1
Chapter-2 Probab ility Distribution Functions
The square of the quantum mechanica l wave function P(x) = Ψ∗Ψ is interpreted as a probability
distribution in quan tum mechanics.
More formally a probability distribution function (PDF), P( x), can be descr ibed in terms of the number of
similar systems
Ω
(x) in a large ensemble of systems Ω(N), where P(x) is the probability of obtaining
outcome ‘x’ when
Ω
(N) is sampled , tested measured, etc.
P(x)=lim
!(N)"#
!(x)
!(N)
Normalization
The PDF must be normalized over it's domain of definition
.
Pi(x)
! =1 or P(x)=1
a
b
"
Discrete PDFs
A discrete PDF may occur from random sampling of
Ω
(N) to obtain similar outcomes
Ω
(x).
Consider the discrete distribution of th e # of events vs x=0-10 eg.
* N = 19
* * * P(0) =1/19 P(5) = 3/19
* * * * * * P(1) =2/19 P(6) = 2/19
* * * * * * * * * P(2) =4/19 P(7) = 1/19
0 1 2 3 4 5 6 7 8 9 P(3) =3/19 P(8) = 1/19
x P(4) =2/19 P(9) = 0/19
Normalization = 1/N
P(x)=
0
9
!
P(0) + P(1) + ...........+P(9) = 1
Continuous Distributions
Continuous PDFs may come from theoretical predictions or fitting a discrete PDF to a continuous form. In
the later case one may argue that we have used our best estimate of the PDF and we will have to consider it
being approximate.
Consider the norm alization of the Boltzmann distribution
P
Boltzmann(E)=Ae!E/kT 0 "E" #
A e!E/kT dE =
0
"
# 1 , A!kT
( )
e!E/kT |0
"=1
A=1
kT
Moments of a PDF
The moments of a PDF are the weighted average of powers of variable ‘x’ taken with the PDF.
< xn> =
<xn>= xn!
"P(x)
or
xn!P(x) dx
"
The zeroth moment is the normalization condition:
1)
<x0>= 1
P(x)
