Quantum Mechanics, part 3Quantum Mechanics, part 3Trapped electronsTrapped electrons

• Infinite Potential WellInfinite Potential Well• Finite Potential WellFinite Potential Well

– Quantum TrapsQuantum Traps• NanocrystallitesNanocrystallites• Quantum DotsQuantum Dots• Quantum corralsQuantum corrals

– 2-D and 3-D Traps2-D and 3-D Traps

• Hydrogen AtomHydrogen Atom– Bohr TheoryBohr Theory– Solution to Schrödinger EquationSolution to Schrödinger Equation

A quantum corral of iron atomsA quantum corral of iron atoms

““Confinement leads to quantization”Confinement leads to quantization”

Electron Trap

Energy Level Diagrams---DISCRETE LEVELS NOT CONTINUOUS!!!!!!!!!!!11

Particle in a Box by analogyParticle in a Box by analogy(Infinite Potential Well)(Infinite Potential Well)

• Classically - Classically - any energy and any energy and momentum just momentum just like a free like a free particleparticle

Standing waves in a stringStanding waves in a string

L

xnA

kxAx

sin

sin

Particle in a BoxParticle in a Box• QM - Boundary conditions for the QM - Boundary conditions for the

matter wavematter wave

L

xnAx

sin

n

L

and

L

xnAx

so

nkL

kLAL

kxAx

2

sin

0sin

sin

22

2

8n

mL

hhcE

Particle in a BoxParticle in a Box

L

xnAx

sin22

2

8n

mL

hE

Introduction to Wave Mechanics Introduction to Wave Mechanics (review)(review)

• The wave functionThe wave function

– Interpretation - Probability function and Interpretation - Probability function and

density density

– NormalizationNormalization

– Probability of locating a particleProbability of locating a particle

– Expectation valueExpectation value

2

12

dxP

b

a

ab dxP2

b

a

dxxx2

Infinite Potential Well Boundary conditions are Infinite Potential Well Boundary conditions are everything!!!!!!!!!!!!!!everything!!!!!!!!!!!!!!

Solution using Schrödinger Wave EquationSolution using Schrödinger Wave Equation

2 2

22

dU E

m dx

2 2

2, 0 x L

2

dE

m dx

U = 0 inside the well U = 0 inside the well and and everywhere else, everywhere else,so so = 0 if x < 0 or x > L. = 0 if x < 0 or x > L.

2

2 2

2d mE

dx

22

2

dk

dx

22

2mEk

wherewhere

The general solution isThe general solution isthat of SHM; i.e.,that of SHM; i.e.,

Apply the boundaryApply the boundaryConditions at x = 0Conditions at x = 0

0B

AlsoAlso

requiresrequireskL n

nk

L

oror

1,2,3,n

kxBkxAx cossin)(

0

00cos0sin)0(

B

BA

0sin)( kLAL

Determining the constant Determining the constant AA in the Infinite in the Infinite Potential Well Potential Well

Solution using Schrödinger Wave Equation example prob 39-2Solution using Schrödinger Wave Equation example prob 39-2

sinn

A xL

Normalize the probabilityNormalize the probability

2 2 2

0

sin 1L nA x dx

L

2 2

0

sin 1L n

A x dxL

2 1 1sin cos 2

2 2

2

0

1 1 2cos 1

2 2

L n xA dx

L

2 2

0

2sin 1

2 4

LL L n x

A An L

2 2 2 0 2sin sin 1

2 4

L L n n LA A

n L L

2 2 0 12 4

L LA A

n

2A

L

Infinite Potential Well Infinite Potential Well Solution using Schrödinger Wave EquationSolution using Schrödinger Wave Equation

2 22

22E n

mL

1,2,3,n

22

28

hE n

mL

nk

L

oror

2

sin2

..

sin

k

L

xn

Lx

CB

kxAx

Infinite Potential Well Infinite Potential Well Solution using Schrödinger Wave EquationSolution using Schrödinger Wave Equation

2

2 2

2d mE

dx

Verify that the above isVerify that the above isa solution to the differentiala solution to the differentialequation.equation.

2

2 2

2sin

2 2sin

nd x

L L mE nx

dx L L

2

2

2 2 2sin sin

n n mE nx x

L L L L L

2

2

2n mE

L

2 2 2

22

nE

mL

2 22

22E n

mL

1,2,3,n

Why isn’t n = 0 a valid Why isn’t n = 0 a valid quantum number?quantum number?

Infinite Potential Well Infinite Potential Well Solution using Schrödinger Wave EquationSolution using Schrödinger Wave Equation

sin2

L nx

L

22

28

hE n

mL

1,2,3,n

Energy level transitionsEnergy level transitions

2

2 228f i f i

hE E E hf n n

mL

2b

ab

a

P dx

2 2sinb

ab

a

nP A x dx

L

Particle Finite Potential WellParticle Finite Potential WellRegions of the potential wellRegions of the potential well

Wave function and Wave function and probability functionsprobability functions

Energy level diagram for Energy level diagram for L = 100 pm and UL = 100 pm and Uoo = 450 eV = 450 eV

Matter wave leaks into the walls. For any quantum state the wavelength is longer so the corresponding energy is less for the finite well than the infinite trap/well.

mEh 2/

Finite Well Cont.• Given U0=450 eV, L=100 pm• Remove the portion of the energy diagram of the infinite well above E=450

eV and shift the remaining levels (three in this case) down.

Examples of quantum electron trapsExamples of quantum electron traps

NanocrystallitesNanocrystallites

Quantum DotQuantum DotA quantum corral of iron atomsA quantum corral of iron atoms

2 D and 3 D rectangular corrals2 D and 3 D rectangular corrals

2 22 2

, 8 8x yn n x yx y

h hE n n

mL mL

2 2 22 2 2

, , 8 8 8x y zn n n x y zx y z

h h hE n n n

mL mL mL

Simple Harmonic OscillatorSimple Harmonic Oscillator21

( )2

U x kx

1

2nE n hf

The Nature of the Nuclear AtomThe Nature of the Nuclear AtomRutherford 1911 (w/grad students Geiger and Rutherford 1911 (w/grad students Geiger and

Marsden) ScatteringMarsden) Scattering– Scattering alpha particles from gold foilScattering alpha particles from gold foil– Some alphas bounced back as if “a cannonball Some alphas bounced back as if “a cannonball

bouncing off tissue paper”bouncing off tissue paper”– Established the nuclear atomEstablished the nuclear atom– Electron outside a very small positive nucleusElectron outside a very small positive nucleus– Classical theory leads to contradictionClassical theory leads to contradiction

An electron would An electron would spiral into the spiral into the nucleus in a timenucleus in a time

st 610

AAargh….AAargh….

Electrons are trapped by the Nucleus

• Could the energy states be discrete?• Stability of the atom is due to quantization of

energy much like the trapped electron in the finite well!!!!

• Bohr postulates that angular momentum and thus energy is quantized in units of Planck's constant

• There is a hint from the signature of atomic spectra…this week’s lab…….

Hydrogen Line SpectraHydrogen Line Spectra

2i

2H n1

21

R1

1H m73297310R ,,

Johannes Balmer 1897: Balmer Series SpectrumJohannes Balmer 1897: Balmer Series Spectrum

Atomic Line SpectraAtomic Line Spectra

• Rydberg FormulaRydberg Formula

2i

2f n

1n1

R1

21

2 e

eE k

r

Bohr Model of the Atom (1913)Bohr Model of the Atom (1913)• Semiclassical nuclear modelSemiclassical nuclear model

– Assumes Electrostatic ForcesAssumes Electrostatic Forces– Stationary Orbits hypothesizedStationary Orbits hypothesized

2 2

2e

e vF k m

r r

221

2 e

eE K U mv k

r

2 21

2 e e

e eE k k

r r

2h

nnmvrL

prL

22 2

02ne

r n n amk e

NoteNote0

1

4ek

Bohr Model of the AtomBohr Model of the Atom2

0 2e

amk e

0 0.0529a nm2

2 20

1 113.6

2ek eE eVa n n

2 2

1 113.6

f i

E eV hfn n

2

2 20

1 1 1

2e

f i

k e

a hc n n

Bohr Model of the AtomBohr Model of the Atom

2

0

1%!2e

H

k eR

a hc

2 2

1 113.6

f i

E eVn n

2

2 20

1 113.6

2ek eE eVa n n

2

2 20

1 1 1

2e

f i

k e

a hc n n

The Bohr Model and Standing Electron WavesThe Bohr Model and Standing Electron Waves(Arthur Sommerfeld)(Arthur Sommerfeld)

2n r

2h

nh r

2nh rp

2nh rmvn rmv L

n L

2h

nnmvrL

prL

ResultsResults

• Consistent with basic Hydrogen Consistent with basic Hydrogen spectrumspectrum

• Explains origin of photonsExplains origin of photons

• Fails to explain more complex Fails to explain more complex spectra and fine points of Hydrogen spectra and fine points of Hydrogen spectrumspectrum

The Solution to the The Solution to the SchrödingerSchrödinger Equation for HydrogenEquation for Hydrogen

• Predicts 3 quantum numbers –n,Predicts 3 quantum numbers –n,,m,m

• Successfully describes atomic spectraSuccessfully describes atomic spectra

1,2,3,...

1

n

n

m

2 2 2

2 2 2 2 2

2 1 1sin

2 sin sinU E

m r r r r r

R r

Solution is the product of 3 functionsSolution is the product of 3 functions

4

2 2 2 2

1

32 o

meE

n

20

0 2

4a

me

NoteNote

0

1

4ek

The Solution to the The Solution to the Schrödinger Equation for Schrödinger Equation for

HydrogenHydrogen

1 3

1o

ra

s

o

r ea

224P r r

The Solution to the The Solution to the Schrödinger Equation for Schrödinger Equation for

HydrogenHydrogen

Correspondence PrincipleCorrespondence Principle

•For large quantum For large quantum numbers, the results of numbers, the results of quantum mechanical quantum mechanical calculations approach calculations approach those of classical those of classical mechanicsmechanics