In 3D: If the amplitude of the wavefunction of a particle at some point (x,y,z) is Y(x,y,z) then the probability of finding the particle between x and x+dx, y+dy and z+dz, ie, in a volume dV = dx dy dz is given by
It follows therefore that
is a probability density.
In spherical coordinates
More important to know probability of finding electron at a given distance from nucleus!
not very useful to describe orbitals!
Ö in a shell of
dA = r2sinqdfdq
dV = r2sinqdfdqdr
For spherically symmetric orbitals:
the radial distribution function is defined as
P(r)= 4pr2 Y(r)
and P(r)dr is the probability of finding the electron in a shell of radius r and thickness dr
For spherically symmetrical orbitals
prob = Y2dt
P(r) = r2R(r)2
(for spherical symmetry)
So what do we learn?
The 2p orbital is on average closer to the nucleus, but note the 2s orbitalís high probability of being
3d vs 4s
The Ylm(,) angular part
of the solution form a set
of functions called the
n.b. These are generally
imaginary functions -we use
real linear combinations to
Shapes arise from combination of radial R(r) and angular parts Ylm(,) of the wavefunction.
Usually represented as boundary surfaces which include of the probability density. The electron does not undergo planetary style circular orbits.