Outline of solution for H atom system, using Schrodinger Wave Equation
1.
Ψ in terms of Cartesian coordinates H$ Ψ( x ,y ,z ) = EΨ( x ,y ,z )
$ as sum of kinetic and potential energy terms:
and define the Hamiltonian operator H
express the wavefunction
so
$ + PE
$
H$ = KE
becomes
− h2  ∂ 2
∂2
∂ 2  Ze2
$
H = 2  2 + 2 + 2 +
8π µ  ∂x ∂y ∂z 
r
The wave equation now takes the form of
− h2 $
$ Ψ = EΨ
( 2 ) KEΨ + PE
8π µ
often represented by the symbol
2.
The potential energy operator is simple and it is
V , leading to
− h2 $
( 2 ) KEΨ + VΨ = EΨ
8π µ
convert from Cartesian coordinates (x, y, z) to spherical coordinates
equivalents:
x = r sinθ cosφ
y = r sinθ sin φ
( r, θ , φ ) , using the
z = r cosθ
− h2  1 ∂ 2 ∂
1
∂
∂
1
∂2 
$
H = 2  2 (r
)+
(sinθ ) + 2 2
+V
8π µ  r ∂r
∂r r 2 sinθ ∂θ
∂θ
r sin θ ∂φ 2 
After a simple rearrangement, the wave equation takes the form of
− 8π 2 µ
$
KEΨ( r , θ , φ) =
( E − V )Ψ ( r , θ , φ )
h2
3.
Note three second-order differential operators in the kinetic energy portion of the Hamiltonian:
2
(1) they all contain a common (therefore factorable) 1 / r term, but more importantly, (2) each
term is a differential with respect to only one coordinate variable. This leads to a separation of
the wave equation into three parts, each associated with only one coordinate variable, as shown:
a.
b.
c.
the phi term:
d 2Φ
= − m2 Φ
2
dφ
the theta term:
1 ∂
∂Θ
m2
(sinθ
)−
+ βΘ = 0
sinθ ∂θ
∂θ
sin2 θ
the r term:
1 ∂ 2 ∂R
β
8π 2 µ
(r
) − 2 R + 2 (E − V )R = 0
r 2 ∂r
∂r
r
h
Note that this "separation of variables" leads to a new set of wavefunctions, represented by
R, Θ , Φ . They are related to the original total wavefunction Ψ as a triple
product, i.e.,
Ψ( r , θ , φ) = R( r ) • Θ (θ) • Φ ( φ)
Also note that both phi and theta terms contain a quantum number m , and that both involve
angular coordinates, φ and θ . They are often taken together and called the ANGULAR
portion of the wavefunction. In the text their product is represented by the symbol Y (p. 27).
upper case
4.
a.
The phi portion has the same form as found earlier for a particle-in-a-box system, except
that the electron is now confined to a closed loop rather than a one-dimensional line. So
2π
its range extends from beginning to end of a closed loop, or from zero to
in terms
of radians. Boundary conditions require that the values at these two points be the same:
Φ @φ= 0 = Φ @φ= 2π
This condition is satisfied if
m
m
are
that proper values for
b.
β sin m( 0) = β sin m( 2π )
or
is zero, or a positive or negative integer. We conclude
0, ± 1, ± 2, ± 3, ±...
Solution of the theta portion generates an additional quantum number
form:
Θ (m , l ) =
(2l + 1)(l −|m|)!
Pl (cosθ )
2(l+|m|)!
l
(l−|m|)!
note the term
|m|
m
and has the
l
What relation must exist between quantum numbers
and
, in order for this factorial
term to be valid?
(Recall: (1) factorials of negative numbers are not defined, and that
(2) the factorial of zero is…)
This term informs that the value of
can be equal to
m
l , but cannot be larger than l . So the range of values possible for m
extend from a minimal value of m = −l to a maximum value of m = +l , or
m = 0, ± 1, ± 2, ± 3, ..., ± l . Note l can be zero, but not negative.
the value of
c.
Solution of the radial portion also generates an additional quantum number
the form:
n
and has
3
R(n , l )
 2Z  (n − l − 1)! − ρ/ 2 l 2 l + 1
= 
ρ Ln+ 1

3 e
 na0  2n[ (n + 1)!)]
(n − l − 1)!
note
n must be larger than l . This
defines the range of possible values for l as, 0, 1, 2, 3, ..., and defines the
maximum value of l as (n − 1) . Note that n must be non-zero and positive.
In order for this term to be non-negative the value of
5.
Given the above limits on ranges and maximum values for quantum numbers
unique numerical scheme involving
m , l , and n
m
and
l, a
can be generated as shown below:
When quantum number
n is…
then q.n.
l
m
can have values
1
2
0
0
1
+1, 0, -1
0
0
1
+1, 0, -1
2
+2, +1, 0, -1, -2
0
0
1
+1, 0, -1
2
+2, +1, 0, -1, -2
3
+3, +2, +1, 0, -1, -2, -3
etc.
etc.
3
4
etc.
Summary:
and q.n.
has values
0
quantum number
0
n
can have any non-zero positive integer value.
l
can be zero or any positive integer to a
maximum of
m
(n − 1) ; l
n.
can be zero or any positive or negative integer,
to a maximum of
Letter Equivalents:
is limited by
± l; m
is limited by
l.
Quantum mechanics (as theory) and spectroscopy (as the related experiment)
l
blend together in the labeling of values for quantum number
. Early
spectroscopists developed descriptive names to characterize certain sets of spectral lines. They
observed "sharp" series of lines, as well as "principal", "diffuse" and "fundamental" series of lines.
Eventually these line types were associated with specific values for quantum number
leading to the following letter equivalents for numerical values of
when
letter equivalent
Nomenclature:
l
=0
s
=1
p
=2
d
l:
l,
=3
f
The 3 quantum numbers have both "common" and "formal " names as follows:
n
shell
principal quantum number
l
subshell
orbital angular momentum quantum number
m
orbital
projection of orbital angular momentum on a component of an
externally applied field. To avoid a future conflict of symbols,
this quantum number will be henceforth written as
ml .