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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
A Course in Quantum Mechanics Unique graduate-level textbook on quantum mechanics by John David Jackson, author of the renowned Classical Electrodynamics A Course in Quantum Mechanics is drawn directly from J. D. Jackson's detailed lecture notes and problem sets. It is edited by his colleague and former student Robert N. Cahn, who has taken care to preserve Jackson's unique style. The textbook is notable for its original problems focused on real applications, with many addressing published data in accompanying tables and figures. Solutions are provided for problems that are critical for understanding the material and that lead to the most important physical consequences. Overall, the text is comprehensive and comprehensible; derivations and calculations come with clearly explained steps. More than 120 figures illustrate underlying principles, experimental apparatus, and data. In A Course in Quantum Mechanics readers will find detailed treatments of: * Wave mechanics of de Broglie and Schrödinger, the Klein-Gordon equation and its non-relativistic approximation, free particle probability current, expectation values. * Schrödinger equation in momentum space, spread in time of a free-particle wave packet, density matrix, Sturm-Liouville eigenvalue problem. * WKB formula for bound states, example of WKB with a power law potential, normalization of WKB bound state wave functions, barrier penetration with WKB. * Rotations and angular momentum, representations, Wigner d-functions, addition of angular momenta, the Wigner-Eckart theorem. * Time-independent perturbation theory, Stark, Zeeman, Paschen-Back effects, time-dependent perturbation theory, Fermi's Golden Rule. * Atomic structure, helium, multiplet structure, Russell-Saunders coupling, spin-orbit interaction, Thomas-Fermi model, Hartree-Fock approximation. * Scattering amplitude, Born approximation, allowing internal structure, inelastic scattering, optical theorem, validity criterion for the Born approximation, partial wave analysis, eikonal approximation, resonance. * Semi-classical and quantum electromagnetism, Aharonov-Bohm effect, Lagrangian and Hamiltonian formulations, gauge invariance, quantization of the electromagnetic field, coherent states. * Emission and absorption of radiation, dipole transitions, selection rules, Weisskopf-Wigner treatment of line breadth and level shift, Lamb shift. * Relativistic quantum mechanics, Klein-Gordon equation, Dirac equation, two-component reduction, hole theory, Foldy-Wouthuysen transformation, Lorentz covariance, discrete symmetries, non-relativistic and relativistic Compton scattering.
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John David Jackson
A Course in Quantum Mechanics
Edited by
Robert N. Cahn
This edition first published 2024
© 2024 John Wiley & Sons, Inc.
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Contents
Cover
Title Page
Copyright Page
Preface
About the Companion Website
1 Basics
1.1 Wave Mechanics of de Broglie and Schrödinger
1.2 Klein-Gordon Equation
1.3 Non-Relativistic Approximation
1.4 Free-Particle Probability Current
1.5 Expectation Values
1.6 Particle in a Static, Conservative Force Field
1.7 Ehrenfest Theorem
1.8 Schrödinger Equation in Momentum Space
1.9 Spread in Time of a Free-Particle Wave Packet
1.10 The Nature of Solutions to the Schrödinger Equation
1.11 A Bound-State Problem: Linear Potential
1.12 Sturm-Liouville Eigenvalue Problem
1.13 Linear Operators on Functions
1.14 Eigenvalue Problem for a Hermitian Operator
1.15 Variational Methods for Energy Eigenvalues
1.16 Rayleigh-Ritz Method
Problems
2 Reformulation
2.1 Stern-Gerlach Experiment
2.2 Linear Vector Spaces
2.3 Linear Operators
2.4 Unitary Transformations of Operators
2.5 Generalized Uncertainty Relation for Self-Adjoint Operators
2.6 Infinite-Dimensional Vector Spaces - Hilbert Space
2.7 Assumptions of Quantum Mechanics
2.8 Mixtures and the Density Matrix
2.9 Measurement
2.10 Classical vs. Quantum Probabilities
2.11 Capsule Review of Classical Mechanics and Conservation Laws
2.12 Translation Invariance and Momentum Conservation
2.13 Dirac’s p’s and q’s
2.14 Time Development of the State Vector
2.15 Schrödinger and Heisenberg Pictures
2.16 Simple Harmonic Oscillator
Problems
3 Wentzel-Kramers-Brillouin (WKB) Method
3.1 Semi-classical Approximation
3.2 Solution in One Dimension
3.3 Schrödinger Equation for the Linear Potential
3.4 Connection Formulae for the WKB Method
3.5 WKB Formula for Bound States
3.6 Example of WKB with a Power Law Potential
3.7 Normalization of WKB Bound State Wave Functions
3.8 Bohr’s Correspondence Principle and Classical Motion
3.9 Power of WKB
3.10 Barrier Penetration with the WKB Method
3.11 Symmetrical Double-Well Potential
3.12 Application of the WKB Method to Ammonia Molecule
Problems
4 Rotations, Angular Momentum, and Central Force Motion
4.1 Infinitesimal Rotations
4.2 Construction of Irreducible Representations
4.3 Coordinate Representation of Angular Momentum Eigenvectors
4.4 Observation of Sign Change for Rotation by 2
π
4.5 Euler Angles, Wigner d-functions
4.6 Application to Nuclear Magnetic Resonance
4.7 Addition of Angular Momenta
4.8 Integration Over the Rotation Group
4.9 Gaunt Integral
4.10 Tensor Operators
4.11 Wigner-Eckart Theorem
4.12 Applications of the Wigner-Eckart Theorem
4.13 Two-Body Central Force Motion
4.14 The Coulomb Problem
4.15 Patterns of Bound States
4.16 Hellmann-Feynman Theorem
Problems
5 Time-Independent Perturbation Theory
5.1 Time-Independent Perturbation Expansion
5.2 Interlude: Spectra and History
5.3 Fine Structure of Hydrogen
5.4 Stark Effect in Ground-State Hydrogen
5.5 Perturbation Theory with Degeneracy
5.6 Linear Stark Effect in Hydrogen
5.7 Perturbation Theory with Near Degeneracy
5.8 Zeeman and Paschen-Back Effects in Hydrogen
Problems
6 Atomic Structure
6.1 Parity
6.2 Identical Particles and the Pauli Exclusion Principle
6.3 Atoms
6.4 Helium Atom
6.5 Periodic Table
6.6 Multiplet Structure, Russell-Saunders Coupling
6.7 Spin-Orbit Interaction
6.8 Intermediate Coupling
6.9 Thomas-Fermi Atom
6.10 Hartree-Fock Approximation
Problems
7 Time-Dependent Perturbation Theory and Scattering
7.1 Time-dependent Perturbation Theory
7.2 Fermi’s Golden Rule
7.3 Scattering Amplitude
7.4 Born Approximation
7.5 Scattering Theory from Fermi’s Golden Rule
7.6 Inelastic Scattering
7.7 Optical Theorem
7.8 Validity Criterion for the First Born Approximation
7.9 Eikonal Approximation
7.10 Method of Partial Waves
7.11 Behavior of the Cross Section and the Argand Diagram
7.12 Hard Sphere Scattering
7.13 Strongly Attractive Potentials and Resonance
7.14 Levinson’s Theorem
Problems
8 Semi-Classical and Quantum Electromagnetic Field
8.1 Electromagnetic Hamiltonian and Gauge Invariance
8.2 Aharonov-Bohm Effect
8.3 Semi-Classical Radiation Theory
8.4 Scalar Field Quantization
8.5 Quantization of the Radiation Field
8.6 States of the Electromagnetic Field
8.7 Vacuum Expectation Values of E, E ⋅ E over Finite Volume
8.8 Classical vs. Quantum Radiation
8.9 Quasi-Classical Fields and Coherent States
Problems
9 Emission and Absorption of Radiation
9.1 Matrix Elements and Rates
9.2 Dipole Transitions
9.3 General Selection Rules
9.4 Charged Particle in a Central Field
9.5 Decay Rates with
LS
Coupling
9.6 Line Breadth and Level Shift
9.7 Alteration of Spontaneous Emission from Changed Density of States
Problems
10 Relativistic Quantum Mechanics
10.1 Klein-Gordon Equation
10.2 Dirac Equation
10.3 Angular Momentum in Dirac Equation
10.4 Two-Component Equation and Plane-Wave Solutions
10.5 Dirac’s Treatment of Negative Energy States
10.6 Heisenberg Operators and Equations of Motion
10.7 Hydrogen in the Dirac Equation
10.8 Foldy-Wouthuysen Transformation
10.9 Lorentz Covariance
10.10 Discrete Symmetries
10.11 Bilinear Covariants
10.12 Applications to Electromagnetic Form Factors
10.13 Potential Scattering of a Dirac Particle
10.14 Neutron-Electron Scattering
10.15 Compton Scattering
Problems
A Dimensions and Units
B Mathematical Tools
B.1 Contour Integration
B.2 Green Function for Helmholtz Equation
B.3 Wigner 3-
j
and 6-
j
Symbols
C Selected Solutions
C.1 Chapter 1
C.2 Chapter 2
C.3 Chapter 3
C.4 Chapter 4
C.5 Chapter 5
C.6 Chapter 6
C.7 Chapter 7
C.8 Chapter 8
C.9 Chapter 9
C.10 Chapter 10
Bibliography
Index
List of Tables
CHAPTER 03
Table 3.1 Comparison of the wave function at...
Table 3.2 Alpha decays of Po isotopes, with...
Table 3.3 Alpha decays of Pu isotopes...
CHAPTER 07
Table 7.1 Representative cross sections...
List of Illustrations
CHAPTER 01
Figure 1.1 Connection between the...
Figure 1.2 Numerical integration...
Figure 1.3 Examples showing the homogeneous...
CHAPTER 03
Figure 3.1 The quantities and, which...
Figure 3.2 The linear potential, leading...
Figure 3.3 Contours used find solutions to...
Figure 3.4 The Airy functions, and, which...
Figure 3.5 Contours used find the asymptotic...
Figure 3.6 Potential with turning points for...
Figure 3.7 The WKB quantization condition follows...
Figure 3.8 The semi-classical and quantum pictures...
Figure 3.9 The connection formulae for WKB match...
Figure 3.10 Representation of barrier penetration...
Figure 3.11 A symmetric double-well potential.
Figure 3.12 The extreme case of the symmetric...
Figure 3.13 The double-well problem labeled by...
Figure 3.14 The odd solutions to the double-well...
Figure 3.15 A particle placed in the right-hand...
Figure 3.16 The ammonia molecule and its fundamental...
Figure 3.17 Simple model for barrier penetration...
Figure 3.18 Simple model for field emission. The...
CHAPTER 04
Figure 4.1 An infinitesimal rotation of...
Figure 4.2 Schematic representation of...
Figure 4.3 Data showing the difference...
Figure 4.4 The Euler angles, define...
Figure 4.5 Schematic representation...
Figure 4.6 The configuration of the magnetic...
Figure 4.7 The effective magnetic field...
Figure 4.8 The probability that the value...
Figure 4.9 Data from Rabi, et al., 1939 / American...
Figure 4.10 For a photon emitted in...
Figure 4.11 Combining the potential...
Figure 4.12 Radial wave functions for...
Figure 4.13 For case (b), there are no...
Figure 4.14 For case (a), there are branch...
Figure 4.15 For case (c), there is a pole...
Figure 4.16 The Grotrian diagram shows the...
Figure 4.17 The Grotrian diagrams for the...
CHAPTER 05
Figure 5.1 The potential energy for an...
Figure 5.2 The energy of the four states...
Figure 5.3 The energies of the states...
Figure 5.4 Stark effect in hydrogen. The...
CHAPTER 07
Figure 7.1 An incident beam of particles...
Figure 7.2 The differential cross section...
Figure 7.3 The differential cross sections...
Figure 7.4 In the eikonal approximation, the...
Figure 7.5 An attractive square-well...
Figure 7.6 These graphs might represent...
Figure 7.7 The Argand diagram. For...
Figure 7.8 Differential cross section...
Figure 7.9 Potential that can produce...
Figure 7.10 Argand diagram and cross section...
Figure 7.11 The phase shift for a three-dimensional...
CHAPTER 08
Figure 8.1 The Aharonov-Bohm effect results...
Guide
Cover
Title Page
Copyright Page
Table of Contents
Preface
About the Companion Website
Begin Reading
A Dimensions and Units
B Mathematical Tools
C Selected Solutions
Bibliography
Index
End User License Agreement
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Preface
Prof. John David Jackson is known throughout the physics community for his text Classical Electrodynamics, which first appeared in 1962. More than 60 years later, in its third edition, it remains the standard text for graduate students and a frequent reference for researchers. As a faculty member at the University of California beginning in 1967, Jackson taught many other courses, from first-year physics to graduate courses in particle physics. It was this latter course that led me to become his graduate student. From time to time, he taught a year-long graduate course in quantum mechanics. His meticulous notes from these courses in 1978-1979 and 1988-89 were given to me by his daughter, Nan Jackson, after his death in 2016. Nearly the entirety of the text here is taken directly from his notes, supplemented occasionally with explanations he might have added in class, but which do not appear in the notes.
There is no lack of outstanding texts on quantum mechanics, starting with Dirac’s book first published in 1930. Other classics include Schiff’s book, based on the lectures of J. Robert Oppenheimer, and the Landau and Lifshitz book, part of their extraordinary series covering all of theoretical physics. Many other fine texts have appeared in the last 60 years. References to those by J. J. Sakurai, K. Gottfried, E. Merzbacher, and by H. Bethe and R. Jackiw and others appear in these lectures. Still more texts have appeared since Jackson delivered these lectures. Why, then, should we have another text? Relativity and quantum mechanics are the two transcendent achievements of twentieth-century physics. Quantum mechanics has more profoundly affected science and humanity generally. By the late 1920s, quantum mechanics provided a complete understanding of the immediate world around us, the world of atoms and molecules, an understanding beyond the reach of classical physics. The unique significance of quantum mechanics justifies the variety of treatments by exceptional physicists, each providing new insights, just as each new fine recording of the Beethoven string quartets provides truly new pleasures.
The problems in Jackson’s Classical Electrodynamics are legendary. On the occasion of his 60th birthday, a group of Cornell physics graduate students composed a song to a tune from “The Sound of Music” with the line “How do you solve a problem out of Jackson in any less than geologic time?” Included here are many of the problems Jackson assigned in the quantum mechanics courses. In some instances, the problems are so essential to the development that I have included Jackson’s solutions, as well. These are marked with an asterisk. In the other instances I have maintained Jackson’s own policy of never making the solutions public. The students in the two courses were also assigned some problems from texts mentioned above. The diligent reader may also wish to seek additional problems there, though the ones included here will push the student to a firm understanding of the material.
Many students find Classical Electrodynamics particularly challenging because of the demands it places on mathematical preparation. Of course, part of the value of that text is that it provides the means to achieve a higher level of mathematical facility. Jackson’s quantum mechanics courses also made mathematical demands, but rather fewer than in Classical Electrodynamics. Fourier transforms and integration by parts are the most frequent devices. As an aid, I have added a brief appendix giving the rudiments of complex analysis and contour integration. But just as in Classical Electrodynamics, the mathematical tools are applied for the purpose of addressing real and important physical problems. While Jackson’s own research was primarily in particle physics, atomic physics plays a more central role in the text, with multiple references to two classic works: Condon and Shortley, Theory of Atomic Spectra, and Bethe and Salpeter, Quantum Mechanics of One- and Two-electron Atoms.
As the title suggests, this book is a course, not a comprehensive treatment of quantum mechanics. The goal is not so much to present physics, as it is to show how to do physics. For this reason, many derivations are given in greater detail than is conventional. A preference is given to concrete applications of quantum mechanics over formal development. References to complementary treatments are provided. It is assumed that the reader has some familiarity with quantum mechanics at the undergraduate level and with the associated mathematical techniques. Throughout, the intention is to emulate the informality of the actual lectures Jackson delivered to the Berkeley students. One manifestation of that informality is that in some instances ħ and c appear, while in many others they do not. Knowing how to restore ħ and c is just one aspect of the topic. No doubt some typographic errors or worse will have evaded scrutiny in my transcription of Jackson’s notes. For these, I alone am responsible.
I hope this presentation of Jackson’s notes will provide some of the joy and pleasure that he experienced himself and transmitted to his students in seeing the beauty of this subject, which is at once elegant, abstract, and essential to understanding the tangible world around us.
R. Cahn
March 2023
About the Companion Website
This book is accompanied by a companion website:
www.wiley.com/go/Jackson/QuantumMechanics
This website includes solutions to the problems, available to registered instructors
1 Basics
In 1900, Planck found that he could explain the spectrum of blackbody radiation by introducing a new physical constant with the dimensions of angular momentum. The Planck constant is the basis of quantum mechanics, determining when classical concepts no longer apply, specifying the inevitable uncertainties introduced by measurement, and fixing the size of atoms. The gradual development of wave mechanics led by Niels Bohr over the next twenty years was superseded by the extraordinarily rapid progress that followed de Broglie’s proposal that particles were also waves. By 1927, the canon was largely in place and was codified by Dirac’s magisterial text in 1930. Suddenly there was a comprehensive theory of atoms and molecules and their electromagnetic interactions: our material world.
We review some of the essential aspects of quantum mechanics, the wave function and Born’s interpretation of its absolute square as a probability density. The Schrödinger equation prescribes the non-relativistic behavior of the wave function in space and time in terms of a Hamiltonian, which combines the kinetic and potential energies. A free particle already evidences the effects of quantum mechanics by gradually spreading from any initially confined configuration. The behavior of a particle in the field of a constant conservative force is closely related to what would be expected classically, with motion that is bounded or unbounded depending on the potential and the particle’s energy. The Schrödinger equation brought with it the power of the well-developed mathematics of differential equations. A recurrent concept is that of an eigenvalue, a value of a parameter that allows the solution of a differential equation subject to boundary conditions. The Rayleigh-Ritz method provides a means of determining a good approximation to the smallest eigenvalue, generally the lowest energy, by varying parameters in a trial function.
1.1 Wave Mechanics of de Broglie and Schrödinger
Louis duc de Broglie, in his 1924 Ph.D. thesis, proposed that material particles of mass , momentum and energy have associated with them a wave vector and angular frequency through the Planck constant
This leads us to associate with a particle of momentum and energy a wave amplitude
de Broglie’s insight led to the rapid development of quantum mechanics by Werner Heisenberg, Erwin Schrödinger, Paul Dirac, Max Born, and others. We review here the elements of quantum mechanics from the wave perspective. Quantum mechanics, like any physical theory, cannot be derived as if it were a mathematical theorem. Its validity can be judged only by its ability to predict correctly what is observed. Later, we will show how a quantum mechanical prescription can be derived from some reasonable assumptions. But such derivations are not proofs, but rather logical arguments in favor of a theory that must face the test of experiment.
The idea of localized particles (“at , with momentum ”) plus experience with superposition of waves, lead to consideration of a general wave amplitude
in association with a free particle of mass . For a one-dimensional case, as we show more completely below, if is narrowly confined about with a spread , will describe a particle with a spread . We shall return to the uncertainty principle later.
With classical ideas (sound intensity, Poynting vector, etc.) to guide us, we are led fairly directly to Born’s probability interpretation of , i.e. the probability of finding a particle “at ” at time in a volume is proportional to . Thus we speak of as the probability density at.
1.2 Klein-Gordon Equation
With Einstein’s relation we see that
But we can write
Thus, independent of
This is the Klein-Gordon equation, a relativistic wave equation for free spinless particles, e.g. mesons. We note that this equation allows solutions with in place of .
1.3 Non-Relativistic Approximation
If describes a state of motion of a particle, so also does , where is independent of the motion of the particle. For example, we could take or . This is permissible because only is relevant as a probability density.
Consider the non-relativistic limit
so that
Thus, by superposition, in the non-relativistic limit
Because the choice of phase is at our disposal, we define the non-relativistic wave amplitude
Now dropping the subscript NR, we have
so that
and as before
Thus we obtain the general equation describing non-relativistic motion of free particles, the Schrödinger equation,
1.4 Free-Particle Probability Current
If is a density, there should be a probability current and a continuity equation to connect them because probability is conserved.
Consider
and use the Schrödinger equation to get
So if we write
we have a continuity equation,
Note that is arbitrary to the extent of adding a curl of some vector function, since the divergence of a curl is always zero. This is analogous to the arbitrariness of the Poynting vector in electromagnetism.
To find for our general free-particle , consider
Combining this with its complex conjugate, we find
The presence of the factor shows that each component wave of definite momentum contributes a factor of to the flux, as expected. Note that there is a correspondence
The identification by de Broglie of the momentum with , where occurs in a plane wave as , leads through the Fourier transform with the identification of with .
1.5 Expectation Values
If is the probability density in space, then it is natural to define averages of coordinates and functions of coordinates for the particular state of motion as integrals over all space, weighted with . Suppose is a reasonable real function of and and assume has a finite integral over all space. Then the average of (or expectation value of) is
In general, is a function of time through both the explicit time dependence of and the time dependence of . For example,
This all seems plausible, even obvious and necessary. If is big somewhere, the will reflect that. If the place where is big moves in time, then will move in time. This leads to the question: what is the expression for ?
We look at this in two ways: what is and what is in terms of ? First, using the continuity equation, Eq. (1.19)
The last step follows from integration by parts if the current is confined to some finite portion of space:
The use of integration by parts is ubiquitous. This is inevitable because, when confronted with a formal integral expression, there is no way to proceed other than integration by parts.
Returning to the calculation at hand,
Once again invoking integration by parts,
We conclude that
On the other hand, if we think of as the probability density in momentum space, then we would define
where our normalization has been
To check whether this is all consistent, we evaluate
This confirms our picture that
and allows us to identify
Using this identification, we can find the expectation value of a function of as
To give this unambiguous meaning, must be well-behaved, e.g. developable in a power series in .
1.6 Particle in a Static, Conservative Force Field
So far we have just considered the de Broglie relation and Born’s probability interpretation of for free particles. Suppose a particle is moving in a potential . What equation governs the motion? Motivated by classical treatments of the index of refraction, Fermat’s principle, etc. we modify our equations for a free particle:
where we used our identification , with the substitution
In this way, we arrive at the time-dependent Schödinger equation
This familiar equation will occupy us for some time. We note that the picture developed for the free particle requires no significant modification. The probability interpretation of Born is unaffected. The argument for the conservation of probability and the construction of the current survive as long as the potential is real. The expression of expectation values and the representation of momentum as an operator are unchanged.
1.7 Ehrenfest Theorem
Consider the motion of a particle of mass described by , with satisfying
We want to see to what extent we can make a connection between the Schrödinger equation and Newton’s Laws, so let’s calculate
Integration by parts, assuming the wave function vanishes fast enough at infinity, enables us to resolve this. Terms with any combination of and like
are zero as long as vanishes fast enough at infinity. This is just the divergence theorem as is apparent if we introduce a constant vector
where the surface elements are at infinity. Thus
Thus, the term in square brackets in Eq. (1.40) vanishes with integration by parts. The last two pieces combine with an integration by parts to leave us with
which looks a bit like Newton’s equation of motion. A more direct connection would have in place of . If changes slowly enough in and is reasonably localized, then we can show the correspondence with
Consider the force at evaluated by expanding about .
Then
So if doesn’t vary too much over the spread of the wave amplitude, our correspondence holds.
1.8 Schrödinger Equation in Momentum Space
We can define a momentum-space wave amplitude by
What equation does satisfy? We compute
where we integrated by parts to get the first term. Now suppose that could be written in a series expansion
where summation over repeated indices is assumed. Then
We see, then, that in momentum space the operators are . So we can write the momentum-space Schrödinger equation as
1.9 Spread in Time of a Free-Particle Wave Packet
If we represent a particle by a wave packet, quantum mechanics, as represented by the Schrödinger equation, results in the spreading of that wave packet. If, in particular, we start with a normalized Gaussian wave packet,
so that
we find (see Problem 1.6) that at time
More generally, without assuming a Gaussian wave packet, for a given , there is an uncertainty in the corresponding momentum . This means that . There will be built up in time a smearing in of an amount . Thus, the new spreading in the packet is . If the original and new ’s are combined quadratically (appropriate for uncorrelated effects), we get essentially the above result.
1.10 The Nature of Solutions to the Schrödinger Equation
A free particle can have any momentum and its corresponding energy. If the particle is otherwise free, but confined to a rectangular box by infinitely high potential walls, we constrain the function to vanish at the walls. The result is that only certain momenta are allowed. There are states of arbitrarily high energy, but the states can be counted one-by-one, from the lowest energy on up. The spectrum of states is discrete, whereas, for the completely free particle, any positive energy is possible and the spectrum of states is continuous. A simple inference can be drawn by considering classical analogs. If the motion is bounded, as for the particle in the box, the spectrum of states is discrete. If the motion is unbounded, the spectrum is continuous.
These simple observations can be extended to consider more complex potentials. This is illustrated in Figure 1.1.
Figure 1.1 Connection between the classical motion of a particle in different potentials and its corresponding eigenvalue spectra. (a) Classical motion bounded in space, denumerable discrete energy eigenvalues with . (b) For , classical motion bounded in space, so denumerable discrete eigenenergies. For , unbounded classical motion, continuum for . (c) Classical motion is bounded between and , but same has “orbits” that are unbounded. The energy spectrum is continuous. Figures adapted from Gottfried, Quantum Mechanics: Fundamentals, p. 50.
1.11 A Bound-State Problem: Linear Potential
We consider a quite general Schrödinger equation defined by a Hamiltonian operator
and its associated differential equation
A solution that is physically reasonable in the sense described below determines an eigenvalue, . For any eigenstate , the expectation value of any operator not containing time dependence explicitly is independent of time. We therefore speak of a stationary state. There is no dispersion in energy since .
How should the problem of solving the time-independent Schrödinger equation be posed? We want to be “reasonable.” One criterion might be because is a probability density. This is often true, but it is too restrictive. For example, is allowed in some circumstances. However, is not allowed because it says the particle is mostly at infinity.
Some experience can be found in determining numerically solutions to the Schrödinger equation for a linear potential: for . This is a potential we shall encounter again.
With the substitutions
we have the dimensionless form
There are two classes of solutions: ones that are even under () and ones that are odd under this symmetry. The ones that are odd must have . To solve the differential equation numerically, one must in this case take some value for and then integrate in steps of . If the solution is even, then . In either case, there is just one parameter to set at the origin and this is equivalent to setting the scale for . Thus, the only open issue is the value of : For what values of do we get physically acceptable solutions? In particular, the solution must vanish in the limit .
It is convenient to think of the second-order equation as a pair of first-order equations. Without loss of generality, one can take , which is to say that is measured in units of .
Whatever the assumed value of , upon starting the integration at , at some point the integration reaches the value , where the motion is forbidden classically. In general, beyond this point the solution will be the sum of two pieces, one of which dies exponentially in and the other of which grows exponentially in . This latter is unacceptable physically. Only for certain values of will the solution be purely exponentially dying. These values are the eigenvalues. By varying and seeing whether the solution diverges to or to , one can narrow down the range for until a good approximation for the eigenvalue is obtained. See Figure 1.2.
Figure 1.2 Numerical integration of the linear potential, Eq. (1.59), shows that the first eigenvalue lies between 2.338 and 2.339 if the function is odd in . The equation can also be solved analytically in terms of the Airy function , which will be encountered in a later chapter.
1.12 Sturm-Liouville Eigenvalue Problem
The time-independent Schrödinger equation is ubiquitous. It is therefore worthwhile to consider more generally the properties of the class of differential equations that arise from it.
Sturm-Liouville theory considers the ordinary differential equation in one dimension for :
for with real and positive on this interval and a parameter. We impose boundary conditions at and at , with independent of . These are homogeneous boundary conditions in that if is a solution, so is any constant times (See Figure 1.3.). The Schrödinger equations we will encounter will often have and as special examples of the more general theory.
Figure 1.3 Examples showing the homogeneous boundary conditions at and : at , and at , The chosen values have and . With each increasing eigenvalue, the number of nodes of the eigenfunction increases by one.
One can prove the following (see, e.g. Margenau and Murphy, The Mathematics of Physics and Chemistry, Vol.1, 2nd edition, pp. 270 ff., Morse and Feshbach, Methods of Theoretical Physics, pp. 736–739 or pp. 720–24, Courant and Hilbert, Methods of Mathematical Physics, Chap. V):
If and are positive on the interval, solutions exist only for a countably infinite discrete set of eigenvalues with a smallest non-negative member . The eigenvalues have no upper bound as . The solutions are called eigenfunctions with eigenvalues . Eigenfunctions with increasing energy have an increasing number of nodes. We shall see this is a recurring feature of relevant solutions to the Schrödinger equation.
Moreover, if we order the solutions by , then the number of nodes (zeros) of in the interval is . Writing the equation twice and multiplying by and we find
Now subtracting
Integration by parts gives
The vanishing of the term at the end points is a consequence of the boundary conditions at and at , It follows that the eigenfunctions for different eigenvalues are orthogonal on the interval with the weight . If instead , we see that , that is, the eigenvalues are real. It can be proved that the eigenfunctions in fact are complete: any “reasonable” function satisfying the same boundary conditions can be expanded as a linear combination of the eigenfunctions. For a proof, see, for example, Morse and Feshbach, Methods of Theoretical Physics, pp. 738–9.
1.13 Linear Operators on Functions
The Sturm-Liouville problem is an example of a more general eigenvalue problem with linear operators, i.e. an operator on functions such that for constants and
Examples:
Another particular example is the integral operator
Suppose we define a scalar product of two functions by
Given a linear operator , we define its adjoint by
In the case of our integral operator we see that we need to take
This is analogous to the usual meaning of adjoint for matrices: . An operator is self-adjoint or Hermitian if . In this case
that is, the expectation value of a Hermitian operator is real. Conversely, if we want an operator to represent a physical quantity, it must be Hermitian. For an anti-Hermitian operator we have . If is Hermitian, then is anti-Hermitian.
1.14 Eigenvalue Problem for a Hermitian Operator
We suppose is a Hermitian operator on functions on an interval with some boundary conditions. If we indicate a function by , then the eigenvalue problem is
We know the are real because
and both the numerator and denominator are real. If is Hermitian then
Since the eigenvalues are real
Thus if the eigenvalues are different, then the eigenfunctions are orthogonal. In general, it is possible that there are linearly independent eigenfunctions with the same eigenvalue, in which case orthogonality doesn’t follow. This won’t happen in the specific instance at hand, a one-dimensional differential equation with boundary conditions.
We now show how this formalism can incorporate the Sturm-Liouville problem. We take
and ask under what conditions is Hermitian.
We want the first term to vanish and the second term to be equal to . The latter is achieved if are all real. For the former, it suffices for at and , or alternatively, for
These are precisely the conditions we postulated in the treatment above of the Sturm-Liouville problem.
1.15 Variational Methods for Energy Eigenvalues
A powerful method of generating approximate energies of quantum mechanical systems is the so-called variational technique – Rayleigh-Ritz method is another term. Rayleigh used it in acoustics. We phrase it terms of the Schrödinger equation and energy eigenvalues, but it has very general applicability.
Consider a well-posed energy eigenvalue problem:
with a spectrum of eigenvalues , with corresponding eigenfunctions , but suppose we do not know the eigenvalues or eigenfunctions. Suppose that we are somehow given a function that is not too far from the correct wave function . We can form the quantity
where might be one-dimensional or three-dimensional and is some trial function, obeying the boundary conditions. What can we say about as an estimate of ? Two things:
If differs from the true solution by a small amount, characterized by , so then .
This means that is stationary with respect to small variations away from the true eigenfunction, .
To prove this, expand in the basis of true eigenfunctions
where we assume orthonormality of the eigenfunctions
and that is normalized:
Then immediately
Now consider a trial wave function that is near the true one
where and are normalized: . It suffices to consider only a that is orthogonal to since any part of that is not can be absorbed into , after which we can renormalize so that the coefficient of is unity. Now expand the expression for , but impose the normalization by dividing by .
Then to order we have
so that
where the inequality follows from the observation that if it were not true then would have lower energy than the ground state. If we can make a good guess at with , then we will get a good estimate of . This approach is systematized in the Rayleigh-Ritz method.
1.16 Rayleigh-Ritz Method
We construct a “trial” eigenfunction , with a set of variational parameters. Then we form the functional
of the parameters . We know that , with equality only for . We therefore vary the parameters to find a minimum, where we must have
The solution to these simultaneous equations yields the optimal value for possible within the limitations of the form of our trial function.
As a very simple example, let us seek the energy of the ground state in a one-dimensional box, and take Then
Since
we find
With the standard integral
where the gamma function
has the properties
we find
The requirement gives
and
while the exact result is , a error. The moral is that even simple choices for the trial wave function can give remarkably good results for the ground state energy.
References and Suggested Reading
There are numerous accounts of the history of quantum mechanics. Some particularly notable ones written by physicists are:
Gamow, G.
Thirty Years that Shook Physics
. New York: Doubleday.
Pais, A.
Inward Bound: Of Matter and Forces in the Physical World
. Oxford University Press.
Segrè, G.
Faust in Copenhagen: A Struggle for the Soul of Physics, Viking
. New York.
Every student of quantum mechanics should thrill to the remarkable story told in these books.
For classic presentations of mathematical physics:
Courant and Hilbert,
Methods of Mathematical Physics
.
Margenau and Murphy,
The Mathematics of Physics and Chemistry
, Vol.1, 2
nd
edition.
Morse and Feshbach,
Methods of Theoretical Physics
.
Problems
1.1 A one-dimensional wave amplitude is given by
with
Find and sketch and . What is the significance of and ?
Find ,, , and .
Show from the results of (b) that the r.m.s. deviations and satisfy , independent of .
1.2 A particle is in the ground state of a one-dimensional box of length . with wave amplitude given by
At the walls are suddenly dissolved and the particle subsequently moves freely. What is the probability that, after , the particle’s momentum is between and ? Sketch fairly accurately versus .
1.3 An anti-proton can be bound to a proton by electromagnetic interaction to form a hydrogenic atomic system. At short distances ( cm) the interaction is augmented by strong forces, including annihilation.
Neglecting annihilation and other strong-interaction effects, scale the system appropriately from the atom and compute the Bohr radius, and the spacing between the and levels, in keV.
From the known transition probability in hydrogen () and the scaling properties of the Larmor (electric dipole) radiation formula, find the corresponding transition probability for the atom.
1.4 A particle with mass and charge interacts with a real potential and also an external electromagnetic field described by the real potentials and . The Schrödinger equation is
Show that the normal expression for the probability current is augmented by the term
Interpret.
1.5*
Show that a free-particle wave amplitude evolves in time from the amplitude according to
where the kernel (propagation function) is given by the Fourier integral
As a preliminary to evaluating this integral, prove that
Show that
1.6 Suppose that at , a one-dimensional wave amplitude is
Using the one-dimensional analog of the solution to Problem 1.5, show that the amplitude at a later time has the form
where
is the r.m.s. spread in .
1.7
By explicit construction with the Schrödinger equation, show that for any wave amplitude
where .
Similarly, show that
This is the basis for the quantum mechanical virial theorem (for energy eigenstates only).
1.8 The power of the variational method can be illustrated by estimating the ground state energies of the hydrogen atom and the isotropic harmonic oscillator, using trial wave functions that are appropriate to the corresponding potential. For the ground state, the angular momentum is zero and the wave function depends only on . We can write the Schrödinger equation as
It is conventional to introduce a radial wave function so that the equation is
and must vanish at .
Use a variational principle for the radial wave-function in an s-wave state.
For the hydrogen atom, take a trial wave function of the form
where is the parameter to be varied. Find the best value of and the corresponding binding energy. How accurate is the estimate?
For the three-dimensional isotropic oscillator, take a trial wave function
where is the variational parameter. Find the optimal value of and the corresponding energy. How accurate is the estimate?
1.9 A hydrogen atom is confined in a spherical box of radius , with the proton at the center. In atomic units, the radial equation for s-states is
where and . Make a variational estimate of the ground state energy as a function of using the trial wave function
where is a variational parameter. Find the value of R when the atom has zero binding energy. Find the values of for and , and compare with known results. For positive binding energies, plot the energy as a function of .
Hint: the variational energy takes the form, . The stationary point is specified by a relation of the form . Rather than solving for , choose a series of values of , determine each corresponding and .
2 Reformulation
In developing quantum mechanics, we can continue relying on wave functions and operators acting on them, but there are conceptual advantages to working in a more abstract fashion. The basic tool is linear vector spaces. The general principle of quantum mechanics is that a physical system can be represented by linear combinations of basis states, which can be viewed as elements of a linear vector space. A given state of the system is represented by a ray in the vector space. By a ray, we mean a quantity defined up to an overall constant phase, which is inessential because it always drops out in computing a physical quantity.
Quantum mechanics requires complex numbers and thus we consider complex linear vector spaces. Observable quantities are represented by linear operators on the vector space with the special property of being Hermitian (or self-adjoint), which guarantees that they produce real, not complex, results. The rules for calculating the probabilities of physical processes in quantum mechanics are clear. Interference phenomena are characteristic of quantum mechanics and are quite different from those of classical mechanics, though they are familiar from electrodynamics. Observables whose operators commute can be measured simultaneously, or more precisely can be measured in either order with identical results. Observables all of which commute are termed compatible. A complete set of compatible observables is one that does not admit an additional independent operator. A complete set of observables can label a set of linearly independent vectors that is a complete basis for the states.
Classical mechanics provides a basis for postulating quantum mechanical relations. Whether the resulting theory is correct is a matter to be tested experimentally. The results over nearly one hundred years have been entirely positive. Symmetries of the Lagrangian in classical mechanics lead to conservation laws in accordance with Noether’s theorem. The same is true in quantum mechanics, where symmetries play an even greater role, as they are represented in linear vector spaces. Dirac developed a particularly powerful representation of eigenstates of position and momentum , which facilitates many analyses. The Schrödinger and Heisenberg pictures provide complementary views of quantum mechanics. In the Schrödinger picture, the states evolve, while in the Heisenberg picture the states are fixed but the operators evolve in time. Often a physical circumstance involves not a single state, but a superposition of states; then analysis can be facilitated with a density matrix describing the mixture of states, either initial or final.
2.1 Stern-Gerlach Experiment
The paradigm for measurement in quantum mechanics was established in 1922 when the “old quantum theory” still prevailed. In an experiment proposed by Otto Stern and executed by Walther Gerlach, a beam of neutral silver atoms was passed through a region of intense and inhomogeneous magnetic field, with the field gradient transverse to the line of flight. The force in such a region, classically, is given by , where is the magnetic moment of the atom. The electronic configuration of the silver atom has a single s-wave electron outside a closed shell, so its magnetic moment is precisely that of the electron. Classically, if the magnetic moments of the atoms were randomly oriented, a band would be observed in the sensors stretching between the locations where atoms would end up if their moments were aligned with the field gradient to the point where they would end up if anti-aligned. What Stern and Gerlach observed in 1922 was two distinct, separated spots from their beam of silver.
The theory of linear vector spaces provides a particularly convenient description of such phenomena, and indeed all quantum mechanical phenomena.
2.2 Linear Vector Spaces
We begin by considering N-dimensional complex spaces whose members we can represent by an N-tuple of complex numbers . Clearly multiplying by a complex number gives another such element. Similarly, the sum of two elements is itself an element.
We define an inner product of two elements and by
which is a complex number. The inner product has the properties
We see that is real and greater than zero unless . We define the norm of by
If we have a set of N-tuples , we say that the are linearly independent if the relation (here means the N-tuple ) for complex requires that for every . It is clear that if we take etc., these are indeed linearly independent, but not every set of N-tuples will be linearly independent. A basis set for the linear space is a collection of linearly independent vectors. Any vector can be expressed as a linear combination of the basis vectors.
Our reason for focusing on linear vector spaces is that they enable us to represent a fundamental aspect of quantum mechanics: particles and systems of particles show interference between distinct components that exist simultaneously and thus must be represented as linear combinations of basis states. We will adopt the notation introduced by Dirac, representing a vector by the “ket” . The reason for calling this a ket will become apparent shortly. Following the example of N-tuples, we assume that we can add two vectors and multiply them by complex numbers and the result is also a vector. Moreover, there is a zero vector with the property
for every and . In our explicit example, of course, is the N-tuple .
We proceed by analogy with the N-tuple example. We suppose we have basis states , the analogs of above. We continue the analogy by defining a scalar product of two vectors:
In the Dirac notation, we take this expression apart. Then we define the “bra” states as operators that act on “kets” to give complex numbers. Explicitly, we set
If we take all linear combinations of the , we get a linear vector space called the dual space. If we have a vector
we define its dual by
This generates the required scalar product. Note that the meaning of is derived from the definition of the scalar product. If the basis states are to represent a physical situation, then the choice of basis states is not completely arbitrary as we shall see below.
The abstract vectors behave much like ordinary vectors in three-dimensional space. An example is the Cauchy-Schwarz inequality. Since for any , consider
and in particular take . Then
The Cauchy-Schwarz relation allows us to think in geometrical terms. If then we can think of as the cosine of the “angle” between and . If , then and are orthogonal.
Any vector can be expanded in our basis
But then
so that
Similarly,
Evidently, the sum of bras and kets (in the opposite to the usual order) can be identified as the unit operator or , i.e.
This is called the completeness relation and is dual to the orthogonality relation
Suppose there are two orthonormal sets of basis vectors, and . We can expand one set in terms of the other:
The and are N N matrices. Substituting one relation into the other, we have
The sum over is just matrix multiplication:
so that
or as matrices
and similarly . Now if we compute
we find that in matrix notation, if we define
then
Thus is a unitary matrix – its adjoint is its inverse – as is . We see that a change of basis is accomplished by a unitary transformation.
2.3 Linear Operators
Consider a relation that assigns to every vector , another vector in the same linear space, and call it . This relation is a linear operator if it obeys the rules
We say that two operators and are equal if for every state . We define a null operator by for every and similarly an identity operator by . We define the sum of linear operators in the obvious fashion: . The commutation and associative properties follow:
It is clear that multiplication of operators is associative , but there is no guarantee that it is commutative, .
Just as there are operators that act on the space generated by the basis vectors , there are linear operators that act on the dual space, generated by the . In fact, if we have an operator acting on the original space, this will create an associated operator acting on the dual space defined by the rule that if
then we define by
where is the dual of and is the dual of . This abstract definition can be made concrete by considering
where the are complex numbers, the matrix representation of , and
where similarly the are the representation of . But if is the dual of , then
Thus, our abstract definition is equivalent to
We can write, quite generally,
which we might equally as well have taken as the definition of . From the rules of matrix multiplication it is apparent that .
If an operator is equal to its adjoint , the operator is said to be self-adjoint or Hermitian, just as in the previous chapter, where we considered linear operators, , on a space of functions. Of course, not every operator is Hermitian. If is Hermitian, is anti-Hermitian:
