# ε π ε λ - La Salle University by linfengfengfz

VIEWS: 0 PAGES: 7

• pg 1
```									                            CHEM 301: QUANTUM MECHANICS
Chapter 1: From Classical to Quantum Mechanics

LIGHT
Waves                           c = λν        (ν = frequency)
Energy of photons              E = hν
Mass of photon                 m = E/c2 = hν/c2
Momentum                       p = mc = hν/c = h/λ
Blackbody (Planck)                             8π hc
ρ (λ ) =
λ (e hc / λkT − 1)
5

MATTER
Classical Mechanics
Momentum                    p = mv  (v = velocity)
Kinetic Energy              KE = ½ mv2 = p2/2m
Wave Mechanics
λ = h/p
deBroglie Relationship
BOHR MODEL OF ATOM
Energy of Hydrogen nth                 e 4 me
En = − 2 2 2
orbital                           8ε o h n
th
ε on2h2
Radius of H n orbital          r=          = 52.9 pm for n = 1
π me e 2
1       1   1          
= RH  2 − 2
n               
       n1, n2 =1,2,3, … n1 < n2
Rydberg-Ritz Equation           λ       1 n2           
(Hydrogen emission)
e 4 me
RH =      2
= 1.097 x 107        m-1
2
CONSTANTS (SI UNITS)

Planck’s Constant              h          = 6.626 x 10-34 J·s
ħ = h/2π = 1.055 x 10-34 J·s
Boltzmann Constant             k          = 1.380 x 10-23 J/K
Bohr Radius                    ao         = 5.292 x 10-11 m-1
Speed of Light                 c          = 3.000 x 108 m/s
Electron Mass                  me         = 9.109 x 10-31 kg
Proton Mass                    mp         = 1.673 x 10-27 kg
Electron Charge                e          = 1.602 x 10-19 C
Energy Conversions             1 eV       = 1.602 x 10-19 J
-1
1 cm       ~ 1.986 x 10-23 J
= energy corresponding to light of 1/λ (λ in units of cm)

Gentry, 2012
CHAPTER 1: FROM CLASSICAL TO QUANTUM MECHANICS
Lecture Outline

DUAL NATURE OF LIGHT
Light sometimes behaves like waves - Maxwell’s Equations
- Light can be described as electro-magnetic waves
Gives rise to interference patterns
- Wavelength (λ) and frequency (ν) depend on speed of light (c)
c = λν , where c is independent of λ and ν
BUT… Light sometimes behaves like particles (photons ) - Photoelectric Effect
Einstein introduced concept of photons (light particles) in 1905
Discrete particles with defined energy: E = hν
Photoelectric Effect – light striking a metal electrode in a vacuum tube knocks an
electron off the metal, and that ejected electron has kinetic energy.
1) Energy from light is used to ionize the electron.
2) Any excess energy becomes kinetic energy (KE) of the electron
V
hν

e–
electrode
vacuum tube

Dilemmas of Photoelectric Effect
- There is minimum frequency for the light, below which there is no ejection of
electrons – this is due to quantum nature of atomic orbitals
- Kinetic energy of ejected electrons depends only on frequency, not intensity of light
Particle model says KE of each ejected electron is due to incoming energy of a
single light particle striking the electrode and kicking off an electron.
Higher light intensity means more photons, but each of those photons still
have only the one energy and so ejected electrons have only one KE.
Wave model would allow range of kinetic energies.
- Ejection is instantaneous with light hitting the electrode
Particle model says electron ejected as soon as particle hits
Wave model requires several wavelengths to hit electrode -not instantaneous
Mass of photon particle
Photons have no “rest mass”, but do have relativistic mass due to velocity
From relativity: E = mc2      Applied to photon: E = hν therefore m = hν/c2
Wave model is not wrong, it is just incomplete – need both models
Particle model does not explain interference patterns
Wave model does not explain momentum of light striking matter
“Light is considered to act as a photon when interacting with matter, but to propagate
through space as a wave.” (LaPaglia)
Can never measure both types of attributes at same time – only one or the other

Gentry, 2012
Energy of Light is Quantized - Blackbody Radiation
Blackbody radiation = light given off from a cavity inside a heated block of material
Radiation depends only on temperature (T), not on the material
ρ(λ) = radiation density of emitted radiation as function of wavelength, λ
= radiated energy per unit volume of material and per unit wavelength
5000K
Rayleigh-

Radiation density, ρ
Radiation density, ρ
bl. body                                                                                                  Jeans law
radiation.
4500K

4000K

3500K                                                  Experimental

heated block
of material                                       0           1000       2000    3000
Wavelength (nm)                                      Wavelength

Rayleigh-Jeans Law - Classical Model
Assumed
(a) Blackbody radiation due to collection of independent oscillators operating at
different wavelengths (or frequencies)
(b) Classical electromagnetism: number of oscillator modes per frequency is
proportional to square of the frequency
(c) Each oscillator has average thermal energy, Eave = kT
8π                8π kT
Rayleigh-Jeans: ρ ( λ ) =                                   4
E ave =
λ           λ4
k = Boltzmann’s Constant = 1.380x10-23 J/K
Works for long wavelengths
... BUT fails for short wavelengths (“Ultraviolet Catastrophe”)

Planck’s Law (1901) - Quantum Model
Planck empirically found that get correct result if assume light energy is quantized
Oscillators at given frequency can only absorb and emit discrete packets of energy
Average energy may not be surrounding thermal energy if energy steps too big

Planck’s Law: E = n hν, n=0,1,2,...   h = Planck’s constant = 6.626 x 10-34 J·s
8π hc
Planck’s Blackbody Radiation: ρ (λ ) = 5 hc / λkT
λ (e        − 1)

Planck’s law becomes equivalent to Rayleigh-Jeans at long λ since
x2 x3
ex = 1+ x +     +    + ⋅ ⋅ ⋅ and only (1+x) are important for small x
2! 3!
Planck’s solution to blackbody problem was totally empirical at the time,
For years he resisted believing that energy was truly quantized
He thought the error was in his equation, not in his classical picture of nature

Ch 1: Classical vs. Quantum Mechanics                                             -3-
Why did Rayleigh-Jeans fail and Planck succeed?
Classical model
Assumes that light can have any specified amount of energy no matter what
the wavelength or frequency – so average energy is just available kT
Thus, thermal energy of blackbody is able to transfer energy to all the oscillators
no matter what their frequency, even if energy transfer is only small
Once the oscillators absorb energy, the ultraviolet catastrophe is a result of the
1/λ4 mathematical artifact of applying Maxwell’s equations.

Planck’s quantum model
Assumes high frequency (small wavelength) oscillators are limited in their
ability to absorb energy from the thermal body
Planck’s Law, E = nhν, n=0,1,2,... says that light energy can only be present in
discrete multiples.
Can’t wait around for more light to strike the oscillator to add up to minimum
energy. Not like holding a match for a longer time under a pot of water.
Planck’s Law consistent with photon particle model. Single photons have specific
energy, E = hν. Additional energy due to multiple photons each with same hν.
Classical   Quantum   Allowed Energy Levels                         Quantum Energy Levels
Model       Model    (gap depends on freq.)
En = n·hv
E

kT
E2 = 2·hv
0
Energy

ρ(λ)

Blackbody Radiation
E1 = 1·hv

λ

Actually have spread of probability for populating each of these discrete levels.
Thermal energy is probabilistic - not a simple yes/no cutoff

Boltzmann distribution: e-∆E/kT                    k = Boltzmann’s Constant = 1.38x10-23J/K

WAVENUMBERS, cm-1 [a practical note for spectroscopy]
Wavenumbers ≡ cm-1 ≡ 1 / λ (with λ measured in units of cm)
Wavenumbers formally have units of inverse distance, BUT used as equivalent to energy
It is the energy that corresponds to a photon of wavelength λ.
Energy of one photon, E = hν = hc/λ
1 cm-1 → 1.986 x 10-23 Joules

Ch 1: Classical vs. Quantum Mechanics                      -4-
DUAL NATURE OF MATTER

Matter, like light, can also
- be described alternatively as particles or waves,
- and is quantized.
Particle Nature of Matter
This is the classical Newtonian model. Things of mass have substance and moving
objects follow classical trajectories.
Deterministic model      (quantum mechanics is probabilistic model)
Wave Nature of Matter
Matter can be bent, diffracted, and have interference patterns in the same way that light
waves can be manipulated.

1925 – Davisson and Germer Diffraction Experiment
The two observed interference pattern of electrons in crystalline nickel, similar to x-
ray diffraction patterns. Consistent with waves
Today – Electron Microscopes
Electron microscopes use magnetic fields as lenses to bend and focus electron waves
similar to optical lenses focusing light waves

Double-Slit Diffraction
Pass electron beam through either single or double slit
Like light waves, an electron beam will radiate in all directions if passes through
narrow slit
In case of double slits:
Get constructive interference if both waves simultaneously have a peak (in phase)
Get destructive interference if two waves are 180º out of phase
Can view diffraction pattern with a TV phosphorescent screen
NOTE: Only get interference if electron wave simultaneously passes through
BOTH slits!
No interference if electron particle passes through one slit or the other slit
Only would see two sets of straight-through particle streams if have particles

Single Slit                Double Slit                  Double Slit Interference

Constructive
Interference
(both waves peak)
Destructive
Interference
(waves out of phase)

Ch 1: Classical vs. Quantum Mechanics          -5-
deBroglie Relationship (1924)
deBroglie reasoned that matter could show same duality as light
No experiments existed in 1924 to say electrons act like waves.
Proposed relationship between particle (momentum, p) and wave (wavelength, λ)
a.k.a. the “Rosetta Stone” of waves vs. particles for matter
This is a postulate – not explicitly proved – but 80 years later continues to be consistent
with experimental observations
h                 h = Planck’s constant = 6.626 x 10-34J·s,
λ=                      p = classical momentum = m·v
p
Relationship relates classical mechanics (particle momentum) to waves (have no mass)
deBroglie relationship was postulated from light equations:
h
E = hν =          and Einstein’s theory of special relativity (1905) E=mc2
cλ
Setting the two energies equal to each other and rearranging gives:
h
λlight =
mc
deBroglie postulated that matter has same equation as light, but with speed of
particle (v) rather than speed of light (c)
h
λ matter =
mv
Helpful relationships from classical mechanics
p = mv and KE = ½ mv2 = p2/2m,
where KE = kinetic energy, v = velocity, m = mass,
Examples:
Electron with energy = 1.5eV,            wavelength = 1.0 nm
H2 molecule with KE = kT at 300K,        wavelength = 120 pm
Tennis ball (57g) at 80km/h,             wavelength = 5.2x10-34 m
Hint: when in doubt when doing calculations, convert everything to SI units.

Ch 1: Classical vs. Quantum Mechanics              -6-
Quantized Nature of Hydrogen Emission
Rydberg-Ritz Equation
Empirical observation of emission lines from hydrogen atoms

1         1   1 
cm −1 =     = RH ⋅  2 − 2  , n1, n2 = 1,2,3, …, n1 < n2
n      
λ         1 n2 
RH = Rydberg constant = 1.097x107 m-1 = 109,700 cm-1

Prior models of the Atom
Thomson (1904) – atom was solid sphere including positive protons and negative
electrons all distributed throughout the sphere - “plum pudding model”
Rutherford (1911) – atom had positive nucleus surrounded by mostly empty space with
electrons moving anywhere within that space.
Neither model explains Rydberg emission – Why get discrete lines?
Bohr model of Atom (1913)
Electrons occupy well-defined orbits around the hydrogen nucleus
Attracted to nucleus by classical electrical attraction forces:
qq
E elect = 1 2    q1 = charge on electron, q2 = charge on nucleus, r = separation
4πε o r
But electrons would spiral inwards if not for orbits being quantized – can only be a given
distances from nucleus
Accurately predicted Rydberg-Ritz hydrogen spectrum

Summary of Bohr Atom
e4 m
Energy = E n = − 2 2e 2
8ε o h n
ε on2h2
Radius = r =
π me e 2
Bohr Radius = ao = radius for lowest H orbital (n=1)
= 52.9 pm
e 4 me
Rydberg Constant = R H =        2
= 1.097*105 cm-1
2
n    =   quantum number for orbital, n = 1,2,3…
me   =   mass of electron
e    =   electrical charge of electron
εo   =   vacuum permittivity

Problems with Bohr Model
It only works for atoms having 1 electron
Has non-zero angular momentum for ground state
Doesn’t account for wave nature of electrons !

Ch 1: Classical vs. Quantum Mechanics              -7-

```
To top