VIEWS: 1 PAGES: 30 POSTED ON: 11/27/2012
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 • 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 dP( x ) P( xi ) where lim dx xi 0 xi Probability Calculations The probability of finding the particle is given by f f P( x1 x x2 )= ( xk )x P ( xi ) for discrete variable xi k i k i 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 x a 2 1 A sin dx A a 1 2 2 0 a 2 2 A a 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 x a 2 a x x sin dx 0 a a 2 Standard Deviation Standard deviation (σ) tells us how much deviation to expect from the average value. Variance (σ2) also gives us the spread. Pi xi x ( xi x )2 2 = for discrete variable xi i xi dP xi x dx( x x ) 2 2 , for continuous variable x dx (x x ) x x 2 2 2 2 x x 2 2 2 Gaussian Distribution Binomial distribution, Gaussian distribution and Poisson distribution are the most common distributions Star encountered in physics. t We will discuss only the x 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. 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 1 ( x) e ( x a )2 (2 ) 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 Gaussian Distribution x- σ<x<x+σ 0.07 The probability of 0.06 the particle to be in 0.05 this region is 68.3% Probability 0.04 Purple dash line 0.03 x- 2σ<x<x+2σ 0.02 The probability of 0.01 the particle to be in 0 0 5 10 15 20 25 30 35 40 45 50 this region is 95% x Thermal Physics Thermal physics is the study of • 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 3 Ek 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 (8.62 105 eV / K )(293K ) 0.038eV 2 2 It is useful to remember that at room temperature of 300 K the value of kT is 1 kT 0.02585eV 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 V P At T=273, the value of kT is 273K kT (0.0585) 0.0235 K 300 K (6.02 1023 )(0.0235eV ) 19 V (1.60 10 J / eV ) 0.0224m 3 1.01105 N / m 2 V 22.4 liters