Spectroscopy
Symbols, Terminology and Constants in Science and Mathematics
Science and Mathematics uses a large range of units, symbols and terms, and initially it can seem like a foreign language. Here we have an inexhaustive list of commonly used mathematical terms and symbols that appear commonly in all sciences  especially physical and inorganic chemistry, spectroscopy and analytical chemistry. Click on these links to jump to The Greek Alphabet  Factor Prefixes  Constants  Formulae
The modern alphabet on which most Western alphabets are based is a combination of the Latin and Greek alphabets. Characters from the Greek alphabet appear in most science and mathematics. They can be used to represent more than one quantity. Usually the use is specified. However, some common occurrences are listed below. NOTE: (upper) and (lower) indicate the upper case and lower case characters respectively.
Lower case  Upper case  Name  Common use 
α  Α  alpha  Used as a general variable in numerous situations. Always specified. Used often in quantum mechanics. 
β  Β  beta  Used as a general variable in numerous situations. Always specified. used often in quantum mechanics. 
γ  Γ  gamma  An irrep in group theory (upper). 
δ  Δ  delta  Indicates an infinitesimal change (lower) or general finite change (upper). Lower case is also used to signify a partial derivative. 
ε  Ε  epsilon  Permitivity of free space (ε_{0}). 
ζ  Ζ  zeta  Effective nuclear charge in quantum chemistry (lower). 
η  Η  eta  Used to represent many coefficients. Commonly, viscosity (lower). 
θ  Θ  theta  A contextspecific angle (lower), or heat (upper). 
ι  Ι  iota  
κ  Κ  kappa  Numerous uses. In chemistry  the compressability of a compound (lower). 
λ  Λ  lambda  Wavelength (lower) of a wave or particle. 
μ  Μ  mu  Symbol for "micro" (1×10^{6}) (lower). Also the symbol for "reduced mass"  see below. 
ν  Ν  nu  Frequency, Hz or s^{1} (lower) 
ξ  Ξ  xi  Numerous uses. Used instead of Q for the partition function in a grand canonical ensemble in statistical mechanics (upper). 
ο  Ο  omicron  
π  Π  pi  Important irrational number in mathematics (lower). Mathematical operator for "product of all terms over a specified region" (upper). 
ρ  Ρ  rho  Density (lower). Can be physical density (kg m^{3}), or some specified density, such as electron density (ρ(e^{})). 
σ  Σ  sigma  Numerous uses. Indicates sigmabonds, collision crosssection etc. (lower). Mathematical operator for "sum of all terms over a specified region" (upper). 
τ  Τ  tau  Some specific time not denoted by t (lower). Usually the "1/e" time. 
υ  Υ  upsilon  Uncommonly used. Usage usually specified. 
φ  Φ  phi  Numerous uses in the mathematics of quantum mechanics. Can indicate a specific eigenstate (lower), a general eigenstate (upper). Used in geometry to indicate an angle in the spherical polar coordinate system (lower). 
χ  Χ  chi  The character of an irrep in group theory (upper). 
ψ  Ψ  psi  Used extensively in quantum mechanics. Represents wavefunctions. Component of the Schrödinger equation. Lower case is usually a specific wavefunction, whereas upper case might be a general wavefunction. 
ω  Ω  omega  Angular frequency (lower) in rotation. Unit of electrical resistance (upper). Can be adapted to be used in rotationvibration spectroscopy (e.g ϖ, the central point between the P and Q branches in rotvib spec.) 
Often  rather than using a number in standard mathematical form it is preferable to use a quantity's units, with a suitable factor prefix to indicate scale. For example, it is much quicker and more simple to write λ = 4.00×10^{7} m as 400 nm. The n, for "nano", is a prefix used to indicate a factor of 1×10^{9}.
Prefix 
Name 
Scale 
E 
exa 
10^{18} 
P 
peta 
10^{15} 
T 
tera 
10^{12} 
G 
giga 
10^{9} 
M 
mega 
10^{6} 
k 
kilo 
10^{3} 
d 
deci 
10^{1} 
c 
centi 
10^{2} 
m 
milli 
10^{3} 
μ 
micro 
10^{6} 
n 
nano 
10^{9} 
p 
pico 
10^{12} 
f 
femto 
10^{15} 
a 
atto 
10^{18} 
z 
zepto 
10^{21} 
y 
yocto 
10^{24} 
Below are listed some useful constants that are important in physical chemistry and spectroscopy.
Constant 
Value 
Units 
Equivalency 
h  Planck's constant 
6.626068×10^{34} 
J s 

ħ  reduced Planck's constant 
1.054571×10^{34} 
J s 
h / 2π 
amu  atomic mass unit 
1.660539×10^{27} 
kg 

k_{B}  Boltzmann's constant 
1.380650×10^{23} 
J K^{1} 

R  molar gas constant 
8.314472×10^{0} 
J mol^{1} K^{1} 
N_{A}k_{B} 
N_{A}  Avogadro's number 
6.022141×10^{23} 
(atoms, particles etc.) mol^{1} 

c  speed of light in a vacuum 
2.997925×10^{8} 
m s^{1} 

m_{e}  mass of an electron 
9.109382×10^{31} 
kg 

e  elementary charge 
1.602177×10^{19} 
C 

ε_{0}  permittivity of free space 
8.854188×10^{12} 
F m^{1} 
1 / μ_{0}c_{0}^{2} 
μ_{0}  vacuum permeability 
4π×10^{7} 
J s^{2} / C^{2} m 

Below are listed some common formulae used in the theory behind basic spectroscopic techniques, divided into relevant subsections.
Basic Equations
Momentum: p = mv
Kinetic Energy: E_{k} = ½mv^{2} = p^{2}/2m
Speed of light: c = ν×λ
Energy of a wave: E = hν
de Broglie formula: λ = h/p
Boltzmann Statistics
where N_{upper} is the population of the upper state, N_{lower} is the population of the lower state, d_{upper} and d_{lower} are the degeneracies of the upper and lower states, ΔE is the energy gap between the two states (in Joules, not kJ), k_{B} is Boltzmann's constant and T is the absolute temperature of the system (in K)
Rotational Spectroscopy
where F(J) is the energy of rotational level J and B is the rotational constant
where h is Planck's constant, I is the moment of inertia, c is the speed of light in vacuo, in cm s^{1}
where μ is the reduced mass of the molecule
where degeneracy of a rotational energy level J is given by
The primary selection rule for pure rotational spectroscopy is
The line spacing in a pure rotation spectrum is simply given by
Vibrational Spectroscopy
The energy of a level in a vibrational system is given by
where ω_{e} is the harmonic band centre, given by
where k is the bond force constant, μ is the reduced mass of the molecule and c is the speed of light in a vacuum. The usual selection rule of Δv = ±1 applies.