February 29, 2000
Notes for Lecture #20
Magnetic eld due to electrons in the vicinity of a nucleus
According to the Biot-Savart law, the magnetic eld produced by a current density J(r0 ) is
given by:
Z
J(r0 ) (r r0 )
0
B(r) =
d3 r 0
(1)
4
jr r0j3
In this case, we assume that the current density is due to an electron in a bound atomic state
with quantum numbers jnlml i, as described by a wavefunction nlm (r), where the azimuthal
quantum number ml is associated with a factor of the form eim . For such a wavefunction
the quantum mechanical current density operator can be evaluated:
l
l
J(r0) =
eh
nlml
2mi
r0
nlml
nlml
r0
nlml :
(2)
Since the only complex part of this wavefunction is associated with the azimuthal quantum
number, this can be written:
J(r ) =
0
eh
2mir0 sin 0
nlml
@
nlml
@0
@
nlml
!
@0 nlml
^0
=
eh
ml ^0
mr 0 sin 0
j
nlml
j2 :
(3)
We need to use this current density in the Biot-Savart law and evaluate the eld at the
nucleus (r = 0). The vector cross product in the numerator can be evaluated in spherical
polar coordinates as:
^0
( r0) = r0 ( x^ cos 0 cos 0 y^ cos 0 sin 0 + ^z sin 0)
(4)
Thus the magnetic eld evaluated at the nucleus is given by the integral:
B(0) =
0 eh
ml
4m
Z
d3 r 0
j
nlml
0
0
0
0 sin 0 + ^
z sin 0 )
j2 r ( x^ cos cos r0 siny^cos
:
0 r0 3
(5)
In evaluating the integration over the azimuthal variable 0 , the x
^ and y^ components vanish
leaving the simple result:
B(0) =
0 eh
ml^z
4m
Z
3 0
d r
j
nlml
j
2
1
r0 3
0 e
4m
L
1
r0 3
:
(6)