CHAPTER 4 QUANTUM PHYSICS

Transcription

1 CHAPTER 4 QUANTUM PHYSICS INTRODUCTION Newton s corpuscular teory of ligt fails to explain te penomena like interference, diffraction, polarization etc. Te wave teory of ligt wic was proposed by Huygen in 1679, explain tese penomena. However, new penomena like Compton effect, potoelectric effect, Zeeman effect, emission of ligt, absorption of ligt etc., cannot be explained by te above teories. Te failure of tese teories leads to te discovery of a new teory, called quantum teory of radiation of ligt. BLACK BODY RADIATION Definition A body wic can absorb all wavelengt of an electromagnetic radiation and also can emit all wavelengt of radiation wen it is eated to a suitable temperature is called a black body. Te radiation emitted from black body is known as black body radiation or total radiation. Emissive power: Energy emitted per unit area per unit time. Absorptive power: Energy absorbed per unit area per unit time. 138

2 Caracteristics of Black Body Radiation Black body radiations are caracterized by Stefan s law, Wein s law and Rayleig-Jean s law. According to tese laws, black body radiation is stated as follows: 1. Stefan-Boltzmann s law Te radiation energy (E) emitted per unit time per unit area of a perfect black body is directly proportional to te fourt power of its absolute temperature T. E = σ T 4 Were, σ is te proportionality constant known as Stefan s constant and its value is 5.78 X 10-8 Wm - K -4.. Wein s displacement law Wein s displacement law states tat te wavelengt corresponding to te maximum energy is inversely proportional to absolute temperature T. λ m T = Constant Tis law sows tat as te temperature increases te wavelengt corresponding to maximum energy decreases. Tis law olds good only for sorter wavelengts and not for Longer wavelengts. 3. Rayleig-Jean s law According to tis law, te energy distribution is directly proportional to te absolute temperature and is inversely proportional to te fourt power of te wavelengt. It is governed by te equation. Tis law olds good only for longer wavelengt regions and not for sorter wavelengts. 139

3 Energy Distribution of a Black Body Te distribution of energy for different wavelengt at various temperatures of te source is as sown in following fig. From te above grap, te following observations are made: Te distribution of te energy is not uniform. For a particular temperature, te intensity of radiation increases up to a particular wavelengt and ten it is found to decrease wit increase in wavelengt. As temperature increases te peak energy sifts towards sorter wavelengts. Based on te applications of Wein and Rayleig equation, te energy spectrum of black body radiation cannot be explained completely. In order to explain te distribution of energy in te spectrum of a black body, Planck suggested a new ypotesis in te year 1900 and terefore, e derived te new radiation law wit some assumptions. Planck s Quantum teory of Black Body Radiation Planck derived te expression for te energy distribution based on te following ypotesis: 140

4 Planck s Hypotesis: Te black body radiation camber is filled up not only wit radiations but also wit a large number of oscillating particles. Te particles can vibrate in all possible frequencies. Te frequency of radiation emitted by an oscillator is te same as tat of te frequency of te vibrating particles. Te oscillatory particles cannot emit energy continuously. Radiation is emitted (or) absorbed by a body in an integral multiple of a fundamental quantum of energy called poton. Te vibrating particles can radiate energy wen te oscillators move from one state to anoter. Te radiation of energy is not continuous, but discrete in nature. Te values of te energy of te oscillators are like 0, γ, γ, 3γ.nγ Planck s Quantum Teory of radiation Law Let us consider te number of vibrating particles in te body as N 0,N 1,N,.N n. According to Planck s ypotesis, te energy of te above particles can be written as 0, ε,ε, 3ε, 4ε. nε. Terefore, te total number of vibrating particles is given as, N = N 0 +N 1 +N +.+N n (1) Similarly, te total energy of te body is given as, E = 0+ ε+ε+ 3ε+ 4ε+. +nε () Terefore, te average energy of te particle is given as, (3) According to Maxwell s distribution formula, te number of particles in te n t oscillatory system can be written as, N n = N 0 e nε/kt (4) Were, ε is te energy per oscillator, k is te Boltzmann s constant and T is te absolute temperature. According to Maxwell s distribution function, te total number of particles N can be written as, N = N 0 + N 0 e ε/kt + N 0 e ε/kt + N 0 e 3ε/kT

5 Or N = N 0 [1+ e ε/kt + e ε/kt + e 3ε/kT +.] (5) We know, 1+x+x +x =. Terefore, we can write equation (5) as, (6) Similarly, te total energy of te body can be written as, E= 0+ εn 0 e ε/kt + εn 0 e ε/kt +3ε N 0 e 3ε/kT +. Or E= εn 0 e ε/kt [1+ e ε/kt + 3e ε/kt +.] (7) (7) as, We know, 1+x+3x +4x 3 +.+nx n-1 =. Terefore, we can write equation (8) Or (9) 14

6 Substituting te values of ε = γ in equation(9), we ave (10) as, If γ and γ+dγ is te frequency range, te number of oscillations can be written (11) Terefore, te total energy per unit volume for a particular frequency can be obtained by multiplying equations (11) and (10) as, Or (1) frequencies. Equation (1) is known as Planck s equation for radiation law interms of It can also be written interms of wavelengt as, 143

7 Or (13) Equation (13) represents te Plank s radiation law interms of wavelengt. i. Deduction of Wien s displacement law Wen λ is very small, γ is very large Or (14) Tis is Wien s displacement law. ii. Deduction of Rayleig-Jean s law Wen λ is very large, γ is very small te iger orders] [ ence, we neglect Terefore, Or (15) 144

8 Equation (15) represents Rayleig-Jean s law. Tus, Wien s and Rayleig- Jean s formula are te special cases of Plank s formula on black body radiation. Planck s formula as been found to agree remarkably well wit experimental observations of Lammer, Pringseim, Kurlbaum and oters as sown in Fig and tus establised one for all te validity of te quantum ypotesis. Poton & Its Properties Poton Te discrete energy values in te form of small packets or quanta s of definite frequency or wavelengt are called potons. Potons propagated like a particle wit speed of ligt as 3 X 10 8 ms -1. Properties of Poton (i) Te existence of poton and electron are same in nature. (ii) Potons will not ave any carge. So tey are not affected by magnetic & electric field. (iii) Tey donot ionize gases. (iv) Te energy of one poton is given by E=γ. (v) Te mass of a poton is given by, m = /Cλ. 145

9 (vi) Momentum of te poton is given by P= /λ. Derivation of De-Broglie wavelengt Energy of a poton can be written as E = ν c λ = (1) Were ν is te frequency of te poton λ is te wavelengt If te poton possesses some mass by virtue of its motion ten according to te teory of relatively its energy is given by E = (mass of a poton) c E = mc () From (1) & () c mc λ = Mass of te poton, c 1 m = λ c momentum of te poton c p =. c λc p = λ De-Broglie carried tese considerations over to te dynamics of a particle and said tat te wavelengt λ of te wave associated wit a moving mattered particle aving a momentum mv is given by λ = (3) mv Tis equation is called De-Broglie wave equation 146

10 3. Expression for wavelengt associated wit te electron accelerated by potential difference Consider an electron of carge e accelerated by a potential difference of V volts. If te velocity acquired by te electron of mass m is v ten 1 mv mv = ev m v = ev = mev mv = mev (4) de Broglie wavelengt associated wit te electron, Substituting eqn.(4) in eqn.(5) λ = (5) mv λ = mev Substituting = e Js; 19 = C and m = kg we get, λ = V λ = V 10 m ` 1.5 λ = Å 147 V

11 1.5 λ = nm V 4.7 Scrödinger wave equation Te equation tat describes te wave like beavior of electrons wit te appropriate potential energy and boundary conditions is called te Scrödinger s equations. It is a differential equation capable of describing te motion of an electron. It forms te foundations for quantum mecanics witout tis equation if could not be possible to understand te principles and operations of many some conductor devices. 1. Derivation of time independent wave equation A traveling wave in te positive x direction resulting from sinusoidal oscillations of a particle can be described by a traveling wave equation i t y = Ae ω were y te displacement at time t and at distance x from te equation and ω te angular frequency. Let x,y,z be te coordinates of te particle and Ψ, te wave displacement for te de Broglie waves at any time t. Te classical differential equation of a wave motion is given by Ψ Ψ Ψ Ψ = v v + + = Ψ t x y z (1) Here, v is te wave velocity. Ten Ψ = 1 v Ψ t () Te solution of eqn. (1) gives Ψ as a periodic displacement in terms of time. ( r, t) ( ) i t Ψ = Ψ 0 r e ω (3) Differentiating eqn. () twice wit respect to t, we get 148

12 Ψ = Ψ t 0 e iωt ( iω ) Ψ iωt = Ψ 0 t ( ω)( ω) e i i Ψ = ω Ψ = Ψ 0, t iωt ( r) e ω ( r t) Q i = 1 Substituting te value of Ψ t in eqn. (1), we get Ψ Ψ Ψ ω Ψ = 0 x y z v (4) ω v Ψ + Ψ = 0 But, ω = πν = π v (Here, ν is te frequency) λ ω π = v λ Substituting te value of v ω in eqn.(4) Ψ Ψ Ψ 4π Ψ = 0 x y z λ 4π λ Ψ + Ψ = (5) From de Broglie relation, if a particle is beaving as a wave ten λ = mv 4π Ψ + m v Ψ = (6) A particle can beave as a wave under motion only if its is kept in a potential field. Let E = total energy of te particle 149

13 V = Potential energy of te particle Te kinetic energy of te particle is 1 E = V + mv 1 ( ) mv = E V or m v m( E V ) = (7) Substituting tis in eqn.(6), we get 4π Ψ + m ( E V ) Ψ = or ( V ) 0 8π m Ψ + 0 E Ψ = (8) Eqn.(8) is called Scrödinger time independent wave equation. Te quantity Ψ is called wave function Let us now substitute in eqn. (8) = π Ten te Scrödinger time-independent wave equation, in usually used form, may be written as m E ( V ) Ψ + Ψ = (9) In one dimension Ψ = ( ) Ψ = 0 dx d m E V For a free particle te potential energy, V=0 Ten te Scrödinger time independent wave equation becomes 150

14 d Ψ m 0 = EΨ = dx (10). Derivation of time dependent Scrödinger wave equation Te Scrödinger time dependent wave equation is obtained from Scrödinger time independent wave equation by eliminating E. Te classical differential equation of a wave motion is given by Ψ Ψ Ψ Ψ = v v + + = Ψ t x y z (11) Te solution of eqn. (11) gives Ψ as a periodic displacement in terms of time. ( r, t) ( ) i t Ψ = Ψ 0 r e ω (1) Here, Ψ 0 is te amplitude at te point considered. It is function of position r i.e., of co-ordinates (x,y,z) and not of time t. Differentiating eqn. (1) wit respect to t, we get Ψ = Ψ t 0 iωt ( r) e ( iω ) ( πν ) ( ) i t = i Ψ 0 r e ω (since ω = π iν Ψ πν = ) E = πi Ψ E ( Q E = ν or ν = ) EΨ = Ψ Ψ = πi t i t EΨ = i Ψ t (13) Scrödinger time independent wave equation (9) is m E ( V ) Ψ + Ψ = (14) 151

15 Substituting te value of EΨ in eqn.(14), we get m i Ψ t Ψ + V Ψ = Ψ t Ψ = m i V Ψ 0 m Ψ + V Ψ = i Ψ t (15) Eqn. (15) is called time dependent Scrödinger equation. Eqn. (15) can be written as + V Ψ = i Ψ m t or H Ψ = EΨ (16) Here, H m = + V and E = i t Eqn. (16) describes te motion of a non-relativistic material particle. 4.8 Pysical significance of te wave function Ψ 1. Ψ is a wave function and used to identify state of te particle.. Te wave function Ψ measures te variations of te matter wave. Tus it 3. connects te particle and its associated wave statistically. Ψ is a measure of te probability of finding particle at a particular position and does not give exact location of te particle. 4. Ψ is a complex quantity and we cannot measure it exactly. 15

16 5. Te probability of finding out a particle in a particular volume element dτ is given by * P r d = Ψ Ψ d = Ψ d τ τ τ were * Ψ is called te complex 6. conjugate of Ψ. Ψ dτ = 1, means were te particles presence is certain in space. 7. Te wave function Ψ as no pysical meaning, wereas te probability density as pysical me a0ning. 4.9 Applications of Scrödinger equation 1. Particle in a box Consider te beavior of an electron wen it is confined to a certain region 0<x<L. Te potential energy of te electron is zero inside tat region and infinite outside as sown in fig. Te electron cannot escape because it would need an infinite potential energy. function is In terms of te boundary conditions imposed by te problem te potential V(x) = 0, 0 < x < L and V(x) =, L x 0, Scrodinger wave equation is, m ψ + ( E V ) ψ = 0 Inside te potential well, equation becomes ψ ( x) m + ( ) 0 E ψ x = x ψ ( x) x + k ψ x = ( ) (1) me were k = () 153

17 Tis is te wave equation for a free particle inside a potential well. A possible solution to equation () is ψ ( x) = Asin kx + B cos kx (3) were A and B are constants. Since te particle cannot penetrate an infinitely ig potential barrier for ψ ( x) = 0 at x=0, B must be zero for ψ ( x) = 0 at x=l, kl must be an integral multiple of π. Terefore, ψ ( x) = Asin kx were k = nπ /L, n = 1,,4 ( ) sin nπ x ψ n L = A L (4) Wic is called Eigen function or caracteristic function. All ψ n for n = 1,,4. Constitute te Eigen functions of te system. Eac Eigen function identifies a possible state for te electron. For eac n value, tere is one special k value. Substituting for k from equation () and (4) we get me 1 = nπ L or E n n 8mL = (5) Te energies E n defined by te above equation wit n= 1,,3,.. are called Eigen energies of te system. We still ave not completely solved te problem, because A as yet to be determined. To find A, we use wat is called te normalization condition. Te total probability of finding te electron in te wole region 0 < x < L is unity because we know te electron is somewere in tis region. Terefore ψ summed between x=0 and x=l must be unity or dx L 0 ψ x dx = (6) 154

18 L A L nπ x A sin dx 1 = (7) L 0 A L 0 1 nπ x 1 cos dx 1 = L (since θ = θ (cos 1 sin ) A L π nx x sin 1 = π n L 0 L i.e. A L = i.e. 1 A = L Te normalized wave function is ψ n nπ x = sin L L (8) we can now summarize te beavior of an electron in one dimensional potential energy well. Te wave functions ψ n and te corresponding energies E n, wic are often called Eigen function and Eigen values sown in fig. Bot depend on te quantum number n. Te energy of te electron increases wit n, so te minimum energy of te electron corresponds to n=1. Tis is called ground state and te energy in te ground state is te lowest energy te electron can possess. Te energy of te electron in te potential well cannot be zero even toug te PE is zero. Tus te electron always as KE even wen it is in te ground state. Te node of a wave function is defined as te point were ψ = 0 inside te well. function. Te energy is found to increase as te number of nodes increases in a wave. Tree dimensional potential well To examine properties of a particle confined to a region of space, we take tree dimensional space wit a volume marked by a, b, c along te x, y, and z axis respectively. Te PE is zero inside and is infinite on te outside as sown in Fig. Tis is a tree dimensional potential well. 155

19 Te solution for a one-dimensional well can be extended quite easily to study te beavior of te electron in suc tree dimensional well. If we assume tat all te sides of te box are of te same lengt, te eigen functions are given by 1 8 ψ n = sin k 3 1xsin k y sin k3z L (1) were, n1π nπ n3π k1 = ; k = ;& k3 = L L L we can write tis energy in terms of n, n & n as follows 1 3 E n n 8mL = () were, n = n + n + n 1 3 conclusion is tat in tree dimension (i) Tere are tree integers n 1, n, and n 3 called quantum number wic are required to specify completely eac energy state. (ii) Te energy E now depends on tree quantum numbers. (iii) Several combinations of tree quantum numbers may give different wave functions, but of te same energy value. Suc states and energy levels are said to be degenerate. For example tree Independent states aving quantum numbers (1,1,), (1,,1) and (,1,1) ave te same energy E n1n n3 ( ) n n n = (3) 8mL E n1 nn3 6 = 8mL Tese levels are tree fold degenerate. Tat is te number of states tat ave te same energy is termed te degeneracy of tat energy level. Te second level E 11 is tus tree-fold degenerate. It must be noted tat states suc as (1,1,1), (,,)etc, are non degenerate Degeneracy It sould be noted tat te energy E depends only on te sum of te squares of n x, n y and n z. 156

20 Consequently tere will be in general several different wave functions aving te same energy. For example te tree independent stationary states aving quantum numbers (,1,1), (1,,1) and (1,1,) for n x,n y, n z ave te same energy value 6. Suc states and energy levels are said to be degenerate and te 8mL corresponding wave functions are ψ 11, ψ 11, & ψ 11. On oter and if tere is only one wave function corresponding to a certain energy, te state and te energy level are said to be non-degenerate. For example te ground state wit quantum numbers (1,1,1) as te energy 3 8mL and no oter state as tis energy. Te degeneracy breaks down on applying a magnetic field or electric field to te system. Electron Microscope: Electron microscope is like te optical microscope. Tis is an instrument primarily used for magnifying small objects to suc an extent tat teir minute parts may be observed and studied in detail. Wit te electron microscope, magnifications of 10 to 100 times tat of finest optical microscope make some of teir detailed structure, visible to te eye. Principle: A stream of electrons can be focused by suitable electric and magnetic fields and are possible troug te object. Tese electrons wic carries information s about te object. Te resolving power of on optical instrument like a microscope is directly proportional to te aperture of te lens used and inversely proportional to te wavelengt of te ligt used. So te smaller te wavelengt, te greater sall be te magnifying power. If we accelerate te electrons troug a potential difference of say 60,000 V, te velocity v attained by tem can be calculated from te relation, ev = (1/)mv 157

21 Were m is te mass of electron = 9x10-31 kg. calculating λ from te de-broglie wave equation, λ= /mv, te wavelengt comes out to be about 5x 10-1 m wic is smaller tan tat of te visible ligt. Various types of electron microscopes are 1. Transmission Electron Microscope(TEM). Scanning Electron Microscope (SEM) 3. Scanning Transmission Electron Microscope (STEM) Applications Te electron microscope as a very large magnifying power of te order of 50Å in size as been made visible. An electron microscope is put to valuable uses in all te fields. In medicine and Biology, it is used to study virus, te disease causing agent beyond reac of ordinary microscope. Te bacteria are sown in greater detail. So tat suitable means may be used for teir destruction. It as been used in te investigation of atomic structure and structure of crystals in details. Small particles forming colloids ave become open to study and analysis. It as also been used in te study of te structure of Textile fibers, purification of lubricating oils, compassion of paper and paints, surfaces of metals and plastics. Scanning Electron Microscope Principle: Wen te accelerated from te electron gun strike te sample, it produces secondary electrons. Tese secondary electrons are collected by an electron detector wic in turn gives a 3- dimensional image of te sample. 158

22 Description: It consists of an electron gun to produce ig energy electron beam. A set of magnetic conducting lenses are used to condense te electron beam and a scanning coil is arranged between magnetic condensing lens and te sample. Te sample is placed over a sample are collected by electron detector and are converted into electrical signals. Tese electrical signals can be fed into CRO troug video amplifier. Working: Te electrons produce by te electron gun called primary electrons are accelerated by te anode plate. Tese accelerated primary electrons are made to incident on te sample troug magnetic condensing lenses and scanning coil. Te primary electrons wile falling on te sample produces secondary electrons of lower energy. Te low energy secondary electrons are collected by a 159

23 collector and ten converted into electrical signals. Te weak electrical signals are amplified by te amplifier and are fed into CRO. Ten te wole image of te sample is obtained in te CRO screen. Advantages: 1. Specimen of large tickness can be examined.. It can be used to get 3- dimensional image of an object. Te magnification may be up to times greater ten te size of te object. Transmission Electron Microscope (TEM) Principle A stream of electrons wile passing troug te object carries information about te object is focused by electric and magnetic fields and ten studied. Description Te TEM consists of an electron gun made of tungsten filament. Wen eated, te filament emits electrons and are accelerated wit an energy of 100 kev. Tese electrons coming out of te electron gun are focused on te specimen by a set of condensing lenses. Te specimen is eld in a special older wic can be moved or tilted so tat te specimen can be studied at different angles. All te lenses used in transmission electron microscope are of magnetic type called magnetic lenses. Te electrons emerging out of electron gun passes troug te lenses and gets diverged or converged. Te filaments are extremely sensitive to oxidation, ence it must be protected wit vacuum. Te vacuum required in tis electron microscope is in te order of 10-6 torr. 160

24 At te bottom of te microscope, te fluorescent screen is provided over wic te unscattered electrons forms te image of te specimen. Working A stream of electrons emitted from te electron gun is made to pass troug te centre of te magnetic condensing lens(l1). Te electrons are made as parallel beam and are directed to fall on te object (AB) under study. Wile electrons passing troug te specimen, it is noted tat te electrons will be transmitted more troug te transparent part of te object and transmitted lesser troug te denser portions. 161

25 Transmission Electron Micrscope Te transmitted electron beam on falling over te magnetic objective lens (L), produces enlarged image of te recent screen (F) wic is kept at te bottom of te,microscope by means of tird electromagnet called projector lens (L3). Te image obtained on te screen can be potograped wit a suitable potograpic plate for permanent record. Clear focusing of te image is obtained by adjusting te intensity of te magnetic field. Advantages 1. Te magnification as ig as times as tat of te object can be produced.. Te focal lengt of te microscopic system can be varied. 16

26 Applications 1. It is used in te study of atomic structures.. It is used in te study of structure of textile fibers, material surface, composition of paints etc. 3. In medical field it is used to study te structure of virus, bacteria, etc. 163

27 Comparison between optical and electron microscope: S.No Optical microscope Electron microscope 1. It is made of ordinary lens system. It is made of magnetic system.. Magnification is 000 times as tat of te object. Magnification is more tan times as tat of te object. 3. It as a large aperture. It as a small aperture. 4. Focal lengt of an optical lens is fixed. Focal lengt of an electron lens can be canged. 5. It can give only two dimensional effects. It can give only tree dimensional effects No vacuum is required. Cost is low. Vacuum is required. Cost is ig. 8. Compact in size. Larger in size. 164

28 3.Quantum Pysics & Applications Objective Type Questions: 1. Te wavelengt of electron accelerated by Potential V is A. 1.0 B. 0 C. 1 D. 10. Heisenberg s uncertainty Principle rates A. Momentum & displacement B. Time & displacement C. Momentum & Time D. Energy & displacement 3. Wic of te following are true for a matter wave A. Larger te mass of particle, smaller te wavelengt B. smaller te mass of particle, smaller te wavelengt C. Larger te velocity of particle, Larger te wavelengt D. smaller te velocity of particle, Larger te wavelengt A. 1 & B. 1 & 4 C. & 3 D. 1 & 3 4. Wic of te following statements are true? 1. Any particle in motion exibit wave nature. Only sub atomic particles in motion are accompanied by matter waves 3. Matter wave travels wit a velocity less tan tat of ligt 4. Te product of te uncertainty in te Position of a particle at some instant and te uncertainty in its momentum is at best equal to / П A. 1 & B. & 3 C. 3 &4 D. 1 & 4 5. Te uncertainty position of an electron is 4 x m, ten te uncertainty in momentum is A x 10-5 Kgm/s B x 10-9 Kgm/s C x 10-4 Kgm/s D x 10-4 Kgm/s 6. Wic is te de-broglie wavelengt associated wit an electron if it is accelerated by a potential of V volts? A. λ= / (mev) B. ( (mev) /) σ C. λ= /pv D. 1/ (mev) 7. A quantum particle is confined to move in a dimensional potential box of dimension L wen te particle is in te ground state, te probability of finding te particle is maximum at a. x=0 b. x= α c. x= L/ d. X= L 8. Te wavelengt associated wit a particle at rest is a. 0 b. sligtly greater tan 1 c. α d. less tan 165