CHAP. 12: BASIC TERMS OF CHEMICAL PHYSICS [CONTENTS] 421
12.2 Fundamentals of quantum mechanics
12.2.1 Schr ̈odinger equation
The starting point for describing a system on the microscopic level is the description of a set of
particles (elementary particles, atoms and molecules) using theirwavefunctionΨ. The wave-
function is obtained by solving theSchr ̈odinger equation. It is written for time-independent
case as follows
−∑Ni=1h^2
8 π^2 mi∇~^2 iΨ(~v) +V(~v)Ψ(~v) =EΨ(~v), (12.19)where~v = (~r 1 , ~r 2 ,... , ~rN) is the generalized vector giving the position of all particles in
the system, andV(~v) is the potential energy of the system. The symbol∇~^2 i = ∂^2 /∂~ri^2 =
∂^2 /∂x^2 i+∂^2 /∂yi^2 +∂^2 /∂zi^2 denotes theLaplace operator, andhis thePlanck constant
(h= 6. 626 × 10 −^34 J s). The Schr ̈odinger equation has a stable solution only for certain val-
ues ofE, called the eigenvaluesof energy, with the corresponding wavefunctions Ψ called
eigenfunctions. The second power of the absolute value of the wavefunction,|Ψ|^2 = Ψ∗Ψ (∗
denotes complex conjugate), gives the probability density of the occurrence of a system in the
state defined by the vector~v. These solutions are characterized by one or several integers known
asquantum numbers^5. If the same energy corresponds to several different combinations of
quantum numbers, we say that the given energy level isdegenerate. The number of these
combinations is called thedegree of degeneracy(g) of the energy level.
12.2.2 Solutions of the Schr ̈odinger equation
The general solutions of the Schr ̈odinger equation for systems containing more particles is
usually very complicated. However, it is often possible to obtain its solution by partitioning
the total potential energy of a system into energies corresponding to individual molecules of
the system
V(~v) =∑Ni=1Vi, (12.20)(^5) This applies only to the so called bound states. Continuum states, as e.g. freely moving particle, do not
have discrete energy levels.