ρ () tdv$$$$scienide2.uwaterloo.ca/~nooijen/chem356/chem+356+pdf/ch_1.pdf ·...

Fall 2014 Chem 356: Introductory Quantum Mechanics

8

Chapter 1 – Early Quantum Phenomena ..................................................................................................... 8

Early Quantum Phenomena ..................................................................................................................... 8

Photo-‐electric effect .............................................................................................................................. 11

Emission Spectrum of Hydrogen ............................................................................................................ 13

Bohr’s Model of the atom ...................................................................................................................... 14

De Broglie Waves ................................................................................................................................... 17

Double slit experiment .......................................................................................................................... 18

Chapter 1 – Early Quantum Phenomena

Early Quantum Phenomena Blackbody Radiation 1900 (Planck)

the first evidence of quantization

Tiny hole, emits radiation

Body at temperature T (in equilibrium)

Approximate black bodies: stars, a stove, flame or furnace

( )v T dvρ in J m-‐3


Chapter 1 – Early Quantum Phenomena 9

radiant energy density (intensity) between v and v + dv

-‐ Same for any material -‐ Maximum in distribution shifts to higher v with increasing T

Wien: λmaxT = 2.9 ⋅10−3

m K

Stefan-‐Boltzman (see McQuarrie):

R = c

4Ev (T ) = c

4ρν (T )dv

0

∞

∫

=σT 4 , σ ≈ 5.6697 ⋅10−8 J m-‐2 k-‐4 s-‐1

Planck took experimental curves and fitted them to a formula:

ρv (T )dv = 8πh

c3 ⋅ v3

ehv/kBT −1dv

c : speed of light ~ 3 x 108 m s-‐1

h : Planck’s constant ~ 6.6 x 10-‐34 J s Bk : Boltzmann ~ 1.4 x 10-‐23 J k-‐1

Works Perfectly. Planck’s problem: How can one derive it from known classical physics?

Classical Physics → ρv (T ) =

8πkBTc3 v2 Exact as v → 0

Disaster as v large Something was VERY WRONG with classical physics

Planck made a quantum hypothesis, a particular oscillator of frequency v could only have energies nhv quantized: n is an integer. With this ad hoc assumption, he could derive his formula. Nobody understood what this meant. Some more details: ρ(ν ,T ) can be converted to ρ(λ,T )dλ

v = c

λ

dv = − c

λ 2 dλ



8πhc3 ⋅ v3

ehv/kBT −1dv →

8πhc3 ⋅

cλ

⎛⎝⎜

⎞⎠⎟

3

ehc/kBTλ −1− cλ 2

⎛⎝⎜

⎞⎠⎟

dλ

also 2

1

v

v

dv∫ → 2

1

dλ

λ

λ∫ → 1

2

dλ

λ

λ−∫

2 1v v> → 2 1λ λ< (exercise)

ρ(λ,T )dλ = 8πhcλ5

1

ehc

λkBT −1

⎛

⎝⎜⎜

⎞

⎠⎟⎟

dλ

Einstein later on (1915) gave an insightful derivation. Let us discuss: Consider equilibrium between any 2 quantum levels (micro-‐balance)

2

1

2

1

~

~

B

B

Ek T

Ek T

N e

N e

−

−

BN1ρ(ν ,T )− BN2ρ(ν ,T )− AN2 = 0 Stimulated ~ ρ(v,T ) spontaneous

ρ(ν ,T )[

N1

N2

−1]= AB

2 1

1

2

B B

E E hvk T k TN

e eN

−

= = 33

8A h vB c

π= ⋅ Property of radiation

i.e. does not depend on material. [Derived much later by Dirac, Quantum electrodynamics]

ρν (T ) = 8πhc3 ⋅ v3

ehv

kBT −1

Black body: all quantum levels in equilibrium



Black: To every frequency there is a set of levels 2 1E E hv− → (dense) all hv occur.

Planck was first, but it is hard to see that Blackbody radiation implies quantization of energy levels. Few people understood the implications. Einstein did! Photo-‐electric effect (Einstein 1905)

Light: oscillating electromagnetic field jiggles electrons → provides kinetic energy, enough such that electron can leave the metal. Classically one expects:

-‐ Kinetic energy of electron depends on the intensity of the light -‐ Electrons can be ejected for any frequency of light, if intensity is high enough.

What is observed (Lenard, Millikan, later on)

-‐ Kinetic energy of electron does not depend on intensity of light -‐ Only above threshold frequency do you see any electrons -‐ # of ejected electrons depends on the intensity of field -‐ Kinetic energy depends linearly on v

Einstein: light is absorbed in discrete packets of energy that depend on the frequency (called photons, much later). The photon density is proportional to intensity of the light.



Ephoton = hv

Ekin = hv −φ

= hv − hv0

φ : workfunction: energy to liberate electron from the metal

Intensity of light ~ number of photons

This picture explains all aspects of the experiment. Some aspects were predicted by Einstein, and verified later. → Wave nature of light was well established. Now light looks like particles! Modern version of photoelectric effect:

Photoelectron Spectroscopy

Molecular orbital picture

Different binding energies → different onsets for kinetic energy



Emission Spectrum of Hydrogen

Consists of discrete lines, sharp frequencies with a regular pattern. Balmer (a school teacher) found the first set of lines

!v wavenumber !v = v

c= 1λ in cm-‐1

The number of crests in per cm

Balmer !vn = R( 1

22 −1n2 ) 3,4,5...n =

Lyman !vn = R( 1

12 −1n2 ) 2,3,4n =

Paschen !vn = R( 1

32 −1n2 ) 4,5,6n =

Picture: Hydrogen atom has discrete set of energy level 2

Rn

−



−R( 1

ni2 −

1nf

2 ) = R( 1nf

2 −1ni

2 )

ni = 1,2,3 etc. f in n>

Simple convincing picture

Why do energy levels go as 2

1n

?? What do integers have to do with it??

Bohr’s Model of the atom (1913)

A first attempt. The concept was not quite right, but the numbers came out very well. Let us see what he did: Electron moves in circle around nucleus



!r = x(t) = Rcosωt

!v = dr

dt= −ωRsinωt

( ) siny t R tω= cosR tω ω+

!a = dv

dt= −ω

2Rcosωt !F = m!a directed inwards

−ω2Rsinωt

F = mRω 2 = mv2

R v = Rω

F = e2

4πε0R2 Coulomb Attraction

Bohr’s vital insight concerned the angular momentum

!L = !r × !p = !r × m!v

!Lz = m(rxvy − ryvx ) = mωR2(cos2ωt + sin2ωt)

= mωR2 = mvR ( v =ωR )

Bohr postulated that the angular momentum is quantized…

Lz = mvR = n! = n h

2π

....and that electrons can move in stationary orbits. [glaring conflict with classical physics: accelerating charge radiates and loses energy → Falls into the nucleus] Let us put it together..

mv2

R= e2

4πε0R2 (1)

mvR = n! (2)



En =

12

mv2 − e2

4πε0R(kinetic energy + Coulomb potential)

= 1

2e2

4πε0R− e2

4πε0R= − 1

2e2

4πε0R (3)

Use (1)

(mvR)2

R⋅ 1mR2 = e2

4πε0R2

(n!)2 = e2m

4πε0

⋅R

1R= e2m

4πε0n2!2

En = − 1

2e4m

(4πε0 )2!2 ⋅1n2 (use in (3))

(Ek − En ) = hv = hc !v = − 1

8me4

ε02h2 ( 1

k 2 −1n2 )

RH = 1

8me4

ε0ch3 ! 109757 cm−1

Bohr later found that em m= the mass of the electron should be replaced by the reduced mass

µ = me

mp

me + mp

… then HR is even closer to experiment.

Repeating the analysis for a nuclear charge of Z (change Coulomb potential term)

RH = Z 2e4µ

8ε02ch3

µ = me

mp

me + mp

This gave excellent agreement for He+, Li+ etc., using the appropriate mass

mp → mnucleus

He+ measured from the solar spectrum (was assigned using Bohr’s formula)



De Broglie Waves

In 1923 de Broglie proposed that particles have a wave length λ = h

p

His line of thought (more or less) of photons (light particles) is that they have no mass, but they do have momentum.

Scattered light imparts momentum to electron (Compton)

Relativistically (see problem assignment)

E = hv = pc λ = c

v= h

p

De Broglie: Generalize to all particles, not only

photons: λ = h

p

We can even assign direction

!k is wave number

2kπλ = , 2 2 2

x y zk k k k= + +

k = 2π

λ

2πh!p =!k

2kπλ =

!p = "!k

! = h

2π

Wavelength electron: 319 10m −= ⋅ kg

v ~

3kBTm

from 21 32 2 Bmv k T=

4~ 10 m s-‐1 thermal energy (translational)

p = mv , λ = h

mv~ 6.00−8 m 60 nm=



One can use electrons, speed v , to diffract like a wave Electron microscopy Shorter wavelengths then visible light. Larger wavelengths than x-‐rays. All of this is well confirmed by experiment:

-‐ Today Zeilinger (Vienna) confirmed wave-‐character of C60 Bucky balls! -‐ Electron microscopy widely used

De Broglie relation sheds light on Bohr quantization rate

L = mvR = pR = n! = nh

2π

Or 2πR = n h

p= nλ

How to interpret?

If wavelength fits on circle 2 Rπ → constructive interference

Otherwise mismatch → not a stationary solution.

De Broglie opened up the perspective to view electron as a wave. Double slit experiment (take 1) a.1 Shine light through a small hole and detect beam.

Big hole → ‘straight line’



a.2 do the same experiment with tiny bullets

individual holes in screen collect # of bullets in buckets

Now repeat the experiment with 2 holes:

Bullets: individual bullets sum of patterns

Repeat with light:

Dark and light bands. Interference effect



What is the explanation see McQuarrie:

Extra length d sinθ = nλ → constructive interference

d sinθ = (n+ 1

2)λ → destructive interference

How do electrons behave? a) Like bullets in that they hit the screen as discrete units b) As a wave: electrons, (and neutrons, bucky balls) show dispersion and

interference effect How do light/photons behave?

Lower intensity of the light: photons will hit the screen as isolated specks!

→ photons (moving at the speed of light) and electrons (quantum particles) behave very alike. Even if one electron or one photon at a time goes through → interference pattern builds up (no prediction for individual particles, only statistics is predicted)



Electron propagates like wave, is detected like particle when visualized macroscopically.

Through which slit does the electron go? Measure it!

Experiment works beautifully. Each electron passes through one hole, only and then hits the screen (also for photon!) → Correlate slit ↔ spot on the screen. Pattern builds up over time, but…it is like a bullet pattern, no interference. Looking at which slit the electron takes destroys the interference pattern.

ρ () tdv$$$$scienide2.uwaterloo.ca/~nooijen/chem356/chem+356+pdf/ch_1.pdf ·...

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