Physics 214

1
Physics 214
5 Quantum Physics

• Photons and Electromagnetic Waves
• The Particle Properties of Waves
• The Heisenberg Uncertainty Principle
• The Wave Properties of Particles
• A Particle in a Box
• The Schroedinger Equation
• The Simple Harmonic Oscillator

2
Photons and Electromagnetic Waves

• Long wavelength electromagnetic radiation acts as
  waves
• consider a radio wave with ?? 2.5 Hz
• E 10-8 eV too small be detected as a single
  photon......
• a detectable single would require 1010 such
  photons, which on the average act as a continuous
  wave
• Short wavelength electromagnetic radiation acts
  as particles
• consider a X ray wave with ?? 1018 Hz
• E 103 eV, which can easily be detected as a
  single photon

3
How should we think of light?
4
Remember Beats
5
A solitary pulse can be produced by mixing
together waves of infinitely many different
harmonic waves of different frequencies, such
pulses are called wave-packets. These pulses
exhibit properties of both particles and waves

• particle
• localized
• finite size
• wave
• delocalized

Photon
6
Particles Behaving Like Waves
7
The wavepacket as whole moves with a velocity
vG ---the group velocity The waves in the
wavepacket move with a velocity vp----the
phase velocity
8
BEATS
(
)
(
)
(
)
y
x
,
t

y
x
,
t

y
x
,
t
total
1
2
(
)
(
)
(
)
y
x
,
t

A
sin
k
x
-
w
t

f

A
sin
k
x
-
w
t

f
1
1
1
1
2
2
2
2
total
let
A

A

A


f

f

0
1
2
1
2








æ
æ
k
-
k
ö
k

k
ö
w
-
w
w

w
ç
ç


t
x
y

2
A
cos
x

sin
-
t
2
1
1
2
2
1
1
2
2
2
2
2
total
è
ø
è
ø
(
)
(
)

2
A
cos
k
x

w
t
sin
k
x
-
w
t
1
2
4
4
4
3
4
4
4
2
4
4
3
4
4
1
b
b
s
s
modulated amplitude
interference wave
beat
w
-
w
D
w
D
k
w




k

2
1

2
2
2
b
b

Beats
9
Dispersion
10
Different Phase and Group Velocities
11
Intensity of Particle Streams and Probability
12
1
(
)
2
u
r
,
t

e
,
(
)

The instantaneous energy density of a light wave
E
r
t
0
2
E
If the frequency of the light is
n

,

(E is energy
) this must
h
(
)
u
r
,
t
correspond to
photons per unit volume at that point and time
E

(
)
Thus the photon density at
r
,
t
is proportional to the square
(
)
of the amplitude of the electromagnetic wave
amplitude at
r,t

13
?x
the wave
If there is only one photon,
(
)
packet model implies that the photon density at
r
,
t
has
to be interpreted as the probability that a
photon is at
the position
r
at the time
t
.
This interpretation also works
even if there are many photons

14
A wave-packet has an average position,

which corresponds to the average position of
the photon
ò
x

x
Prob(
x
)
dx
but has no exact position that can be measured
The width of the wave-packet corresponds to
the dispersion of
the positions of the photons
ò
(
)
(
)
2
2
D
x

x
-
x

x
-
x
Prob(
x
)
dx


15
Prob
Noting that
(
)
x
,
t
µ
E
(
x
,
t
)
2
Prob
and the normalization condition for probability
distributions
ò
(
)
x
,
t
dx

1
we can define
2
E
(
x
,
t
)
(
)
Prob
x
,
t


ò
2
E
(
x
,
t
)
dx

16
The Heisenberg Uncertainty Principle
A wave-packet has width ?x It is made up with a
range ?? of wavelength waves

D
l


l
-
l

range of wavelengths of
max
min
harmonic oscillators making up wave-packet
º
proportional to the dispersion of the EXPECTED
observed value of a wavelength in the wave-packet
º
UNCERTAINTY in the observed value of
l
D
l
i
.
e
l

l


2

17

D
x

x
-
x

range of spatial positions covered
max
min
by wave-packet
º
proportional to the dispersion of
the EXPECTED observed value of the position of
the wave-packet
º
UNCERTAINTY in the observed value of
x
D
x
x

x


2
The difference of the wavelengths,

Dl
,
of the waves contained
in the wave-packet cannot be greater than the
width of the
wave-packet
,
otherwise waves with wavelengths larger than
that of the wave-packet would be in the
wave-packet hence

Þ
D
x
³
D
l

18
h

D
l

D
p
h
Þ
D
x
³
D
p
Þ
D
x
D
p
³
h
Heisenberg's Uncertainty Principle
A more accurate analysis shows that
h
h
D
x
D
p
³


2
4
p
æ
m
D
x
p
D
p
D
x
ö
æ
æ
h
h
h
ö
ö
ç
(
)
(
)
(
)
Þ
³
Û
D
E
³
Û
D
t
D
E
³
è
ø
è
ø
è
ø
p
m
2
v
2
2
p
p
D
p
D
x
é
ù
2
E

Þ
D
E



v


2
m
m
D
t
ë
û
h
i
.
e
.

D
t
D
E
³

2


19
Complex Number Representation of Waves
Complex number definition
i

-1
i

2
-1
z

x

iy
(
)
(
)
(
)
z

z

x

iy

x

iy

x
x

i
y
x

x
y
-
y
y
1
2
1
1
2
2
1
2
1
2
1
2
1
2
(
)

x
x
-
y
y

i
y
x

x
y
1
2
1
2
1
2
4
3
4
1
2
1
2
1
2
4
4
3
4
4
x
iy
3
3
complex conjugate
z

x
-
iy
magnitude squared
z
2


z

z
x
2
y
2


20
e
i
q

cos
q

i
sin
q
1
(
)
Þ
e
-
e

sin
q
i
q
-
i
q
2
i
1
(
)
Þ
e


cos
q
e
q
-
i
i
q
2
Thus
E
(
)
(
)
(
)
E
k
,
t

E
sin
kx
-
w
t

e
(
)
-
e
(
)
max
-
w
-
-
w
kx
t
kx
t
i
i
max
2
i

21
The wave-packet is a linear combination
(integral
)
of infinitely many waves,
thus has a wave-function

2
p
æ
é
ù
ö
(
)
(
)
E
l
,
x
,
t

E
l
sin
x
-
w
(
l
)
t
ê
ú
è
ø
max
l
ë
û
in general is a complex valued function
and its magnitude squared at the point
x
at time
t
(
)
(
)
(
)
(
)
2
Y
Y
x
,
t

Y
x
,
t
x
,
t

Prob
x
,
t
when K
is chosen so that
ò
2
(
)
Y
x
,
t
dx

1

22
The Wave Properties of Particles
For massless particles using
The Special Theory of Relativity
E
E

pc
Þ
p


c
and
Plancks Hypothesis
hc
E
h
E

h
n

Þ

l
c
l
h
h
hk
\
p

Û
l




p


kh
l
p
2
p
de Broglie hypothesized that this was valid for
particles with mass also
Þ
h
h
l





p

kh
p
m
v
g


23
comparison of group velocity to phase velocity
for a free particle with mass
E
n


h
E
v

ln

p
m
v
g
p
2
for a free particle
E

2
m
1
m
v
2
v
g
2
g
Þ
ln


m
v
2
g
v
g
\
v

p
2

24
de Broglies explanation of the Bohr model using
matter waves
Electrons are waves,
but they are restricted to the
one dimensional Bohr orbits
They only way they can exist in such a
restricted
region of space is as standing wave patterns
In order to fit into orbits without destructive
interference
one has to have an integer number of standing
wave patterns in one orbit
h
\
2
p
r

n
l

n
v
m
h
v

Þ
m
r

n


nh
2
p


25
This condition is exactly
Bohrs Angular Momentum Quantization!!!!
Localization of waves
Standing waves
Þ
Þ
Quantized energies
Electron diffraction from Nickel crystals
confirmed de Broglies ideas
26
The diffraction pattern produced by electrons
passing through 2 slits can be viewed as the
probability distribution of the electrons
hitting the screen

• If ? ltlt distance between slits then
• Slits act as single slits
• If ? ltlt slit width then
• Electrons act as particles
• If ? slit width or ? gt slit width electron acts
  as
• WAVE !!!!

photons display the same behavior
27
A Particle in a Box
U0
28
1
3
L

l


l


l


2
l
2
1
2
2
3
4
n
2
L
L

l
Û
l

2
n
n
n
Stationary States for Electron in Box
n
p
æ
ö
(
)
(
)
(
)
(
)
Y
x
,
t

A
sin
k
x
cos
w
t

A
sin
x
cos
w
t
è
ø
n
n
L
2
p
n
p
as
k


n
l
L
n
(
)
(
)
Boundary Conditions
Y
0
,
t

0

Y
L
,
t

29
n
p
æ
ö
(
)
(
)
(
)
2
Prob
x
,
t

Y
x
,
t

A
2
sin
2
x
cos
2
w
t
è
ø
L
h
h
nh
p



,

n

1
,
,

K
2
L
l
2
L
n
n
n
i
.
e
.
Momentum is Quantized
n
h
2
2
2
p
h
æ
2
ö
4
L
2
K



n
2

E
è
ø
m
2
2
m
2
8
mL
n
n
Kinetic energy

Total energy as electric
é
ù
ê
ú
potential is zero inside box
ë
û
Kinetic

Total energy are Quantized
E

n
E
2
n
1
2
h
æ
ö
E

gt
0
Zero Point Energy
K
è
ø
2
8
mL
1
If electron drops from energy level
b
to
energy level
a
the frequency of light emitted is
h
æ
ö
(
)
n

b
-
a
Hz
2
2
è
ø
ba
8
mL
2


30
The Schrödinger Equation
Schrödinger first guessed that the
(
)
matter wave
Y
x
,
t
would satisfy the
linear wave equation

Y
w

Y

Y
2
2
2
2


v

2

t
2
k
2

x
2

x
2
just as string waves do,
however this
did not give the correct non relativistic
energy for a free
(traveling)
electron
p
2
i
.
e
.

E



K
2
m
nor
did it give the correct spectrum for
the hydrogen spectrum
(
bound
\
localized electron
)


31
The Equation for matter
(in particular electrons
)

that does give the correct energy is
(
)
(
)

Y
x
,
t

Y
x
,
t
2
h
2
i
(
)

-

U
(
x
)
Y
x
,
t
h

t
2
m

x
2
which is called
Schrödinger's Time Dependent Wave Equation


U
(
x
)
is the P
.
E
.
of the particle





For free particles
U
(
x
)

0
this equation has
solutions of the form
æ
ö
(
)
-
i
px
Et
ç

ç

x
t
(
)
(
)

e
-
h
Y

e
,
ikx
i
w
t
è
ø


E

w


p

k
h
h


32
æ
ö
(
)
-
i
px
Et
ç

ç

substituting

e
into
(
)
-
h
,
(
)
w
ikx
i
t
è
ø
Y
x
t

e
(
)
(
)

Y
x
,
t

Y
x
,
t
2
h
2
i

-
gives
h

t
2
m

x
2
æ
ö
æ
ö
(
)
(
)
-
-
i
px
Et
2
i
px
Et
iE
ip
ç

h
2
ç

ç

ç

-
i
e


-
e
h
h
h
è
ø
è
ø
2
m
h
h
æ
ö
æ
ö
-
(
)
-
(
)
i
px
Et
i
px
Et
p
2
ç

ç

ç

ç

Û
Ee

e
h
h
è
ø
è
ø
2
m
p
2
Þ
E


2
m
the non relativistic K.
E
.
for a free particle


33
(
)
(
)
(
)
Notice that
Y
x
,
t

e
(
)

F
x
W
t
-
w
ikx
i
t
plugging this product into
(
)
(
)

Y
x
,
t

Y
x
,
t
2
h
2
(
)

-

U
(
x
)
Y
x
,
t
gives
ih

t
2
m

x
2
(
)
(
)
d
W
t

F
x
2
é
ù
h
2
(
)
(
)
(
)
x

-

U
(
x
)
F
x
W
t
F
ih
ê
ú
dt
2
m

x
ë
û
2
(
)
(
)
1
d
W
t
1

F
x
é
ù
h
2
2
(
)
Û

-

(
x
)
F
x
U
ih
ê
ú
(
)
(
)
W
t
dt
F
x
2
m

x
ë
û
2
This equation can only have a solution if both
sides are constant
(
)
(
)
1
d
W
t
1

F
x
é
ù
h
2
2
(
)



-

U
(
x
)
F
x
ê
ú
ih
(
)
(
)
W
t
dt
F
x
2
m

x
ë
û
2

E
E
34
(
)
1
d
W
t

E

ih
(
)
W
t
dt
(
)
d
W
t
(
)
ih

E
W
t
Þ
dt
iEt
-
(
)
Þ
W
t

Ce


C
is a constant
h
(
)
1
h
2

2
F
x
é
ù
(
)

-

U
(
x
)
F
x

E
ê
ú
(
)
F
x
2
m

x
ë
û
2
h

2
2
é
ù
(
)
(
)
Þ
-

U
(
x
)
F
x

E
F
x
2
m

x
ë
û
2
Time Independent Schrödinger Equation


35
Solutions of differential equations are
not completely characterized by the equation
alone
.
The functions must also satisfy some
boundary conditions,
such as having a specified
value at
t

0
.
For the Schrödinger Equation the
boundary conditions are

ò
(
)
2
(
1)
Y
x
,
t
dx

1
-

(
)
(2
)

Y
x
,
t
is a continuous function in x
(
)
d
Y
x
,
t
(3
)

is a continuous function in x
dx
(
)
(4
)

Y
x
,
t

0
where
U
(
x
)



36
Particle in a Box -- Again
The potential energy of a particle in a box is

for
x
lt
0
ì
ï


U
(
x
)

0
for
x
Î
0
,
L

í
ï

for
x
gt
L
î
This gives the Time Independent Schroedinger
Equation

2
é
ù
h
2
(
)
(
)
-

U
(
x
)
F
x

E
F
x
2
m

x
ë
û
2
2
æ
h
ö
Which has solutions when
E

n


n

1
,
2
,
2
K
è
ø
2
n
8
mL
For
n

1
one can then solve the equation

é
ù
æ
h
2
h
2
2
ö
(
)
(
)
-

U
(
x
)
F
x

F
x
è
ø
2
2
m

x
ë
û
1
8
mL
1
2
n
p
x
æ
ö
(
)
Þ
F
x

A
sin


A
is a constant that can be chosen to
è
ø
1
L
(
)
normalize
F
x


1
37
The Simple Harmonic Oscillator
The potential energy of a particle that moves in
SHM is
1
1
U
(
x
)

k
x
2

m
w
2
x
2

2
2
k
where
w



k is the force constant
m
This gives the Time Independent Schroedinger
Equation

1
2
é
ù
h
2
(
)
(
)
-

m
w
2
x
2
F
x

E
F
x
2
m

x
2
ë
û
2
1
æ
ö
w
Which has solutions when
E

n



n

0
,
1
,
2
,
h
K
è
ø
n
2
For
n

0
one can then solve the equation
w

1
2
é
ù
h
2
h
(
)
(
)
-

m
w
x
F
x

F
x
2
2
2
m

x
2
ë
û
2
2
æ
ö
w
m
-
ç

(
)
x
2
Þ
F
x

Be


B
is a constant that can be chosen to
è
ø
2h
(
)
normalize
F
x