lecture 43 electron spin and multi-electron atoms.€¦ · · 2016-12-08you already know a lot...

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: 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


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


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


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


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



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



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:


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


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

Good place for a break and for

some questions!

1s

2s

2p

3s

3p

3d

To

tal E

nerg

y

e e

e e

e ee e



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.

Can Schrodinger make sense of the periodic table?

YES!! …with a but.

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….


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)


1s

2s

2p

3s

3p

3d

To

tal E

nerg

y

e e

e e

e ee e



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



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??


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.

Good place for a break.

lecture 43 electron spin and multi-electron atoms.€¦ · · 2016-12-08you already know a lot...

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