26 The hydrogen atom
Fig. 2.1Polar plots of the squared modulus of the angular wavefunctions for the hydrogen atom withl= 0 and 1. For each
value of the polar angleθa point is plotted at a distance proportional to|Y(θ, φ)|^2 from the origin. Except for (d), the plots
have rotational symmetry about thez-axis and look the same for any value ofφ.(a)|Y 0 , 0 |^2 is spherical. (b)|Y 1 , 0 |^2 ∝cos^2 θ
has two lobes along thez-axis. (c)|Y 1 , 1 |^2 ∝sin^2 θhas an ‘almost’ toroidal shape—this function equals zero forθ=0. (|Y 1 ,− 1 |^2
looks the same.) (d)|Y 1 , 1 −Y 1 ,− 1 |^2 ∝|x/r|^2 has rotational symmetry about thex-axis and this polar plot is drawn forφ=0;
it looks like (b) but rotated through an angle ofπ/2. (e)|Y 2 , 2 |^2 ∝sin^4 θ.
Any linear combination of these is also an eigenfunction ofl^2 ,e.g.Y 1 ,− 1 −Y 1 , 1 ∝
x
r
=sinθcosφ, (2.14)Y 1 ,− 1 +Y 1 , 1 ∝y
r=sinθsinφ. (2.15)These two real functions have the same shape asY 1 , 0 ∝z/rbut are
aligned along thex-andy-axes, respectively.^14 In chemistry these dis-(^14) In the absence of an external field
to break the spherical symmetry, all
axes are equivalent, i.e. the atom does
not have a preferred direction so there
is symmetry between thex-,y-and
z-directions. In an external magnetic
field the states with different values of
m(but the samel) are not degenerate
and so linear combinations of them are
not eigenstates of the system.
tributions forl= 1 are referred to as p-orbitals. Computer programs
can produce plots of such functions from any desired viewing angle (see
Blundell 2001, Fig. 3.1) that are helpful in visualising the functions with
l>1. (Forl= 0 and 1 a cross-section of the functions in a plane that
contains the symmetry axis suffices.)
2.1.2 Solution of the radial equation
An equation forR(r) is obtained by setting eqn 2.4 equal to the constant
b=l(l+ 1) and putting in the Coulomb potentialV(r)=−e^2 / 4 π 0 r.It