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Physics 2130: General Physics 3
Reading: Read Chapter 9 on Electron Spin
Homework: HWK14 due today at 5PM.
Lecture 43
Electron spin and multi-electron atoms.
Solved S’s equation for hydrogen:
wave functions, energies, angular momentum
In atoms with multiple electrons, what do you expect to change
in the way you set up the problem
and would then change the solutions?
Student Ideas:
A. Potential energy is more complicated
B. Now have electron-electron coulomb
C. V will likely depend on time.
D. Nuclear charge is different for each atom.
E. Pauli Exclusion Principle is important
F. Does nuclear structure matter? Yes!
G. And many more interesting things….
Last Time: Multi-electron atom thoughts
1s
2s
2p
3s
3p
3d
To
tal E
nerg
y
e e
e e
e ee e
Shell not full – reactive
Shell full – stable
HHeLiBeBCNO
ELECTRONS HAVE SPIN (intrinsic angular momentum)
Pauli’s Exclusion Principle: only one electron
allowed per given set of quantum numbers.
Last Time: Multi-electron atom thoughts
Last Time: Crash course in spin 1/2
We say particles have spin ½ when:
If you measure the angular momentum along any
axis, you get ONE OF TWO answers: +/- h/2
If there are only two eigen-values, how many
eigen-states are likely available for this quantity?
A) 0 B) 1 C) 2 D) 3 E) Infinite number
We will call these states and
The quantum behavior of spin ½ is described by
matter waves that live in a 2-d vector space, a Hilbert
Space with complex-valued vector components.
TRUE(A) or FALSE(B) The general spin ½ particle
CAN be in a linear superposition of these spin ½
eigen-states.
2 21
general
Last Time: Crash course in spin 1/2
You already know a lot about 2-d vectors!
Many ways to talk about vector
math:
ˆ ˆ
ˆ ˆ
x y
x y
x y
a a a
a a x a y
a a i a j
Example: Position vector in 2-d real space
You already know a lot about 2-d vectors!
Or perhaps: ,x ya a a
Or perhaps:x
y
aa
a
A two-
component
column
vector.
Components list.
Column vectors are useful:x
y
aa
a
Example: Position vector in 2-d real space
You already know a lot about 2-d vectors!
Example: Find new components if you
rotate the vector by q.
Rotateda R aq
“Rotation operator”
cos sin
sin cosR
q qq
q q
The operators are all two-by-two matrices and you
use matrix multiplication to get the new components.
Here is one way to talk about the quantum MATH:
TRUE(A) or FALSE(B) The general spin ½ particle
CAN be in a linear superposition of these spin ½
eigen-states.
1
0
0
1
general
Last Time: Crash course in spin 1/2
Crash course in spin 1/2
If we choose to use column vectors for the spin
states:
What do the operators look like?
1
0
0
1
A) Totally and completely confused.
B) Must be functions of position
C) Must be derivatives of some type
D) Must be 2 by 2 matrices
E) Something else
Crash course in spin 1/2
If we choose to use column vectors for the spin
states and measuring the spin component SZ can
only yield:
Which matrix is the operator for SZ ?
1 0
0 12
0
1
A)
B)
2ZS 1
0
0 1
1 02
C)
D)
1 0
0 12
0
02
i
i
Crash course in spin 1/2
If we choose to use column vectors for the spin
states and measuring the spin component SZ can
only yield:
Which matrix is the operator for SZ ?
22 1 03
0 14S
0
1
2ZS 1
0
0 1
1 02XS
1 0
0 12ZS
0
02Y
iS
i
And, also:
Crash course in spin 1/2
All spin ½ particles including electrons, protons,
neutrons, muons, all the quarks, and more have
the same measurable values of components of S:
2ZS 2XS 2YS
Along with the spin comes a magnetic moment,
with ‘gyromagnetic ratio’, g, unique to each type
of particle. Different for electron, different for
proton, etc.
S g
Particle energy in a B-field is: H B S B g
Crash course in spin 1/2
If we put a spin ½ particle in a B-field that is size, B,
pointed in the z-direction, then the energy operator
is:
Which matrix is the operator for H ?
0
1
A)
B)
1
0
C)
D)
1 0
0 12Bg
1 0
0 12Bg
1 0
0 12ZS
ZH B S B g
1 0
0 12
1 0
0 12g
1s
2s
2p
3s
3p
3d
To
tal E
nerg
y
e e
e e
e ee e
Shell not full – reactive
Shell full – stable
HHeLiBeBCNO
Why do the electrons fill the orbitals in this way?
ELECTRONS HAVE SPIN (intrinsic angular momentum)
Pauli’s Exclusion Principle: only one electron
allowed per given set of quantum numbers.
Solved S’s equation for hydrogen:
wave functions, energies, angular momentum
In atoms with multiple electrons, what do you expect to change
in the way you set up the problem
and would then change the solutions?
Student Ideas:
A. Potential energy is more complicated
B. Now have electron-electron coulomb
C. V will likely depend on time.
D. Nuclear charge is different for each atom.
E. Pauli Exclusion Principle is important
F. Does nuclear structure matter? Yes!
G. And many more interesting things….
Last Time: Multi-electron atom thoughts
V (for q1) = kqnucleus*q1/rn-1 + kq2q1/r2-1 + kq3q1/r3-1 + ….
Schrodinger’s solution for multi-electron atomsNeed to account for all the interactions among the electrons
Must solve for all electrons at once! (use matrices)
Can Schrodinger make sense of the periodic table?
1s
2s
2p
3s
3p
3d
To
tal E
nerg
y
e e
e e
e ee e
Shell not full – reactive
Shell full – stable
HHeLiBeBCNO
Will the 1s orbital be at the same energy level for
each atom in the Periodic Table? YES(A) or NO(B)
1s
2s
2p
3s
3p
3d
To
tal E
nerg
y
e e
e e
e ee e
Shell not full – reactive
Shell full – stable
HHeLiBeBCNO
NO. Change number of protons … Change potential energy in
Schrodinger’s equation … 1s held tighter if more protons.
The energy of the different orbitals depends on the atom.
Will the 1s orbital be at the same energy level for
each atom in the Periodic Table? YES(A) or NO(B)
What’s different for these cases?
Potential energy (V) changes!
(Now more protons AND other electrons)
Need to account for all the interactions among the electrons
Must solve for all electrons at once! (use matrices)
Schrodinger’s solution for multi-electron atoms
V (for q1) = kqnucleusq1/rn-1 + kq2q1/r2-1 + kq3q1/r3-1 + ….
Gets very difficult to solve … huge computer programs!
Solutions change:
- wave functions change
higher Z more protons electrons in 1s more strongly
bound radial distribution quite different
general shape (p-orbital, s-orbital) similar but not same
- energy of wave functions affected by Z (# of protons)
higher Z more protons electrons in 1s more strongly
bound (more negative total energy)
In particular, why do we get columns of
similar chemical behavior??
Can Schrodinger make sense of the periodic table?
n=1
n=2
For a given atom, Schrodinger predicts allowed wave functions
and energies of these wave functions.
Why would behavior of Li be similar to Na?
a. because shapes of outer most orbitals are similar.
b. because energies of outer most electrons are similar.
c. both a and b
d. some other reason
1s
2s
3s
l=0 l=1 l=2
4s
2p
3p
4p3d
En
erg
y
Li (3 e’s)
Na (11 e’s)
m=-1,0,1
m=-2,-1,0,1,2
2s
2p
1s
3s
In case of Na, what will energy of outermost electron be and WHY?
a. much more negative than for the outermost electron in Li
b. similar to the energy of the outermost electron in Li
c. much less negative than for the outermost electron in Li
Li (3 e’s)
Na (11 e’s)
Wave functions for Li vs Na
2s2p
1s
3s
In case of Na, what will energy of outermost electron be and WHY?
b. very similar to the energy of the outermost electron in Li
AND somewhat (within a factor of 3) of the ground state of H
Wave functions for sodium
What affects total energy of outermost electron?
1. The effective charge (force) it feels towards center
of atom.
2. It’s distance from the nucleus.
What effective charge does 3s electron feel
pulling it towards the nucleus?
Close to 1 proton… 10 electrons closer in
shield (cancel) a lot of the nuclear charge.
What about distance?
In H, 3s level is on average 9x further than 1s, so 9*Bohr radius.
In Na, 11 protons pull 1s, 2s, 2p closer to nucleus
distance of 3s not as far out.
Electron in 3s is a bit further than 1s in H, but ~same as 2s in Li.
Proximity of electrons in 1s, 2s, 2p is what makes 3s a bit bigger.
Quantum mechanics DOES make sense of the atoms
And: Molecules, solids, liquids,
chemical reactions, etc.