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Physics 334
Modern Physics
Credits: Material for this PowerPoint was adopted from Rick Trebino’s lectures from Georgia Tech which were based on
the textbook “Modern Physics” by Thornton and Rex. Many of the images have been used also from “Modern Physics” by
Tipler and Llewellyn, others from a variety of sources (PowerPoint clip art, Wikipedia encyclopedia etc), and contributions
are noted wherever possible in the PowerPoint file. The PDF handouts are intended for my Modern Physics class, as a
study aid only.
Chapter 2 Statistics and Thermodynamics
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Why Statistics
Probability Distribution
Gaussian Distribution Functions
Temperature and Ideal Gas
The Maxwell-Boltzmann Distribution
Density of States
What has Statistics to do with Modern Physics
In physics and engineering courses we talk about motion of
baseballs, airplanes, rigid bodies etc. These descriptions are from a
macroscopic point of view
In Kinetic theory of gases, the macroscopic properties of gases for
example, pressure, volume and temperature are based on
microscopic motion of molecules and atoms that make up the gas.
All matter is made of atoms and molecules, so it make sense to talk
about macroscopic quantities in light of microscopic quantities.
Therefore it is necessary to combine mechanics (classical or
quantum) with statistics. This is called Statistical Mechanics.
The main concept that will be used is the concept of probabilities,
since in quantum mechanics the interest is in measuring the
probabilities of a physical quantity (position, momentum, energy etc.)
Probability Distribution Function
A distribution function f(x) or Ф(x) is a function that gives
the probability of the particle found in a certain range of
allowed values of an event.
An event is an act of making a measurement of an
observable. Say if our measurement is of position x than
the probability function of finding the particle in the range
x and x+dx is
P( xi )
 ( x) 
, for discrete variable xi
xi
dP( x )
 ( x) 
, for continuous variable x
dx
 P( xi ) 
dP( x )
where
 lim 

xi 0
dx

x
i


Probability Calculations
The probability of finding the particle is given by
f
f
k i
k i
P( x1  x  x2 )= ( xk )x   P( xi ) for discrete variable xi
xf
P( x1  x  x2 )=   ( x )dx, for continuous variable x
xi
where the region is defined between x i and x f
Normalization
The particle has to be some where. Normalization
means that the sum or integral should be equal to 1.

P(  x  )=  P( xi )=1 for discrete variable xi
k 
+
P(  x  )=   ( x)dx  1, for continuous variable x
-
un normalized distribution functions can always be normalized
For a 1-D function of x, the normalization condition is
*

 ( x) ( x)dx  1
Normalization
Example: Find the normalizing constant A for the
function
x 
 ( x)  A sin  
 a 
*

 ( x) ( x)dx  1
a

0
2 1
x 
A sin   dx  A a  1
2
 a 
2
2
A
a
2
Expectation (Average) Value
Average value of a physical quantity is called expectation
value. If several measurements are made, the value that
is expected on the average is the expectation value even
though no single measurement may be the expectation
value. This is also the mean value.

x =x =  xi P( xi )=1 for discrete variable xi
i 
+
x =x =

i - 
x ( x)dx  1, for continuous variable x
Expectation Value
Example: Find the expectation value of the function
2 x 
 ( x)  sin  
a  a 

x    * ( x) x ( x)dx

2
a
2 x 
x   x sin   dx 
a
2
 a 
0
a
Standard Deviation
Standard deviation (σ) tells us how much deviation to
expect from the average value. Variance (σ2) also gives
us the spread.


 xi  x 
2
 xi  x 
2

 ( xi  x )2

i 

dP
  dx( x  x )
, for continuous variable x
dx

2
  (x  x )  x  x
2
2
  
2
Pi
= for discrete variable xi
xi
2
x  x
2
2
2
Gaussian Distribution
Binomial distribution, Gaussian
distribution and Poisson
distribution are the most
common distributions
encountered in physics.
We will discuss only the
Gaussian distribution as it
occurs widely and is used in
many different fields. Gaussian
distribution is also called Bell
curve or standard distribution.
Consider a drunk undergoing a
random walk.
Star
t
x
Gaussian Distribution
Displacement is the sum of several random steps. The
probability of small steps to cancel each other out is
greater than the probability of large steps which might be
in the same direction.
The distribution function is given as
 ( x) 
1
2
2
e
 ( x  a )2
(2 )
2
Bell Curve
The solid green line shows that
1. The “most probable” value (peak) is at x=25
2. The median (50% level) is at x=25
3. The mean is at x=25
σ=6
Red dash line
x- σ<x<x+σ
The probability of
the particle to be in
this region is 68.3%
Purple dash line
x- 2σ<x<x+2σ
The probability of
the particle to be in
this region is 95%
Gaussian Distribution
0.07
0.06
Probability
0.05
0.04
0.03
0.02
0.01
0
0
5
10
15
20
25
x
30
35
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45
50
Thermal Physics
Thermal physics is the study of
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Temperature
Heat
How these affect matter
How heat is transferred between systems and to the
environment
Heat
It is a process in which energy is exchanged because of
temperature differences.
Thermal Contact
Objects are said to be in thermal contact if energy can be
exchanged between them.
Thermal Equilibrium
Energy cease to exchange between objects
Zeroth Law of Thermodynamics
If objects A and B are separately in thermal equilibrium
with a third object, C, then A and B are in thermal
equilibrium with each other.
Allows a definition of temperature
Temperature is the property that determines whether or not an
object is in thermal equilibrium with other objects
Thermometers
Used to measure the temperature of an object or a
system
Make use of physical properties that change with
temperature
Many physical properties can be used
volume of a liquid
length of a solid
pressure of a gas held at constant volume
volume of a gas held at constant pressure
electric resistance of a conductor
color of a very hot object
Thermometers, cont
A mercury thermometer is
an example of a common
thermometer
The level of the mercury
rises due to thermal
expansion
Temperature can be
defined by the height of the
mercury column
Temperature Scales
Thermometers can be calibrated by placing them in thermal
contact with an environment that remains at constant
temperature
Environment could be mixture of ice and water in thermal
equilibrium
Also commonly used is water and steam in thermal equilibrium
Celsius Scale
Temperature of an ice-water mixture is defined as
0º C
This is the freezing point of water
Temperature of a water-steam mixture is defined
as 100º C
This is the boiling point of water
Distance between these points is divided into 100
segments or degrees
Fahrenheit Scales
Most common scale used in the US
Temperature of the freezing point is 32º
Temperature of the boiling point is 212º
180 divisions between the points
Kelvin Scale
When the pressure of a gas goes to zero, its
temperature is –273.15º C
This temperature is called absolute zero
This is the zero point of the Kelvin scale
–273.15º C = 0 K
To convert: TC = TK – 273.15
The size of the degree in the Kelvin scale is the same as
the size of a Celsius degree
Pressure-Temperature Graph
All gases extrapolate to
the same temperature at
zero pressure
This temperature is
absolute zero
Modern Definition of Kelvin Scale
Defined in terms of two points
Agreed upon by International Committee on Weights and
Measures in 1954
First point is absolute zero
Second point is the triple point of water
Triple point is the single point where water can exist as
solid, liquid, and gas
Single temperature and pressure
Occurs at 0.01º C and P = 4.58 mm Hg
Modern Definition of Kelvin Scale, cont
The temperature of the triple point on the Kelvin scale is 273.16
K
Therefore, the current definition of the Kelvin is defined as
1/273.16 of the temperature of the triple point of water
Some Kelvin
Temperatures
Some representative
Kelvin temperatures
Note, this scale is
logarithmic
Absolute zero has
never been reached
Comparing Temperature Scales
Converting Among Temperature Scales
TC  TK  273.15
9
TF  TC  32
5
5
TC  TF  32 
9
9
TF  TC
5
Temperature and Kinetic Energy
The average kinetic energy of a molecule in thermal equilibrium
with its surrounding is given by
Ek 
3
kT , where k is the Boltzmann constant = 8.617 10-5eV / K
2
Example
Calculate the average kinetic energy of a gas molecule at room
temperature T=20 degree Celsius
T  20  273  293K
3
3
Ek  kT  (8.62 105 eV / K )(293K )  0.038eV
2
2
It is useful to remember that at room temperature of 300 K the
value of kT is
kT  0.02585eV 
1
eV
40
Ideal Gas
A gas does not have a fixed volume or pressure
In a container, the gas expands to fill the container
Most gases at room temperature and pressure behave
approximately as an ideal gas.
Exercise: Consider a container of gas that has a volume V in
thermal equilibrium at a temperature T. Show that the pressure P
is given by PV=NkT, where N is the Avogadro's number and k is
the Boltzmann constant.
Example: Calculate the volume occupied by one mole of
molecules at a pressure of one atmosphere (1.01x105N/m2) and
a temperature of 273 K. This condition is called Standard
Temperature and Pressure (STP)
Ideal Gas
Example: Calculate the volume occupied by one mole of
molecules at a pressure of one atmosphere (1.01x105N/m2) and
a temperature of 273 K. This condition is called Standard
Temperature and Pressure (STP).
N A kT
P
At T=273, the value of kT is
V
 273K 
kT  (0.0585) 
  0.0235 K
 300 K 
 (6.02 1023 )(0.0235eV ) 
19
3
V 
(1.60

10
J
/
eV
)

0.0224
m

1.01105 N / m 2


V  22.4 liters