Transcript Chemistry 330 Chapter 11

Chemistry 330 Chapter 11
Quantum Mechanics
– The Concepts
The Early Years

Before 1890 – there were two camps
–
Classical Physics – thermodynamics, mechanics,
kinetic theory, optics, and electricity and magnetism.
Matter
Waves
Energy and Momentum (Classical)


Successful with large particles
Total energy consists of
–
–
Kinetic term
Potential term
Kinetic energy
Potential energy
Ek  1 mv 2
2
V  mgh
kq1q2
or V 
r
Energy and Position

The application of Newtonian physics gives an
exact solution for
–
–
–
Energy
Position
Momentum
The Walls Come Tumbling Down!!




The Photoelectric effect (Heinrich Hertz –
1887)
X-rays (Wilhelm Roentgen – 1895)
Radioactivity (Henri Becquerel – 1896)
The Discovery of the electron (J.J. Thomson –
1897)
Tumblin’ Down (cont’d)





Nature of Radioactivity (Marie Curie – 1899)
Quantum hypothesis (Max Planck – 1900)
Explanation of the photoelectric effect (Albert
Einstein – 1905)
The discovery of the -particle (Geiger,
Marsden, Rutherford – 1909)
The Bohr Model of the Atom (Neils Bohr –
1913)
Blackbody Radiation

Example – a pinhole in an otherwise closed
container.
–
–
The radiation is reflected many times within the
container and comes to thermal equilibrium with the
walls at a temperature T.
Radiation leaking out through the pinhole is
characteristic of the radiation within the container.
An Example of a Blackbody
The Energy Distribution



Energy density increases in the visible region
as the temperature is raised
Maximum shifts to shorter wavelengths.
The total energy density (the area under the
curve) increases as the temperature is
increased (as T4).
The Spectral Distribution
Blackbody Radiation

Rayleigh and Jeans
–
Electromagnetic field was a collection of oscillators
with their own characteristic frequency
8k B T
u 
4


Wien’s displacement law
m T  2.8979 x 103 m K
Predictions vs. Experimental
The Planck Hypothesis



Energy of the oscillators has only certain
allowed values!
Set the energy of the oscillator inversely
proportional to the wavelength
Energy gap
–
–
0 in classical physics
Finite – E = h  = h c/ in Planck’s treatment
The Planck Hypothesis (cont’d)

The Planck distribution accounts very well for
the experimentally determined distribution of
radiation.

8hc 
1
u  5  hc

  e k B T  1 


The Photoelectric effect

Electrons are ejected from a metal when the
incident radiation has a frequency above a
value characteristic of the metal
E T  1 mv2  
2
 – the binding energy of the electron
Einstein Applies Planck’s
Hypothesis

Incident radiation is composed of photons that
have energy proportional to the frequency of
the radiation.
  h o
o – the threshold frequency
The Photoelectric effect
Heat Capacities

Low-temperature
molar heat capacities
–
–
Experimental (points)
Einstein (solid curve)
The Discrete Nature of Atomic
Spectra

Atomic spectrum for hydrogen consisted of
discrete, sharp lines!
1
1
1
 RH 2  2 

n 
k
Balmer Equation – 1885
RH = Rydberg Constant
= 109 678 cm-1
The Bohr Model of the Hydrogen
Atom


Neils Bohr – laws of classical electrodynamics
did not apply to systems at the atomic level
Postulates
–
–
–
–
Energy of the H atom is quantized
Electron is promoted from a low to high energy level
by the absorption of a photon
The amount of energy absorbed and emitted by the
atom is quantized
Only orbits of certain angular momenta are allowed
Bohr and Balmer

The Bohr Model successfully explains the
Balmer equation
2
1 2 me

3

ch
4
1
1
 2  2 
 nf ni 
The De Broglie Hypothesis



Planck and Einstein are correct!
Particles must have wave-like properties
De Broglie’s matter waves
h
h
 

p
mv
An Illustration of the de Broglie
Relation


The wave is associated
with a particle
A particle with high
momentum has a
wavefunction with a
short wavelength, and
vice versa.
The Davisson-Germer Experiment


The scattering of an electron beam from a
nickel crystal shows a variation of intensity
Characteristic of a diffraction experiment in
which waves interfere constructively and
destructively in different directions.
Experimental Verification of WaveParticle Duality
The Classical Wave Equation

From classical physics
2
2
2
2
     
1  



2
2
2
2
2
x
y
z
v t
 = (x,y,z) eit
The Square of the Wavefunction

For stationary states
2
2
2
2
      
 2  2  2 0
2
x
y
z
v
The function  is called the amplitude
of the wave.
Schrödinger and de Broglie

Combine the de Broglie hypothesis with the
classical wave equation
2
2
2

      
 2 

E 

2m  x
y 2
z2 
 V x , y , z 
2
The Energy Operators

The kinetic energy operator

2
2
2







ˆK 
 2  2  2 
2m  x
y
z 
The potential energy operator ˆV
2
The Hamiltonian

The sum of the operators for the kinetic and
the potential energy yields the Hamiltonian
Hˆ  Kˆ  Vˆ
ˆH  E
The Born Interpretation



Max Born – the wavefunction  is a probability
amplitude
Square modulus (* or 2) is a probability
density.
The probability of finding a particle in the
region dx located at x is proportional to 2
dx.
The Born Interpretation in 3D Space


Relates the probability of
finding the particle in the
volume element d = dx
dy dz
At location r –
proportional to the
product of d and the
value of 2 at that
location.