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Phy208 Exam 3

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MTE 3 Results

Average 79.75/100std 12.30/100

A 19.9% AB 20.8%B 26.3%BC 17.4%C 13.1%D 2.1%F 0.4%

FinalMon. May 12, 12:25-2:25, Ingraham B10

Get prepared for the Final!

Remember Final counts 25% of final grade!It will contain new material and MTE1-3 material

(no alternate exams!!! but notify SOON any potential and VERY serious problem you have with this time)

3

Atomic Physics

Previous Lecture:

Particle in a Box, wave functions and energy levels

Quantum-mechanical tunneling and the scanning tunneling microscope

Start Particle in 2D,3D boxes

This Lecture:

More on Particle in 2D,3D boxes

Other quantum numbers than n: angular momentum

H-atom wave functions

Pauli exclusion principle

HONOR LECTURE

! PROF. R. Wakai (Medical Physics)

! Biomagnetism

Biomagnetism deals with the registration and analysis of magnetic fields which

are produced by organ systems in the body.

Classical: particle bounces back and forth. Sometimes velocity is to left, sometimes to right

Quantum mechanics: Particle is a wave: p = mv = h/! standing wave: superposition of waves traveling left and right => integer

number of wavelengths in the tube

From last week: particle in a box

Energy

n=1

n=2

n=3

n=4

n=5!

En =p2

2m= n

2 h2

8mL2

"

# $

%

& '

Summary of quantum information

Energy is quantized

the larger the box the lower the energyof the particle in the box

A quantum particle in a box cannot beat rest! Fundamental state energy is not zero:En=1 = 0.38 eV for an electronin a quantum well of L = 1 nm Consequence of uncertainty principle:

!

"x = L#"px $1/L % 0!

Classical/Quantum Probability

Similar when n $%

n=3

n=2

n=1

Tunneling: nonzero probability of escaping the box. Tunneling Microscope: tunneling electron current from sample to probesensitive to surface variations

Probability(2D)

(nx, ny) = (2,1) (nx, ny) = (1,2)

Particle in a 2D box

Ground state: same wavelength

(longest) in both x and y

Need two quantum #’s,

one for x-motion

one for y-motion

Use a pair (nx, ny)

Ground state: (1,1)

x

y

Same energybut different probability in space

! Ground state

surface of constant

probability

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

All these states have the same energy, but different probabilities

(211) (121) (112)

Particle in 3D box

(221)

(222)

!

px =h

"nx

= nxh

2L

!

E =px

2

2m+py

2

2m+pz

2

2m= E

onx

2 + ny

2 + nz

2( )

!

E = Eonx

2 + ny

2 + nz

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

With increasing energy...

same for y,z

quantum states with same nx, ny, nz have same E

Eg: how many 3D particle states have 18E0?

!Bohr model fails describing atoms heavier than H

!Does it violate the Heisenberg uncertainty principle? A) YES

B) No

! Schrödinger: Hydrogen atom is 3D structure.Should have 3 quantum numbers.

!Coulomb potential (electron-proton interaction) is spherically symmetric.

x, y, z not as useful as r, !, #

! Modified H-atom should have 3 quantum numbers

Other quantum numbers? H-atom

radius and energy of electron cannot be exactly known at the same time!

!

En

= "13.6

n2

eV

!

rn

= n2ao

Bohr

!

L = h l l +1( )

Quantization of angular momentum

" is the orbital quantum number

States with same n, have same energy and can have " =

0,1,2,...,n-1 orbital quantum number

" =0 orbits are most elliptical

" =n-1 most circular

The z component of the angular

momentum must also be quantized

!

Lz

= mlh

m ℓ ranges from - ℓ, to ℓ integer

values=> (2ℓ+1) different values

!

r µ

!

µ =e

T"r2 =

ev

2"r"r2 =

e

2m(mvr) =

e

2mL

!

r µ = µ

Bl l +1( )

The experiment: Stern and Gerlach

It is possible to measure the number of possible values of Lz respect to the axis of the B-field produced by the electron current

e-

The electron moving on the orbit is like a current thatproduces a magnetic momentum µ=IA

Current

electron

Orbital

magnetic

dipole

! For a quantum state with " = 2, how many different orientations of the orbital angular momentum respect to the z-axis are there?

! A. 1

! B. 2

! C. 3

! D. 4

! E. 5

s: ℓ=0

p: ℓ=1

d: ℓ=2

f: ℓ=3

g: ℓ=4

“atomic shells”

! n : describes energy of orbit

! !"describes the magnitude of orbital angular momentum

! m ! describes the angle of the orbital angular momentum

For hydrogen atom:

Summary of quantum numbers

!

Lz

= mlh

!

L = h l l +1( )

!

En

= "13.6

n2

eV

! Spherically symmetric.

! Probability decreases

exponentially with radius.

! Shown here is a surface

of constant probability

!

n =1, l = 0, ml

= 0

3D Surfaces of constant prob. for H-atom

Electron cloud: probability density in 3D of electronaround the nucleus

!

P(r," ,#)dV = $(r," ,#)2dV

!

n = 2, l =1, ml

= 0

!

n = 2, l =1, ml

= ±1

2s-state

2p-state2p-state

Same energy, but different probabilities

Next highest energy: n = 2

!

n = 2, l = 0, ml

= 0

!

n = 3, l =1, ml

= 0

!

n = 3, l =1, ml

= ±1

3p-state

!

n = 3, l = 0, ml

= 0

n = 3: 2 s-states, 6 p-states and...

3s-state

3p-state

!

n = 3, l = 2, ml

= 0

!

n = 3, l = 2, ml

= ±1

!

n = 3, l = 2, ml

= ±2

3d-state 3d-state3d-state

...10 d-states Radial probability

! For 1s, 2p, 3d, rpeak = a0, 4a0, 9a0

! These are the Bohr orbit radii!

! most probable distance of electron

from nucleus!

! They behave like Bohr orbits

because for states with same E,

larger angular momentum

corresponds to more spherical orbits,

orbits are elliptical for small "

!

"n,l,m

(r,# ,$) = Rn,l(r)Y

l,m(# ,$)

Radial Angular

!

rn

= n2ao

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

Electron spin

z-component of spin angular momentum

!

Sz

= msh

Described byspin quantum number ms

! Quantum state specified by four quantum numbers:

! Three spatial quantum numbers (3-dimensional)

! One spin quantum number

!

n, l, ml, m

s( )

How many different quantum states exist with n=2?

A. 1

B. 2

C. 4

D. 8

! = 0 :

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

2s2

! = 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

2p6

All quantum numbers of electrons in atoms

! Electrons obey Pauli exclusion principle

! Only one electron per quantum state (n, !, m!, m

s)

Hydrogen: 1 electron

one quantum state occupied

occupied

unoccupied

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( )

Pauli exclusion principle

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)

Building Atoms

H

1s1He

1s2

The periodic table

! Atoms in same column

have ‘similar’ chemical properties.

! Quantum mechanical explanation:

similar ‘outer’ electron configurations.

Be

2s2Li

2s1N

2p3

C

2p2

B

2p1

Ne

2p6

F

2p5

O

2p4

Mg

3s2Na

3s1P

3p3

Si

3p2

Al

3p1

Ar

3p6

Cl

3p5

S

3p4

As

4p3

Ge

4p2

Ga

4p1

Kr

4p6

Br

4p5

Se

4p4

Sc

3d1

Y

3d2

8 moretransition

metals

Ca

4s2K

4s1