last time… bohr model of hydrogen atom wave properties of matter energy levels from wave...

Last Time…

Bohr model of Hydrogen atom

Wave properties of matter

Energy levels from wave properties

Thu. Nov. 29 2007 Physics 208, Lecture 25 2

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 ‘Particle in a box’

Particle confined to a fixed region of spacee.g. ball in a tube- ball moves only along length L

Classically, ball bounces back and forth in tube. This is a ‘classical state’ of the ball.

Identify each state by speed, momentum=(mass)x(speed), or kinetic energy.

Classical: any momentum, energy is possible.Quantum: momenta, energy are quantized

L


Classical vs Quantum

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

L

Quantum mechanics: Particle represented by wave: p = mv = h / Different motions: waves traveling left and right Quantum wave function:

superposition of both at same time


Quantum version

Quantum state is both velocities at the same time

Ground state is a standing wave, made equally of Wave traveling right ( p = +h/ ) Wave traveling left ( p = - h/ ) Determined by standing wave condition L=n(/2) :

€

=2LOne half-wavelength

€

p =h

λ=

h

2L

momentum

L

Quantum wave function: superposition of both motions.

€

ψ x( ) =2

Lsin

2π

λx

⎛

⎝ ⎜

⎞

⎠ ⎟


Different quantum states p = mv = h /

Different speeds correspond to different subject to standing wave condition

integer number of half-wavelengths fit in the tube.

€

=LTwo half-wavelengths

€

p =h

λ=

h

L= 2po

momentum

€

=2LOne half-wavelength

€

p =h

λ=

h

2L≡ po

momentumn=1

n=2

€

ψ x( ) =2

Lsin

2π

λx

⎛

⎝ ⎜

⎞

⎠ ⎟Wavefunction:


Particle in box question

A particle in a box has a mass m. Its energy is all kinetic = p2/2m. Just saw that momentum in state n is npo. It’s energy levels

A. are equally spaced everywhere

B. get farther apart at higher energy

C. get closer together at higher energy.


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


Question

A particle is in a particular quantum state in a box of length L. The box is now squeezed to a shorter length, L/2.

The particle remains in the same quantum state.

The energy of the particle is now

A. 2 times bigger

B. 2 times smaller

C. 4 times bigger

D. 4 times smaller

E. unchanged


Quantum dot: particle in 3D box

Energy level spacing increases as particle size decreases.

i.e

CdSe quantum dots dispersed in hexane(Bawendi group, MIT)

Color from photon absorption

Determined by energy-level spacing

Decreasing particle size

€

En +1 − En =n +1( )

2h2

8mL2−

n2h2

8mL2


Interpreting the wavefunction

Probabilistic interpretationThe square magnitude of the wavefunction ||2 gives the

probability of finding the particle at a particular spatial location

Wavefunction Probability = (Wavefunction)2


Higher energy wave functions

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


Probability of finding electron

Classically, equally likely to find particle anywhere QM - true on average for high n

Zeroes in the probability!Purely quantum, interference effect


Quantum Corral

48 Iron atoms assembled into a circular ring. The ripples inside the ring reflect the electron quantum states of a

circular ring (interference effects).

D. Eigler (IBM)


Scanning Tunneling Microscopy

Over the last 20 yrs, technology developed to controllably position tip and sample 1-2 nm apart.

Is a very useful microscope!

Tip

Sample


Particle in a box, againL


Particle contained entirely within closed tube.

Open top: particle can escape if we shake hard enough.

But at low energies, particle stays entirely within box.

Like an electron in metal (remember photoelectric effect)


Quantum mechanics says something different!

Quantum Mechanics:some probability of the

particle penetrating walls of box!

Low energy Classical state

Low energy Quantum state

Nonzero probability of being outside the box.


Two neighboring boxes When another box is brought nearby, the

electron may disappear from one well, and appear in the other!

The reverse then happens, and the electron oscillates back an forth, without ‘traversing’ the intervening distance.


Question

‘low’ probability

‘high’ probability

Suppose separation between boxes increases by a factor of two. The tunneling probability

A. Increases by 2

B. Decreases by 2

C. Decreases by <2

D. Decreases by >2

E. Stays same


Example: Ammonia molecule

Ammonia molecule: NH3

Nitrogen (N) has two equivalent ‘stable’ positions.

Quantum-mechanically tunnels

2.4x1011 times per second (24 GHz) Known as ‘inversion line’ Basis of first ‘atomic’ clock (1949)

N

H

H

H


Atomic clock question

Suppose we changed the ammonia molecule so that the distance between the two stable positions of the nitrogen atom INCREASED.The clock would

A. slow down.

B. speed up.

C. stay the same.

N

H

H

H


Tunneling between conductors Make one well deeper:

particle tunnels, then stays in other well. Well made deeper by applying electric field. This is the principle of scanning tunneling microscope.


Scanning Tunneling Microscopy

Over the last 20 yrs, technology developed to controllably position tip and sample 1-2 nm apart.

Is a very useful microscope!

Tip

Sample

Tip, sample are quantum ‘boxes’

Potential difference induces tunneling

Tunneling extremely sensitive to tip-sample spacing


Surface steps on Si

Images courtesy M. Lagally, Univ. Wisconsin


Manipulation of atoms Take advantage of tip-atom interactions to

physically move atoms around on the surface

D. Eigler (IBM)

This shows the assembly of a circular ‘corral’ by moving individual Iron atoms on the surface of Copper (111).

The (111) orientation supports an electron surface state which can be ‘trapped’ in the corral


Quantum Corral

48 Iron atoms assembled into a circular ring. The ripples inside the ring reflect the electron quantum states of a

circular ring (interference effects).

D. Eigler (IBM)


The Stadium Corral

Again Iron on copper. This was assembled to investigate quantum chaos. The electron wavefunction leaked out beyond the stadium too much to to observe

expected effects.

D. Eigler (IBM)


Some fun!

Kanji for atom (lit. original child)

Iron on copper (111)

D. Eigler (IBM)

Carbon Monoxide man

Carbon Monoxide on Pt (111)


Particle in box again: 2 dimensions

Same velocity (energy), but details of motion are different.

Motion in x direction Motion in y direction


Quantum Wave Functions

Ground state: same wavelength (longest) in both x and y

Need two quantum #’s,one for x-motionone for y-motion

Use a pair (nx, ny)

Ground state: (1,1)

Probability(2D)


One-dimensional (1D) case


2D excited states

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

These have exactly the same energy, but the probabilities look different.

The different states correspond to ball bouncing in x or in y direction.


Particle in a box

What quantum state could this be?

A. nx=2, ny=2

B. nx=3, ny=2

C. nx=1, ny=2


Next higher energy state

The ball now has same bouncing motion in both x and in y.

This is higher energy that having motion only in x or only in y.

(nx, ny) = (2,2)


Three dimensions

Object can have different velocity (hence wavelength) in x, y, or z directions. Need three quantum numbers to label state

(nx, ny , nz) labels each quantum state (a triplet of integers)

Each point in three-dimensional space has a probability associated with it.

Not enough dimensions to plot probability But can plot a surface of constant probability.


Particle in 3D box Ground state

surface of constant probability

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

2D case


(211)

(121)

(112)

All these states have the same energy, but different probabilities


(221)

(222)


The ‘principal’ quantum number

In Bohr model of atom, n is the principal quantum number.

Arise from considering circular orbits.

Total energy given by principal quantum number

€

En = −13.6

n2 eV

•Orbital radius is

€

rn = n2ao


Other quantum numbers?

Hydrogen atom is three-dimensional structure Should have three quantum numbers

Special consideration: Coulomb potential is spherically symmetric x, y, z not as useful as r, ,

Angular momentum warning!


Sommerfeld: modified Bohr model Differently shaped orbits

Big angular momentumSmall

angular momentum

All these orbits have same energy…… but different angular momenta

Energy is same as Bohr atom, but angular momentum quantization altered


Angular momentum question

Which angular momentum is largest?


The orbital quantum number ℓ

In quantum mechanics, the angular momentum can only have discrete values , 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

These states all have the same energy

€

L = h l l +1( )


Orbital mag. moment

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

… it produces a loop of current,and hence a magnetic dipole field, very much like a bar magnet or a compass needle.

Directly related to angular momentum

Current

electron

Orbital magnetic moment


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,


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

Example: For ℓ=1, m ℓ = -1, 0, or -1, corresponding to three different directionsof orbital bar magnet.

N

S

m ℓ = +1

S

N

m ℓ = -1ℓ=1 gives 3 states: m ℓ = 0

S N


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


Example: For ℓ=2, m ℓ = -2, -1, 0, +1, +2 corresponding to three different directionsof orbital bar magnet.


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


Summary of quantum numbers

n describes the energy of the orbit ℓ describes the magnitude of angular momentum m ℓ describes the behavior in a magnetic field due to the

magnetic dipole moment produced by orbital motion (Zeeman effect).


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


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.


Electron spin orientations

Spin up

Spin down

N

S

N

S

Only two possible orientations


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.


Include spin

We labeled the states by their quantum numbers. One quantum number for each spatial dimension.

Now there is an extra quantum number: spin.

A quantum state is specified four quantum numbers:

An atom with several electrons filling quantum states starting with the lowest energy, filling quantum states until electrons are used.

€

n, l , ml , ms( )


Quantum Number Question

How many different quantum states exist with n=2?

1248

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


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.


Number of electrons

Which of the following is a possible number of electrons in a 5g (n=5, l=4) sub-shell of an atom?

222017

l=4, so 2(2l+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

17 will fit. And there is room left for 1 more !


Putting electrons on atom

Electrons are obey exclusion principle Only one electron per quantum state

Hydrogen: 1 electron one quantum state occupied

occupiedunoccupied

n=1 states

Helium: 2 electronstwo quantum states occupied

n=1 states


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 Elements are arranged in the periodic table so that

atoms in the same column have ‘similar’ chemical properties.

Quantum mechanics explains this by similar ‘outer’ electron configurations.

If not for Pauli exclusion principle, all electrons would be in the 1s state!

H1s1

Be2s2

Li2s1

N2p3

Mg3s2

Na3s1

C2p2

B2p1

Ne2p6

F2p5

O2p4

P3p3

Si3p2

Al3p1

Ar3p6

Cl3p5

S3p4

H1s1

He1s2


Wavefunctions and probability Probability of finding an electron is given by the

square of the wavefunction.Probability large here

Probability small here


Hydrogen atom: Lowest energy (ground) state

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


Wave representingelectron

Electron wave around an atom

Wave representingelectron

Electron wave extends around circumference of orbit.

Only integer number of wavelengths around orbit allowed.


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


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)

last time… bohr model of hydrogen atom wave properties of matter energy levels from wave...

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