Hermitian linear operator Real dynamical variable –
represents observable quantity
Eigenvector
A state associated with an
observable
Eigenvalue
g
Value of observable associated
with a particular linear operator
and eigenvector
Copyright – Michael D. Fayer, 2007
Momentum of a Free Particle
F
Free
particle
i l – particle
i l with
i h no forces
f
acting
i on it.
i
The simplest particle in classical and quantum mechanics.
classical
rock in interstellar space
Know momentum, p, and position, x, can
can predict exact location at any subsequent time.
Solve Newton's equations of motion.
p = mV
What is the quantum mechanical description?
photons,, electrons,, and rocks.
Should be able to describe p
Copyright – Michael D. Fayer, 2007
Momentum eigenvalue problem for Free Particle
P P  P
momentum
operator
Know operator
momentum
eigenvalues
momentum
eigenkets
want: eigenvectors
eigenvalues
Schrödinger Representation – momentum operator

P  i
x
Later – Dirac’s Quantum Condition shows how to select operators
p
form different representations
p
of Q
Q. M.
Different sets of operators
Copyright – Michael D. Fayer, 2007
Try solution
P  ce i x / 
eigenvalue
Eigenvalue – observable.
Proved that must be real number.
Also: If not real, function would blow up.
Function represents probability (amplitude)
of finding particle in a region of space.
Probability can’t be infinite anywhere. (More later.)
Function representing state of system in a particular
representation and coordinate system called wave function.
Copyright – Michael D. Fayer, 2007
P  i

x
P  ce i x / 
Schrödinger Representation
momentum operator
Proposed
p
solution
Check
i 

 ce i  x /     i  i  /   ce i  x /  
 x
   ce i  x /  
P P  P
Operating on function gives
identical function back
times a constant.
continuous range of eigenvalues
Copyright – Michael D. Fayer, 2007
P   p  ce ipx /   ce ikx
momentum momentum
eigenkets
wave functions
Have called eigenvalues
p  k
=p
k is the “wave
wave vector.
vector ”
c is the normalization constant.
c doesn
doesn’tt influence eigenvalues
direction not length of vector matters.
Normalization
a b
scalar product
a a
scalar p
product of a vector with itself
aa
1/ 2
1
normalized
Copyright – Michael D. Fayer, 2007
Normalization of the Momentum Eigenfunctions – the Dirac Delta Function
b a   b a d
a a

scalar
l product
d
off vector functions
f
i
(linear
(li
algebra)
l b )
*

 a a d .
scalar
l product
d
off vector function
f
i with
i h itself
i lf
1 if P   P
P  P     P dx  
0 if P   P


*
P
Usingg
normalization P '  P
orthogonality P '  P
P  k






c 2  e  ik ' x e ikx dx  c 2  e i ( k  k ') x dx
Can’t do this integral with normal methods – oscillates.



e i ( k  k ') x dx 

 cos(k  k ') x  i sin(k  k ') x  dx .

Copyright – Michael D. Fayer, 2007
Dirac  function
Definition
 ( x )  0 if x  0

d 1
  ( x )dx

For function f(x) continuous at x = 0,


f ( x ) ( x )dx  f (0)


or
 ( x  a )  0 if


xa
f ( x ) ( x  a )dx  f (a )

Copyright – Michael D. Fayer, 2007
Physical illustration
unit area
x
x=0
double height
half width
still unit area
Limit width
0,
area remains 1
height

Copyright – Michael D. Fayer, 2007
Mathematical representation of  function
1
08
0.8
sin( g x )
x
g any real number
0.6
0.4
zeroth order spherical
Bessel function
-40
0.2
-20
Has value g/ at x = 0.
Decreasing oscillations for increasing |x|.
Has unit area independent of the choice of g.
As
g
g=

sin gx
 ( x )  lim
g   x

20
40
-0.2
1. Has unit integral
2. Only non-zero at point x = 0
3. Oscillations infinitely fast, over
finite interval yield zero area.
Copyright – Michael D. Fayer, 2007
Use (x) in normalization and orthogonality of momentum eigenfunctions.
p p 1
Want
p p'  0

c
2
e
 ik ' x ikx
e

1 if k  k '
dx  
0 if k  k '
Adjust c to make equal to 11.
E l t
Evaluate


e i ( k  k ') x dx 

Rewrite

 cos(k  k ') x  i sin(k  k ') x  dx

g
 lim
g 

 cos(k  k ') x  i sin(k  k ') x  dx
g
 sin( k  k ') x   i cos( k  k ') x  


 lim 
g  
(
k

k
')
k

k
'

 

g
g
Copyright – Michael D. Fayer, 2007
 sin( k  k ') x   i cos( k  k ') x  

lim 

g  
(
k

k
')
k

k
'




g
2 sin g( k  k ')
g 
( k  k ')
 lim
g
 function multiplied by 2.
 2  ( k  k ')
2  ( k  k '))  0 if k  k '
 function
The momentum eigenfunctions are
orthogonal.
o
t ogo a . We knew
ew this
t s already.
a eady.
Proved eigenkets belonging to different
eigenvalues are orthogonal.
To find out what happens for k = k' must evaluate 2  ( k  k ') .
But, whenever you have a continuous range in the variable of a vector function
((Hilbert Space),
p ), can't define function at a p
point. Must do integral
g about p
point.
k ' k  
   (k  k ')dk '  1
if
k  k
k ' k 
Copyright – Michael D. Fayer, 2007
Therefore
 2 if P '  P
1
P' P  
2
c
 0 if P '  P
P
are orthogonal and the normalization constant is
c2 
1
2
1
c
2
The momentum eigenfunction are
 p ( x) 
1
2
e ikx
k  p
momentum eigenvalue
Copyright – Michael D. Fayer, 2007
Wave Packets
P P p P
momentum
operator
momentum
eigenvalue
momentum
eigenket
Free particle wavefunction in the Schrödinger Representation
P   p ( x) 
1
2
e ikx 
1
2
 cos kx  i sin kx 
State of definite momentum
k  p k 
2

wavelength
Copyright – Michael D. Fayer, 2007
Born Interpretation of Wavefunction
Schrödinger
Concept of Wavefunction
Solution to Schrödinger Eq
Eq.
Eigenvalue problem (see later).
Thought represented “Matter Waves” – real entities.
Born put forward
Like E field of light – amplitude of classical E & M wavefunction
proportional to E field,   E .
Absolute value square of E field
Intensity.
| E |2  E * E  I
B
Born
IInterpretation
t
t ti – absolute
b l t value
l square off Q.
Q M.
M wavefunction
f
ti
proportional to probability.
Probability of finding particle in the region x + x given by
x x
Prob( x , x  x ) 

 *( x )  ( x ) dx
x
Q. M. Wavefunction
Probability Amplitude.
Wavefunctions complex.
Probabilities always real.
Copyright – Michael D. Fayer, 2007
Free Particle
P   p ( x) 
1
2
e ikx 
1
2
 cos kx  i sin kx 
k  p
What is the particle's location in space?
real
imaginary
x
Not localized
Spread out over all space.
space
Equal probability of finding particle from  to   .
Know momentum exactly
No knowledge of position.
Copyright – Michael D. Fayer, 2009
Not Like Classical Particle!
Pl
Plane
wave
spread
d out over all
ll space.
Perfectly defined momentum – no knowledge of position.
What does a superposition of states of definite momentum look like?
 p ( x )   c( p )p ( x ) dp
d
must integrate because continuous
range of momentum eigenkets
p
coefficient – how much of each
eigenket
i
is
i in
i the superposition
ii
indicates that wavefunction is composed
of a superposition of momentum eigenkets
Copyright – Michael D. Fayer, 2009
First Example
Work with wave vectors - k  p/
k0 – k k0
Wave vectors in the range
k0 – k to k0 + k.
In this range, equal amplitude of each state.
Out side this range, amplitude = 0.
k0 + k
k  k0
k
Then
 k 
k0 k
e
ikx
k0 k
dk
Superposition of
momentum eigenfunctions.
Same as integral for normalization but
integrate over k about k0 instead of over x about x = 0.
k ( x ) 
2sin k x ik0 x
e
x
At x  0 have lim x  0
Rapid oscillations in envelope of
slow, decaying oscillations.
2sin( k x )
 2 k
x
Copyright – Michael D. Fayer, 2009
Writing out the e
 k
ik0 x
2sin( kx )

cos k0 x  i sin k0 x 
x
k  k0
Wave Packet
k
X
This is the “envelope.” It is “filled in”
by the rapid real and imaginary
spatial oscillations at frequency k0.
The wave packet is now “localized” in space.
As | x |  large,  k  small.
Greater k
more localized.
Large uncertainty in p
small uncertainty in x.
Copyright – Michael D. Fayer, 2009
Localization caused by regions of constructive and destructive interference.
5 waves
Cos(ax ) a  1.2, 1.1, 1.0, 0.9, 0.8
Sum of the 5 waves
4
2
- 3 0
- 2 0
- 1 0
1 0
2 0
3 0
- 2
- 4
Copyright – Michael D. Fayer, 2009
Wave Packet – region of constructive interference
Combining different P ,
free particle momentum eigenstates
wave packet.
Concentrate p
probability
y of finding
gp
particle in some region
g
of space.
p
Get wave packet from

 k ( x ) 

f ( k  k0 ) e
ik  x  x0 
dk

f ( k  k0 )  weighting function
Wave packet centered around
x = x0,
made up of k-states centered around
k = k0.
Tells how much of each k-state is in superposition.
Nicely behaved
dies out rapidly away from k0.
f(k-k0)
k
k0
Copyright – Michael D. Fayer, 2007
Copyright – Michael D. Fayer, 2009
f ( k  k0 )  weighting function Only have k-states near k = k0.
At x = x0
e
ik  x  x0 
 e0  1
All k-states in superposition
p p
in p
phase.
Interfere constructively
add up.
All contributions to integral add.
For large (x – x0)
e
ik  x  x0 
 cos k  x  x0   i sin k  x  x0 
Oscillates wildly with changing k.
Contributions from different k-states
| x |  x0
 k  0
destructive interference.
Probability of finding particle
k
0
2

wavelength
Copyright – Michael D. Fayer, 2009
Gaussian Wave Packet
1
G( y) 
exp[
p[( y  y0 )2 / 2 2 ]
2 2
  standard deviation
Gaussian Function
Gaussian distribution of k-states
f (k ) 
 ( k  k0 )2 
e p 
exp
2 
2
2(

k
)
2k


1
Gaussian in k, with
k   normalized
Gaussian Wave Packet
 k ( x ) 
1
2k 2

e

( k  ko )2
2( k )2

f ( k  k0 ) e
Written in terms of x – x0
e
ik ( x  xo )
dk
ik  x  x0 
Packet centered around x0.
Copyright – Michael D. Fayer, 2009
Gaussian Wave Packet
 k ( x )  e
1
 ( x  x0 ) 2 (  k ) 2
ik0 ( x  x0 )
2
(H
(Have
left
l ft outt normalization)
li ti )
e
Looks like momentum eigenket at
k = k0
times Gaussian in real space.
Width of Gaussian (standard deviation)
1

k
The bigger k, the narrower the wave packet.
When x = x0, exp = 1.
For large
g | x - x0|, real exp
p term << 1.
Copyright – Michael D. Fayer, 2009
Gaussian Wave Packets

1
k
k small
packet wide.
x = x0

1
k
k large
packet narrow.
narrow
x = x0
To get narrow packet (more well defined position) must superimpose
broad distribution of k states.
Copyright – Michael D. Fayer, 2009
Brief Introduction to Uncertainty Relation
ob b y oof finding
d gp
particle
ce
Probability
between x & x + x
x x
Prob( x ) 

 * ( x ) ( x )dx
x
Written approximately as
  ( x ) ( x )
For Gaussian Wave Packet
 *k  k  exp  ( x  x0 )2 ( k )2 
 k ( x )  e
1
 ( x  x0 ) 2 (  k ) 2
ik0 ( x  x0 )
2
e
(note – Probabilities are real.)
This becomes small when
( x  x0 ) 2 (  k ) 2  1
Call x  x0  x , then
x k  1
Copyright – Michael D. Fayer, 2009
x k  1
k  p
 k  p
momentum
spread or "uncertainty" in momentum
p
k 

x p  1


x p  
Superposition of states leads to uncertainty relationship
relationship.
Large uncertainty in x, small uncertainty in p, and vis versa.
(See later that x p  /2
differences from choice of 1/e point instead of )
Copyright – Michael D. Fayer, 2009
Observing Photon Wave Packets
Ultrashort mid-infrared optical pulses
1.1
1.1
368 cm-1 FWHM
G
Gaussian
i Fit
40 fs FWHM
G
Gaussian
i fit
0.9
0.7
Inttensity (Arb. Un
nits)
Inte
ensity (Arb. Units)
0.9
collinear
autocorrelation
t
l ti
0.5
0.3
0.7
0.5
 p/h
0.3
0.1
0.1
-0.1
-0.3
-0.2
-0.1
-0.1
0
Delay Time (ps)
0.1
0.2
1850
p  h /
2175
2500
frequency (cm -1)
2825
3150
 1  p/h
t  40 fs = 40  10-15 s
x  c t
p  2.4  1029 kg-m/s (FWHM, 14%)
x  1.2
1 2  105 m  12  m (FWHM)
p0  1.7
1 7  1028 kg-m/s
x p   / 2  5.3  1035 Js  (1.2  105 m) / 2.35  (2.4  1029 kg-m/s) / 2.35  5.2  1035 Js
?
FWHM to 
Copyright – Michael D. Fayer, 2009
Electrons can act as “particle” or “waves” – wave packets.
CRT - cathode ray tube
control electrons aimed
grid
cathode
-
Old style TV or computer monitor
electrons
hit screen
e-
filament
+
electrons
boil off
acceleration
grid
screen
Electron beam in CRT.
Electron wave packets like bullets.
Hit specific points on screen to give colors.
Measurement localizes wave packet.
Copyright – Michael D. Fayer, 2009
Electron diffraction
in coming
electron “wave"
crystal
surface
Electron beam impinges
on surface of a crystal
low energy electron diffraction
diffracts from surface.
Acts as wave.
Measurement
determines
momentum eigenstates.
out going
waves
many grating
different directions
different spacings
Peaks
P
k line
li up.
Constructive interference.
Low Energy Electron Diffraction (LEED)
from crystal surface
Copyright – Michael D. Fayer, 2009
Group Velocities
Time independent momentum eigenket
P   p  x 
1
2
e ikx
p  k
Direction and normalization
Can multiply by phase factor e
ket still not completely defined.
i
 real
Ket still normalized.
Direction unchanged.
Copyright – Michael D. Fayer, 2009
For time independent Energy
In the Schrödinger Representation
(will prove later)
Time dependence of the wavefunction is given by
e
i E t / 
e
i  t
  E
Time dependent phase factor.
e
 i t i t
 1,
e
Still normalized
The time dependent
p
eigenfunctions
g
of the momentum operator
p
are
 k  x, t   e
i  k ( x  x0 )   ( k )t 
Copyright – Michael D. Fayer, 2009
Time dependent Wave Packet
 k  x , t  



f ( k )exp  i  k  x  x0    ( k ) t   dk
time dependent phase factor
Photon Wave Packet in a vacuum
Superposition of photon plane waves.
Need 
  ck  c 2

c  
2  
Copyright – Michael D. Fayer, 2009
Using   ck

 k ( x , t ) 

f ( k )exp  ik ( x x0  ct )  dk

Have factored out k.
At t = 0,
Packet centered at x = x0.
Argument of exp. = 0.
Max constructive interference.
At later times,,
Peak at x = x0 + ct.
Point where argument
g
of exp
p=0
Max constructive interference.
Packet moves with speed of light.
Each plane wave (k-state) moves at same rate, c.
P k maintains
Packet
i i shape.
h
Motion due to changing regions of constructive and destructive
interference of delocalized planes waves
Localization and motion due to superposition
of momentum eigenstates.
Copyright – Michael D. Fayer, 2009
Photons in a Dispersive Medium
Photon enters glass, liquid, crystal, etc.
Velocity of light depends on wavelength.
n()
Index of refraction – wavelength dependent.
glass – middle of visible
n ~ 1.5
n()
()
red
blue

 ( k )  2 c / n(( )  kc/n( k )
c

V

n( )
2
Dispersion,
 (k )
  V
k  2 /
no longer linear in k.
Copyright – Michael D. Fayer, 2009
Wave Packet
 k  x , t  



f (k )exp  i  k (x xo )   (k ) t )  dk
Still looks like wave packet.
At t  0,
the peak is at x  x0
Moves but not with velocity of light
Moves,
light.
Different states move with different velocities;
in glass blue waves move slower than red waves.
Velocity of a single state in superposition
Phase Velocity.
i
Copyright – Michael D. Fayer, 2009
Velocity – velocity of center of packet.
t  0,
x  x0
max constructive interference
Argument of exp. zero for all k-states at x  x0 , t  0.
The argument of the exp. is
  k ( x  x0 )   ( k )t
(k) does not change linearly with k.
Argument will never again be zero
for all k in packet simultaneously.
Most constructive interference when argument changes with k
as slowly
l l as possible.
ibl
This produces slow oscillations.
k-states add constructively, but not perfectly.
Copyright – Michael D. Fayer, 2009
No longer perfect constructive interference at peak.
1
In glass
Red waves get ahead.
Blue waves fall behind.
0.5
-3
-2
-1
1
2
3
-0.5
-1
The argument of the exp. is 
 k ( x  x0 )   ( k )t
When  changes slowly as a function of k within range of k determined by f(k)
Constructive interference.
Find point of max constructive interference.
interference   0  x  x  t   ( k ) 



0
 changes as slowly as possible with k.  k
  k k
max of packet
at time, t.
0
  ( k ) 
x  x0  t 

  k  k0
Copyright – Michael D. Fayer, 2009
Distance Packet has moved at time t
  ( k ) 
d  ( x  x0 )  
 t  Vt
  k  k0
Point of maximum constructive interference (packet peak) moves at
  ( k ) 
Vg  
  Group Velocity
  k  k0
Vg
speed
p
of p
photon in dispersive
p
medium.
Vp
phase velocity = /k
only same when (k) = ck
vacuum (approx
(approx. true in
low pressure gasses)
Copyright – Michael D. Fayer, 2009
Electron Wave Packet
de Broglie wavelength
p  h /   k
(1922 Ph.D. thesis)
A wavelength associated with a particle
unifies theories of light and matter.
E  
p2
k 2
 (k )  E /  

2m  2m
Used p 
k
Non-relativistic
N
l ti i ti free
f
particle
ti l
energy divided by  (will show later).
Copyright – Michael D. Fayer, 2009
Vg – Free non-zero rest mass particle
  ( k )   k 2 k
Vg 


 p/m
2m  k
m
k
p/m  mV /m  V
Group velocity of electron wave packet
same as classical velocity.
However, description very different. Localization and motion due to
changing regions of constructive and destructive interference.
p
electron motion and electron diffraction.
Can explain
Correspondence Principle – Q. M. gives same results as classical mechanics
when
hen in “classical regime
regime.””
Copyright – Michael D. Fayer, 2009
Q. M.
Particle Superposition of Momentum Eigenstates
Partially localized
Wave Packet
x p  /2
Photon – Electron
Photon wave packet description of light same as
wave packet description
i i off electron.
Electron and Photon can act like waves – diffract
or act like particles – hit target.
target
Wave – Particle duality of both light and matter.
Copyright – Michael D. Fayer, 2009