalues with multiplicity greater than one are associated with degenerate states. We need a second observation variable to decide in which state the system is exactly. If, for example, we have previously only measured the energy, we now also determine the angular momentum. Here it is important that both observables are “compatible,” that is, can be measured sharply at the same time. The position and the momentum would be an example of where this is not the case. If two quantities can be measured sharply at the same time, it does not matter which quantity we measure first. The measurements do not influence each other. This property can be transferred to the corresponding operators: The operators commute regarding multiplication, which is generally not the case. We consider an example and introduce an additional observable, which is represented by the matrix  D=

 13 . 31

The eigenvalues of D are d1 = −2 and d2 = 4. For matrices C and D, C · D = D · C holds, that is, they commute with respect to matrix multiplication. So we can use D to cancel the degeneration, that is, to

22

3

Mathematical formulation

determine the exact state. For this purpose, we are looking for a common system of eigenvectors to the two matrices, that is, two vectors d1 and d2 , which are eigenvectors to both matrices C and D. The vectors d1 = (−1, 1) T and d2 = (1, 1) T meet this requirement. So by specifying the measurement of D, we can now determine exactly what state our system is in. For example, if we measure c = 3 and d = −2, we know that the corresponding state vector is d1 . If, on the other hand, we measure c = 3 and d = 4, we know that the state d2 is present. We transfer this procedure to general operators. We consider two Hermitian ˆ which commutate, that is, for which the so-called commutator operators Cˆ and D,   Cˆ , Dˆ = Cˆ Dˆ − Dˆ Cˆ = 0ˆ disappears. In this case, we can find a complete set of common eigenstates. The states are usually designated by their associated eigenvalues: If |c is the eigenstate belonging to the eigenvalue c of operator Cˆ and |d corresponding to the eigenvalue ˆ then the common eigenstate is denoted by |c, d . The following holds: d of D, ˆ d = c|c, d C|c,

and

ˆ d = d|c, d . D|c,

In Sect. 3.3, we will use this fact and give an algorithm how we can accurately characterize a quantum mechanical system. Another example to illustrate degeneracy is discussed in Sect. 4.1.

3.2.3

Postulate 3: Probabilities

The third postulate concerns the probabilities with which we obtain the possible measured values. We already know that in the linear combination |u =

∞ 

cn |u n 

n=1

the squares of the coefficients |cn |2 indicate the probability of measuring the associated eigenstate |u n  (see Sect. 2.2.4). We now consider how to determine these coefficients for any given state |u . We consider the following linear combination

3.2 Postulates of Quantum Mechanics

23

      1 0 0, 6 a = 0, 6 · + 0, 8 · = . 0 1 0, 8 We assume that the standard basis vectors e1 and e2 are eigenstates of an observable, which is described by a 2 × 2 matrix. Then we obtain the evolution coefficients in the linear combination by forming the scalar product with the eigenstates. For example, due to the orthonormality of the vectors we obtain by e1T

    1 0 · a = 0, 6 · (1, 0) · +0, 8 · (1, 0) · = 0, 6 0 1  



 =1

=0

the coefficient belonging to e1 . The probability of measuring the state e1

2 is: e1T · a . In quantum mechanics, we can always write general states as linear combinations of orthonormalized eigenstates. Therefore, the following holds in general u k |u =

∞ 

cn u k |u n  = ck ,

n=1

where we use that u k |u n  gives the value one if n = k and otherwise disappears. The probability is then calculated from the square: |u k |u|2 . We have thus derived the statement of the third postulate: Postulate 3 (Probabilities) The probability w with which the measured value a is encountered during a measurement at the observable A in the state |u is calculated by w(a , |u ) = |a|u|2 , where |a is the eigenstate associated with eigenvalue a designated by the ˆ corresponding operator A.

24

3.3

3

Mathematical formulation

Solution Algorithm

We close with a general solution algorithm how to solve quantum mechanical problems. We consider a quantum mechanical system and assume that we can describe it by an observable A. ˆ In the first step, we have to write this observable as Hermitian operator A. Usually, the classical expression describing the observable is used for this. In it, all classical quantities are replaced by the known quantum mechanical operators (correspondence rules). In the second step, we calculate all eigenvalues a and the corresponding eigenstates |a . If these are not degenerate, we are finished and can describe our system uniquely by these eigenstates. If Aˆ is degenerate, that is, has multiple eigenvalues, we must add a second ˆ We first observable B. This is described analogously as Hermitian operator B. check whether the two operators commutate, that is, whether they can be meaˆ B] ˆ is sured sharply at the same time. For this purpose, the commutator [ A, calculated. If the commutator disappears, we calculate the common eigenstates ˆ Again we check whether there are still degenerate eigenvalues. If of Aˆ and B. the degeneration is completely removed and no more degenerate states exist, we are done. In this case, Aˆ and Bˆ are sufficient for the complete description of the system. If there are still degenerate states, we have to add another observable C and proceed analogously. We iterate this process until there is no more degeneration. This way, we can be ˆ B, ˆ C, ˆ etc. describe the system completely. In this case, sure that the operators A, the measurement of the observables uniquely defines the state. The operators then form a complete set of commutating operators. The goal of any description of a quantum system is therefore to find such a complete set.

4

Examples of Use

4.1

A Thought Experiment with Toy Blocks

We begin with a simple thought experiment, in which we look at a toy for children (Fig. 4.1) and describe it mathematically, within the framework of quantum mechanics. The toy consists of toy blocks that differ in shape (round, square and triangular cross section) and color (white and gray). The child tries to put the toy blocks in a box, but of course the opening has to be found to match the shape of the block. How does this toy fit in with quantum mechanics? We interpret the toy blocks as quantum mechanical particles. The properties shape and color then correspond to two observables. The box can finally be seen as a measuring device for the observable shape. We now want to construct a general basis for a suitable Hilbert space. The associated operators Aˆ and Bˆ commute, because the child can determine the properties shape and color independently. So they form a complete system of commutating operators, because there are only two properties. We can therefore find a common basis of eigenvectors to the eigenvalues (measured variables) shape and color. As usual, we denote these with their corresponding measured value in bra-ket notation: ˆ  = r |r  , A|r

ˆ A|s = s|s

and

ˆ A|t = t |t ,

where r stands for round, s for square and t for triangular. |r  is thus, for example, the eigenstate from Aˆ to eigenvalue r, which represents the possible measurement result round.

© Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2021 M. Pieper, Quantum Mechanics, Springer essentials, https://doi.org/10.1007/978-3-658-32645-6_4

25

26

4

Examples of Use

Fig. 4.1 Toy for children with toy blocks

Analogously, we define the eigenstates of the operator Bˆ that belongs to the observable color: ˆ B|w = w|w

and

ˆ B|g = g|g .

Here w stands for the measurement result white and g for gray. The common eigenstates of Aˆ and Bˆ are then designated as follows: |r , w , |r , g , |s, w , |s, g , |t, w

and

|t, g ,

where the first entry indicates the shape and the second the color. What happens now when the child plays with the toy blocks? Quantum mechanically it determines the observable form. If a block fits through the round opening, it is clear that the particle is in a state |r  . However, as long as the child has not yet determined the color, the building block can be either white or gray. So it is open whether the particle is in the state |r , w or |r , g . Therefore, a degenerate state is present. Only when the second observable color is determined, the degeneration is canceled.

4.3 The Stern-Gerlach Experiment

4.2

27

Schrödinger’s Cat

Schrödinger’s cat is a thought experiment proposed by Erwin Schrödinger in 1935: A cat is locked up in a box. Further there is a radioactive preparation, a Geiger counter and a poison vial in the box. The preparation decays with a certain probability and is detected by the Geiger counter. The Geiger counter triggers a device that breaks the poison vial and the cat dies. There are two states for the radioactive preparation: decayed |d or not decayed |nd . These superpose to a general state |u = c1 |d + c2 |nd , where |c1 |2 and |c2 |2 indicate the probabilities for the individual states. Analogously, Schrödinger transfers the situation to the cat: We have the two states of being alive |a or dead |na . The general state is provided by the linear combination |v = c1 |na + c2 |a , with the known probabilities for the decay of the preparation, which correspond to |d and |nd , accordingly. This superposition is present as long as we do not look into the box and “measure” whether the cat is alive or dead. By the measurement, the wave function “collapses” according to the Copenhagen interpretation and we get one of the single states. Especially here, the imagination plays a trick on us: the superposition of both states means that the cat is both alive and dead, which is not possible. The problem with this example is that we transfer quantum mechanical concepts to the macroscopic world. There are different approaches to solve this paradox, for example, the many-worlds interpretation or Bohm’s mechanics. A further explanation can be found in [4] Sect. 10.5. Paul Dirac warns in this context (see [8], Chap. 4) that the superposition of quantum mechanical states is something completely new to which no classical picture should be applied. There is an essential difference to all existing classical images. Ultimately, it is merely a mathematical theory, that is, a tool that is used to describe quantum mechanical processes.

4.3

The Stern-Gerlach Experiment

In the Stern-Gerlach experiment, uncharged particles (e.g., silver atoms) are observed in a vacuum. The particles leave a source and are then focused into a beam, which is detected on a screen. The result is a spot in the middle. The spot splits when the beam passes through a magnetic field. If the field runs in the z-direction, we observe two separate spots at equal distance from the center of the screen, symmetrically shifted up and down in the z-direction (see Fig. 4.2).

28

4

Examples of Use

Fig. 4.2 Experimental set-up for the Stern-Gerlach experiment

This observation cannot be explained by classical physics, because here the particles would be distributed uniformly in z-direction between the two observed spots (see Fig. 4.2). It is therefore a quantum mechanical effect that we want to describe and investigate with our formalism. What do we need for the description? First we have to specify the Hilbert space H to which the states belong. Then we look at the energy and introduce an operator that describes the corresponding observable. We observe two beams: deflected upwards (+) or deflected downwards (−). So we have two states in which a particle can be on the screen. We denote these with |+ and |− . We obtain any state as a superposition of the two states, that is, mathematically as a linear combination |u = u + |+ + u − |− with appropriate coefficients u + , u − ∈ C whose squares indicate the associated probabilities. Two numbers are therefore sufficient to describe the states. We can therefore write each state as a vector with two components. For this purpose, we identify the two basic states |± with the standard basis vectors. We obtain for |u the following:     1 0 |+ = ˆ , |− = ˆ 0 1

 ⇒

u+ |u  = ˆ u−



    1 0 = u+ + u− . 0 1

4.3 The Stern-Gerlach Experiment

29

A simple representation for our Hilbert space H is therefore H = C2 , with appropriate normalization so that the overall probability is one. This observation is important, because we now know that the operators on H, which describe the observables, are 2 × 2 matrices with complex entries. We will now look at the observable energy: The particle with the magnetic moment µ  ∈ R3 classically has the interaction energy H = −µ  · B with the 3  magnetic field B ∈ R (see [9], Chap. 3). The vectors are three-dimensional, that is, µ  = (µ1 , µ2 , µ3 )T and B = (B1 , B2 , B3 )T . The magnetic field is fixed in the experiment. We can therefore only quantize the moment µ,  that is, identify it as an operator within the framework of the correspondence principle. For this, we calculate the scalar product of the two vectors: H = −µ  · B = −(µ1 · B1 + µ2 · B2 + µ3 · B3 ). The associated Hamiltonian is applied to the state vectors |± ∈ C2 :   Hˆ |± = − µˆ 1 · B1 + µˆ 2 · B2 + µˆ 3 · B3 |±   = − µˆ 1 |± · B1 + µˆ 2 |± · B2 + µˆ 3 |± · B3 . The three components µˆ i of the magnetic moment—understood as operators— are thus 2 × 2 matrices with complex entries. In order to be able to determine these matrices in more detail, properties resulting from physical observations and requirements are used. For a detailed calculation, we refer to the literature, for example, [8], Chap. 37, and only consider energy measurement as an example here. It follows from the experiment that we can only measure two discrete values  ∓µ0 B, where we refer B = | B|  to the strength of the magnetic for H = −µ  · B: field and µ0 is derived from the positions of the two observed spots. We have “∓” as sign, because in the formula for H, there is a minus sign. The measured energy values correspond to the eigenvalues of the associated operator. So we have Hˆ |± = ∓µ0 B|±.

(4.1)

In our experiment, the magnetic field runs in z-direction, that is, the first and second components of the magnetic field vector disappear: B1 = B2 = 0. Therefore, the magnetic field is B3 = B and Hˆ is reduced to Hˆ = −µˆ 3 B. We apply this in Eq. 4.1 and cancel −B:

30

4

Examples of Use

(4.2) With the matrix  µˆ 3 := µ0

 1 0 0 −1

the condition from Eq. 4.2 is satisfied for the standard basis vectors. By using further properties, for example, that the operators must be hermitic (µˆ i+ = µˆ i ), the remaining matrices can also be determined:  µˆ 1 := µ0

 01 10

 and

µˆ 2 := µ0

 0 −i . i 0

These matrices (without the factor µ0 ) are the Pauli matrices, a first indication that the spin of the particles is responsible for the observed splitting of the beams. The commutators     µˆ 1 , µˆ 2 = 2iµ0 µˆ 3 , µˆ 1 , µˆ 3 = −2iµ0 µˆ 2

and

  µˆ 2 , µˆ 3 = 2iµ0 µˆ 1 ,

show that the matrices µˆ i do not commute. The result is typical for angular momentum quantities. A comparison with Poisson bracket from classical mechanics shows that the result fits to an angular momentum quantity (see [10], Chap. 2 and [11], Chap. 5). This observation suggests that the unknown property (spin) is an “angular momentum-like” quantity that gives rise to the corresponding magnetic moment. For more details on spin, we refer to the standard textbooks again.

What You Learned From This essential

In this introduction to the mathematical formulation of quantum mechanics, you have… • used examples to analyze discrete and degenerate energy levels • established Hermitian operators according to the correspondence rules for the description of observables • investigated the significance of quantum mechanical state vectors in Hilbert space • interpreted eigenvalues of Hermitian operators as measured values • calculated probabilities for measured values • applied three postulates for the mathematical description of quantum mechanical systems to examples

© Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2021 M. Pieper, Quantum Mechanics, Springer essentials, https://doi.org/10.1007/978-3-658-32645-6

31

Literature

1. Huebener, R.P.; Schopohl, N.: Die Geburt der Quantenphysik. Springer Spektrum: Wiesbaden, 2016. 2. Orzel, C.: How to Teach Quantum Physics to Your Dog. Scribner: New York, 2010. 3. Nolting, W.: Theoretical Physics 6: Quantum Mechanics—Basics. Springer International Publishing: Cham, 2017. 4. Rebhan, E.: Theoretische Physik: Quantenmechanik. Spektrum Akademischer Verlag: Heidelberg, 2008. 5. von Neumann, J.: Mathematical Foundations of Quantum Mechanics. Sixth printing. Princeton Univers. Press: New Jersey, 1971. 6. Gasiorowicz, S.: Quantum Physics. Third edition. John Wiley & Sons: New York, 2003. 7. Großmann, S.: Funktionalanalysis im Hinblick auf Anwendungen in der Physik. Fifth Edition. Springer Spektrum: Wiesbaden, 2014. 8. Dirac, P.A.M.: Principles of Quantum Mechanics. Fourth Edition. Oxford University Press: Oxford, 1958. 9. Nolting, W.: Theoretical Physics 3: Electrodynamics. Springer International Publishing: Cham, 2016. 10. Nolting, W.: Theoretical Physics 2: Analytical Mechanics. Springer International Publishing: Cham, 2016. 11. Nolting, W.: Theoretical Physics 7: Quantum Mechanics – Methods and Applications. Springer International Publishing: Cham 2017.

© Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2021 M. Pieper, Quantum Mechanics, Springer essentials, https://doi.org/10.1007/978-3-658-32645-6

33