Relationship between speed of light (c),
frequency (f) and wavelength (λ)
C=f*λ
1-D plane wave:
E ( x)    e
ikx iwt
k  2 / 
  2f
 / k  f  c
Particle Wave Duality
Einstein’s hypothesis
E  hf  
  h / 2
  2f
p
E  cp
3
h

Particle Wave Duality
de Broglie’s Hypothesis
Einstein’s hypothesis
p
h

p
h

p  mV
E  cp
1
2
E  mV
2
Particle nature of light
Wave nature of a particle
4
Conversion between wavelength and,
energy
Relationship between speed of light (c),
frequency (f) and wavelength (λ)
C=f*λ
1-D plane wave:
E ( x)    e
ikx iwt
Wave nature applied to orbits
• In order for a standing wave to be sustained, there must be an
integral number of wavelengths around the circle’s circumference
2 rn  n , n  1, 2,3,...
nh
2 rn 
mv
nh
mvrn 
n
2
which is Bohr’s Quantum Condition!
• The moral of the story?
– the wave nature of electrons is inescapable It is integral to the
nature of electrons and electron states in the atom!
• This approach began what is now Quantum Mechanics
6
Derive Hydrogen atom energy
spectrum
• Note the force goes as F ~ 1/r2
Wave Equation
 p ( x)    e
p
E
i x i t


   eikxiwt
Is a solution of Schrödinger as well as Maxwell
Equation
d
i  ( x, t )  E ( x, t )  H ( x, t )
dt
H is the Hamiltonian of the particle
8
E (H) can take different forms depending on the
relation ship between energy and momentum.
For a low-energy electron
p2
E
2m
 p ( x)    e
d
i  p ( x)  E p ( x)
dt
d
 i  p ( x)  P p ( x)
dx
 d 2
(
)
d
i  p ( x, t )  i dx  p ( x, t )
dt
2m
9
p
E
i x i t


Generalization to a free particle:
d
 2 d 2
i  ( x, t ) 
 ( x, t )
2
dt
2m dx
For a particle in a potential U(x)
p2
E
 U ( x)
2m
d
 2 d 2
i  ( x, t )  [
 U ( x)] ( x, t )
2
dt
2m dx
The Schrödinger Equation
Other quantum mechanics equations: Dirac Eq. Klein-Gordon Eq.
10
Atomic Spectra reveals atomic structure
11
L/4
L
L/4
L/4
L/4
0
L