Mathematics for physicists - Instructors' manual (solution to even numbered problems)

145 Pages·2019·12.593 MB·English

by  Alexander Altland, Jan von Delft

Mathematics for Physicists:
Introductory Concepts and Methods
Solutions to even-numbered problems
ALEXANDER ALTLAND & JAN VON DELFT
January 9, 2019
Contents
S Even-numbered solutions 3
SL Problems: Linear Algebra 5
S.L1 Mathematics before numbers 5
Sets and Maps 5
Groups 5
Fields 7
S.L2 Vector spaces 8
Vector spaces: examples 8
Basis and dimension 11
S.L3 Euclidean geometry 12
Normalization and orthogonality 12
Inner product spaces 12
S.L4 Vector product 15
Algebraic formulation 15
Further properties of the vector product 16
S.L5 Linear Maps 17
Linear maps 17
Matrix multiplication 18
The inverse of a matrix 19
General linear maps and matrices 22
Matrices describing coordinate changes 24
S.L6 Determinants 27
Computing determinants 27
S.L7 Matrix diagonalization 29
Matrix diagonalization 29
Functions of matrices 34
S.L8 Unitarity and Hermiticity 35
Unitarity and orthogonality 35
Hermiticity and symmetry 36
Relation between Hermitian and unitary matrices 41
S.L10 Multilinear algebra 42
Direct sum and direct product of vector spaces 42
Dual space 43
Tensors 44
i
ii Contents
Alternating forms 45
Wedge product 45
Inner derivative 46
Pullback 47
Metric structures 47
SC Solutions: Calculus 49
S.C1 Differentiation of one-dimensional functions 49
Definition of differentiability 49
Differentiation rules 49
Derivatives of selected functions 49
S.C2 Integration of one-dimensional functions 51
One-dimensional integration 51
Integration rules 52
Practical remarks on one-dimensional integration 56
S.C3 Partial differentiation 59
Partial derivative 59
Multiple partial derivatives 59
Chain rule for functions of several variables 60
S.C4 Multi-dimensional integration 60
Cartesian area and volume integrals 60
Curvilinear area integrals 63
Curvilinear volume integrals 64
Curvilinear integration in arbitrary dimensions 66
Changes of variables in higher-dimensional integration 68
S.C5 Taylor series 70
Complex Taylor series 70
Finite-order expansion 71
Solving equations by Taylor expansion 72
Higher-dimensional Taylor series 73
S.C6 Fourier calculus 74
The δ-Function 74
Fourier series 79
Fourier transform 81
Case study: Frequency comb for high-precision measurements 85
S.C7 Differential equations 87
Separable differential equations 87
Linear first-order differential equations 90
Systems of linear first-order differential equations 91
Linear higher-order differential equations 98
General higher-order differential equations 99
Linearizing differential equations 100
S.C8 Functional calculus 101
Euler-Lagrange equations 101
iii
S.C9 Calculus of complex functions 103
Holomorphic functions 103
Complex integration 104
Singularities 105
Residue theorem 106
SV Solutions: Vector Calculus 112
S.V1 Curves 112
Curve velocity 112
Curve length 112
Line integral 113
S.V2 Curvilinear Coordinates 114
Cylindrical and spherical coordinates 114
Local coordinate bases and linear algebra 117
S.V3 Fields 118
Definition of fields 118
Scalar fields 119
Extrema of functions with constraints 120
Gradient fields 122
Sources of vector fields 123
Circulation of vector fields 125
Practical aspects of three-dimensional vector calculus 127
S.V4 Introductory concepts of differential geometry 134
Differentiable manifolds 134
Tangent space 134
S.V5 Alternating differential forms 135
Cotangent space and differential one-forms 135
Pushforward and Pullback 136
Forms of higher degree 137
Integration of forms 140
S.V6 Riemannian differential geometry 141
Definition of the metric on a manifold 141
Volume form and Hodge star 142
S.V7 Differential forms and electrodynamics 143
Laws of electrodynamics II: Maxwell equations 143
Invariant formulation 144
SL Problems: Linear Algebra
S.L1 Mathematics before numbers
S.L1.1 Sets and Maps
PL1.1.2 Composition of maps
(a) Since02 =0,(±1)2 =1,(±2)2 =4,theimageofSunderAisT =A(S)= {0,1,4} .
√ √ √
(b) Since 0=0, 1=1, 4=2, the image of T under B is U =B(T)= {0,1,2} .
√
(c) The composite map C =B◦A is given by C :S →U, n7→C(n)= n2 =|n| .
(d) A, B and C are all surjective. B is injective and hence also bijective. A and C are
notinjective,since,e.g.,theelements+2and−2havethesameimageunderA,with
A(±2)=4, and similarly C(±2)=2. Therefore, A and C are also not bijective.
S.L1.2 Groups
PL1.2.2 The groups of addition modulo 5 and rotations by multiples of 72 deg
(a) 0 1 2 3 4 (b) • r(0) r(72) r(144) r(216) r(288)
0 0 1 2 3 4 r(0) r(0) r(72) r(144) r(216) r(288)
1 1 2 3 4 0
r(72) r(72) r(144) r(216) r(288) r(0)
2 2 3 4 0 1
r(144) r(144) r(216) r(288) r(0) r(72)
3 3 4 0 1 2
4 4 0 1 2 3 r(216) r(216) r(288) r(0) r(72) r(144)
Theneutralelementis0. r(288) r(288) r(0) r(72) r(144) r(216)
The inverse element of
n∈Z is 5−n. The neutral element is r(0).
q
The inverse element of r(φ) is r(360−φ).
(c) The groups (Z5, ) and (R72, •) are isomorphic because their group composition
tablesareidenticalifweidentifytheelementnofZ withtheelementr(72n)ofR .
5 72
(d) The group (R360/n, •) of rotations by multiples of 360/n deg is isomorphic to the
group (Z , ) of integer addition modulo n.
n
5
6 S.L1 Mathematics before numbers
PL1.2.4 Group of discrete translations on a ring
(a) Consider the group axioms:
(i) Closure: by definition a,b∈Z⇒(a+b)modN ∈ZmodN. Thus: x,y ∈G⇒
∃n ,n ∈ ZmodN : x = λn ,y = λn . It follows that T(x,y) = λ·(n +
x y x y x
n )modN ∈λ·ZmodN =G. (cid:88)
y
(ii) Associativity: The usual addition of integers is associative, m,n,l∈Z ⇒ (m+
n)+l = m+(n+l), and this property remains true for addition modulo N.
Forx,y,z∈Gwethereforehave:T(T(x,y),z)=λ·((n +n )+n )(modN)=
x y z
λ·(n +(n +n ))(modN)=T(x,T(y,z)). (cid:88)
x y z
(iii) Neutral element: The neutral element is 0=λ·0∈G: For all x∈G we have:
T(x,0)=λ·(n +0)(modN)=x. (cid:88)
x
(iv) Inverse element: The inverse element of n ∈ ZmodN is [N +(−n)]modN ∈
ZmodN.Thereforetheinverseelementofx=λ·n∈Gisgivenby−x≡λ·(N+
(−n))∈G,sinceT(x,−x)=λ·(n+(N+(−n)))(modN)=λ·0(modN)=0.(cid:88)
(v) Commutativity(forthegrouptobeabelian):Forallx,y∈GwehaveT(x,y)=
λ·(n +n )modN = λ·(n +n )modN = T(y,x), since the usual addition
x y y x
of real numbers is commutative, and this property remains true for addition
modulo N. (cid:88)
Since (G,T) satisfies properties (i)-(v), it is an abelian group. (cid:88)
(b) The group axioms of (T, ) follow directly from those of (G,T):
(i) Closure: T ,T ∈ T ⇒ T T = T ∈ T, since x,y ∈ G ⇒ T(x,y) ∈ G
x y x y T(x,y)
[see (a)]. (cid:88)
(ii) Associativity: For T ,T ,T ∈T we have: (T T ) T =T T =
x y z x y z T(x,y) z
T (=a)T =T T =T (T T ). (cid:88)
T(T(x,y),z) T(x,T(y,z)) x T(y,z) x y z
(iii) Neutral element: The neutral element is T ∈ T: For all T ∈ T we have:
0 x
T T =T =T =T . (cid:88)
x 0 T(x,0) x+0 x
(iv) Inverse element: The inverse element of T ∈ T is T ∈ T, where −x is the
x −x
inverse element of x ∈ G with respect to T, since T T = T =
x −x T(x,−x)
T =T . (cid:88)
x+(−x) 0
(v) Commutativity(forthegrouptobeabelian):Forallx,y∈GwehaveT T =
x y
T =T =T T ,sincethecompositionruleT inGiscommutative.(cid:88)
T(x,y) T(y,x) y x
Since (T, ) satisfies properties (i)-(v), it is an abelian group. (cid:88)
PL1.2.6 Decomposing permutations into sequences of pair permutations
(a) The permutation [132] is itself a pair permutation, as only the elements 2 and 3 are
exchanged, hence its parity is odd.
(b) To obtain 123 7−[2→31] 231 via pair permutations, we bring the 2 to the first slot, then
the 3 to the second slot: 123 7−[2→13] 213 7−[3→21] 231, thus [231] = [321]◦[213], with even
parity.
Below we proceed similarly: we map the naturally-ordered string into the desired order
one pair permutation at a time, moving from front to back:
7 S.L1.3 Fields
(c) 1234[7−32→14]3214[7−14→32]3412 ⇒ [3412]=[1432]◦[3214] even
(d) 1234[7−32→14]3214[7−14→32]3412[7−21→34]3421 ⇒ [3421]=[2134]◦[1432]◦[3214] odd
(e) 12345[17−5→342]15342[17−3→245]15243[17−2→435]15234 ⇒ [15234]=[12435]◦[13245]◦[15342]
odd
(f) 12345[37−2→145]32145[27−1→345]31245[17−5→342]31542 ⇒ [31542]=[15342]◦[21345]◦[32145]
odd
S.L1.3 Fields
PL1.3.2 Complex numbers – elementary computations
For z =3+ai and z =b−2i (a,b∈R), we find [brackets give results for a=2, b=3]:
1 2
(a) z¯ =3−ai [3−2i]
1
(b) z −z =3−b+(a+2)i [4i]
1 2
(c) z z¯ =(3+ai)·(b+2i)=−2a+3b+i(6+ab) [5+12i]
1 2
z¯ 3−ai (3−ai)(b+2i) 2a+3b+i(6−ab)
(d) 1 = = = [1]
z b−2i (b−2i)(b+2i) b2+4
2
p √
(e) |z |= 9+a2 [ 13]
1
(f) |bz −3z |=|i(ab+6)|=|ab+6| [12]
1 2
PL1.3.4 Algebraic manipulations with complex numbers
(a) (z+i)2 =(x+i(y+1))2 = x2−(y+1)2+i2x(y+1) ,
z z z¯+1 (x+iy) (x+1−iy)
(b) = · = ·
z+1 z+1 z¯+1 (x+1+iy) (x+1−iy)
x(x+1)+y2+i(y(x+1)−xy) x(x+1)+y2+iy
= = ,
(x+1)2+y2 (x+1)2+y2
z¯ z¯ z¯+i (x−iy) (x+i(1−y))
(c) = · = ·
z−i z−i z¯+i (x+i(y−1)) (x−i(y−1))
x2+y(1−y)+i(x(1−y)−yx) x2+y(1−y)+ix(1−2y)
= = .
x2+(y−1)2 x2+(y−1)2
PL1.3.6 Multiplying complex numbers – geometrical interpretation
q
(cid:0) (cid:1)
z1 = √18+√18i7→(√18,√18) ρ1 = 18+18 = 12 φ1 =arctan 1 = π4
√ √ √ (cid:0) (cid:1)
z2 = 3−i7→( 3,−1) ρ2 = 3+1=2 φ2 =arctan −√13 = 116π
√ q (cid:0)√ (cid:1)
z3 =z1z2 =(√18+√18i)( 3−i) ρ3 = 38+18+38+18 =1 φ3 =arctan √33−+11 = 1π2
8 S.L2 Vector spaces
q q q q q q q q
= 3 + 1 +( 3 − 1)i7→( 3 + 1, 3 − 1)
8 8 8 8 8 8 8 8
z = 1 = √8 = √8(1−i) ρ =√2+2=2 φ =arctan(cid:0)−1(cid:1)= 7π
4 √z1 1√+i (1+√i)(1−i√) 4 4 4
= 2− 2i7→( 2,− 2)
q
(cid:0) (cid:1)
z5 =z¯1 = √18 − √18i7→(√18,−√18) ρ5 = 18 + 18 = 21 φ5 =arctan −1 = 74π
As expected, we find: Im(z)
ρ3 =ρ1ρ2 21 z1 1 z3=z1z2
φ3 =φ1+φ2 π/4
π/12
ρ =1/ρ
4 1
π/6 Re(z)
φ =−φ 7π/4
4 1 π/4
ρ5 =ρ1 11π/6
φ5 =−φ1 z =z¯ 2
5 1
2
z
2
z = 1
4 z1
S.L2 Vector spaces
S.L2.3 Vector spaces: examples
PL2.3.2 Vector space axioms: complex numbers
First, we show that (C,+) forms an abelian group. Notation: z =x +iy , j =1,2,3.
j j j
(i) Closure holds by definition: z +z ≡(x +x )+i(y +y )∈C. (cid:88)
1 2 1 2 1 2
(ii) Associativity: (z +z )+z =((x +x )+i(y +y ))+(x +iy )
1 2 3 1 2 1 2 3 3
=((x +x )+x )+i((y +y )+y )
1 2 3 1 2 3
=(x +(x +x ))+i(y +(y +y ))
1 2 3 1 2 3
=z +(z +z ). (cid:88)
1 2 3
(iii) Neutral element: z+0=(x+0)+i(y+0)=x+iy=z. (cid:88)
(iv) Additive inverse: with −z=(−x)+i(−y)
is z+(−z)=(x−x)+i(y−y)=0. (cid:88)
(v) Commutativity: z +z =(x +x )+i(y +y )
1 2 1 2 1 2
=(x +x )+i(y +y )=z +z . (cid:88)
2 1 2 1 2 1
Second, we show that multiplication by a real number, · , likewise has the properties
required for (C,+,·) to form a vector space. For λ,µ∈R, we have
(vi) Closure: λz=λ(x+iy)=(λx)+i(λy)∈C, since λx,λy∈R
9 S.L2.3 Vector spaces: examples
(vi) Multiplication of a sum of scalars and a complex number is distributive:
(λ+µ)·z=(λ+µ)(x+iy)=(λ+µ)x+i(λ+µ)y
=(λx+µx)+i(λy+µy)=λz+µz. (cid:88)
(vii) Multiplication of a scalar and a sum of complex numbers is distributive:
λ(z +z )=λ((x +x )+i(y +y ))=λ(x +x )+iλ(y +y )
1 2 1 2 1 2 1 2 1 2
=(λx +λx )+i(λy +λy )=λz +λz . (cid:88)
1 2 1 2 1 2
(viii)Multiplication of a product of scalars and a complex number is associative:
λ·(µ·z)=λ·(µx+iµy)=(λµ)x+i(λµ)y=λµ(x+iy)=(λµ)·z. (cid:88)
(ix) Neutral element: 1·z=1·(x+iy)=z. (cid:88)
Therefore, the triple (C,+,·) represents an R-vector space.
Remark: This R-vector space is two-dimensional (i.e. isomorphic to R2), since each com-
plex number z=x+iy is represented by two real numbers, x and y. This fact is utilized
when representing complex numbers as points in the complex plane, with coordinates x
and y in the horizontal and vertical directions, respectively.
PL2.3.4 Vector space of polynomials of degree n
(a) The definition of addition of polynomials and the usual addition rule in R yield
p (x)+p (x)=a x0+a x1+...a xn+b x0+b x1+...b xn
a b 0 1 n 0 1 n
=(a +b )x0+(a +b )x1+...(a +b )xn =p (x),
0 0 1 1 n n a+b
since a+b=(a +b ,...,a +b )T ∈Rn+1. Therefore p p =p . (cid:88)
0 0 n n a b a+b
The definition for the multiplication of a polynomial with a scalar and the usual
multiplication rule in R yield
cp (x)=c(a x0+a x1+...a xn)=ca x0+ca x1+...ca xn =p (x),
a 0 1 n 0 1 n ca
since ca=(ca0,...,can)T ∈Rn+1. Therefore c•pa =pca . (cid:88)
(b) We have to verify that all the axioms for a vector space are satisfied. First, (P , )
n
indeed has all the properties of an abelian group:
(i) Closure:addingtwopolynomialsofdegreenagainyieldsapolynomialofdegree
at most n. (cid:88)
(ii,v)Associativity and commutativity follow trivially from the corresponding prop-
erties of Rn+1. For example, consider associativity:
p (p p )=p p =p =p =p p =(p p ) p .(cid:88)
a b c a b+c a+(b+c) (a+b)+c a+b c a b c
(iii) The neutral element is the null polynomial p , i.e. the polynomial whose coef-
0
ficients are all equal to 0. (cid:88)
(iv) The additive inverse of p is p . (cid:88)
a −a
10 S.L2 Vector spaces
Moreover, multiplication of any polynomial with a scalar also has all the properties
requiredfor(Pn, ,•)tobeavectorspace.Multiplicationwithascalarc∈Rsatisfies
closure, since c•pa =pca again yields a polynomial of degree n. (cid:88) All the rules for
multiplication by scalars follow directly from the corresponding properties of Rn+1.
(cid:88)
Each element p ∈P is uniquely identified by the element a ∈Rn+1 – this identi-
a n
ficationis a bijectionbetweenPn andRn+1, hence(Pn, ,•) isisomorphic toRn+1
and has dimension n+1. (cid:88)
(c) The bijection between P and Rn+1 associates the standard basis vectors in Rn+1,
n
namelye =(0,...1,...,0)T (witha1atpositionk and0≤k≤n),withabasisin
k
the vector space (Pn, ,•), namely {pe0,...,pen}, corresponding to the monomials
{1,x,x2,...,xn}, since p (x)=xk. This statement corresponds to the obvious fact
ek
thateverypolynomialofdegreencanbewrittenaslinearcombinationofmonomials
of degree ≤n.
PL2.3.6 Vector space with unusual composition rule – multiplication
(a) First, we show that (V , ) forms an abelian group.
a
(i) Closure holds by definition. (cid:88)
(ii) Associativity: (v v ) v =v v =v =v
x y z x+y−a z (x+y−a)+z−a x+y+z−2a
=v =v v =v (v v ).(cid:88)
x+(y+z−a)−a x y+z−a x y z
(iii) Neutral element: v v =v =v , ⇒ 0=v .(cid:88)
x a x+a−a x a
(iv) Additive inverse: v v =v =v =0, ⇒ −v =v .(cid:88)
x −x+2a x+(−x+2a)−a a x −x+2a
(v) Commutativity: v v =v =v =v v . (cid:88)
x y x+y−a y+x−a y x
(b) To ensure that the triple (V , ,·) forms an R-vector space, scalar multiplication,
a
·,whichbydefinitionsatisfiesclosure,alsohastohavethefollowingfourproperties,
each of which amounts to a condition on the form of f:
(vi) Multiplication of a sum (γ+λ)·v =γ·v λ·v
x x x
of scalars and a vector v =v
(γ+λ)x+f(a,γ+λ) γx+f(a,γ)+λx+f(a,λ)−a
is distributive: f(a,γ+λ)=f(a,γ)+f(a,λ)−a (1a)
(vii) Multiplication of a scalar λ·(v +v )=λ·v λ·v
x y x y
and a sum of vectors v =v
λ(x+y−a)+f(a,λ) λx+f(a,λ)+λy+f(a,λ)−a
is distributive: −λa+f(a,λ)=f(a,λ)+f(a,λ)−a (1b)
(viii)Multiplication of a product (γλ)·v =γ·(λ·v )
x x
of scalars and a vector v =v
(γλ)x+f(a,γλ) γ(λx+f(a,λ))+f(a,γ)
is associative: f(a,γλ)=γf(a,λ)+f(a,γ) (1c)
(ix) Neutral element: 1·v =v
x x
v =v
x+f(a,1) x
f(a,1)=0 (1d)
Evidently,theformoff isfullydeterminedbythedistributivitycondition(vii),since