UNIT-1 MODERN PHYSICS

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1 UNIT- MODERN PHYSICS Introduction to blackbody radiation spectrum: A body wic absorbs all radiation tat is incident on it is called a perfect blackbody. Wen radiation allowed to fall on suc a body, it is neiter reflected nor transmitted. Suc a black body after absorbing te incident radiation gets eated and starts emitting radiation of all possible wavelengts. In practice, perfect blackbody does not exist and we can ave objects tat are only close to a blackbody. A blackbody, on eating, can emit all radiations it as absorbed and is called blackbody radiation. Figure sows te energy distribution curves in wic energy density E (energy emitted by te blackbody per unit area of te surface) is plotted as a function of wavelengt at different temperatures of te blackbody. Te important points tat can be noted down from tese curves are. All curves sows a peak suggesting tat te emitted intensity is maximum at a particular wavelengt. 2. An increase in temperature results in an increase in te energy emitted. 3.As te temperature increases, te peak sifts to lower wavelengt. Laws of blackbody radiation:. Stefan-Boltzmann law It states tat te total energy density E 0 of radiation emitted from a blackbody is directly proportional to te fourt power of its absolute temperature T. i.e., E 0 T 4 Or E 0 T 4 were is a constant called Stefan s constant wit numerical value equal to Wm -2 K -4. Tis law agrees well wit te experimental results. 2. Wien s displacement law It states tat te wavelengt m corresponding to te maximum emissive energy decreases wit increasing temperature. i.e., m T Or m T b, were b is called te Wien s constant and is equal to mk. 3. Wien s law Using laws of termodynamics and classical concepts, Wien developed an expression for energy density as, U(λ)dλ C λ 5 e C 2 λt dλ

2 were C and C 2 are constants. Tis law olds good for smaller values of wavelengt. 4.Rayleig-Jeans law Rayleig derived an expression for te energy density of radiation based on classical teory wic is given by, U(λ)dλ 8πkT λ 4 dλ were k is called Boltzmann s constant and its value is.38 x 0-23 JK - Tis law olds good only for large values of wavelengt. As per te Rayleig-Jeans law, te radiant energy increases wit decreasing wavelengt and a blackbody must radiate all te energy at very sort wavelengt. But in actual practice, it doesn t appen so. Te failure of Rayleig-Jeans law to explain te aspect of very little emission of radiation beyond te violet region towards te lower wavelengt side of te spectrum is referred as ultraviolet catastrope. Bot Wien s law and Rayleig-Jeans law indicates failure of classical teory in explaining blackbody radiation. Planck s radiation law: Max Planck proposed a law based on quantum teory. According to tis, atoms or molecules absorb or emit radiation in quanta or small energy packets called potons. If is te frequency of potons, ten its energy can be explained as E were 6.63x0-34 Js is called Planck s constant. Applying quantum teory, Planck obtained an expression for energy density of blackbody radiation as,

3 U(λ)dλ 8πc λ 5 e c dλ () Tis law agrees well wit te experimental observation of blackbody radiation and is valid for all wavelengts.. For sorter wavelengts: i.e., wen λ is small, eis c very large Or i.e., e>> c ence(e c ) e c Substituting tis in Planck s radiation law i.e., in eq() ten, U(λ)dλ 8πc λ 5 e c i.e.,u(λ)dλ C λ 5 C 2 dλ eλt Were C 8c andc 2 c k Hence at smaller wavelengts, Planck s radiation law reduces to Wien s law. 2. For longer wavelengts: i.e., wen λ is large, e c is very small Expanding te power series, we ave e c + c! + ( c )2 + 2! Since c is very small, its iger power terms can be neglected Ten te above expression becomes e c + c or e c c dλ Substituting tis in Planck s radiation lawi.e., in eq(), U(λ)dλ 8πc λ 5 c dλ i.e.,u(λ)dλ 8πkT λ 4 Hence at longer wavelengts, Planck s radiation law reduces to Rayleig-Jeans law. Potoelectric effect: Emission of electrons from a metal surface wen ligt of suitable energy falls on it is called potoelectric effect. Te experimental setup for observing potoelectric effect consists of a pair of metal plate electrodes in an evacuated discarge tube connected to a voltage source as sown:

4 Wen ligt of suitable energy is incident on te catode, electrons are emitted and a current flows across te discarge tube. Some special features of potoelectric emission are:. It is an instantaneous process-tere is no time interval between te incidence of ligt and te emission of potoelectrons. 2. Tere is a minimum frequency for te incident ligt called tresold frequency, below wic no potoelectric emission occurs. Tis depends upon te nature of te material of te emitter surface. Te energy corresponding to te tresold frequency, called te work function is te minimum energy required to release an electron from te emitter surface. 3. For a given frequency of te incident ligt, potocurrent is directly proportional to te intensity of te incident ligt. 4. Te potoelectron emission can be stopped by applying a reverse voltage to te pototube. ie., by making te emitter electrode positive and te collector negative. Te negative collector potential required to stop te potoelectron emission is called te stopping potential. Einstein s Teory: Potoelectric effect can be explained on te basis of quantum teory of ligt. Wen te energy equal to work function of te metal is incident on te metal surface, te incident poton liberates electrons from teir bound state. Wen te incident poton carries energy in excess of te work function, te extra energy appears as te kinetic energy of te emitted electron. Wen te intensity of ligt increases, te number of potoelectrons emitted increases but teir kinetic energy remain unaltered. Wen a poton of frequency is incident on a metal surface of work function, ten, + ( 2 mv2 ) max were ( mv2 ) max is te kinetic energy of te emitted potoelectrons. Tis is known as Einstein s potoelectric equation. Since 0, it can also be written as, ( mv2 ) max - (- 0 ) If V 0 is te stopping potential corresponding to te incident poton frequency, ten, ( mv2 ) max - Compton Effect: ( mv2 ) max (- 0 ) or ( 2 mv2 ) max ev 0

5 Wen X-rays are scattered by a solid medium, in addition to te scattered X-rays of same frequency, tere exist some scattered X-rays of a sligtly lower frequency (iger wavelengt). Compton observed tis penomenon and is called Compton Effect. Compton Effect can be explained on te basis of te quantum teory and laws of conservation of energy and momentum. Consider an x-ray poton of energy incident on an electron at rest. After te interaction, te X-ray poton gets scattered at an angle wit its energy canged to and te electron wic was initially at rest recoils at an angle. It can be sown tat te increase in wavelengt is given by λ (- cos θ) m 0 C were m 0 is te rest mass of te electron. Wen 90 0, Å. Tis constant value is called Compton wavelengt. m 0 C Wave particle dualism: Te Potoelectric Effect and Compton Effect conclusively establised te particle beavior of ligt. Te penomena of interference, diffraction and polarization give exclusive evidence for te wave beavior of ligt. Hence we ave to conclude tat ligt beaves as an advancing wave in some penomena and it beaves as a flux of particles in some oter penomena. Terefore we say tat ligt exibits wave-particle duality. De-Broglie s ypotesis: De-Broglie extended te wave particle dualism of ligt to te material particles. Tis is known as de- Broglie ypotesis. According to tis ypotesis, material particles in motion possess a wave caracter. Te waves associated wit material particles are called matter waves or de-broglie waves. According to Planck s teory of radiation, E () were is te frequency associated wit te radiation. According to Einstein s mass-energy relation, E mc (2) were m is te mass of te poton and c is te velocity of ligt Combining () and (2), i.e., mc 2 > mc (sinceυ ) λ mc Terefore momentum associatedwit te particle is given by p mc, Or, were is called de-broglie wavelengt. p De-Broglie wavelengt associated wit te accelerated electron:

6 A beam of ig energy electrons can be obtained by accelerating tem in an electric field. Consider an electron starting from rest wen accelerated wit a potential difference V, te kinetic energy (E) acquired by te electron is given by, E 2 mv2 and also E ev Tus, 2 mv2 ev m Or 2 v 2 ev 2m p i.e., 2 ev E 2m were v is te velocity of te electron, m its mass and p te momentum. Now te momentum may be expressed as, p 2mE 2meV Hence te de-broglie wavelengt λ p 2mE 2meV To test de-broglie ypotesis, Heisenberg and Scrödinger formulated teories wereas G.P.Tomson, Davisson and Germer conducted experiments. Davisson-Germer experiment: Te electron diffraction experimental setup used by Davisson and Germer to verify de-broglie s ypotesis is as sown: Te filament F is eated to produce electrons via termionic emission. Tese electrons are passed troug a narrow aperture forming a fine beam of accelerated electrons. Te electron beam was ten made to incident on a single crystalline sample of Nickel. Te electrons scattered at different angles were counted using a detector. Te experiment was repeated by recording te scattered electron intensities at various positions of te detector. A sarp maximum occurred in electron density at an angle (Φ) of 50 0 wit te incident beam for an accelerating potential of 54 V. Te angle of incidence corresponding to tis is 25 0 and from figure, glancing angle θ (angle of diffraction) is From X-ray diffraction experiment, spacing of te planes responsible for diffraction was found to be 0.09 nm. Assuming first order diffraction, Bragg s law can be written as, 2d sin sin nm. By de-broglie relation, λ nm 2meV ( ) Tus Davisson and Germer experiment directly verifies te de-broglie ypotesis. Caracteristic properties of matter waves:

7 . Matter waves are associated wit moving particle. 2. Wavelengt of matter waves is inversely proportional to te velocity wit wic te particle is moving ( ). Hence a particle at rest as an infinite wavelengt. mv 3. Wavelengt of matter waves is inversely proportional to te mass of te particle. Hence wavelike beavior of eavier bodies is not very evident wereas wave nature of subatomic particles could be observed experimentally. 4. Wave function is used to define a matter wave wic is related to te probability of finding a particle at any place at any instant. 5. Matter waves are represented by a wave packet made up of a group of waves of sligtly differing wavelengts. Hence we talk of group velocity of matter waves rater tan te pase velocity (velocity of a single wave). Te group velocity can be sown to be equal to te particle velocity. Pase velocity: General expression for a wave isy A cos (t-kx) were Y Displacement at any instant t, A Amplitude of vibration, 2 is te angular frequency and k is te wave vector or wave number. Pase velocity or wave velocity of a wave is te velocity of te wave wen pase is constant. i.e., t-kx constant or,kx t + constant or, x ωt + constant k Hence Pase velocityv p dx ω dt k Group velocity: Te de-broglie waves are represented by a wave packet and ence we ave group velocity associated wit tem. Group velocity is te velocity wit wic te wave packet travels. Consider two waves aving same amplitude but aving sligtly different frequency and wave number represented by te equations Y A cos (ωt-kx) Y 2 A cos [(ω+δω) t (k+δk) x] Te resultant displacement due to te superposition of te above two waves is, Y Y + Y 2 A cos (ωt-kx) + A cos [(ω+δω) t (k+δk)x] A B A B Since, cos A + cos B 2cos ( ) cos ( ), k k k Y 2A cos {( ) t ( ) x}cos {( ) t ( ) x} As te difference in frequency of te two waves is very small, we can assume, 2+Δω 2 and 2k+Δk 2k k Y 2Acos {( ) t ( ) x} cos (ωt-kx) 2 2 Te velocity of te resultant wave (group velocity) is given by te speed wit wic a reference point, say te maximum amplitude point, moves. Taking te amplitude of te resultant wave as constant, we ave, k 2Acos {( ) t ( ) x} constant 2 2 k or, ( ) t ( ) x constant 2 2

8 or, x + constant Group velocity v g dx ω dt k Wen and k are very small, v g dω dk Relation between group velocity and pase velocity: Pase velocity, v p were ω is te angular frequency of te wave v p k () Group velocity, v g v g (From eq ()) v p + k v p + ( ) ( )were te propagation constant (or wave number) k But ( ) Tusv g v p - λ dv p dλ v p + ( ) ( ) v p + ( ) ( ) v p + ( ) ( ) Terefore v g v p + ( ) Relation between group velocity and particle velocity: Energy of a poton E ν or ν E () We know te angular frequency of te waveω 2ν or ω () dω ( )de (2) Furter, k since dk ( )dp (3) Substituting te value of d and dk from equations (2) and (3) in te expression for group velocity, v g dω (4) de dk dp If a particle of mass m is moving wit a velocity v particle, its energy is given by, E mv2 particle (5) Substituting tis in equation (4), v g de dp d dp Hence v g v particle v particle

9 Relation between velocity of ligt, group velocity and pase velocity: Pase velocity, v p were k is te propagation constant or wave number We knowte angular frequency of te wave ω 2ν or ω () Furter, Wave numberk since were p is te momentum of te wave Tus v p v pase v particle c 2 But v g v particle v pase v g c 2 (Since E mc 2 ) Expression for de-broglie wavelengt using group velocity: We know, te group velocity v g dω, were angular frequency ω 2 and wave number k dk dω 2d and dk 2d( ) Tus v g dω 2d dk 2 d( d( λ ) d λ ) () Or,d( λ ) d v g d v particleg () [since v g v particle ] Total energy of a particle moving under an applied potential V is given by, E mv2 + V de-broglie related E to above expression Ten, mv2 + V Assuming V as a constant potential and differentiating te above equation, d mv particle dv or, d ( )dv Substituting tis in equation (), d( ) Integrating, + constant i.e., + constant Assuming constant of integration to be zero, + constant or, λ, te de-broglie wavelengt. p

10 ************** VTU Model Question Paper.a)) If te momentum of a particle is increased to four times, te de-broglie wavelengt is i) become twice ii) become for times iii) become one-fourt iv) become alf 2) Blackbody radiation spectrum, maximum intensity is sifting towards i) sorter wavelengt ii) longer wavelengt iii) no cange iv)none of tese 3) Group velocity of wave is equal to i) V pase ii) V particle iii) Velocity of ligt iv) none of tese 4) de-broglie wavelengt of an electron accelerated by a potential of 60 V is i).85 Å ii).58 Å iii).589 Åiv).57 Å (4 marks) b) Describe Davisson-Germer experiment to prove te dual nature of matter waves. (8marks) c) Explain pase velocity and Group velocity. Derive de-broglie wavelengt using Group velocity.(8 marks) Dec 08/ Jan 09 a) ) Te de-broglie wavelengt associated wit an electron of mass m and accelerated by a potential V is i) 2mVe ii) 2mVe iii) 2) Davisson and Germer were te first to demonstrate: i) Te straigt line propagation of ligt ii) Te diffraction of potons iii) Te effective mass of electron iv) None of tese 3) Electrons beaves as waves because tey can be: i) Deflected by an electric field ii) Diffracted by a crystal iii) Deflected by magnetic field iv) Tey ionize a gas 4) In Davisson-Germer experiment, te ump is most prominent wen te electron is accelerated by i) 34 volts ii) 54 volts iii) 60 volts iv) 80 volts (04 Marks) b) Define pase velocity and group velocity. Sow tat group velocity is same as particle velocity.(08 Marks) c) Derive de-broglie wavelengt using Group velocity. (04 Marks) d) Compare te energy of a poton wit tat of a neutron wen bot are associated wit wavelengt of Å. Given tat mass of neutron is kg. (04 Marks) iv) June-July 2009 ) a)) An electron and a proton are accelerated troug same potential. Te ratio of de-broglie wavelengt e / p is i) ii) iii) iv) 2) Wavefunction associated wit a material particle is i) Single valued ii) Finite iii) Continuous iv) all te above 3) In a blackbody radiation spectrum, te maximum energy peaks sift towards te sorter wavelengt side wit te increase in temperature. Tis confirms i) Stefan s law ii) Wein s law iii) Rayleig-Jean s lawiv) Planck s law 4) Te group velocity of te particle is m/s, wose pase velocity is i) m/s ii) m/s iii) 3 nm/s iv) m/s (04 Marks) b) Describe Davisson and Germer experiment for confirmation of debroglie ypotesis.(08 Marks) c) Explain pase and group velocity. Calculate te de-broglie wavelengt of a bullet of mass 5 gm moving wita velocity 20 km/r.(08 Marks) Dec.09/Jan.0 a)) Wien s law is deduced from Planck s radiation formula under te condition of i) Very small wavelengt and temperature ii) Large wavelengt and temperature iii) Small wavelengt and ig temperature iv) Large wavelengt and small temperature 2) Te Compton wavelengt is given by

11 i) /m 0 C 2 ii) 2 /m 0 C 2 iii) /m 0 C iv) 2 /2m 0 C 3) Wic of te following relations can be used to determine de-broglie wavelengt associated wit a particle? i) 2mE ii) iii) 2mVe iv) All of tese 4) If te group velocity of a particle is m/s, its pase velocity is i) 00 m/s ii) m/s iii) m/s iv)3 0 0 m/s(04 Marks) b) Wat is Planck s radiation law? Sow ow Wien s law and Rayleig-Jean s law can be derived from it. (06 Marks) c) Define group velocity. Derive relation between group velocity and pase velocity.(06 Marks) d) A fast moving neutron is found to be ave a associated de-broglie wavelengt 2Å. Find its kinetic energy and group velocity of te de-broglie waves.(04 Marks) May/June 200 a) ) In a blackbody radiation spectrum, te Wien s distribution law is applicable only for i) Longer wavelengt ii) Sorter wavelengt iii) Entire wavelengt iv) None of tese 2) Te de-broglie wavelengt associated wit an electron of mass m and accelerated by a potential V is i) 2mVe ii) 2mVe iii) 3) Electrons beaves as a wave because tey can be i) Diffracted by a crystal ii) Deflected by magnetic field iii) Deflected by electric field iv) Ionise a gas 4) If te group velocity of de-broglie wave is m/sec, its pase velocity is i) m/sec ii) m/sec iii) m/sec iv) m/sec (04 Marks) b) Explain duality of matter waves. (04 Marks) c) Define pase velocity and group velocity. Sow tat group velocity is equal to particle velocity.(08 Marks) d)calculate te momentum of te particle and de-broglie wavelengt associated wit an electron wit a kinetic energy of.5 kev. (04 Marks) January 20 a)) Green ligt incident on a surface releases potoelectrons from te surface. If now blue ligt is incident on te same surface, te velocity of electrons i) increases ii) decreases iii) remains same iv) becomes zero 2) Rayleig-Jean s teory of radiations agree wit experimental results for i) all wavelengts ii) sorter wavelengts only iii) longer wavelengts only iv) middle order wavelengts only 3) Te de-broglie wavelengt of an electron accelerated to a potential difference of 00 volts is i).2å ii) 0Å iii) 00Å iv) 2Å 4) Te wave nature associated wit electrons in motion was verified by i) potoelectric effect ii) Compton effect iii) diffraction by crystals iv) Raman effect (04Marks) b) State and explain de-broglie s ypotesis. (04 Marks) c)define pase velocity and group velocity. Obtain te relation between group velocity and particle velocity. Obtain te expression for de-broglie wavelengt using group velocity. (08 Marks) d) Find te kinetic energy and group velocity of an electron wit de-broglie wavelengt of 0.2 nm. (04 marks) ************* iv)