B
,... .
(2.12b)
1
2
2
6
3
4
30
5
6
42
7
8
30
3. Basic trigonometric functions
– Trigonometric functions of the sum and the difference of two arguments: 3
cos a b cos a cos b sin a sin b , (3.1a)
sin a b sin a cos b cos a sin b .
(3.1b)
– Sums of two functions of arbitrary arguments:
a b
b a
cos a cos b 2cos
cos
,
(3.2a)
2
2
a b
b a
cos a cos b 2sin
sin
,
(3.2b)
2
2
a b
b a
sin a sin b 2sin
cos
.
(3.2c)
2
2
– Trigonometric function products:
2cos a cos b cos( a b) cos( a b) , (3.3a)
2sin a cos b sin( a b) sin( a b) , (3.3b)
2sin a sin b cos( a b) cos( a b) ; (3.3c)
for the particular case of equal arguments, b = a, these three formulas yield the following expressions for the squares of trigonometric functions, and their product:
2
1
1
2
1
cos a 1 cos 2 a,
sin a cos a sin 2 a,
sin a 1 cos 2 a .
(3.3d)
2
2
2
– Cubes of trigonometric functions:
3
3
1
3
3
1
cos a cos a cos3 a,
sin a sin a sin 3 .
a
(3.4)
4
4
4
4
– Trigonometric functions of a complex argument:
2 Note that definitions of Bk (or rather their signs and indices) vary even in the most popular handbooks.
3 I am confident that the reader is quite capable of deriving the relations (3.1) by representing exponent in the elementary relation ei( a b) = eiae ib as a sum of its real and imaginary parts, then Eqs. (3.3) directly from Eqs. (3.1), and then Eqs. (3.2) from Eqs. (3.3) by variable replacement; however, I am still providing these formulas to save their time. (Quite a few formulas below are included for of the same reason.)
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sin( a ib) sin a cosh b i cos a sinh b, (3.5)
cos ( a ib) cos a cosh b i sin a sinh .
b
– Sums of trigonometric functions of n equidistant arguments:
n sin
sin
n 1 n
k
sin sin .
(3.6)
k 1 cos
cos
2
2
2
4. General differentiation
– Full differential of a product of two functions:
d( fg) ( df ) g f ( dg).
(4.1)
– Full differential of a function of several independent arguments, f(1, 2,…, n): n
f
df
d k .
(4.2)
k 1
k
– Curvature of the Cartesian plot of a smooth function f():
2
2
d f /
1
d
.
(4.3)
R
1 df / d23/2
5. General integration
– Integration by parts:4
g ( B)
f ( B)
f dg fg B
.
(5.1)
A
g df
g ( A)
f ( A)
– Numerical (approximate) integration of 1D functions: the simplest trapezoidal rule, b
h
3 h
h
N
h
b a
f ( ) d h f a f a
... f b h
f
a nh ,
h
. (5.2)
2
2
2
a
n 1
2
N
has a relatively low accuracy (error of the order of ( h 3/12) d 2 f/ d2 per step), so that the following Simpson formula,
b
h
b a
f ( ) d f ( a) 4 f ( a h) 2 f ( a 2 h) ... 4 f ( b h) f ( b), h
,
(5.3)
3
2 N
a
whose error per step scales as ( h 5/180) d 4 f/ d4, is used much more frequently.5
4 This formula immediately follows from Eq. (4.1).
5 Higher-order formulas (e.g., the Bode rule), and other guidance including ready-for-use codes for computer calculations may be found, for example, in the popular reference texts by W. H. Press et al., cited in Sec. 16
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6. A few 1D integrals6
(i) Indefinite integrals
– Integrals with (1 + 2)1/2:
1
1 2
1/2
d 1 2
1/2 ln 1 2
1/2 ,
(6.1)
2
2
d
ln 1 2
1/2
,
(6.2a)
1 2
1/2
d
.
(6.2b)
2
1
3 / 2
1 2
1/2
– Miscellaneous indefinite integrals:
d
a
1
1
,
(6.3a)
cos
2
2 a
1/2
1
2
a 1/2
1
sin cos2
2
2 sin 2 cos 2 2 1
d
,
(6.3b)
5
4
8
d
2
1
a b
2
2
a b
(6.3c)
a b cos
tan
tan
,
for
2
2
a b 1/2
2
2
a b 1/2
2
1
tan
d
.
(6.3d)
1 2
(ii) Semi-definite integrals:
– Integrals with 1/( e 1):
d
ln
a
1 e
,
(6.4a)
a e
1
d
1
ln
.
a
(6.4b)
a
e
e
0
1
1
(iii) Definite integrals
– Integrals with 1/(1 + 2):7
d
,
(6.5a)
1
2
2
0
below. Besides that, some advanced codes are used as subroutines in the software packages listed in the same section. In some cases, the Euler-Maclaurin formula (2.12) also may be useful for numerical integration.
6 A powerful (and free :-) interactive online tool for working out indefinite 1D integrals is available at http://integrals.wolfram.com/index.jsp.
7 Eq. (6.5a) follows immediately from Eq. (6.3d), and Eq. (6.5b) from Eq. (6.2b) – a couple more examples of the (intentional) redundancies in this list.
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d
;
(6.5b)
3 / 2
2
0
1
1
more generally,
d
2 n
3 !
!
13
...
5 2 n
3
n
(6.5c)
2
n
n
n
0 1
2 2 2!! 2 2 4
...
6 2
2,
for
,
2 ,...
3
– Integrals with (1 – 2 n)1/2:
1
1/ 2
d
1
n 1
,
(6.6a)
1/ 2
2
1
n
2 n
2 n
2 n
0
1
1/ 2
1/ 2
n
1
3 n 1
2
1 d
,
(6.6b)
4 n
2 n
2 n
0
where ( s) is the gamma function, which is most often defined (for Re s > 0) by the following integral:
s1
e d ( s) .
(6.7a)
0
The key property of this function is the recurrence relation, which is valid for any s 0, –1, –2,…:
( s )
1 s( s) .
(6.7b)
Since, according to Eq. (6.7a), (1) = 1, Eq. (6.7b) for non-negative integers takes the form
( n )
1 n ,
!
for n ,
0 ,
1 ,...
2
(6.7c)
(where 0! 1). Because of this, for integer s = n + 1 1, Eq. (6.7a) is reduced to
n
e d !
n .
(6.7d)
0
Other frequently met values of the gamma function are those for positive semi-integer values:
1
1/ 2
3 1 1/2
5 1 3 1/2
7 1 3 5
,
,
,
1/ 2
, ... .
(6.7e)
2
2 2
2 2 2
2 2 2 2
– Integrals with 1/( e 1):
s 1
d
121 s( s)( s), for s 0,
(6.8a)
0 e
1
s 1
d
( s) ( s), for s ,
1
(6.8b)
0 e
1
where ( s) is the Riemann zeta-function – see Eq. (2.6). Particular cases: for s = 2 n,
2 n 1
d
22 n 1
1
2 n
B ,
(6.8c)
2 n
e
n
0
1
2
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2 n 1
2 n
d
(2 )
B
.
(6.8d)
2 n
1
4
0 e
n
where Bn are the Bernoulli numbers – see Eq. (2.12). For the particular case s = 1 (when Eq. (6.8a) yields uncertainty),
d
ln 2
.
(6.8e)
0 e
1
– Integrals with exp{– 2}:
2
s
1 s 1
e
d
,
for s 1
;
(6.9a)
2 2
0
for applications the most important particular values of s are 0 and 2:
2
1 1
1/ 2
e
d
,
(6.9b)
2 2
2
0
2
2
1 3
1/ 2
e
d
,
(6.9c)
2 2
4
0
though we will also run into the cases s = 4 and s = 6:
2
1/ 2
4
1 5 3 1/2
2
6
1 7 15
e
d
,
e
d
;
(6.9d)
2 2
8
0
2 2
16
0
for odd integer values s = 2 n + 1 (with n = 0, 1, 2,…), Eq. (6.9a) takes a simpler form:
2
2 n 1
1
!
n
e
d n
1
.
(6.9e)
2
2
0
– Integrals with cosine and sine functions:
1/
cos
2
d sin
2
2
d .
(6.10)
8
0
0
a
cos
d
e
.
(6.11)
a 2 2
2 a
0
2
sin
d
.
(6.12)
2
0
– Integrals with logarithms:
1
a 1
2
1/2
ln
d a a
a
(6.13)
1/ 2
2 11/2
2
0
a 1
, for
.
1
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1
1 1 1/ 2
ln
d 1
.
(6.14)
1/ 2
0
– Integral representations of the Bessel functions of integer order:
1
i sin
n
i sin
J
( )
e
d ,
e
that
so
J
;
(6.15a)
n
k
ik
e
2
k
1
I
( )
.
(6.15b)
n
cos
e
cos n d
0
7. 3D vector products
(i) Definitions:
– Scalar (“dot-“) product:
3
a b a b ,
(7.1)
j
j
j1
where aj and bj are vector components in any orthogonal coordinate system. In particular, the vector squared (the same as its norm squared) is the following scalar:
3
2
2
2
a a a a a .
(7.2)
j
j 1
– Vector (“cross-”) product:
n
n
n
1
2
3
a b n ( a b a b ) n ( a b a b ) n ( a b a b ) a a
a ,
(7.3)
1
2 3
3 2
2
3 1
1 3
3
1 2
2 1
1
2
3
b
b
b
1
2
3
where {n j} is the set of mutually perpendicular unit vectors8 along the corresponding coordinate system axes.9 In particular, Eq. (7.3) yields
a a 0.
(7.4)
(ii) Corollaries (readily verified by Cartesian components):
– Double vector product (the so-called bac minus cab rule):
a (b c) b(a c) c(a b) .
(7.5)
– Mixed scalar-vector product (the operand rotation rule):
a b c b c a c a b.
(7.6)
8 Other popular notations for this vector set are { e } and { rˆ }.
j
j
9 It is easy to use Eq. (7.3) to check that the direction of the product vector corresponds to the well-known “right-hand rule” and to the even more convenient corkscrew rule: if we rotate a corkscrew's handle from the first operand toward the second one, its axis moves in the direction of the product.
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– Scalar product of vector products:
a b c d a cb d a db c; (7.7a)
in the particular case of two similar operands (say, a = c and b = d), the last formula is reduced to
a b2
2
2
( ab) (a b) .
(7.7b)
8. Differentiation in 3D Cartesian coordinates
– Definition of the del (or “nabla”) vector-operator : 10
3
n
,
(8.1)
j
j1
rj
where rj is a set of linear and orthogonal ( Cartesian) coordinates along directions n j. In accordance with this definition, the operator acting on a scalar function of coordinates, f(r),11 gives its gradient, i.e. a new vector:
3
f
f n
grad f .
(8.2)
j
j 1
rj
– The scalar product of del by a vector function of coordinates (a vector field), 3
f (r) n f (r) ,
(8.3)
j
j
j1
compiled by formally following Eq. (7.1), is a scalar function – the divergence of the initial function: 3 f
f j
f
div ,
(8.4)
j
1
rj
while the vector product of and f, formed in a formal accordance with Eq. (7.3), is a new vector – the curl (in European tradition, called rotor and denoted rot) of f: n
n
n
1
2
3
f
f
f
f
f
f
3
2
1
3
2
1
f
n
n
n
curl f.
(8.5)
1
2
3
r
r
r
r
r
r
r
r
r
1
2
3
2
3
3
1
1
2
f
f
f
1
2
3
– One more frequently met “product” is (f)g, where f and g are two arbitrary vector functions of r. This product should be also understood in the sense implied by Eq. (7.1), i.e. as a vector whose j th Cartesian component is
3
g
f g
f
(8.5)
j
j .
j'
j' 1
rj'
10 One can run into the following notation: /r, which is convenient in some cases, but may be misleading in quite a few others, so it will be not used in this series.
11 In this, and four next sections, all scalar and vector functions are assumed to be differentiable.
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9. The Laplace operator 2
– Expression in Cartesian coordinates – in the formal accordance with Eq. (7.2):
3
2
2
.
(9.1)
2
j1 rj
– According to its definition, the Laplace operator acting on a scalar function of coordinates gives a new scalar function:
3
2
2 f
f f
div
grad f
.
(9.2)
2
j1 rj
– On the other hand, acting on a vector function (8.3), the operator 2 returns another vector: 3
2f n 2 f .
(9.3)
j
j
j1
Note that Eqs. (9.1)-(9.3) are only valid in Cartesian (i.e. orthogonal and linear) coordinates, but generally not in other orthogonal coordinates – see, e.g., Eqs. (10.3), (10.6), (10.9) and (10.12) below.
10. Operators and 2 in the most important systems of orthogonal coordinates12
(i) Cylindrical 13 coordinates {, , z} (see Fig. below) may be defined by their relations with the Cartesian coordinates:
r z
3
r
r cos,
1
r 2 r sin,
(10.1)
0
2
r .
z
3
r 1
– Gradient of a scalar function:
f
1 f
f
f n
n
n
.
(10.2)
z
z
– The Laplace operator of a scalar function:
2
1
f
1
2
2
f
f
f
,
(10.3)
2
2
2
z
– Divergence of a vector function of coordinates (f = n f + n f + n z fz): 1 f
1 f
f
z
f
.
(10.4)
z
12 Some other orthogonal curvilinear coordinate systems are discussed in EM Sec. 2.3.
13 In the 2D geometry with fixed coordinate z, these coordinates are called polar.
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– Curl of a vector function:
1 f
f
1
(
)
z
f f
f
z
f
f n
n
n
.
(10.5)
z
z
z
– The Laplace operator of a vector function:
1
2 f
1
2 f
2
2
f n f
f
2
n f
f
2
n f
.
(10.6)
2
2
2
2
z
z
(ii) Spherical coordinates { r, , } (see Fig. below) may be defined as: r 3
r r sin cos,
1
r r sin sin,
(10.7)
2
r
r
0
2
r r cos.
3
r 1
– Gradient of a scalar function:
f
1 f
1
f
f n
n
n
.
(10.8)
r
r
r
r sin
– The Laplace operator of a scalar function:
2
1
f
1
f
1
f
2
2
f
r
sin
.
(10.9)
2
2
2
2
r r
r r sin
( r sin )
– Divergence of a vector function f = n r fr + n f + n f : 1 r 2 f
1
(
sin )
1
r
f
f
f
.
(10.10)
r 2
r
r sin
r sin
– Curl of the similar vector function:
1
( f sin
) f
1 1 f
( rf )
1
( rf )
f
r
r
f n
n
n
. (10.11)
r
r sin
r sin
r
r r
– The Laplace operator of a vector function:
f
2
2
2
2
2
f n f
f
( f sin )
r
r
2
r
2
r
r sin
2
r sin
fr
f
2
1
2
2cos
n f
f
(10.12)
2
r sin 2
2
2
r
r sin 2
fr
f
2
1
2
2cos
n f
f
.
2
r sin 2
2
r sin
2
r sin2
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11. Products involving
(i) Useful zeros:
– For any scalar function f (r) ,
f curl (
grad f ) 0 .
(11.1)
– For any vector function f (r) ,
f
(
div
curl f ) 0.
(11.2)
(ii) The Laplace operator expressed via the curl of a curl:
2f f f .
(11.3)
(iii) Spatial differentiation of a product of a scalar function by a vector function:
– The scalar 3D generalization of Eq. (4.1) is
f g f g f g.
(11.4a)
– Its vector generalization is similar:
f g f g f g.
(11.4b)
(iv) 3D spatial differentiation of products of two vector functions:
f g f g f g f g g
f ,
(11.5)
f g f g g f f g g f , (11.6)
f g g f f g .
(11.7)
12. Integro-differential relations
(i) For an arbitrary surface S limited by closed contour C:
– The Stokes theorem, valid for any differentiable vector field f(r):
f d 2r f
d 2 r f dr
f dr ,
(12.1)
n
S
S
C
C
where d 2r n d 2 r is the elementary area vector (normal to the surface), and dr is the elementary contour length vector (tangential to the contour line).
(ii) For an arbitrary volume V limited by closed surface S:
– Divergence (or “Gauss”) theorem, valid for any differentiable vector field f(r):
f
d 3 r f d 2r f d 2 r .
(12.2)
n
V
S
S
– Green’s theorem, valid for two differentiable scalar functions f(r) and g(r):
f 2 g
g 2 f d 3 r f g g f
d 2 r .
(12.3)
n
V
S
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– An identity valid for any two scalar functions f and g, and a vector field j with j = 0 (all differentiable):
f (j g) g(j f
d 3
)
r fgj d 2 r .
(12.3)
n
V
S
13. The Kronecker delta and Levi-Civita permutation symbols
– The Kronecker delta symbol (defined for integer indices):
,
1 if j' j
,
(13.1)
jj'
,0 otherwise.
– The Levi-Civita permutation symbol for three integer indices (each taking one of the values 1, 2, or 3):
,
1
if
"
any
in
follow
indices
the
("
correct" even" order
)
1
: 2 3 1 2...,
(13.2)
jj' j"
,
1
if
"
any
in
follow
indices
the
("
incorrect" odd" order
)
1
: 3 2 1 3...,
,0 if
coincide.
indices
any two
– Relation between the products of the Levi-Civita and Kronecker symbols:
δ jl
δjl'
δ jl"
3
jj'j" kk'k"
δj'l δj'l' δj'l" ;
(13.3a)
l, l' , l" 1 δj"l δj"l' δj"l"
the summation of three such relations written for three different values of j = k, yields the so-called contracted epsilon identity:
3
δ δ
δ δ .
(13.3b)
jj'j"
jk'k"
j'k'
j"k"
j'k"
j"k'
j1
14. Dirac’s delta function, sign function, and theta function
– Definition of 1D delta function (for real a < b):
b
f (0), if a 0 b,
f ( ) ( ) d
(14.1)
a
,
0
otherwise,
where f() is any function continuous near = 0. In particular (if f() = 1 near = 0), the definition yields
b
,
1 if a 0 ,
b
( ) d
(14.2)
a
,
0
otherwise.
– Relation to the theta function () and the sign function sgn() d
1 d
( )
( )
sgn( ) ,
(14.3a)
d
2
d
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where
sgn( ) 1 ,
0
if ,
0
,
1 if ,
0
( )
sgn( )
(14.3b)
2
,
1 if ,
1
,
1 if .
1
– An important integral:14
eis ds 2 ( )
.
(14.4)
– 3D generalization: the delta function (r) of the radius-vector is defined as
f
V
3
( ),
0
if 0
,
f r r d r
(14.5)
V
,
0
otherwise;
it may be represented as a product of 1D delta functions of Cartesian coordinates:
(r) ( r ) ( r ) ( r ) .
(14.6)
1
2
3
(The 2D generalization is similar.)
15. The Cauchy theorem and integral
Let a complex function f(z) be analytic within a part of the complex plane z, which is limited by a closed contour C and includes point z ’. Then
f(z) dz 0
,
(15.1)
C
dz
f(z)
2 ifz '
.
(15.2)
z z '
C
The first of these relations is usually called the Cauchy integral theorem (or the “Cauchy-Goursat theorem”), and the second one, the Cauchy integral (or the “Cauchy integral formula”).
16. References
(i) Properties of some special functions are briefly discussed at the relevant points of the lecture notes (in alphabetical order):
– Airy functions: QM Sec. 2.4;
– Bessel functions: EM Sec. 2.7;
– Fresnel integrals: EM Sec. 8.6;
– Hermite polynomials: QM Sec. 2.9;
14 The coefficient in this relation may be readily recalled by considering its left-hand side as the Fourier-integral representation of the function f( s) 1, and applying Eq. (14.1) to the reciprocal Fourier transform: 1
f ( s) 1
is
e
2()
d .
2
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– Laguerre polynomials (both simple and associated): QM Sec. 3.7;
– Legendre polynomials, associated Legendre functions: EM Sec. 2.8 and QM Sec. 3.6;
– Spherical harmonics: QM Sec. 3.6;
– Spherical Bessel functions: QM Secs. 3.6 and 3.8.
(ii) For more formulas, and their discussions, I can recommend the following handbooks (in alphabetical order):15
– M. Abramowitz and I. Stegun (eds.), Handbook of Mathematical Formulas, Dover, 1965 (and numerous later printings);16
– I. Gradshteyn and I. Ryzhik, Tables of Integrals, Series, and Products, 5th ed., Acad. Press, 1980;
– G. Korn and T. Korn, Mathematical Handbook for Scientists and Engineers, 2nd ed., Dover, 2000;
– A. Prudnikov et al., Integrals and Series, vols. 1 and 2, CRC Press, 1986.
The popular textbook,
– G. Arfken et al., Mathematical Methods for Physicists, 7th ed., Acad. Press, 2012, may be also used as a formula manual.
Many formulas are also available from the symbolic calculation parts of commercially available software packages listed in Sec. (iv) below.
(iii) Probably the most popular collection of numerical calculation codes are the twin manuals
– W. Press et al. , Numerical Recipes in Fortran 77, 2nd ed., Cambridge U. Press, 1992;
– W. Press et al. , Numerical Recipes [in C++ – KKL], 3rd ed., Cambridge U. Press, 2007.
These lecture notes include very brief introductions into numerical methods of differential equation solution:
– ordinary differential equations: CM Sec. 5.7, and
– partial differential equations: CM Sec. 8.5 and EM Sec. 2.11,
which include references to the literature for further reading.
(iv) The most popular software packages for numerical and symbolic calculations, all with plotting capabilities (in alphabetical order):
– Maple (http://www.maplesoft.com/);
– MathCAD (http://www.ptc.com/products/mathcad/);
– Mathematica (http://www.wolfram.com/products/mathematica/index.html);
– MATLAB (http://www.mathworks.com/products/matlab/).
15 On a personal note, perhaps 90% of all formula needs throughout my research career were satisfied by a tiny, wonderfully compiled old book: H. Dwight, Tables of Integrals and Other Mathematical Data, 4th ed., Macmillan, 1961, whose used copies, rather amazingly, are still available on the Web.
16 An updated version of this collection is now available online at http://dlmf.nist.gov/.
Selected
Mathematical
Formulas Page
16
of
16
Essential Graduate Physics
Lecture Notes and Problems
Open online access at
http://commons.library.stonybrook.edu/egp/
and
https://sites.google.com/site/likharevegp/
Appendix CA
Selected Physical Constants
according to the 2018 International CODATA recommendation.1
Last corrections: 2021/08/20
Symbol
Constant
SI value
Gaussian value
Relative r.m.s.
and unit
and unit
uncertainty
c
speed of light
2.997 924 58
108
2.997 924 581010
0
in vacuum
m/s
cm/s
(defined value)
N
6.022 140 761023
6.022 140 761023
0
A
Avogadro
constant
1/mol
1/mol
(defined value)
Planck
6.626 070 15×10−34
6.626 070 15×10−27
0
2
constant
J/Hz
erg/Hz
(defined value)
1.380 649 000
k
10–23
1.380 649 00010–16
0
B
Boltzmann
constant
J/K
erg/K
(defined value)
1.602 176 634
e
elementary
10–19
4.803 204 713×10−10
0
electric charge
C
statcoulomb
(defined value)
electric
8.854 187 8128
10–12
0
–
~1.510–10
constant
F/m
magnetic
1.256 637 062 12 10–6
0
–
~1.510–10
constant
N/A2
m
0.910 938 370×10−30
0.910 938 370×10−27
e
electron’s
~310–10
rest mass
kg
g
1.672 621 923×10−27
1.672 621 923×10−24
m p
proton’s
~310–10
rest mass
kg
g
6.674 30×10−11
6.674 30×10−8
G
gravitation
~210–5
constant
m3/kgs2
cm3/gs2
See, e.g., http://physics.nist.gov/cuu/Constants/index.html. CODATA is an interdisciplinary Committee on Data for Science and Technology of the International Council of Science (ISCU). Its recommendations, renewed every four years, are widely accepted by the scientific community.
© K. Likharev
Graduate
Physics
CA: Constant Appendix
Comments:
1. The fixed value of c transfers the legal definition of the second (as “the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium-133 atom”) to that of the meter. These values are back-compatible with the legacy definitions of the meter (initially, as the 1/40,000,000th part of the Earth meridian length) and the second (for a long time, as the 1/(246060) = 1/86,400th part of the Earth rotation period), within the experimental errors of those measures.
2. The exact value of the Avogadro number, prescribed by the last CODATA adjustment of fundamental constants in 2018, fixes 1kg in the atomic units of mass (u), defined as 1/12 of the 12C
atom’s mass, excluding the legacy etalons of the kilogram from the primary metrology – even though their masses are compatible with the new definition within the experimental accuracy.
3. The exact value of , also prescribed by CODATA in 2018, together with the fixed value of the second, enables the fundamental definition of energy units (in the SI system, the Joule) in terms of time/frequency.
4. The only role of the Boltzmann constant k B is to express the kelvin (K) in energy units. If temperature is used in these units (as is done, for example, in the SM part of this series), this constant is unnecessary.
5. 0 and 0 are also not really the fundamental constants; their role is just to fix electric and magnetic units in the SI system. Their product is exactly fixed as 00 1/ c 2, and 0 virtually coincides with the legacy value 410–7. (Before the 2018 adjustment, that value was considered exact, but the exact fixation of e in the new system of constants gives it an experimental uncertainty, if only a very small one – see the table above.)
6. The dimensionless fine structure (“Sommerfeld’s”) constant is numerically the same in any system of units:
2
e / 4 c
units
SI
in
0
-3
1
7.297 352 563 10
×
.
2
e /
c
units
Gaussian
in
137
The relative uncertainty of the first value is smaller than 10–10.2 The accuracy of the second, mnemonic value is better than 0.03%.
7. The listed proton’s rest mass m p is close to 1.007 u, while the neutron’s rest mass is close to 1.009 u; their differences from 1u reflect mostly the binding energy of these baryons in the 12C nucleus.
8. Note the relatively poor accuracy with which we know the Newtonian constant of gravitation
– due to the extreme weakness of gravity on human scales of mass and distance.
2 L. Morel et al., Nature 588, 61 (2020).
Selected Physical Constants
Page 2 of 2
Essential Graduate Physics
Lecture Notes and Problems
Open online access at
https://sites.google.com/site/likharevegp/
and
http://commons.library.stonybrook.edu/egp/
References
(a partial list of textbooks and monographs used at work on the series1)
CM
A. L. Fetter and J. D. Walecka, Theoretical Mechanics of Particles and Continua, Dover, 2003.
H. Goldstein, C. Poole, and J. Safko, 3rd ed., Classical Mechanics, Addison Wesley, 2002.
R. A. Granger, Fluid Mechanics, Dover, 1995.
J. V. José and E. J. Saletan, Classical Dynamics, Cambridge U. Press, 1998.
L. D. Landau and E. Lifshitz, Fluid Mechanics, 2nd ed., Butterworth-Heinemann, 1987.
L. D. Landau and E. Lifshitz, Mechanics, 3rd ed., Butterworth-Heinemann, 1976.
L. D. Landau and E. Lifshitz, Theory of Elasticity, Butterworth-Heinemann, 1986.
H. G. Schuster, Deterministic Chaos, 3rd ed., VCH 1995.
A. Sommerfeld, Mechanics, Academic Press, 1964.
A. Sommerfeld, Mechanics of Deformable Bodies, Academic Press, 1964.
EM
V. V. Batygin and I. N. Toptygin, Problems in Electrodynamics, 2nd ed., Academic Press, 1978.
D. J. Griffiths, Introduction to Electrodynamics, 3rd ed. Prentice-Hall, 2007.
J. D. Jackson, Classical Electrodynamics, 3rd ed., Wiley, 1999.
L. D. Landau and E. Lifshitz, Electrodynamics of Continuous Media, 2nd ed., Reed, 1984.
L. D. Landau and E. Lifshitz, The Classical Theory of Fields, 4th ed., Pergamon, 1975.
W. K. H. Panofsky and M. Phillips, Classical Electricity and Magnetism, 2nd ed., Dover, 1990.
J. A. Stratton, Electromagnetic Theory, Adams Press, 2007.
I. E. Tamm, Fundamentals of the Theory of Electricity, Mir, 1979.
A. Zangwill, Modern Electrodynamics, Cambridge U. Press, 2013.
QM
E. S. Abers, Quantum Mechanics, Pearson, 2004.
G. Auletta, M. Fortunato, and G. Parisi, Quantum Mechanics, Cambridge U. Press, 2009.
L. E. Ballentine, Quantum Mechanics: A Modern Development, 2nd ed., World Scientific, 2014.
1 This list does not include the numerous sources (mostly recent original publications) cited in the lecture notes and problem solutions, the open-access materials mentioned in the Preface, and the mathematics textbooks and handbooks listed in MA Sec. 16.
© K. Likharev
ential Gra
duate P
hysics
K. Likharev
A. Z. Capri, Nonrelativistic Quantum Mechanics, 3rd ed., World Scientific, 2002.
C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, in 2 vols., Wiley-VCH, 2005.
F. Constantinescu, E. Magyari, and J. A. Spiers, Problems in Quantum Mechanics, Elsevier, 1971.
V. Galitski et al., Exploring Quantum Mechanics, Oxford U. Press, 2013.
K. Gottfried and T.-M. Yan, Quantum Mechanics: Fundamentals, 2nd ed., Springer, 2004.
D. Griffith, Quantum Mechanics, 2nd ed., Pearson Prentice Hall, 2005.
L. D. Landau and E. M. Lifshitz, Quantum Mechanics (Nonrelativistic Theory), 3rd ed., Pergamon, 1977.
A. Messiah, Quantum Mechanics, Dover, 1999.
E. Merzbacher, Quantum Mechanics, 3rd ed., Wiley, 1998.
D. A. B. Miller, Quantum Mechanics for Scientists and Engineers, Cambridge U. Press, 2008.
J. J. Sakurai, Modern Quantum Mechanics, Revised ed., Addison-Wesley, 1994.
L. I. Schiff, Quantum Mechanics, 3rd ed., McGraw-Hill, 1968.
R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1980.
F. Schwabl, Quantum Mechanics, 3rd ed., Springer, 2002.
SM
R. P. Feynman, Statistical Mechanics, 2nd ed., Westview, 1998.
K. Huang, Statistical Mechanics, 2nd ed., Wiley, 1987.
R. Kubo, Statistical Mechanics, Elsevier, 1965.
L. D. Landau and E. M. Lifshitz, Statistical Physics, Part 1, 3rd ed., Pergamon, 1980.
E. M. Lifshitz and L. Pitaevskii, Physical Kinetics, Pergamon, 1981.
R. K. Pathria and P. D. Beale, Statistical Mechanics, 3rd ed., Elsevier, 2011.
J. R. Pierce, An Introduction to Information Theory, 2nd ed., Dover, 1980.
M. Plishke and B. Bergersen, Equilibrium Statistical Physics, 3rd ed., World Scientific, 2006.
F. Schwabl, Statistical Mechanics, Springer, 2000.
J. M. Yeomans, Statistical Mechanics of Phase Transitions, Oxford U. Press, 1992.
Multidisciplinary and Specialty
N. W. Ashcroft and N. D. Mermin, Solid State Physics, W. B. Saunders, 1976.
K. Blum, Density Matrix and Applications, Plenum, 1981.
H.–P. Breuer and E. Petruccione, The Theory of Open Quantum Systems, Oxford U. Press, 2002.
S. B. Cahn and B. E. Nadgorny, A Guide to Physics Problems, Part 1, Plenum, 1994.
S. B. Cahn, G. D. Mahan, and B. E. Nadgorny, A Guide to Physics Problems, Part 2, Plenum, 1997.
J. A. Cronin, D. F. Greenberg, and V. L. Telegdi, Graduate Problems in Physics, U. Chicago, 1967.
J. R. Hook and H. E. Hall, Solid State Physics, 2nd ed., Wiley, 1991.
G. Joos, Theoretical Physics, Dover, 1986.
A. S. Kompaneyets, Theoretical Physics, 2nd ed., Dover, 2012.
M. Lax, Fluctuations and Coherent Phenomena, Gordon and Breach, 1968.
N. N. Lebedev et al., Problems in Mathematical Physics, Prentice-Hall, 1965
E. M. Lifshitz and L. P. Pitaevskii, Statistical Physics, Part 2, Pergamon, 1980.
N. Newbury et al., Princeton Problems in Physics, Princeton U., 1991.
L. Pauling, General Chemistry, 3rd ed., Dover, 1988.
M. Tinkham, Introduction to Superconductivity, 2nd ed., McGraw-Hill, 1996.
J. D. Walecka, Introduction to Modern Physics, World Scientific, 2008.
J. M. Ziman, Principles of the Theory of Solids, 2nd ed., Cambridge U. Press, 1979.
References
Page 2 of 2