Feb 29, 2000
Notes for Lecture
Notes for Lecture #20
Magnetic field due to electrons in the vicinity of a nucleus
According to the Biot-Savart law, the magnetic field produced by a
current density J(r¢) is given by:
|
B(r) = |
m0
4 p
|
|
ó
õ
|
d3r¢ |
J(r¢) ×(r - r¢)
|r - r¢|3
|
|
| (1) |
In this case, we assume that the current density is due to an
electron in a bound atomic state with quantum numbers |n l ml ñ, as described by a wavefunction
ynlml(r), where the azimuthal quantum number ml is associated with
a factor of the form ei ml f. For such a wavefunction the quantum
mechanical current density operator can be evaluated:
|
J(r¢) = |
-e (h/2p)
2mi
|
(ynlml*Ѣ ynlml-ynlml Ѣynlml* ). |
| (2) |
Since the only complex part of this wavefunction is associated
with the azimuthal quantum number, this can be written:
|
J(r¢) = |
-e (h/2p)
2mi r¢sinq¢
|
|
æ
ç
è
|
ynlml* |
¶
¶f¢
|
ynlml-ynlml |
¶
¶f¢
|
ynlml* |
ö
÷
ø
|
|
^
f¢
|
= |
m r¢ sinq¢
|
|ynlml|2. |
| (3) |
We need to use this current density in the Biot-Savart law and
evaluate the field at the nucleus (r = 0). The vector
cross product in the numerator can be evaluated in spherical polar
coordinates as:
|
|
^
f¢
|
×(-r¢) = r¢ |
æ
è
|
- |
^
x
|
cosq¢cosf¢ - |
^
y
|
cosq¢sinf¢ + |
^
z
|
sinq¢ |
ö
ø
|
|
| (4) |
Thus the magnetic field evaluated at the nucleus is given by the
integral:
|
B(0) = - |
m0 e (h/2p) ml
4 pm
|
|
ó
õ
|
d3r¢|ynlml|2 |
|
r¢ |
æ
è
|
- |
^
x
|
cosq¢cosf¢ - |
^
y
|
cosq¢sinf¢ + |
^
z
|
sinq¢ |
ö
ø
|
|
r¢ sinq¢ r¢3
|
. |
| (5) |
In evaluating the integration over the azimuthal variable
f¢, the [^(x)] and [^(y)]
components vanish leaving the simple result:
|
B(0) = - |
4 pm
|
|
ó
õ
|
d3r¢|ynlml|2 |
1
r¢3
|
º - |
m0 e
4 pm
|
L |
á
|
1
r¢3
|
ñ
|
. |
| (6) |
File translated from TEX by TTH, version 2.20.
On 29 Feb 2000, 17:06.