The probability of finding the particle in an infinitesimal cube of sidesdx,dy,dz
isc^2 dxdydz, and the probability of finding the particle somewhere in a volumeV
is the integral over that volume ofc^2 with respect todx,dy,dz(a triple integral);
c^2 is thus aprobability density function, with units of probability per unit volume.
Born’s interpretation was in terms of the probability of a particular state, Pauli’s the
chemist’s usual view, that of a particular location.
The Schr€odinger equation overcame the limitations of the Bohr approach (see
the beginning ofSection 4.2.6): the quantum numbers follow naturally from it
(actually the spin quantum numbermsrequires a relativistic form of the Schr€odinger
equation, the Dirac equation, and electron “spin” is apparently not really due to the
particle spinning like a top). The Schr€odinger equation can be solved in an exact
analytical way only for one-electron systems like the hydrogen atom, the helium
monocation and the hydrogen molecule ion, but the mathematical approach is
complicated and of no great relevance to the application of this equation to the
study of serious molecules. However a brief account of the results for hydrogenlike
atoms is in order.
The standard approach to solving the Schr€odinger equation forhydrogenlike
atoms involves transforming it from Cartesian (x,y,z) to polar coordinates (r,y,’),
since these accord more naturally with the spherical symmetry of the system. This
makes it possible to separate the equation into three simpler equations,f(r)¼0,
f(y)¼0, andf(’)¼0. Solution of thef(r) equation gives rise to thenquantum
number, solution of thef(y) equation to thelquantum number, and solution of the
f(’) equation to themm(often simply calledm) quantum number. For each specific
n¼n^0 ,l¼l^0 andmm¼mm^0 there is a mathematical function obtained by combining
the appropriatef(r),f(y) andf(’):
cðr;y;’;n^0 ;I^0 ;m^0 mÞ¼fðrÞfðyÞfð’Þ (4.32)The functionc(r,y, ’) (clearlyccould also be expressed in Cartesians),
dependsfunctionallyonr,y,’andparametrically onn,landmm: for each
particular set (n^0 ,l^0 ,mm^0 ) of these numbers there is a particular function with the
spatial coordinates variablesr,y,’(orx,y,z). A function likeksinxis a function of
xand depends only parametrically onk. Thiscfunction is anorbital(“quasi-orbit”;
the term was invented by Mulliken,Section 4.3.4), and you are doubtless familiar
with plots of its variation with the spatial coordinates. Plots of the variation of
c^2 with spatial coordinates indicate variation of the electron density (recall the
Born interpretation of the wavefunction) in space due to an electron with quantum
numbersn^0 ,l^0 andmm^0. We can think of an orbital as a region of space occupied by
an electron with a particular set of quantum numbers, or as a mathematical function
cdescribing the energy and the shape of the spatial domain of an electron. For an
atom or molecule with more than one electron, the assignment of electrons to
orbitals is an (albeit very useful)approximation, since orbitals follow from solution
of the Schr€odinger equation for a hydrogen atom.
The Schr€odinger equation that we have been talking about is actually thetime-
independent(and nonrelativistic) Schr€odinger equation: the variables in the equa-
tion are spatial coordinates, or spatial and spin coordinates (Section 5.2.3.1) when
4.2 The Development of Quantum Mechanics. The Schr€odinger Equation 101