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Names: Englert, Berthold-Georg, 1953– author.
Title: Lectures on quantum mechanics / Berthold-Georg Englert.
Description: Second edition, corrected and enlarged. | Hackensack : World Scientific Publishing
Co. Pte. Ltd., 2024. | Includes bibliographical references and index. |
Contents: Basic matters -- Simple systems -- Perturbed evolution.
Identifiers: LCCN 2023040959 (print) | LCCN 2023040960 (ebook) |
ISBN 9789811284724 (v. 1 ; hardcover) | ISBN 9789811284984 (v. 1 ; paperback) |
ISBN 9789811284755 (v. 2 ; hardcover) | ISBN 9789811284991 (v. 2 ; paperback) |
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Subjects: LCSH: Quantum theory. | Physics.
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Printed in Singapore
To my teachers, colleagues, and students
This page intentionally left blank
 Preface

This book on the Perturbed Evolution of quantum systems grew out of a

set of lecture notes for a fourth-year undergraduate course at the National
University of Singapore (NUS). The reader is expected to be familiar with
the subject matter of a solid introduction to quantum mechanics, such as
Dirac’s formalism of kets and bras, Schrödinger’s and Heisenberg’s equa-
tions of motion, and the standard examples that can be treated exactly,
with harmonic oscillators and hydrogen-like atoms among them.
After brief reviews of quantum kinematics and dynamics, including dis-
cussions of Bohr’s principle of complementarity and Schwinger’s quantum
action principle, the attention turns to the elements of time-dependent
perturbation theory and then to the scattering by localized interactions.
Fermi’s golden rule, the Born series, and the Lippmann–Schwinger equa-
tion are returning themes.
A chapter on general angular momentum prepares the ground for a dis-
cussion of indistinguishable particles. The scattering of two particles of the
same kind, the basic properties of two-electron atoms, and a glimpse at
many-electron atoms illustrate the matter. Throughout the text, the learn-
ing student will benefit from the dozens of exercises on the way and the
detailed exposition that does not skip intermediate steps.
Two companion books on Basic Matters and Simple Systems cover the
material of the preceding courses at NUS for second- and third-year stu-
dents, respectively. The three books are, however, not strictly sequential
but rather independent of each other and largely self-contained. In fact,
there is quite some overlap and a considerable amount of repeated mate-
rial. While the repetitions send a useful message to the self-studying reader
about what is more important and what is less, one could do without them
and teach most of Basic Matters, Simple Systems, and Perturbed Evolution
in a coherent two-semester course on quantum mechanics.

vii
viii Lectures on Quantum Mechanics: Perturbed Evolution

All three books owe their existence to the outstanding teachers, col-
leagues, and students from whom I learned so much. I dedicate these lec-
tures to them.
I am grateful for the encouragement of Professors Choo Hiap Oh and
Kok Khoo Phua who initiated this project. The professional help by the
staff of World Scientific Publishing Co. was crucial for the completion; I
acknowledge the invaluable support of Miss Ying Oi Chiew and Miss Lai
Fun Kwong with particular gratitude. But nothing would have come about,
were it not for the initiative and devotion of Miss Jia Li Goh who turned
the original handwritten notes into electronic files that I could then edit.
I wish to thank my dear wife Ola for her continuing understanding and
patience by which she is giving me the peace of mind that is the source of
all achievements.
Singapore, March 2006 BG Englert

Note on the second edition

The feedback received from students and colleagues, together with my own
critical take on the three companion books on quantum mechanics, sug-
gested rather strongly that the books would benefit from a revision. This
task has now been completed.
Many readers have contributed entries to the list of errata. I wish to
thank all contributors sincerely and extend special thanks to Miss Hong
Zhenxi and Professor Lim Hock.
In addition to correcting the errors, I tied up some loose ends and
brought the three books in line with the later volumes in the “Lectures
on . . . ” series. There is now a glossary, and the exercises, which were in-
terspersed throughout the text, are collected after the main chapters and
supplemented by hints.
The team led by Miss Nur Syarfeena Binte Mohd Fauzi at World Sci-
entific Publishing Co. contributed greatly to getting the three books into
shape. I thank them very much for their efforts.
Beijing and Singapore, November 2023 BG Englert
 Contents

Preface vii

Glossary xiii
Miscellanea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
Latin alphabet . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv
Greek alphabet and Greek-Latin combinations . . . . . . . . . . xvi

1. Basics of Kinematics and Dynamics 1

1.1 Brief review of basic kinematics . . . . . . . . . . . . . . . 1
1.2 Bohr’s principle of complementarity . . . . . . . . . . . . 7
1.2.1 Complementary observables . . . . . . . . . . . . . 7
1.2.2 Algebraic completeness . . . . . . . . . . . . . . . . 12
1.2.3 Bohr’s principle. Technical formulation . . . . . . . 14
1.2.4 Composite degrees of freedom . . . . . . . . . . . . 15
1.2.5 The limit N → ∞. Symmetric case . . . . . . . . . 17
1.2.6 The limit N → ∞. Asymmetric case . . . . . . . . 22
1.2.7 Bohr’s principle. Quantum indeterminism . . . . . 26
1.3 Brief review of basic dynamics . . . . . . . . . . . . . . . . 27
1.3.1 Equations of motion . . . . . . . . . . . . . . . . . 27
1.3.2 Time transformation functions . . . . . . . . . . . . 29
1.4 Schwinger’s quantum action principle . . . . . . . . . . . . 32
1.4.1 An example: Constant force . . . . . . . . . . . . . 35
1.4.2 Insertion: Varying an exponential function . . . . . 36
1.4.3 Time-independent Hamilton operator . . . . . . . . 38

2. Time-Dependent Perturbations 41
2.1 Born series . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.2 Scattering operator . . . . . . . . . . . . . . . . . . . . . . 43

ix
x Lectures on Quantum Mechanics: Perturbed Evolution

2.3 Dyson series . . . . . . . . . . . . . . . . . . . . . . . . . . 46

2.4 Fermi’s golden rule . . . . . . . . . . . . . . . . . . . . . . 47
2.5 Photon emission by a “two-level atom” . . . . . . . . . . . 52
2.5.1 Golden-rule treatment . . . . . . . . . . . . . . . . 52
2.5.2 A more detailed treatment . . . . . . . . . . . . . . 54
2.5.3 An exact treatment . . . . . . . . . . . . . . . . . . 60
2.6 Driven two-level atom . . . . . . . . . . . . . . . . . . . . 62
2.6.1 Schrödinger equation . . . . . . . . . . . . . . . . . 62
2.6.2 Resonant drive . . . . . . . . . . . . . . . . . . . . 65
2.6.3 Periodic drive . . . . . . . . . . . . . . . . . . . . . 66
2.6.4 Very slow drive: Adiabatic evolution . . . . . . . . 68
2.7 Adiabatic population transfer . . . . . . . . . . . . . . . . 71
2.8 Equation of motion for the unitary evolution operator . . 74

3. Scattering 79
3.1 Probability density, probability current density . . . . . . 79
3.2 One-dimensional prelude: Forces scatter . . . . . . . . . . 82
3.3 Scattering by a localized potential . . . . . . . . . . . . . 86
3.3.1 Golden-rule approximation . . . . . . . . . . . . . . 86
3.3.2 Example: Yukawa potential . . . . . . . . . . . . . 90
3.3.3 Rutherford cross section as a limit . . . . . . . . . 91
3.4 Lippmann–Schwinger equation . . . . . . . . . . . . . . . 92
3.4.1 Born approximation . . . . . . . . . . . . . . . . . . 100
3.4.2 Transition operator . . . . . . . . . . . . . . . . . . 100
3.4.3 Optical theorem . . . . . . . . . . . . . . . . . . . . 102
3.4.4 Example of an exact solution . . . . . . . . . . . . 104
3.5 Partial waves . . . . . . . . . . . . . . . . . . . . . . . . . 105
3.6 s-wave scattering . . . . . . . . . . . . . . . . . . . . . . . 110

4. Angular Momentum 115

4.1 Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
4.2 Addition of two angular momenta . . . . . . . . . . . . . . 119
4.2.1 General case . . . . . . . . . . . . . . . . . . . . . . 119
4.2.2 Two spin- 12 systems . . . . . . . . . . . . . . . . . . 122
4.2.3 Total angular momentum of an electron . . . . . . 123

5. External Magnetic Field 125

5.1 Electric charge in a magnetic field . . . . . . . . . . . . . . 125
5.2 Electron in a homogeneous magnetic field . . . . . . . . . 132
 Contents xi

6. Indistinguishable Particles 139

6.1 Indistinguishability . . . . . . . . . . . . . . . . . . . . . . 139
6.2 Bosons and fermions . . . . . . . . . . . . . . . . . . . . . 141
6.3 Scattering of two indistinguishable particles . . . . . . . . 144
6.4 Two-electron atoms . . . . . . . . . . . . . . . . . . . . . . 148
6.4.1 Variational estimate for the ground state . . . . . . 148
6.4.2 Perturbative estimate for the first excited states . . 155
6.4.3 Self-consistent single-electron wave functions . . . . 156
6.5 A glimpse at many-electron atoms . . . . . . . . . . . . . 159

Exercises with Hints 165

Exercises for Chapters 1–6 . . . . . . . . . . . . . . . . . . . . . 165
Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

Index 187
This page intentionally left blank
 Glossary

Here is a list of the symbols used in the text; the numbers in square brackets
indicate the pages of first occurrence or of other significance.

Miscellanea
0 null symbol: number 0, or null column, or null matrix,
or null ket, or null bra, or null operator, et cetera
1 unit symbol: number 1, or unit matrix, or identity
operator, et cetera
A= bB read “A represents B” or “A is represented by B”
Max{ } , Min{ } maximum, minimum of a set of real numbers
a∗ , a complex conjugate of a, absolute value of a
Re(a) , Im(a) real, imaginary part of a
a= a length of vector a
a · b, a × b scalar, vector product of vectors a and b
†
A adjoint of A [3]
det(A), tr(A) determinant [144], trace of A [14]
| i, h |; |1i, ha| generic ket, bra; labeled ket, bra [1]
|initi, hfin| initial ket, final bra [63]
|ini, |outi kets for incoming, outgoing particles [92]
|j1 , j2 ; j, mi, |(j1 , m1 )(j2 , m2 )i kets for composite angular momentum [120]
h | i, | ih | bra-ket, ket-bra [2,3]
| . . . , ti, h. . . , t| ket, bra at time t [27]
h. . . , t1 |. . . , t2 i time transformation function [29]
hAi mean value, expectation value of A
[A, B] commutator of A and B [19]
 
A(t)A(t0 ) > time-ordered product [46]
↑, ↓ spin-up, spin-down [122]
x! factorial of x [152]
f 2 (x), f −1 (x)
x 7→ f (x):

square, inverse
of the function
f 2 (x) = f f (x) , f f −1 (x) = x, f −1 f (x) = x

xiii
xiv Lectures on Quantum Mechanics: Perturbed Evolution

f (x)2 , f (x)−1 square, reciprocal

2 of the function value:
f (x)2 = f (x) , f (x)−1 = 1/f (x)
f (A; B) ordered function of operators A, B [13]
dt, δt differential, variation of t
d ∂
, total, parametric time derivative
dt ∂t
∇ gradient vector differential operator
(dr ), dS ; dΩ volume element for r , vectorial surface element [79];
solid-angle element [86]
⊗ tensor product: |ai ⊗ |bi = |a, bi

Latin alphabet
a range of the hard-sphere potential [110]
a0 Bohr radius, a0 = 0.529 Å [151]
a(t), b(t) probability amplitudes [70]
a(s) Laplace transform of α(t) [60]
A, aj generic operator, its jth eigenvalue [1]
A(t) collection of dynamical variables [27]
A± , A†± harmonic-oscillator ladder operators [129]
A(r ), B(r ) vector potential, magnetic field at r [125]
Å angstrom unit, 1Å = 10−10 m = 0.1 nm [151]
c speed of light, c = 2.99792 × 108 cm s−1 [125]
cos, sin, . . . trigonometric functions
cosh, sinh, . . . hyperbolic functions
e elementary charge, e = 4.80320 × 10−10 Fr [91]
e; ex = exp(x) Euler’s number, e = 2.71828 . . . ; exponential function
E, En , E energy, nth eigenenergy [48], Lagrange parameter [157]
f (uk , vl
) normalized mixed matrix element [12]
f k 0, k scattering amplitude [98]
F, F (x) force [35,85]
G, G1 , G2 generators [32]
G± (r , r 0 ), G Green’s functions [94], Green’s operator [101]
h = 2π~ Planck’s constant,
~ = 1.05457 × 10−34 J s = 0.658212 eV fs [19]
H, Ht , H Hamilton operator [27], at time t [29], matrix for H [68]
H0 , H1 ; H1 dominant, small part of H [41]; interaction picture [45]
Hatom , Hphot , atom, photon part of H [53]
Hint ; Hrot interaction part of H [53]; H for rotation [115]
 Glossary xv

H⊥ , H k perpendicular, parallel part of H [127]

H(1, 2) two-particle Hamilton operator [139]
HCM , Hrel center-of-mass, relative-motion parts of H [140]
Hkin , HNe , Hee parts of H for many-electron atoms [149]
i imaginary unit, i2 = −1
jl ( ) lth spherical Bessel function [107]
j (r , t), jscat (r ) probability current density [79], scattered current [97]
J , J± total angular momentum vector, ladder operators [117]
k ; k, k(x) wave vector [86]; wave number, with x dependence [83]
L, L angular momentum operator (one axis) [24], orbital
angular momentum vector operator [105]
l, m; j, m angular momentum quantum numbers [106,117]
mod modulo (modular arithmetic) [23]
M mass [30]
N
period of the basic unitary operators [7]
O  terms of oder  or smaller [96]
prob(e) probability for event e [2]
P momentum operator for the x direction [19]
p, P, Pj momentum vector, operator, for the jth particle [80]
Pl ( ) lth Legendre polynomial [107]
P principal value [58]
q change in the wave vector [89]; charge [125]
rj jth eigenvalue of the statistical operator [26]
r , R, Rj position vector, operator, for the jth particle [79]
Ry Rydberg unit of energy, 1 Ry = 13.6 eV [150]
s length of difference vector [94], spin value [142]
S(T ), Sn (T ) scattering operator [44], its nth approximation [46]
S; S scattering matrix [86]; spin vector operator [115]
s, p, d, f spectroscopic state labels [123]
t; t1 , t2 time [27]; final, initial time [32]
T duration [30]; transition operator [100]
U, V ; uk , vk basic unitary operators; their kth eigenvalues [8]
U (T ), U0 (T )
unitary evolution operators [44]
U A(t); t0 , t unitary evolution operator [74]
V (r ), V (x), V0 potential energy [80,82,110]
VLS strength of spin-orbit coupling [137]
V velocity vector operator [125]
W12 action operator [33]
x, y, z cartesian coordinates, components of r
xvi Lectures on Quantum Mechanics: Perturbed Evolution

X position operator for the x direction [19]

Ylm (ϑ, ϕ) spherical harmonics [106]
Z0 , Z ± components of S [134]
Z, Zeff atomic number [91], effective value [151]

Greek alphabet and Greek-Latin combinations

α(t), βν (t) time dependent probability amplitudes [55]
γ, γm ← n transition rate [49], from the nth to the mth state [51]
Γ complex decay rate [56]
δjk , δ(x, y) Kronecker’s delta symbol [3,13]
δ, δa, δa variation [29], of variable a, with respect to a
δl lth scattering phase [108]
δ(x − x0 ) Dirac’s delta function [20]
∆; ∆p detuning [66]; finite difference (of variable p) [21]
 small increment
θ scattering angle [90]
κ wave number (integration variable) [94]
λ formal expansion parameter [41]
µB ; µ Bohr magneton; magnetic dipole moment vector [132]
Aν , A†ν ; a0ν , a∗ν ladder operators for photons of the νth kind [52]; their
amplitudes [63]
π Archimedes’s constant, π = 3.14159 . . .
ρ, ρ(r , t) statistical operator [26], probability density [79]
ρ(E) density of states [52]
σ, σ † atomic ladder operators [52]
σ; σx , σy , σz ; Pauli vector operator; cartesian components [115]
dσ
,σ differential cross section [88], total cross section [104]
dω
τ ; t(τ ), X(τ ) path parameter; path variables [33]
φ, ϕ azimuth [23], azimuthal shift [24]
φ(k, x), φ± (k, x) wave-number amplitudes in ψ(x, t) [83,84]
ϕ, ϑ angular parameters [68], azimuth, polar angle [106]
ψ(r , t) position wave function at time t [79]
ψ±,scat (r ) scattered wave function [97]
ω, ∆ω transition frequency [49], frequency shift [58]
ω effective transition frequency [68]
f (ω) state-density averaged squared Rabi frequencies [54]
ω, ωcycl angular velocity vector [115], cyclotron frequency [128]
Ων , Ω0 , Ω νth [53], reduced [65], modified Rabi frequency [67]
Ω1 (t), Ω2 (t) time-dependent Rabi frequencies [71]
 Chapter 1

Basics of Kinematics and Dynamics

1.1 Brief review of basic kinematics

In quantum mechanics, the physical quantities are symbolized by linear

operators A, B, . . . that act on vectors — elements of a vector space, that
is, not physical vectors in the three-dimensional space of our experience. We
often speak of observables when referring to these linear operators, which is
a sloppy use of terminology because, more precisely, the operators are the
mathematical symbols that represent the physical “observable” properties
or simply “observables.” The vectors they act on come in two kinds: ket
vectors |. . .i and bra vectors h. . .| (or right vectors and left vectors). The
mathematical operation of hermitian conjugation, or as the physicists say,
“taking the adjoint,” relates them to each other,

† †
... = ... , ... = ... , (1.1.1)

where it is understood that the ellipses indicate identical sets of quantum

numbers, which serve as the labels that identify the kets and bras. We
write | i and h | for the generic, unlabeled ket and bra.
A measurement of an observable A yields one of the possible measure-
ment results a1 , a2 , a3 , . . ., which are complex numbers in general. If it is
known that a measurement of A will surely return the value aj , then we
say that the quantum mechanical system is in the state |aj i,

aj : A aj = aj aj , (1.1.2)

which — mathematically speaking — is an eigenvector equation, here: an

eigenket equation. Here, we choose to label the eigenkets by their eigen-
values; this is not necessary and not always convenient and other labeling

1
2 Basic Kinematics and Dynamics

conventions are possible. There is also the corresponding eigenbra equation,

aj A = aj aj . (1.1.3)
The measurement result aj is the eigenvalue of A in both the eigenket
equation (1.1.2) and the eigenbra equation (1.1.3).
Under these circumstances, namely the system is in the state described
by the ket |aj i, the probability of finding the value bk upon measuring
observable B is
2
prob(bk ← aj ) = bk aj . (1.1.4)

The complex number hbk |aj i is the probability amplitude for the measure-
ment result bk in state |aj i; its absolute square is the associated probability.
This amplitude has all properties that are required of an inner product, in
particular
a = a0 + a00 : b a = b a0 + b a00 ,
a = α λ: b a = b α λ, (1.1.5)

where λ is any complex number and

∗
ab = ba ,
a a ≥ 0 with “=” only if a = 0 . (1.1.6)
In mathematical terms, these properties characterize the kets as elements
of an inner-product space or Hilbert∗ space. There is a Hilbert space for
the bras as well, related to that of the kets by hermitian conjugation.
The mathematical property ha|bi = hb|ai∗ has the very important phys-
ical implication that the two probabilities prob(b ← a) and prob(a ← b) are
equal,

prob(bk ← aj ) = prob(aj ← bk ) . (1.1.7)

The probabilities for these related, yet different, physical processes,
on the left: the probability of finding bk if aj is the case,
on the right: the probability of finding aj if bk is the case,

are therefore always equal. There is, of course, a lot of circumstantial ev-
idence for the validity of this fundamental symmetry, but — elementary
situations aside — there does not seem to be a systematic direct experi-
mental test.
∗ David Hilbert (1862–1943)
 Brief review of basic kinematics 3

Different measurement results for the same quantity A exclude each

other. This physical fact is expressed by the mathematical statement of
orthogonality,
aj ak = 0 if aj 6= ak or j 6= k . (1.1.8)
Inasmuch as prob(aj ← aj ) is the probability that a control measurement
confirms what is known, we must have
prob(aj ← aj ) = 1 (1.1.9)
so that haj |aj i = 1 must hold. Thus,
 
0 if j 6= k
aj ak = = δjk , (1.1.10)
1 if j = k
where we employ Kronecker’s∗ delta symbol for a compact presentation of
this statement of orthonormality.
Each measurement has a result. This physical fact has a mathematical
analog as well, the completeness relation
X
aj aj = 1 (= identity operator) (1.1.11)
j

so that the kets |aj i make up a basis for the ket space and the bras haj |
compose a basis for the bra space. As an immediate consequence, we note
that the eigenket equation
A aj = aj aj , (1.1.12)
multiplied by haj | on the right and then summed over j, yields
X
A= aj aj aj , (1.1.13)
j

the so-called spectral decomposition of A. We get the spectral decomposi-

tion of A† ,
X
A† = aj a∗j aj , (1.1.14)
j

by making use of the familiar product rule for the adjoint,

 †
1 λ 2 = 2 λ∗ 1 (1.1.15)

for any ket-bra |1ih2| and complex number λ.

∗ Leopold Kronecker (1823–1891)
4 Basic Kinematics and Dynamics

An apparatus that measures the physical property A, in fact measures

all functions of A,

f (A) aj = aj f (aj ) , (1.1.16)

because you just evaluate the function f (aj ) after finding the jth outcome.
Put differently, it is our choice whether we want to call the result aj or f (aj )
when the jth outcome is found. It follows that the spectral decomposition
of f (A) is given by
X
f (A) = aj f (aj ) aj . (1.1.17)
j

It makes consistent sense to regard f (A) thus defined as an operator-valued

function of operator A. For example, consider the simple function A2 ,
X 2
2
f (A) = A = aj aj aj
j
X
= aj aj aj ak ak ak
j,k
| {z }
= δjk
X X
= aj a2j aj = aj f (aj ) aj , (1.1.18)
j j

indeed. Similarly, you easily show that it works for other powers of A, then
for all polynomials, then for all functions that can be approximated by, or
related to, polynomials, and so forth. But what is really needed to ensure
that f (A) is well defined is that the numerical function f (aj ) is well defined
for all eigenvalues aj . As a consequence, two functions of A are the same
if they agree for all aj ,

f (A) = g(A) if f (aj ) = g(aj ) for all j . (1.1.19)

Exercise 1 provides an example.
We recall that operators of two particular kinds play special roles in
quantum mechanics. These are the hermitian ∗ operators, which are equal
to their adjoints,
hermitian: H = H † , (1.1.20)

and the unitary operators,

−1
unitary: U = U † , U U † = 1 = U †U , (1.1.21)
∗ Charles Hermite (1822–1901)
 Brief review of basic kinematics 5

for which the inverse equals the adjoint; see Exercises 2 and 3 for properties
of hermitian and unitary operators and the link between them.
Several observables A, B, C, . . . have their state kets |aj i, |bk i, |cl i, . . .
with probability amplitudes haj |bk i, hbk |cl i, hcl |aj i, . . . . These amplitudes
are not independent, however, but must obey the composition law
X
aj bk = aj cl cl bk , (1.1.22)
l

an immediate consequence of the completeness of the |cl i kets. The self-

suggesting interpretation
“First there is |bk i, eventually |aj i, and in between |cl i,
but we do not know which C value was actually the case (1.1.23)
and so we must sum over all cl .”
is wrong. The assumption of an actual C value at an intermediate stage
leads to logical contradictions.
There are two main reasons for this. First, the l sum is not a sum
of products of probabilities but of probability amplitudes. The resulting
statement about probabilities reads
X
prob(aj ← bk ) = prob(aj ← cl ) prob(cl ← bk )
l
X
+ cl0 aj aj cl cl bk bk c l 0 , (1.1.24)
l6=l0

where the appearance of the l 6= l0 terms signifies the possible occurrence

of quantum mechanical interferences. Only when the l 6= l0 sum happens
to vanish, which is an exceptional situation, the interpretation in (1.1.23)
is justified.
Second, there is the fundamental aspect that some observables exclude
each other mutually. This feature of quantum mechanics has no true ana-
log in classical physics. In particular, there are pairs of complementary
observables. The pair A, B is complementary if the probabilities in (1.1.4),
2
prob(bk ← aj ) = bk aj , (1.1.25)

do not depend on the quantum numbers aj and bk . Physically speaking, if

the system is prepared in a state in which the value of A is known, that is,
we can predict with certainty the outcome of a measurement of property
A, then all measurement results are equally probable in a measurement of
B, and vice versa.
6 Basic Kinematics and Dynamics

Now, if C and D are complementary, we have the sum over intermediate

C values of (1.1.22) supplemented by a sum over intermediate D values,
X X
aj bk = aj cl cl bk = aj dm dm bk . (1.1.26)
l m

The wrong interpretation after (1.1.22) would then imply that both C and
D have definite, though unknown, values at the intermediate stage because
the two sums are on equal footing. But this is utterly impossible.
Given operator A with its (nondegenerate) eigenvalues aj and the kets
|aj i, can we always find another observable, B, such that A, B are a pair
of complementary observables? Yes, we can by an explicit construction, for
which
N
1 X 2π
bk = √ aj ei N jk (1.1.27)
N j=1

is the basic example; more about this in Section 1.2.1. It is here assumed
that we deal with a quantum degree of freedom for which there can be at
most N different values for any measurement.
We need to verify that the B states of this construction are orthonormal.
Indeed, they are,
1 X −i 2π jk 2π
bk bl = e N aj am ei N lm
N j,m | {z }
= δjm
N
1 X −i 2π j(k − l)
= e N = δkl . (1.1.28)
N j=1

Then,
X
B= bk bk bk (1.1.29)
k

with any convenient choice for the nondegenerate B values bk will do. By
construction, we have
2 2
1 2π 1
aj bk = √ ei N jk = (1.1.30)
N N
so that A, B are a complementary pair, indeed. We note that this property
is actually primarily a property of the two bases of kets (and bras) associ-
ated with the pair of observables. A common terminology is to call such
pairs of bases unbiased.
 Bohr’s principle of complementarity 7

In passing, it is worth mentioning that there are quite basic questions

about sets of bases that are pairwise unbiased — referred to as mutually
unbiased bases — that do not have a known answer. Quantum kinematics
is not a closed subject but still the object of research despite the profound
understanding that has resulted from a century of intense studies.

1.2 Bohr’s principle of complementarity

1.2.1 Complementary observables

We consider the situation where we can have at most N different outcomes
of a measurement, that is, there are no more than N pairwise orthogonal
states available. One such set is composed of all the eigenstates of some
observable A, with the respective kets denoted by |a1 i, |a2 i, . . . , |aN i, which
make up a basis of orthonormal kets. Another set is obtained immediately
by a cyclic permutation, effected by the unitary operator U ,

a1 −→ a2 = U a1 ,
a2 −→ a3 = U a2 ,
..
.
aN −→ a1 = U aN , (1.2.1)

generally

U aj = aj+1 , (1.2.2)

where the index is to be understood modulo N so that |aN +1 i = |a1 i, for

example. Applying U twice shifts the index by 2,

U 2 aj = aj+2 , (1.2.3)

and N such shifts amount to doing nothing,

U N aj = aj+N = aj . (1.2.4)

Accordingly, we have

UN = 1 (1.2.5)

so that U is a unitary operator of period N .

8 Basic Kinematics and Dynamics

The eigenvalues of U must obey the same equation

uN = 1 if U u = u u (1.2.6)

for which
2π
uk = ei N k , k = 1, 2, . . . , N (1.2.7)

are the possible solutions, all of which occur. We can, therefore, write the
equation for U also in the factorized form

U N − 1 = (U − u1 )(U − u2 ) · · · (U − uN )
N
Y
= (U − uk ) . (1.2.8)
k=1

Let us isolate one factor,

Y
U N − 1 = (U − uk ) (U − ul ) , (1.2.9)
l(6=k)

and note the following:

(
Y 0 if m 6= k ,
(U − ul ) um = (1.2.10)
l(6=k)
uk α if m = k,

with some complex number α 6= 0, because one of the factors

U − ul → um − ul vanishes if m 6= k but all are nonzero if m = k. We con-
clude that the operator acting on |um i in (1.2.10) is a numerical multiple of
|uk ihuk |, the projector on the kth eigenstate. This product of N − 1 factors
is a polynomial in U of degree N − 1, for which we can also give another
construction. We apply the familiar identity
 
X N − 1 = (X − 1) 1 + X + X 2 + · · · + X N −1
N
X −1
= (X − 1) Xl (1.2.11)
l=0

to X = U/uk ,
N
X −1
N N
U − 1 = (U/uk ) − 1 = (U/uk − 1) (U/uk )l
l=0
N
X
= (U/uk − 1) (U/uk )l , (1.2.12)
l=1
 Bohr’s principle of complementarity 9

where the first step exploits uNk = 1 and the last step makes use of
(U/uk )0 = 1 = (U/uk )N . Now, for U → uk , the sum equals N , and so we
arrive at
N
1 X
l
uk uk = U/uk . (1.2.13)
N
l=1

So, we know the eigenvalues of U and have an explicit construction for

the projectors on those eigenvalues as a function of U itself, and now we
find out how the eigenkets of U are related to the original set of kets |aj i.
We begin with
N
1 X −l l
uk uk aN = uk U aN (1.2.14)
N | {z }
l=1
= al

and then apply haN | from the left,

2 1 −N 1
aN uk uk aN = uk aN = u = . (1.2.15)
N k N
We make use of the freedom to choose the overall complex phase of |uk i
and agree on
1
uk aN = √ , (1.2.16)
N
with the consequence
N
1 X
uk = √ al u−l
k
N l=1
N
1 X 2π
=√ al e−i N kl (1.2.17)
N l=1

and, after taking the adjoint,

N
1 X i 2π kl
uk = √ e N al . (1.2.18)
N l=1

We read off that

1 2π
uk al = √ ei N kl (1.2.19)
N
of which the l = N case is (1.2.16).
10 Basic Kinematics and Dynamics

We have now a second set of bras and kets, for which we can repeat the
story of cyclic permutations, effected by the unitary operator V ,

uk V = uk+1 ,
uk V 2 = uk+2 ,
..
.
uk V N = uk . (1.2.20)

In full analogy with what we did above for U , we conclude here that

VN =1: V is unitary with period N , (1.2.21)

2π
that the eigenvalues of V are vl = ei N l , and that the projector on the lth
eigenvalue is

N
1 X k
vl v l = (V /vl ) (1.2.22)
N
k=1

and are led to

N
1 X −k
uN vl vl = vl uk (1.2.23)
N
k=1

and then
2 1
uN vl vl uN = uN vl = . (1.2.24)
N
1
Here, too, we choose huN |vl i = √ and establish
N

N
1 X −i 2π kl
vl = √ e N uk (1.2.25)
N k=1

as well as
N
1 X 2π
vl = √ uk ei N kl . (1.2.26)
N k=1
 Bohr’s principle of complementarity 11

Can we continue like this and get more and more sets of kets? No!
Because the kets |vl i are identical with the kets |al i; see
N
X 1 2π
vl = uk √ ei N kl
N
k=1 | {z }
= uk al
!
X
= uk uk al = al . (1.2.27)
k
| {z }
=1

We have been led back to the initial set of kets.

In summary, we have a pair of reciprocally defined unitary operators,

U vl = vl+1 , uk V = uk+1 , (1.2.28)

which are of period N ,

UN = 1 , V N = 1. (1.2.29)

Their eigenstates are related to each other by the probability amplitudes

1 2π
uk vl = √ ei N kl (1.2.30)
N
so that the probabilities
2 1
uk vl = (1.2.31)
N
do not depend on k and l. Accordingly, the two bases are unbiased and U
and V are a pair of complementary observables.
Being complementary partners of each other, U and V should have a
simple commutation relation. We find it by considering the effect of U V
and V U upon huk |,

uk U V = uk uk V = uk uk+1 ,
uk V U = uk+1 U = uk+1 uk+1 . (1.2.32)
2π
Since uk+1 = uk ei N , this establishes
2π 2π
uk V U = ei N uk uk+1 = ei N uk U V , (1.2.33)
12 Basic Kinematics and Dynamics

and the completeness of the bras huk | implies

2π 2π
U V = e−i N V U , V U = ei N U V . (1.2.34)

The generalization to
2π 2π
U k V l = e−i N kl V l U k , V l U k = ei N kl U k V l (1.2.35)
is immediate. These are the Weyl∗ commutation relations for the comple-
mentary pair U, V .

1.2.2 Algebraic completeness

Now, all functions of U are polynomials of degree N − 1, and all functions
of V are also such polynomials. Therefore, a general function of both U
and V is always of the form
N
X −1 N
X
f (U, V ) = fkl U k V l = fkl U k V l (1.2.36)
k,l=0 k,l=1

or can be brought into this form. It is written here such that all U s are
to the left of all V s in the products, but this is no restriction because the
relations (1.2.35) state that other products can always be brought into this
U, V -ordered form.
In fact, all such functions of U and V make up all operators for this
degree of freedom, which is to say that the complementary pair U, V is
algebraically complete. To make this point, we consider an arbitrary opera-
tor F and note that then the numbers huk |F |vl i are known. We normalize
these mixed matrix elements by dividing by huk |vl i, thus defining the set
of N 2 numbers
uk F vl
f (uk , vl ) = . (1.2.37)
uk vl
Multiply by |uk ihuk | from the left and by |vl ihvl | from the right and sum
over k and l,
X X
uk uk f (uk , vl ) vl vl = uk uk vl f (uk , vl ) vl
k,l k,l
| {z }
= uk F vl
X X
= uk uk F vl vl = F . (1.2.38)
k l
| {z } | {z }
=1 =1
∗ Claus Hugo Hermann Weyl (1885–1955)
Another random document with
no related content on Scribd:
father scolded me for giving the bear hide away. My little bear hide
was of good size. My father had it tanned for me, and the hide was
also decorated with paint. The bear hide also had a soft, feathery
appearance about its head. I wore it in dances, and kept it by my
pillow in our lodge. Only a few years ago I was visiting the Sioux,
and while I was gone some white man came to our village. He saw
the bear robe in our lodge. He asked how much they wanted for the
hide, and my bear was sold to some white man. When I came back
home I missed my bear, and asked where it was. My folks said, “We
sold it to a white man.” I was sorry, but it was all right, for we do not
have any more Bear dances.

FOOTNOTES:
[84] Told by White-Bear.
 ABSTRACTS.
1. THE WOLF AND LUCKY-MAN CREATE LAND.
Wolf and Lucky-Man meet on shore of big lake, where two ducks
are swimming. Wolf challenges Lucky-Man to see who can endure
rain longest. Lucky-Man wins. Wolf sends Duck down to fetch dirt
from bottom of lake. Duck brings up mud, which Wolf throws in north
and forms into prairie. Lucky-Man sends Duck for more mud, which
he throws on south side of Wolf’s land. Hills and mountains are
formed and buffalo are on land. There is channel between two
countries created, occupied by Missouri River.

2. THE SPIDERS GIVE BIRTH TO PEOPLE.

Wolf and friend change Spider-Man and Woman by rubbing them
with wild sage dipped in water and teach them how to lie together.
Their progeny are human beings.

3. THE ORIGIN OF THE ARIKARA.

Large people on earth long ago destroyed by flood, by Nesaru.
People turn into corn and are put into cave with animals. Nesaru
turns ear of corn into woman and sends her to bring people from
earth. People and animals know her. Badger, Mole, and long-nosed
Mouse offer to help her to take people out. They dig in turns.
Thunder opens earth. People go out upon earth, journey west,
leaving behind Badgers, long-nosed Mice, Moles, and some people
who turn back into earth and become animals. People come to great
basin, which Kingfisher fills up by striking bill into banks. Journey is
continued until people stopped by timber, which is removed by Owl.
They come to big lake. Loon parts waters. Mother-Corn returns to
heavens. People here make games, first shinny and then javelins, to
catch ring with. Winners kill those of other side. Mother-Corn returns
to give people rules to go by. Man is selected as chief. He instructs
people as to scalping. Mother-Corn makes bundle, songs, ritual, and
ceremonies. Man instructs medicine-man, teaches them sleight-of-
hand, and tells them to make village. Mother-Corn leads people to
Republican River, Kansas. Awaho people come last and receive
ceremonies from Mother-Corn. They offer smoke to gods. Dog
comes to village and complains that Mother-Corn has left out Dog
and Whirlwind. Dog has come from Sun, who has given it curative
power. Whirlwind is disease, and if dog meat first offered as sacrifice
gods will send storm to drive away disease. Whirlwind comes and
Dog appeases gods and says he will be people’s guardian. Mother-
Corn says gods in heavens are four world-quarters. They will send
storm if smoke not given to them first. Mother-Corn is Cedar-Tree in
front of lodge and Stone at right of her is man who established office
of chief. Nesaru watches over them and gives them long life.

4. ORIGIN OF THE ARIKARA.

Mother-Corn is assisted by Badger, Gopher, long-nosed Mouse,
and Mole to get people out of ground, as in No. 3. People see where
other people helped out of ground by Buffalo. They start on journey
and are stopped by obstacles, as in No. 3, and are helped by
Kingfisher, Owl, and Loon. Some people stay behind as Worms,
Birds, Fish, and Loons. [Mother-Corn offers smoke and sends
animals for offerings to gods.] Prairie-Chicken kills wild-cat, which
represents heavens, and brings it to Mother-Corn for offering. Three
Stars in East bring Mother-Corn stone for pipe to form smoke. Pipe
is made and filled with native tobacco. Prairie-Chicken takes pipe in
succession to gods in Southeast, Southwest, Northwest, and
Northeast, and to Nesaru, all of whom smoke the pipe. Prairie-
Chicken says sand blown by wind made white spots on its feathers.
Smoking by Nesaru is to show consent to Mother-Corn having
people on earth and that gods are to protect them. Dog comes and
tells Mother-Corn that Whirlwind is angry for being slighted in smoke
ceremony. Mother-Corn appeals to Nesaru and the gods for
assistance. Woman says she will protect the people, and turns into
Cedar-Tree. Big-Meteoric-Star falls from heavens by Cedar-Tree to
assist. Whirlwind comes and people all run in all directions, and
when Whirlwind strikes them it changes their language. People who
stand on Cedar-Tree and Rock are Arikara. Wind strikes Mother-
Corn and she vomits four times, water and ears of corn of different
color. Whirlwind tells Mother-Corn it has left behind diseases, but
says when they offer smoke to the gods they are to give it smoke
last, that it may not come very often. Cedar-Tree asks Mother-Corn
that it may be known as “Wonderful Grandmother” and be placed in
front of the medicine-lodge. Big-Meteoric-Star asks to be known as
“Wonderful Grandfather” and sit by Wonderful Grandmother in front
of medicine-lodge. Dog asks, as he brought the news, to guard
camps and villages and to be offered in ceremonies, and his fat to be
used by medicine-men. Mother-Corn gives corn for seeds that corn
may be offered to gods. People who scattered to be their enemies—
to the southwest, “Sahe;” to northeast, “Pechea;” to the east,
“Wooden-Faces;” to south, “Witchcraft-People.” Mother-Corn stays
with people until she has taught them bundle ceremonies. She tells
them to tie all children’s moccasins together on her back. Then they
are to take her to river and throw her in. People do not understand
and keep up singing in night. At daylight they find Mother-Corn has
turned to ear of corn, with buffalo robe tied to it. People place
children’s moccasins with corn and throw them with Mother-Corn
and robe into river. Many years afterwards Mother-Corn returns and
teaches more bundle ceremony songs and finally disappears.

5. THE ORIGIN OF THE ARIKARA.

 Many people in cave under ground with Corn, Mother of tribe.
Mother-Corn sends four birds to find better world, but they are
unsuccessful. Long-nosed Mouse, or Mole, Skunk, and Badger work,
and at last Badger goes through hole, but falls asleep. Returns in
morning and Mother-Corn forces her way through hole followed by
all people. They march westward. They come to wide water, thick
forest, deep ravine in succession, which Fish, Owl, and Kingfisher
help them to cross. They see Buffalo on open prairie and are afraid,
but Mole, Skunk, and Badger make holes all around animal. His
blood sinks into ground and becomes stone, from which pipes were
made. Buffalo butchered and flesh divided among different sacred
bundles, with animal’s joints. People again go on westward and
fowls, fishes, and animals separate from them and give Mother-Corn
power. Mother-Corn separates from animals.

6. THE ORIGIN OF THE ARIKARA.

[Man Bear’s-Tail relates killing of buffalo cow by father, who calls
old woman and keeper of bundle, and describes ceremony of
untying bundle. Old man tells origin of bundle and of people.] Nesaru
makes giants, but being displeased with them turns them into
stones. Nesaru again makes people, small and wonderful. They
displease Nesaru, who tells animals to hide. He is going to make
water rise from earth. Animals give power to Bear to take people
under ground, with assistance of Badger, Mole, and long-nosed
Mouse. Fox acts as runner and errand man. People live under
ground many years. Animals decide to dig upward for land. Bears,
Badgers, Moles, and long-nosed Mice dig and Mole first to get his
head through. Badger enlarges hole. Fox goes through and reports
what he sees outside. Bear makes hole larger and animals go
through, followed by people. Woman, who says she is grain of corn,
tells man they are on island. People taken under ground by Mice
were grains of corn and now turned to people. Mouse leader. They
cross water by aid of woman, who becomes gar-pike. Some fall into
water and become fish. People pick up stones to cut with. Mouse
leads people through thick timber. Some turn to owls. Earthquake
forms deep chasm, which Bear enables people to cross. Whirlwind
makes pathway through thick timber. People come to muddy water in
“Pawnee” country. They find things to wear and eat. First bow made.
Long-nosed Mouse, Bear, Mole, Badger, and Fox die, and their skins
with skulls are wrapped in bundles. They receive ceremony from
Pawnee. Each bundle receives different ritual. Arikara dress ear of
corn as woman and throw it into river. Many years afterwards
strange woman comes into lodge where bundle ceremony. People
take no notice of her and she goes to other bundle lodges. In last old
man recognizes her and Muddy-River-Country ceremony performed.
Woman says that four world-quarters are her father, and that she will
come to them in dreams and tell them about things in bundle. They
are to tie her on bundle and clothe ear of corn. She turns into ear of
corn. They send for other old man and tie ears of corn upon the
bundles.

7. THE ORIGIN OF THE ARIKARA.

Arikara live under ground. Long-nosed Mouse, Mole, Badger, and
Fox agree to take people to top of earth. Mole digs first. Arikara
come out, Fox leading. Earthquake, and other people held fast.
People journey west and come to chasm caused by earth shaking,
but Badger makes pathway. Mother-Corn in heavens asking gods to
let people live. Obstructions arranged by being known as Sickness.
People come to deep river and Loon sent by gods. Loon flies across
river and back and dives. River is open and people cross over.
Waters come together again and some people left on other side.
Mother-Corn stops and says Black-Wind is angry, but Black-
Meteoric-Star will help them. Tells people to get under cedar tree.
Black-Wind comes and takes many people. They go on and come to
steep mountain bank. Bear digs steps on both sides and people go
across. Dog comes up and says his meat shall be offered to gods.
His father is Sun, who has given him power.

8. THE ORIGIN OF THE AWAHO-BUNDLE

PEOPLE.
 People come out of ground, but some are cut off by earthquake.
Heavens hear crying and send Mother-Corn to them. Badger digs
through earth. People come out and walk westward until they come
to thick timber. Screech-Owl flies through and makes pathway. Owl
and Whirlwind are enemies. People followed by “Cut-Nose,” an
animal with long horns. People run until they come to chasm, which
Badger enables them to cross. They then come to thick ice and deep
water, which Loon enables them to cross. Mother-Corn teaches
people ceremonies and rituals and gives them things to put in
bundles. Mother-Corn disappears by ear of corn wrapped in her robe
under bundle. Awaho last people to come out of ground, and where
other bands have camped they find bits of meat offered to gods,
which they use for food. They know all ceremonies and teach them
to others. Nearly all are killed by enemies, but bundle hid under
bank. Women go for bundle and contents are purified. Sacrifices of
meat made the next day. Nesaru made animals to take kernels of
corn under ground. They were people turned to corn by Nesaru. This
is why animals brought them out of ground and why Mother-Corn
was sent by gods in heavens, who had field of corn.

9. MOTHER-CORN’S VISIT TO THE ARIKARA.

Mother-Corn tells Arikara when journeying west to dress her up
and put her in river. When Arikara make permanent village upon
Missouri River old men think it time to send Mother-Corn down
stream. She is taken from bundle, painted, and dressed. After
reciting rituals, Mother-Corn, with children’s moccasins tied about
her waist, is thrown by priests into river, her head up stream. Many
years afterwards woman comes to village and is recognized by man
as Mother-Corn. She teaches them ceremonies and songs and that
night disappears.

10. MOTHER-CORN’S VISIT TO THE ARIKARA.

In olden times, old man made offerings to gods and Mother-Corn.
Mother-Corn is pleased to have smoke with people and starts from
east to visit them. She goes into medicine-lodge. She stays many
days and teaches them many lessons, but people are hungry for
meat. Mother-Corn asks woman to make moccasins for her. She
puts on moccasins and they wear out when she walks slowly twenty
steps. This takes place four times, but fourth pair brings her back to
altar. Her walk means that she has walked long way off in west, and
way very hard. She tells people she has seen buffalo and that they
will be seen in four days. In morning of fourth day they kill many
buffalo, but while they are away, enemies attack village and Mother-
Corn is killed. They bury her and from place where she is laid, grass,
etc., springs up.

11. HOW THE PEOPLE ESCAPED THE BUFFALO.

When people came out through ground they were led by woman,
“Mother.” Among them were all kinds of animals except buffalo.
Monster with horns like buffalo comes out of lake. They call him
“Cut-Nose.” As it comes along, buffalo come from under him. Buffalo
catch up with people and kill some of them. People make canyons
behind, which buffalo can not cross. Whirlwind comes. Mother tells
people to give presents and smoke to it. Whirlwind scatters some of
people. Buffalo with Cut-Nose come behind and people come to big
timber. Owl and Badger try to make path through timber, but fail.
Coyote and Dog come and open way through. Buffalo and Cut-Nose
come again and kill people. They come to deep water. Dogs fail to
make pathway, but Loons make opening through waters. They come
to canyon and Badger makes banks fall, after Kingfisher and Mole
have failed. They cross and make village near canyon. Mother holds
ceremonies for different bundles. Awaho-bundle people come last,
and they receive all ceremonies. Awaho had been left behind when
people came out of ground, and they pick up meat offerings to gods
left behind.

12. WHY THE BUFFALO NO LONGER EAT

PEOPLE.
 Young man goes to village at night and finds people are Buffalo.
They are talking about killing people. He finds human head and
meat. Hears people are to be got out of ground and killed. Near by
sees hole cut in side of hill where bulls circle around and drive
people into cut. He sees people running to cut from out of ground.
He goes among hills. Strange man gives him bow and arrows and
tells him to take young man with bows and arrows to kill and scatter
Buffalo. They go to place and attack Buffalo and kill and scatter
them, so that they become buffalo and never eat people any more.

13. WHY THE BUFFALO NO LONGER EAT

PEOPLE.
People hungry and chief priest opens bundle and offers gifts to
gods for them to send buffalo. Buffalo come three days after
ceremony and old priest tells story. Buffalo are human, but have
horns. When they want meat they recite ritual. When hollow tree is
struck with pole four times people led by Cut-Nose come out and are
killed, except Cut-Nose, who re-enters tree. Boy chased by Buffalo
cow. He sees fine-looking woman wearing white buffalo robe. She
goes west and boy follows. He finds woman at tipi. Woman says she
has selected him to turn her people into real buffalo, so as not to eat
his people. They go through four circles of Buffalo bulls stationed as
sentinels and enter tipi, where woman’s father lives. She covers
young man with her robe. Buffalo are human, but have horns and
tails. They cook and eat human meat. Girl shows him arbors with
human bodies, and hollow cottonwood tree, with long stick, and tells
him its use. Takes him to timber, where during three days he makes
bows and arrows. Next morning they place bows and arrows at foot
of tree. Woman tells young man what to do and they hide. When
Buffalo come towards tree, young man jumps out. Cut-Nose comes
out, and then people. Young man gives men bows and arrows and
tells them to shoot and kill Buffalo. Buffalo run towards village,
chased by people, and they finally become buffalo. Young man and
Buffalo woman take bundle from tipi. They marry and teach people
songs and ceremony of bundle. People become part of Arikara.
 14. THE GIRL WHO MARRIED A STAR.
Girl says she likes Red-Star and would marry him if on earth. In
morning girl sees Porcupine and climbs after it in cottonwood tree.
Tree grows higher and girl reaches another world. Porcupine turns
into man and says he is Star. She stays with him, but cries every
night. She gives birth to male child, who has star on forehead. Son
wants wild turnips and man tells her not to dig for them in valleys.
She digs in valley and stick runs through earth. She looks down and
sees she is far away from her people. Woman tells her to get from
husband sinews of whole buffalo and she will make sinew string to
reach ground below. Girl gets sinew from husband, who forgets two
sinews in shoulder. Old woman makes string and girl also makes
long sinew string. They go to valley and girl takes child on back
under robe, slips down string fastened to stick across hole. She
reaches height of highest tree from ground. Husband sees her
hanging and kills her with stone. Boy slips out of robe and falls on
ground, but is not hurt. Boy nurses at dead mother’s breast. He goes
to cornfield. Old woman catches him and takes him home as
grandson. Grandmother scatters corn in lodge for blackbirds and
places mush behind curtain. Boy calls blackbirds and kills them all
with club. Grandmother brings them to life again and tells them to fly
all over the world. She tells boy to throw wood into pond and next
morning finds black bow and four black arrows. Boy sees big serpent
behind curtain and kills it with bow and arrow and serpent slips into
pond. Serpent is grandmother’s husband. Next day old woman tells
boy not to go to dangerous place. He goes and sees mountain-lion,
which obeys him. He leads lion to old woman’s lodge. The same
occurs with a cinnamon bear. Boy sees four wonderful men killing
buffalo. They frighten him with fœtus of calf. He climbs tree and they
place fœtus in fork. They offer to take calf down if he will give
grandmother to them. He returns and tells her he is satisfied, but
says they would have to give him something in return. They promise
him bow and arrows and old woman tells him to take middle bow of
five leaning against wall of lodge. Men go to grandmother’s lodge
and stay with her. Old woman sends boy with flute to play around
men’s lodge. Men all scared and close up lodge with earth. They die
of hunger. Boy goes to den of snakes. Snakes give him long gut to
eat, but it is snake, and he twists its head off. Snakes go into ground
and try to get into boy’s rectum, but hit rock on which he sits. They
tell stories. Snakes all go to sleep on long circular stick around den.
Boy with flint knife cuts heads on stick, but last one wakes up and
disappears in hole. When boy sleeps he places arrows so that they
can fall on him when Snake approaches him. Boy is very sleepy and
arrows cannot awaken him. Snake goes into his mouth and nestles
in his skull, where it remains until boy becomes skeleton. Boy’s
father sends storm and skull is filled with water, but this does not
drive out Snake. Father gets Sun to move nearer earth and heats
skull until water boils, and Snake crawls out. Boy catches Snake by
neck, hits its snout with stone, and rubs its teeth upon rock. He lets it
go on promise not to bother people after. Boy returns to grandmother
and tells her country is free from wild animals. She disappears, and
boy goes to village and tells his story. He dies after clearing country
of all wild animals.

15. THE GIRL WHO MARRIED A STAR.

Girl taken up to heavens by star digs turnip and sees people on
earth. Old woman makes sinew rope and lets her and child down
through hole, but rope too short. Husband kills her with stone, but
boy safe. He goes to cornfield and is caught by old woman, who
takes him home. He shoots huge serpent behind curtain, who was
woman’s husband. She plans for bear to kill him, but he captures
bear. Boy finds tipi with four strong men playing dice game. He
shoots through hole and cleans man’s nose with arrow. He goes with
them to hunt and they annoy him with elk’s fœtus. He climbs tree
and men remove fœtus from tree only on his promising them his
grandmother. She goes with him to men’s tipi and they teach boy
ceremony of catching eagles and of hunting. Boy meets camp of
Snakes, all of whom but one he kills, as in No. 14. Surviving Snake
enters anus while he sleeps and gets into head, from which it is
driven by water boiling. Boy seizes it and knocks its head on flat
rock. Boy afraid of fœtus because cluster of stars to which boy’s
father belonged did not come up at that time with rest; so father not
present to help him.

16. NO-TONGUE AND THE SUN AND THE MOON.

Young man goes upon high hill to mourn. Little bird takes him to
another place. Man, painted red, comes and says he is going to be
his son and asks for his tongue. Young man cuts off his tongue and
gives it to man and then falls dead. Moon sees him and goes and
touches his feet. Young man sits up and Moon tells him man to
whom he had given tongue is Sun. Moon makes him his own son
and warns him that when Sun offers him choice of weapons he is to
take old ones. Sun takes him to sky in morning and cries because
No-Tongue takes best things, as these give boy life. Sun asks No-
Tongue to send him white buffalo robe. Moon tells him to get dark-
brown robe for Sun and powder it with white clay. Sun hangs up robe
and wind shakes all white clay out of it. Sun tells Moon his Little-Sun
is going to kill No-Tongue. Moon warns No-Tongue and advises him
what to do. No-Tongue goes with party on war-path and Sun plans
for Little-Sun to kill him. Little-Sun with enemy and in morning asks
No-Tongue to shake hands with him. No-Tongue goes and kills Little-
Sun and his people defeat enemy. Sun sends son Big-Sun to kill No-
Tongue, but is killed himself. Sun becomes Buffalo to kill No-Tongue,
but falls into mud hole. No-Tongue makes fire on his back and
Buffalo burns up. Sun tells Moon he will scalp No-Tongue. Moon tells
No-Tongue to put false scalp over head with dog’s blood inside. Sun
comes and takes scalp. Seeing that No-Tongue is not really scalped,
Sun leaves him alone. When old and blind No-Tongue goes to top of
hill and makes circle of red sticks for Sun and circle of white sticks
for Moon. Sun and Moon come and Sun takes old man to his home.

17. HOW BURNT-HANDS BECAME A CHIEF.

Poor boy, Burnt-Hands, lives with grandmother outside of village.
Last-Child, daughter of chief, brings them food. Burnt-Hands follows
trail of wounded elk and finds it dead. Chiefs Red-Bear and Black-
Bear come. Red-Bear shoots boy and drops him into air-hole in ice.
White-Bear’s cub takes boy to father. Father pities and adopts him
as son and teaches him Bear ceremony. Burnt-Hands receives
bundle of medicine and goes home. Notice given for buffalo hunt and
that Red-Bear wants hide of white buffalo. Burnt-Hands goes with
young men to chase. He gets white buffalo robe, as Red-Bear afraid
of him. When he reaches camp he eats meat prepared for Red-Bear.
Burnt-Hands takes white buffalo hide to grandmother, who gives it to
Last-Child. Elk chase is made to get teeth for Red-Bear. Burnt-
Hands promises grandmother elk-tooth dress and tells her in case of
trouble to flee to timber. Burnt-Hands goes to chase and collects-
many elk teeth and so does Red-Bear. They meet at last elk. Burnt-
Hands strikes Red-Bear on head with war-club and drags him to air-
hole. Burnt-Hands finds grandmother and they perform Bear
ceremony. They turn into Bears and attack warriors, killing many.
Others send peace-pipe by Last-Child and it is accepted. Burnt-
Hands makes grandmother thirty-eight years old and himself twenty-
two, and marries Last-Child. Burnt-Hands becomes chief and has
Black-Bear as slave.

18. HOW BURNT-HANDS BECAME A CHIEF.

Poor boy goes on war-path with warriors. Grandmother says he is
not to tell coyote stories and gives him round burnt clay ball that has
handle. When hungry he is to put kernels of corn on ball and roast
them. Boy asked to tell coyote stories, but refuses. He roasts corn
upon clay ball and then tells stories. Enemy comes and men are
scared. When boy has finished eating corn he attacks enemy with
clay ball, which is war-club, and kills many. Enemy run away. Burnt-
Hands made chief and given good tipi and wife.

19. HOW BURNT-HANDS BECAME A CHIEF.

Poor boy tells grandmother to make him bow and arrows that he
may join buffalo hunt. He says he will bring back some tongues and
hearts. Boy sings about being selected to stand in front and make
motions to direct hunters, and he is selected. He kills buffalo and
turning back pulls out buffalo beards and bunch of hair from
shoulder. His robe is taken and he sings about snowstorm coming.
He goes to grandmother and throws hairs on ground and several
tongues and hearts appear. Blizzard kills many men who had made
fun of young man. On next buffalo chase he again stands in front
and is first to kill buffalo. He takes hair as before and it becomes
tongues and hearts. People find out boy is wonderful, and give him
pony. He marries chief’s daughter, and becomes great warrior and
chief.

20. THE TWO BOYS AND THE WATER-SERPENT.

Two boys are accused of eating up pots of corn. They watch at
night near inclosure surrounding village and see long serpent come
and stick its head into smokehole of lodges. Next day they make
many arrows and at night when serpent has its head in lodge they
shoot at it. Serpent goes to river, water of which roars and rises, and
serpent is found dead when river goes down.

21. THE BOY WHO BEFRIENDED THE

THUNDERBIRDS, AND THE SERPENT.
Boy gifted with powers by four-world-quarter gods kills so many
antelope he is called Antelope-Carrier. Wood-Rats have given him
bow and four differently colored arrows. He wanders from home, and
while asleep two Thunderbirds carry him up high mountain. He finds
nest with four young Thunderbirds. Mother Thunderbird comes and
tells him of serpent with two heads that lives in lake and eats her
young. She promises him lightning and control of all birds if he will
help to kill monster. He promises and Thunderbird, after telling him
when serpent would come out of lake, flies away. Fog rises from lake
one day and boy sees monster with two heads crawling out of lake.
Storm comes from west and Thunderbirds return, making lightning,
which strikes serpent. Lightning throws it back, but it again crawls
up. Monster opens its mouth to swallow boy. He shoots black arrow
into its mouth. Monster falls and bursts open. Other head comes and
boy shoots red arrow into its mouth and head broken in pieces.
Thunderbirds come with all kinds of birds, which feast upon serpent.
They give boy power as objects which he swallows. Boy chief of all
birds and kills all bad animals. Two boys, joined together with
rawhide, go to shoot birds. One shoots at white object, like
mushroom, moving up and down and strong wind carries them far
away to an island. They go west and come to lodge of old woman.
She makes cakes, four for the great serpent, who will carry them
across by water. Serpent comes and carries them across, stopping
each day when hungry. They give it cake and soft-shell turtle (lice)
from its head. Wild boy jumps before they come to land and is
swallowed by serpent. Other boy asks serpent to open its mouth
wide and he drags swallowed boy out. Boys travel to Missouri River
bottom. They put log of wood on fire and it is serpent. Foolish boy
eats chunk of meat and he gradually turns to serpent. Other boy
takes him to Missouri River and turns him loose there. Antelope-
Carrier hears of serpent and hunts him with all his birds. Serpent
uses his power and carries him into his den. Antelope-Carrier is
made to vomit up all his power, except lightning in his eyes. Serpent
remains in river and gives its powers to people, and songs and
medicine-men’s ceremony.

22. THE BOY WHO TURNED INTO A SNAKE.

Idiot boy and son of chief go on war-path. They have to return
through want of food, and come to water-serpent. It is so big they
can not get around it, and idiot proposes to burn it. Serpent burns in
two. Idiot eats of serpent meat and his body gradually becomes
colored red and blue. By fourth day his legs are grown together and
become snake’s tail. Other boy carries him to lake, where fishes
object to him, and finally they come to the Missouri River. He rests in
middle of river and people by giving him presents cross over without
danger of drowning.
 23. THE BOY WHO RECEIVED THE MOUSE
POWER.
Young man stays behind when people go hunting. He goes
through village and hears crying. He goes to lodge and sees woman
wrapped in buffalo robe, who tells him people have taken her
children. She says they are in sacred bundle robe, and asks him to
go and bring her children back. He does so and gives nest with
children to woman. She tells him to return at night and then becomes
mouse. Young man goes to lodge at night and finds woman there.
Rats come in human form and priest gives him war-club and power
to become mouse at any time, and little box of medicine. Woman
tells him he is now her son and says they are not to kill mice as they
are his relatives. Young man becomes great warrior. In enemy’s
camp he turns into mouse and drives ponies out of camp after
cutting ropes. He becomes so bold that people become afraid of him,
but finally he and young man who has power of Bear fight and kill
one another.

24. THE BOY AND THE YOUNG HAWKS.

Small boy discovers hawk’s nest with four eggs. Eggs are hatched
and boy feeds birds with insects. Boy goes to take birds home when
he sees man who calls birds his sons and says he will be rewarded
for taking care of them. Boy takes feathers from young birds to put
on his arrows. He becomes good hunter and on war-path fights
where the arrows are thickest. He becomes known as brave, but
finally does wrong among his people. Many try to kill him, but always
forget, until one man capable of killing him does so.

25. THE END OF THE ELK POWER.

Four strong young men, of whom only oldest is married, go to trap
eagles, leaving woman and child at home. On their return woman is
missing. Eldest unmarried brother is filled with pity for child and goes
to cry near timber, where is old skull of buck elk. On second night
voice tells him woman and three others captured by Bear and that he
has received Elk power. He is to go again and receive instructions.
Pretty-Voice goes again and learns ceremony of Elks. He is to blow
whistle and all females will come to him. He goes near Bear’s home
and whistles four times. Women run out of den and they go away
with Pretty-Voice. Bear follows and he orders party to stop. Pretty-
Voice shoots arrows at Bear without effect. He then throws himself
on ground and becomes Elk. Elk and Bear fight, and Bear admits his
defeat. Elk again becomes man and Pretty-Voice wins great honor
by capture of women. He causes ill-feeling by using his magic
whistle to attract girls and then married women. Men shoot at him,
but nothing can harm him. Sioux attack village, but they can do
nothing while Pretty-Voice is living. Men come on friendly visit and
Pretty-Voice secures Sioux girl by his ceremony. She gets to know
secret of his power and then runs away. She obtains necessary
things and then starts at head of war-party to kill Pretty-Voice.
Inhabitants of village are defeated and Pretty-Voice finally falls. His
mother wishes to collect his flesh, as he had told her, but men will
not let her. They make big fire and destroy his body. White fog seen
to arise from place for many days after.

26. THE ELK RESCUES A WOMAN FROM THE

BEAR.
Poor young man and chief’s daughter run away together. They live
alone and man kills deer and elk. He goes to catch eagles and while
away Bear comes and takes wife away. Elk tells man and teaches
him how to transform himself into Elk. Gives him whistle to attract
female elk. Bear leaves den and man blows whistle. Wife and other
women rush out to him. Bear comes and attacks Elk, which puts its
head down and sticks horns into body. Man shoots and kills Bear.
Man takes his wife and Elk other women, who become Elk.

27. THE BOY AND THE ELK.

 Young man goes to place where animal skull near lake to cry
because no girl will marry him. He hears flute and Elk comes. Elk
tells boy to take teeth from skull and gives him flute which will attract
girls to him. He goes home, tries flute, and girls come. After he is
married, women also come and men kill him. One of his relatives
takes teeth and flute. Boy is left unburied and several days
afterwards he goes to mother’s tipi. He sends mother to society of
Young-Dogs for tobacco. Men afraid of him. Boy goes away followed
by relatives. They go into river and all turn into animals. Young man
who had flute and elk teeth does not go and is the only one who
lives.

28. THE COYOTE, THE GIRL, AND THE MAGIC

WINDPIPE.
Beautiful girl lives alone in timber. Has plenty of buffalo meat and
some wonderful bundles. Coyote becomes her errand man. When
out of meat girl tells Coyote to cover his head up as her brothers are
coming. Girl waves buffalo windpipe over smoke and dust in it turns
to her seven brothers. They take bows and arrows and girl goes on
to lodge, yells and waves towards west and south. Buffalo come and
brothers kill them. They return to lodge and girl puts them again into
windpipe as dust. Coyote sees performance and decides to steal
windpipe. Coyote goes away with windpipe, and while he sleeps girl
has brothers bring him back again. This occurs three times. Fourth
time girl lets Coyote carry thing off. He goes up hill near village and
howls for people to come and kick with him. Several young men go
and Coyote turns windpipe upside down, but, instead of dust and
boys, swarm of bumblebees come out. Young men run into timber,
bees go into hollow tree, and Coyote goes away as coyote.

29. THE BUFFALO-WIFE AND THE JAVELIN

GAME.
 Young man out hunting dreams of two buffalo bulls turning into
sticks and of buffalo cow turning into ring. In morning he sees cow
and lies with her. Finds ring in grass and wears it on his wrist. He
makes sticks and plays game with young men, winning many things.
Goes hunting and sees old woman, who induces him to carry her
across river on his back. He can not throw her off and he goes home
with her fast to his back. Medicine-men are sent for, but they can do
nothing. Poor boy puts on old robe and goes to young man’s lodge
with bow and four arrows of different colors. He shoots black arrow
and splits woman in two. With red arrow he takes her off boy. The
other arrows he places on boy’s back to remove sore place. Old
woman is then burned. Next day crying and voice are heard near
where woman burned. Young man finds ring has gone. White tipi
with woman and child inside appears where others were. Young man
goes to see it and woman with new buffalo robe passes by him,
having child. Young man makes bundle of eagle feathers and follows
them. They become buffalo. Calf communicates with father, and
woman finally becomes reconciled to him. They come to hill on
which Buffalo bull, boy’s grandfather, is waiting for them. Man puts
two eagle feathers on his horns. He sends them on to next hill and at
last they come to hill with four Buffalo bulls, chiefs of Buffalo camp.
Man puts feathers on their heads. They are sent into village and
Buffalo become mad because man has not feathers enough to go
around. Man made to sit on hill until they decide what to do with him.
He sticks flint knife into ground and asks gods to form stone around
where he sits. Buffalo devise various ways for killing him, but do not
succeed in doing so. They decide to send man with Buffalo cow and
calf to Indian village for presents. Buffalo bull turns man into Buffalo.
Buffalo follow them. Man finds village and tells errand. People bring
eagle feathers and native tobacco, which man takes to Buffalo.
Buffalo willing to be slaughtered and man tells chiefs. Four times
people go and kill Buffalo. Leader of Buffalo gives man sticks to play
with. Sticks and ring different kinds of people. Man lives long life.
Buffalo calf starts Buffalo ceremony among people.

30. THE ORIGIN OF THE WOLF DANCE.

 Young man, son of chief, refuses to marry and seven girls plan to
put him into hole. They spread weeds over hole and young man falls
in. Girls promise to take him out if he does certain things, but finally
they leave him. He cries and gray Wolf hears. Wolf says he will help
him, and while he is gone Bear comes. Wolf returns and they quarrel
about boy, but finally agree that whoever digs through to boy first
shall claim him. Wolf gets to boy first, but Bear says he shall be his
son. Wolf takes boy among Wolves and he comes to act like wolf.
Afterward Buffalo hunters see him, but they cannot catch him. They
make trap and place buffalo meat inside inclosure. Wolves are run
into trap and four strong men with rawhide leggings are put in. Other
Wolves are let out, but Wolf man caught. They tie him, put him into
sweat-lodge, and make him vomit. Wolf man recovers and has tipi
made. Seven girls who had put boy into hole are invited. Man goes
and calls for Wolves and Bears. They come, and he places them
about tipi. He tells girls, who try to escape, but Wolves eat them.
Father tells people boy’s story and girls’ relatives do not offer to save
them. Young man finally becomes chief. He starts Wolf dance.

31. MEDICINE DANCE OF THE BEAVER, TURTLE,

AND WITCH-WOMAN.
Animals meet for sleight-of-hand performances. Only Beaver, soft-
shell Turtle, and Witch-Woman are to perform. Beaver gnaws nearly
through three of lodge posts and people ask him to stop, as they
think lodge will fall. Turtle sticks knife near left collar-bone and water
pours out all over lodge. People are afraid and Turtle takes all water
back again. Witch-Woman plays with gun, but calls for help and
gives birth to child, who is to be great medicine-man.

32. THE VILLAGE-BOY AND THE WOLF POWER.

Four girls are made fun of for dancing with their brother. “Village-
Boy” has never gone on war-path. Boy goes to graveyard to mourn.
Wolf comes and asks why he is crying. Wolf tells boy to join next
war-party and he will lead him to enemy’s camp. War-party starts