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Magnetic moment
Bohr magneton
Magnetic moment
gyromagnetic ratio (g factor)
determined by details of charge distribution
(elementary particle e.g. electron)
(QED corrections for electron)
7-31. Assuming the electron to be a classical particle, a sphere of radius 10-15 m and a
uniform mass density, use the magnitude of the spin angular momentum | S | = [s (s+1)]1/2
ħ=(3/4)1/2 ħ to compute the speed of rotation at the electron’s equator. How does your
result compare with the speed of light?
Magnetic moment
Stern-Gerlach experiment
Inhomogeneous magnetic field
Stern-Gerlach experiment
Inhomogeneous magnetic field
Stern-Gerlach experiment
Inhomogeneous magnetic field
Classical picture – continuum of possible orientations
Quantum mechanics– 2l +1 deflections ?
Stern-Gerlach experiment
Total Angular Momentum
total angular momentum
or
If J1 is one angular momentum (orbital, spin, or a combination) and J2 is
another, the resulting total angular momentum J = J1 + J2 has the value
[ j ( j + 1)]1/2 ħ for its magnitude, where j can be any of the values
j1 + j2, j1 + j2 - 1, . . . , | j1 - j2 |
7-34. (a) The angular momentum of the yttrium atom in the ground state is characterized
by the quantum number j = 3/2. How many lines would you expect to see if you could
do a Stern-Gerlach experiment with yttrium atoms? (b) How many lines would you expect
to see if the beam consisted of atoms with zero spin, but l= 1?
a)
b)
7-37. A hydrogen atom is in the 3d state (n = 3, l = 2). (a) What are the possible values
of j? (b) What are the possible values of the magnitude of the total angular momentum?
(c) What are the possible z components of the total angular momentum?
a)
b)
c)
Spectroscopic Notation
Single electron
l
s p d f g h
0 1 2 3 4 5
n
K L M N O
1 2 3 4 5
Atomic state
total spin
n
Hydrogen ground state
total orbital angular momentum
total angular momentum
Identical Particles in Quantum Mechanics
Non-interacting particles
e.g.
Identical Particles in Quantum Mechanics
Symmetric wavefunctions –bosons (e.g photons)
Antisymmetric wavefunctions –fermions (e.g electrons)
Symmetric wavefunctions –bosons (e.g photons)
Antisymmetric wavefunctions –fermions (e.g electrons)
Pauli Exclusion Principle
No more than one electron may occupy a given
quantum state specified by a particular set of singleparticle quantum numbers n, l, ml ms.
Ground State of Atoms
He (Z=2)
1s2 – ground state
(more accurate calculations to be used)
He+
Energy needed to remove the first electron (first ionization) potential is 24.4 eV
Ground State of Atoms
l=0
m=0
ms=±1/2
2 electrons
l=1
m=-1,0,1
ms=±1/2
6 electrons
l=2
m=-2,-1,0,1,2
ms=±1/2 10 electrons
ground state
H : 1s1
He: 1s2 (filled shell n=1)
last electron
n=1, l=0
n=1, l=0
Li: 1s2 2s1
n=2, l=0
Be: 1s2 2s2
n=2, l=0
B : 1s2 2s2 2p1
n=2, l=1
C : 1s2 2s2 2p2
n=2, l=1
N: 1s2 2s2 2p3
n=2, l=1
O: 1s2 2s2 2p4
n=2, l=1
F: 1s2 2s2 2p5
n=2, l=1
Ne: 1s2 2s2 2p6 (filled shell n=2)
n=2, l=1
Ground State of Atoms
ground state
last electron
Na: 1s2 2s2 2p6 3s1
n=3, l=0
Mg: 1s2 2s2 2p6 3s2
n=3, l=0
Al: 1s2 2s2 2p6 3s2 3p1
n=3, l=1
Si : 1s2 2s2 2p6 3s2 3p2
n=3, l=1
P: 1s2 2s2 2p6 3s2 3p3
n=3, l=1
S: 1s2 2s2 2p6 3s2 3p4
n=3, l=1
Cl: 1s2 2s2 2p6 3s2 3p5
n=3, l=1
Ar: 1s2 2s2 2p6 3s2 3p6
n=3, l=1
K: 1s2 2s2 2p6 3s2 3p6 3d1
n=3, l=2
K: 1s2 2s2 2p6 3s2 3p6 4s1
n=4, l=0
Ground State of Atoms
2p state is almost always outside 1s electrons - sees the effective charge Zeff =+1
2s state penetrates the shielding of 1s electrons more positive charge –lower energy for 2s
Ground State of Atoms
Li: 1s2 2s1
Greater l smaller involves penetration effect and large energies
The large penetration effect makes the energy 4s lower than 3d.
7-46. Write the ground-state electron configuration of (a) carbon, (b) oxygen, and (c)
argon.
C : 1s2 2s2 2p2
O: 1s2 2s2 2p4
Ar: 1s2 2s2 2p6 3s2 3p6
7-50. If the 3s electron in sodium did not penetrate the inner core, its energy would be
-13.6 eV/32 = -1.51 eV. Because it does penetrate, it sees a higher effective Z and its
energy is lower. Use the measured ionization potential of 5.14 V to calculate Zeff for the 3s
electron in sodium.
7-41. The Lamb shift energy difference between the 22S1/2 and 22P1/2 levels in atomic
hydrogen is 4.372 x 10-6 eV. (a) What is the frequency of the photon emitted in this
transition? (b) What is the photon’s wavelength? (c) In what part of the electromagnetic
spectrum does this transition lie?
a)
b)