tue. dec. 4 2007physics 208, lecture 261 last time… 3-dimensional wave functions decreasing...

Tue. Dec. 4 2007 Physics 208, Lecture 26 1

Last Time…

3-dimensional wave functions

Decreasing particle size

Quantum tunneling

Quantum dots (particle in box)

This week’s honors lecture: Prof. Brad Christian, “Positron Emission Tomography”


Exam 3 results

Exam average = 76% Average is at B/BC

boundary

0

5

10

15

20

25

30

10 20 30 40 50 60 70 80 90 100

Phy208 Exam 3

Range

A

AB

B

BCC

DF

Average

Course evaluations:

Rzchowski: Thu, Dec. 6

Montaruli: Tues, Dec. 11


3-D particle in box: summary

Three quantum numbers (nx,ny,nz) label each state nx,y,z=1, 2, 3 … (integers starting at 1)

Each state has different motion in x, y, z

Quantum numbers determine Momentum in each direction: e.g.

Energy:

Some quantum states have same energy

€

px =h

λ nx

= nx

h

2L

€

E =px

2

2m+

py2

2m+

pz2

2m= Eo nx

2 + ny2 + nz

2( )


Question

How many 3-D particle in box spatial quantum states have energy E=18Eo?

A. 1

B. 2

C. 3

D. 5

E. 6

€

E = Eo nx2 + ny

2 + nz2

( )

nx,ny,nz( ) = 4,1,1( ), 1,4,1( ), (1,1,4)


3-D Hydrogen atom Bohr model:

Restricted to circular orbits

Found 1 quantum number n

Energy , orbit radius

€

En = −13.6

n2 eV

From 3-D particle in box, expect that H atom should have more quantum numbers

Expect different types of motion w/ same energy

€

rn = n2ao


Modified Bohr model

Different orbit shapes

Big angular momentumSmall

angular momentum

These orbits have same energy, but different angular momenta:

Rank the angular momenta from largest to smallest:

AB

C

a) A, B, C

b) C, B, A

c) B, C, A

d) B, A, C

e) C, A, B

€

rL =

r r ×

r p ( )


Angular momentum is quantized orbital quantum number ℓ

Angular momentum quantized , ℓ is the orbital quantum number

For a particular n, ℓ has values 0, 1, 2, … n-1ℓ=0, most ellipticalℓ=n-1, most circular

€

L = h l l +1( )

For hydrogen atom, all have same energy


Orbital mag. moment

Orbital charge motion produces magnetic dipole

Proportional to angular momentum

Current

electron

Orbital magnetic dipole

€

rμ =μB l l +1( )

€

μB ≡eh

2m= 0.927 ×10−23 A ⋅m2

€

rμ

€

rμ =μB

r L /h( )

Each orbit has Same energy: Different orbit shape

(angular momentum): Different magnetic moment:

€

n,l( )

€

En = −13.6 /n2 eV

€

L = h l l +1( )

€

rμ =μB

r L /h( )

€

rμ


Orbital mag. quantum number mℓ

Directions of ‘orbital bar magnet’ quantized. Orbital magnetic quantum number

m ℓ ranges from - ℓ, to ℓ in integer steps (2ℓ+1) different values Determines z-component of L: This is also angle of L

For example: ℓ=1 gives 3 states:

€

Lz = ml h


Question

For a quantum state with ℓ=2, how many different orientations of the orbital magnetic dipole moment are there?

A. 1B. 2C. 3D. 4E. 5


Summary of quantum numbers

n : describes energy of orbit ℓ describes the magnitude of orbital angular momentum m ℓ describes the angle of the orbital angular momentum

For hydrogen atom:


Hydrogen wavefunctions Radial probability Angular not shown For given n,

probability peaks at ~ same place

Idea of “atomic shell” Notation:

s: ℓ=0 p: ℓ=1 d: ℓ=2 f: ℓ=3 g: ℓ=4


Full hydrogen wave functions: Surface of constant probability

Spherically symmetric. Probability decreases

exponentially with radius. Shown here is a surface

of constant probability

€

n =1, l = 0, ml = 0

1s-state


n=2: next highest energy

€

n = 2, l =1, ml = 0

€

n = 2, l =1, ml = ±1

€

n = 2, l = 0, ml = 0

2s-state

2p-state2p-state

Same energy, but different probabilities


€

n = 3, l =1, ml = 0

€

n = 3, l =1, ml = ±1

3s-state

3p-state

3p-state

€

n = 3, l = 0, ml = 0

n=3: two s-states, six p-states and…


…ten d-states

€

n = 3, l = 2, ml = 0

€

n = 3, l = 2, ml = ±1

€

n = 3, l = 2, ml = ±2

3d-state 3d-state3d-state


Electron spin

New electron property:Electron acts like a bar magnet with N and S pole.

Magnetic moment fixed…

…but 2 possible orientations of magnet: up and down

Spin down

N

S

€

ms = −1/2

Described by spin quantum number ms

N

S

€

ms = +1/2 Spin up

z-component of spin angular momentum

€

Sz = msh


Include spin Quantum state specified by four quantum numbers:

Three spatial quantum numbers (3-dimensional)

One spin quantum number

€

n, l , ml , ms( )


Quantum Number Question

How many different quantum states exist with n=2?

A. 1B. 2C. 4D. 8

ℓ = 0 :ml = 0 : ms = 1/2 , -1/2 2

statesℓ = 1 :

ml = +1: ms = 1/2 , -1/2 2 states

ml = 0: ms = 1/2 , -1/2 2 states

ml = -1: ms = 1/2 , -1/2 2 states

2s2

2p6

There are a total of 8 states with n=2


Question

How many different quantum states are in a 5g (n=5, ℓ =4) sub-shell of an atom?

A. 22B. 20C. 18D. 16E. 14 ℓ =4, so 2(2 ℓ +1)=18.

In detail, ml = -4, -3, -2, -1, 0, 1, 2, 3, 4and ms=+1/2 or -1/2 for each.

18 available quantum states for electrons


Putting electrons on atom

Electrons obey Pauli exclusion principle Only one electron per quantum state (n, ℓ, mℓ, ms)

Hydrogen: 1 electron one quantum state occupied

occupiedunoccupied

n=1 states

Helium: 2 electronstwo quantum states occupied

n=1 states

€

n =1,l = 0,ml = 0,ms = +1/2( )

€

n =1,l = 0,ml = 0,ms = +1/2( )

€

n =1,l = 0,ml = 0,ms = −1/2( )


Atoms with more than one electron Electrons interact with

nucleus (like hydrogen) Also with other electrons Causes energy to

depend on ℓ


Other elements: Li has 3 electrons

n=1 states, 2 total, 2 occupiedone spin up, one spin down

n=2 states, 8 total, 1 occupied

€

n =1

l = 0

ml = 0

ms = +1/2

⎛

⎝

⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟

€

n =1

l = 0

ml = 0

ms = −1/2

⎛

⎝

⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟

€

n = 2

l = 0

ml = 0

ms = +1

2

⎛

⎝

⎜ ⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟ ⎟

€

n = 2

l = 0

ml = 0

ms = −1

2

⎛

⎝

⎜ ⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟ ⎟

€

n = 2

l =1

ml = 0

ms = +1

2

⎛

⎝

⎜ ⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟ ⎟

€

n = 2

l =1

ml = 0

ms = −1

2

⎛

⎝

⎜ ⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟ ⎟

€

n = 2

l =1

ml =1

ms = +1

2

⎛

⎝

⎜ ⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟ ⎟

€

n = 2

l =1

ml =1

ms = −1

2

⎛

⎝

⎜ ⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟ ⎟

€

n = 2

l =1

ml = −1

ms = +1

2

⎛

⎝

⎜ ⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟ ⎟

€

n = 2

l =1

ml = −1

ms = −1

2

⎛

⎝

⎜ ⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟ ⎟


Atom Configuration H 1s1

He 1s2

Li 1s22s1

Be 1s22s2

B 1s22s22p1

Ne 1s22s22p6

1s shell filled

2s shell filled

2p shell filled

etc

(n=1 shell filled - noble gas)

(n=2 shell filled - noble gas)

Electron Configurations


The periodic table Atoms in same column

have ‘similar’ chemical properties. Quantum mechanical explanation:

similar ‘outer’ electron configurations.

Be2s2

Li2s1

N2p3

C2p2

B2p1

Ne2p6

F2p5

O2p4

Mg3s2

Na3s1

P3p3

Si3p2

Al3p1

Ar3p6

Cl3p5

S3p4

H1s1

He1s2

CaK As4p3

Ge4p2

Ga4p1

Kr4p6

Br4p5

Se4p4

Sc3d1

Y3d2

8 moretransition

metals

Ca4s2

K4s1

Na3s1


Excited states of Sodium Na level structure

11 electrons Ne core = 1s2 2s2 2p6

(closed shell) 1 electron outside

closed shell Na = [Ne]3s1

Outside (11th) electron easily excited to other states.


Emitting and absorbing light

Photon is emitted when electron drops from one quantum state to another

Zero energy

n=1

n=2

n=3

n=4

€

E1 = −13.6

12 eV

€

E2 = −13.6

22 eV

€

E3 = −13.6

32 eV

n=1

n=2

n=3

n=4

€

E1 = −13.6

12 eV

€

E2 = −13.6

22 eV

€

E3 = −13.6

32 eV

Absorbing a photon of correct energy makes electron jump to higher quantum state.

Photon absorbed hf=E2-E1

Photon emittedhf=E2-E1


Optical spectrum

Optical spectrum of sodium Transitions from

high to low energystates

Relatively simple 1 electron

outside closed shell

Na

589

nm

, 3

p -

> 3

s


How do atomic transitions occur?

How does electron in excited state decide to make a transition?

One possibility: spontaneous emission

Electron ‘spontaneously’ drops from excited state Photon is emitted

‘lifetime’ characterizes average time for emitting photon.


Another possibility: Stimulated emission

Atom in excited state. Photon of energy hf=E ‘stimulates’ electron to drop.

Additional photon is emitted,

Same frequency,

in-phase with stimulating photon

One photon in,two photons out:

light has been amplified

E

Before After

hf=E

If excited state is ‘metastable’ (long lifetime for spontaneous emission) stimulated emission dominates


LASER : Light Amplification by Stimulated Emission of Radiation

Atoms ‘prepared’ in metastable excited states…waiting for stimulated emission

Called ‘population inversion’ (atoms normally in ground state)

Excited states stimulated to emit photon from a spontaneous emission.

Two photons out, these stimulate other atoms to emit.


Ruby Laser

• Ruby crystal has the atoms which will emit photons

• Flashtube provides energy to put atoms in excited state.

• Spontaneous emission creates photon of correct frequency, amplified by stimulated emission of excited atoms.


PUMP

Ruby laser operation

Metastable state

Relaxation to metastable state(no photon emission)

Transition by stimulated emission of photon

Ground state

1 eV

2 eV

3 eV


The wavefunction

Wavefunction = = |moving to right> + |moving to left>

The wavefunction is an equal ‘superposition’ of the two states of precise momentum.

When we measure the momentum (speed), we find one of these two possibilities.

Because they are equally weighted, we measure them with equal probability.


Silicon

7x7 surface reconstruction

These 10 nm scans show the individual atomic positions


Particle in box wavefunction

x=0 x=L

€

ψ x( )2dx = Prob. Of finding particle in region dx

about x

Particle is never here

€

ψ x < 0( ) = ?

Particle is never here

€

ψ x > L( ) = ?


Making a measurement

Suppose you measure the speed (hence, momentum) of the quantum particle in a tube. How likely are you to measure the particle moving to the left?

A. 0% (never)

B. 33% (1/3 of the time)

C. 50% (1/2 of the time)


Interaction with applied B-field

Like a compass needle, it interacts with an external magnetic field depending on its direction.

Low energy when aligned with field, high energy when anti-aligned

Total energy is then

€

E = −13.6

n2eV −

r μ •

r B

= −13.6

n2eV − μ zB

= −13.6

n2eV − ml μB B

This means that spectral lines will splitin a magnetic field


Orbital magnetic dipole moment

Dipole moment

µ=IA

Current =

€

chargeperiod

=e

2πr /v=

ev

2πr

Area =

€

πr2

Can calculate dipole moment for circular orbit

€

μ =evr

2=

eh

2mmvr /h( ) ≡ μB L /h( )

€

μB ≡eh

2m= 0.927 ×10−23 A ⋅m2

= 5.79 ×10−5eV /Tesla

€

L = h l l +1( )

€

μ =μB l l +1( ) magnitude of orb. mag. dipole moment

In quantum mechanics,


Electron magnetic moment

Why does it have a magnetic moment?

It is a property of the electron in the same way that charge is a property.

But there are some differences Magnetic moment has a size and a direction It’s size is intrinsic to the electron,

but the direction is variable.

The ‘bar magnet’ can point in different directions.


Additional electron properties

Free electron, by itself in space, not only has a charge, but also acts like a bar magnet with a N and S pole.

Since electron has charge, could explain this if the electron is spinning.

Then resulting current loops would produce magnetic field just like a bar magnet.

But… Electron in NOT spinning. As far as we know,

electron is a point particle.

N

S


Spin: another quantum number

There is a quantum # associated with this property of the electron.

Even though the electron is not spinning, the magnitude of this property is the spin.

The quantum numbers for the two states are+1/2 for the up-spin state

-1/2 for the down-spin state The proton is also a spin 1/2 particle. The photon is a spin 1 particle.


Orbital mag. moment

Since Electron has an electric charge, And is moving in an orbit around nucleus…

produces a loop of current,and a magnetic dipole moment ,

Proportional to angular momentum

Current

electron

Orbital magnetic moment

€

rμ =μB l l +1( )

magnitude of orb. mag. dipole moment

€

μB ≡eh

2m= 0.927 ×10−23 A ⋅m2

= 5.79 ×10−5eV /Tesla

€

rμ

€

rμ =μB

r L /h( )

Make a question out of this


Orbital mag. quantum number mℓ Possible directions of the ‘orbital bar magnet’ are

quantized just like everything else! Orbital magnetic quantum number

m ℓ ranges from - ℓ, to ℓ in integer steps Number of different directions = 2ℓ+1

N

S

m ℓ = +1

S

N

m ℓ = -1

For example: ℓ=1 gives 3 states:

m ℓ = 0

S N


Particle in box quantum states

n=1

n=2

n=3

Wavefunction Probability

L

€

h

2L€

2h

2L€

3h

2L

€

h2

8mL2€

22 h2

8mL2€

32 h2

8mL2

n p E


Particle in box energy levels

Quantized momentum

Energy = kinetic

Or Quantized Energy

€

E =p2

2m=

npo( )2

2m= n2Eo

€

En = n2Eo

€

p =h

λ= n

h

2L= npo

En

erg

yn=1n=2

n=3

n=4

n=5

n=quantum number


Hydrogen atom energies

Zero energy

n=1

n=2

n=3

n=4

€

E1 = −13.6

12 eV

€

E2 = −13.6

22 eV

€

E3 = −13.6

32 eV

En

erg

y

€

En = −13.6

n2 eV

Quantized energy levels: Each corresponds to

different Orbit radius Velocity Particle wavefunction Energy

Each described by a quantum number n


Quantum numbers

Two quantum numbers


Pauli Exclusion Principle

Where do the electrons go?

In an atom with many electrons, only one electron is allowed in each quantum state (n, ℓ, mℓ, ms).

Atoms with many electrons have many atomic orbitals filled.

Chemical properties are determined by the configuration of the ‘outer’ electrons.


Atomic sub-shells

Each

tue. dec. 4 2007physics 208, lecture 261 last time… 3-dimensional wave functions decreasing...

Documents

energy slide

quantum number n energy

b slide

quantum numbers n x

summary of quantum numbers

z quantum numbers

different angular momenta

energy of orbit