What are matter waves?
Recall that for a wave:
•
Intensity ~ (amplitude of the wave)2
EM waves is a good example, since we know that they can be •
considered also as beams of particles – photons, and the
intensity in the photon model is
(# photons)(h )
I
( A)( t)
If N is the [average] density of photons, in the beam (density •
of photons is the number of photons per unit volume)
(# photons)  N  V  N(c)( t )( A)
(# photons)( h ) N(c)( t )( A)( h )
I

 N(c)( h )
( A)( t )
A( t )
What are matter waves?
I  N(c)( h )  (amplitude of the wave)
2
Here •
Speed of light, c, is a constant –
For a given EM wave, photon energy, hν, is a constant –
The only variable “responsible” for change in the light –
intensity is the average density of photons, N
Therefore, •
N  (amplitude of the wave)
2
What are matter waves?
N may also be considered from a statistical •
point of view, by asking the question:
What is the probability of finding a given number –
of photons in a given volume of the beam?
The answer is of course: •
The probability of finding X number of photons –
in the volume V is given by ratio X/V
The average photon density in this case is •
calculated in the same way: N = X/V !
Definition of Ψ(r,t)
The probability P(r,t)dV to find a particle associated •
with the wavefunction Ψ(r,t) within a small volume
dV around a point in space with coordinate r at some
instant t is
P(r , t)dV  (r , t) dV
2
P(r,t) is the probability density –
For one-dimensional case •
P( x, t)dV  ( x, t) dx
2
Here |Ψ(r,t)|2 = Ψ*(r,t)Ψ(r,t)
Since, N is also proportional to the square of
the amplitude of the corresponding wave, we
conclude that
The probability of finding
photon within a given volume
of the beam is proportional to
the square of the amplitude of
the wave associated with this
beam
What are the matter waves?
Thus, a wave (function describing the wave) can •
be considered as a mathematical function that
measures the photon probability density in the
beam of light
Since not only EM radiation has a dual nature, •
but also matter, Max Born extended this
interpretation to the matter waves proposed by
De Broglie, by assigning a mathematical
function, Ψ(r,t), called the wavefunction to every
“material” particle
Ψ(r,t) is what is “waving”
Definition of Ψ(r,t)
The probability of funding a •
2
PV   P(r , t )dV   (r , t ) dV
particle somewhere in a
V
V
volume V of space is
Since the probability to find –
particle anywhere in space is 1,
we have condition of
normalization
 ( r , t )
2
dV  1
all space
b
For one-dimensional case, •
the probability of funding
the particle in the arbitrary
interval a ≤ x ≤ b is
Pab 

2
( x , t ) dx
a

Peverywhere 
 ( x , t )

2
dx