Transcript Slide 1

PHYS 420
SPRING 2006
Dennis Papadopoulos
LECTURE 15
THE WAVEFUNCTION
minima occur at
D sin    / 2
electronwavelength:   h / p
for small :
sin    
h
2 pD
Electron localized in the hole and indivisible.
Clearly goes through slit #1 or slit #2
Fig. 5-30, p. 182
Sum of those going
through slit#1 and slit #2
Assumption that
electron goes through
either slit#1 or slit#2
must be wrong.
Electron must be going
through both slits at
the same time !!! to
exhibit interference
Fig. 5-31, p. 183
Fig. 5-29, p. 181
Watching Electrons
minima occur at
D sin    / 2
Need to measure the position of
the electron with accuracy Dy<D
electronwavelengt h:   h / p
for small :
sin    
py
h

2 pD p x
y  L
2D
To avoid destroying the interference
pattern
Dp y / p x    h / 2 p x D
Dp y  h / 2 D
Since the same Dpy applies to the detecting photon we
need
DpyDy<h/2
Fig. 5-33, p. 184
•A large number of electrons going through a double
slit will produce an interference pattern, like a wave.
•However, each electron makes a single impact on a
phosphorescent screen-like a particle.
•Electrons have indivisible (as far as we know) mass
and electric charge, so if you suddenly closed one of
the slits, you couldn’t chop the electron in halfbecause it clearly is a particle.
•A large number of electrons fired at two
simultaneously open slits, however, will eventually,
once you have enough statistics, form an
intereference pattern. Their cumulative impact is
wavelike.
•This leads us to believe that the behavior of electrons
is governed by probabilistic laws. --The wavefunction
describes the probability that an electron will be found
in a particular location.
Given the Uncertainty Principle, how do you write an equation of motion
for a particle?
•First, remember that a particle is only a particle sort of, and a wave sort of,
and it’s not quite like anything you’ve encountered in classical physics.
We need to use Fourier’s Theorem to represent
the particle as the superposition of many waves.
wavefunction
of the
electron
adding varying amounts of
an infinite number of waves sinusoidal expression
for harmonics

( x,0)   a(k )e dk
ikx

amplitude of wave with
wavenumber k=2p/
•We saw a hint of probabilistic behavior in the double slit experiment.
Maybe that is a clue about how to describe the motion of a “particle” or
“wavicle” or whatever.
We can’t write a deterministic equation of motion as in
Newtonian Mechanics, however, we know that a large number of
events will behave in a statistically predictable way.
probability for an
electron to be found
between x and x+dx
(x,t)
b
2
dx
P   ( x, t ) dx
2
a
Assuming this particle exists, at any given time it must be
somewhere. This requirement can be expressed mathematically as:
If you search from
hither to yon



( x, t ) dx  1
2
you will find your particle
once, not twice (that would be
two particles) but once.
P(x,t)dx|(x,t)|2dx
b
P(a  x  b)    ( x, t ) dx
2
a
Fig. 6-1, p. 193
Normalization

C 2  Exp[

2x
xo
]dx  1
C  1 / xo
1
x
 ( x) 
Exp[
]
x
o
xo
C/e
Fig. 6-2, p. 193
Conceptual definition
When two or more wave moving through the same region of space,
waves will superimpose and produce a well defined combined effect.
Mathematical definition
For a linear homogeneous ordinary differential equation,
if
and
are solutions, then so is
.
When two waves of equal amplitude and frequency
but opposite directions of travel superimpose, you
get a standing wave- a wave that appears not to
move-its nodes and anti-nodes stay in the same
place. This happens when traveling waves on a
guitar string get to the end of the string and are
reflected back.
Let’s try a typical classical wavefunction:
( x, t )  A sin(kx  t )
For a particle propagating in the +x direction.
Similarly, for a particle propagating in the –x direction:
( x, t )  A sin(kx  t )
We also know that if 1 and 2 are both allowed waves,
then 1+2 must also be allowed (this is called the
superposition principle).
1( x, t )   2( x, t )
 A1 sin(kx  t )  A2 sin(kx  t )
 2 sin kx cost
Oops, the particle vanishes at integer multiples of p/2, 2p/3, etc.
and we know our particle is somewhere.
( x, t )  Aei ( kxt )  A{cos(kx  t )  i sin(kx  t )}
Euler's equation
T rigonometric functions:
ei  cos  i sin 
e i  e  i
cos 
2
e i  e  i
sin  
2i
e i  cos  i sin 
ii  1
d u
du
e  eu
dx
dx
conjugat e( a  ib)  ( a  ib)
if z  a  ib, t hen:
z * z  ( a  ib)(a  ib)
 a i b
2
2 2
 a2  b2
Graphical representation of
a complex number z as a
point in the complex plane.
The horizontal and vertical
Cartesian components give
the real and imaginary parts
of z respectively.
Note that we can construct a wavefunction only if the
momentum is not precisely defined. A plane wave is
unrealistic since it is not normalizable.



e
i ( kxt )
2

dx   1 dx  

You can’t get around the uncertainty principle!
Table 5-1, p. 184
Fig. 5-32, p. 183
Alright, we think we might have an acceptable wave-function. Let’s give it a whirl… If
we think we know what our wave-function looks like now, how do we propagate it
through time and space?
The Schrodinger Equation:
•Describes the time evolution of your wave-function.
•Takes the place of Newton’s laws and conserves energy of the system.
•Since “particles” aren’t particles but wavicles, it won’t give us a precise position
of an individual particle, but due to the statistical nature of things, it will precisely
describe the distribution of a large number of particles.
2
p
E
2m
2

k
 (k ) 
2m
 (k ) / k  k 2m

 ( x,0)   a(k ) exp(ikx)dx


 ( x, t )   a(k ) exp[i (kx   (k )t )]dx

Fig. 6-3, p. 195
a(k )  (C / p ) Exp( 2 k 2 )
( x,0)  CExp[( x
2
)2 ]
t 2
Dx(t )    [
]
2m
2
C  1/ 2 p
Fig. 6-4, p. 196
If you know the position of a particle at time t=0, and you constructed a localized wave
packet, superimposing waves of different momenta, then the wave will disperse
because, by definition, waves of different momenta travel at different speeds.
the position
of the real
part of the
wave…
the
probability
density