“Black body radiat Peduction of Wiens ~ Planck’s then Geeee Jeans’ Law from P| displacement law ene) e and experimental a theory — Connie’ oe ie waves ~ GP Thomann ect on ~ Properties of tice equation - Time inde, ee - Schrédinger’s ae i . pent ets coatings = Phvtica sigutcutoebon srtacSeci . Scanning Sie mona box - Meats nishoone microscope. croscope - Transmission electron Introduction The classical mechanics is not adequate to explain the motion of atomic particles like electrons protons etc. Many examples for the failure of classical mechanics are (i) black body radiation (ii) specific heat of solids at low. temperature (iii) theory of atomic structure (iv) photo - electric effect and (v) Compton effect. To explain the failures of classical mechanics, the quantum theory was developed. 4a) BLACK BODY RADIATION Black Body In practice, a perfe showing close approximation to a perfect black body c ct black body is not available. The body an be constructed. A hollow copper spherical shell is coated with lamp black on its inner surface. In this, a fine hole is made and a pointed projection is provided just in front of the hole. (Fig. 4.1(a)). enter into this spherical shell ffer multiple reflections this body acts as an When the heat radiations through the hole, the heat radiations su and they are completely absorbed. Now, absorber.Engineering Physics - | 4.2 When this body is placed in a constant aan path at high temperature, the heat radiations ee It is to hole (Fig. 4.1(b)). Now, this hole acts as a of the body acts be noted that only the hole and not the walls as the radiator. Pointed projection 7 Heat Hole (a) Absorber _ (b) Radiator Fig. 4.1 Black body Perfect black body A perfect black body is one which absorbs all the heat radiations (all the wavelengths) incident on it, Further, when such a body is placed at constant high temperature, it emits radiation of all the wavelengths. Black body radiation The heat radiation emitted from a black body is known as black body radiation. The wavelength at which the maximum energy of radiation emitted depends only on temperature of the black body and it does not depend on the nature of the material. Any object coated with a dull black pigment is a good approximation to a perfect black body.quantum Physics f SS Laws of black body rq diatio, ee cy . ion Wien’s displacement 1 law This law State, S th, corresponding ty 44. “Mt the product of wavelength (4) ‘m absolute temperature of energy of radiati d the black lation an . body (T) is @ constant. je mT = Constant Ay = Sastant It is found that the : Wavelength corr i _ esponding to the maximum energy of black body Tadiation ig inversly proportional to absolute temperature. ee As the temperature of black body ; on corresponding to ody increases, the maximum energy decreases, Wien’s radiation law Wien deduced the law for the energy emitted by a black body at a given wavelength (A) and temperature (T) to explain the blackbody spectrum. It is known as Wien’s radiation law. The energy density in the wavelength range 4 and i+d) is given by 8nhe 1 8nhe 1 [ : 5 = sh E. “hvikT ~ 5 hele - 5 hvikT 3 helMRF we” Ne -5 ~he/MkT E, = 8nhch e -5 -C,/aT Le \ 1 nts where C, and C, are constaEngineerin| Physics - | 44 : body. T is the temperature of the black values, The constants C, and Cp have the he C, = 8nhe and C,= Limitation length, This law holds good only for short wavelengths and not for longer wavelengths. Rayleigh - Jean’s law This law states that the energy distribution of a black body is directly proportional to the absolute temperature (T) and inversely proportional to the fourth power of the wavelength (i). ie., E,« T i uy 4 where k is Boltzmann’s constant. Limitation This law holds good only for longer wavelength regions and not for shorter wavelengths. It is found that both Wien’s and Rayleigh - Jeans laws do not agree with the experimental results for entire wavelength range. Therefore, it is concluded that classical theory failed to explain the emission of black body radiation.quantum Physics —_ ———— AS Thus, Max - Planck fe oo jaws of the black body radiation ‘d quantum theory to explain TY of black body radiation revoluti, ; tion was introd. ‘Planck hypothesis’ of black body radia ned by Max Planck in the year 1900. js theory successgf, : es ey oplain ee laws of black body yadiation. (42) PLANCK’S THEORY filled up with the radiations a large number of tiny oscillators. They are of atomic dimensions, Hence, they are known as atomic Oscillators or Planck’s oscillators. Each of these oscillators jis vibrating with a characteristic frequency, 2. The frequency of radiation emitted by oscillator is same as that of oscillator frequency. 3. The oscillator cannot absorb or emit energy in a continuous manner. It can absorb or emit energy in multiples of small units called quantum. This quantum of radiation is called photon. The energy of the photon (€) is directly proportional to the frequency of radiation (V), Le., Eocy vhere h is a proportionality constant. It is known as Planck’s constant. 4. The oscillator vibrating with frequency v ae only om energy in quantum of values hv. It indicates : a e the oscillators vibrating with frequency v can only hav discrete energy values £,,.En ineerin Physics - | ing Pr = It is given by where n is a positive integer 1, 2, oo stems It means that the energy of the a quantized and integer n is known number, Planck’s law of Radiation Statement The energy density of heat radiation emitted from a black body at temperature T in the wavelength range from ). to A+dd is given by Here, h - Planck's constant ¢ — Speed of the light V ~ Frequency of radiation k — Boltzmann’s constant T ~ Temperature of the blackbody PLANCK’S LAW OF RADIATION (Derivation) Consider a black body with a large number of atomic oscillators, Average energy E per oscillator ig given by E- . (2) 2|byto all the oscillators N-1 ‘otal number of oscillators Number of atom: in ground ae °scillators 0 xwell’; Ae ji of oscillators with a energy distribution law, the number £, 18 given by N= Nye c/kT oe where T — Absolute temperature of the black body k- Boltzmann’s constant. If N is total num] ber of oscillators and No Nj, Ng, «+, are the number of oscillators with energies ¢,,¢., ¢. then 0» Epp Ege N=N, +N, + \ From the eqn. (2), we have N=N, oN eT ay, et a) From Planck’s quantum theory, ¢ can take only a quantum of values Av. Therefore, the possible values of € are 0, lAv, 2hv, 3Av, ete., (Fig. 4.2) ie., €,=nhv, n=0,1,2 E p= 9, &=hv, &,=2hv Substituting these values in eqn (3), we have 0 = hy/kT ~ Qhv/kT N=Nye + Noe + Noe _AVIAT ~ 2hv/kT w (4) tae N=N,+Noe + Noej= lo ENERGY 4 Fig. 4.2 Energy diagram for a Cee N= N= N= N= Total energy given by E Substituting we have Engineering Physies — | ee = Number of : oscillator n=5 Ng ne s oscillator of frequency V ~ hy/kT put x=e N/K we get 3 = Ny + Nyx + Nox + No¥ +-- G) 2 Nj Wb+xtx ted 1 -» (6) | a =| 1 = ooo = (leap = Lex gets... 1% by using binomial series. No (1-x) of the black body due to all the oscillators is = &)N, + €,N, + _ AT) for €), &, &, and 7 -.)hUre ()),49 quantum Physics E = 0xN, + hy Noe AVIA 2hv Noe Bhv/kT E = AvNge. PUT So hy Nel BVT (8) put =o ET. have E = AvNox + 2hvNox' +... @) E = AVNG Ix + 2x74...) E = AVNgx [1+2x+...] B= Ava] 1 | = (10) (1 -x) = (l-xy 2 = 14+2x+... (1-x)? by using binomial series. Substituting eqns (6) and (10) in eqn (1), we get hv Nox —j x) B= +7 No (1 -x) i- Av Nox x (1-x) (1-2) No ea hAvx B a-® = hv x iD ti | x z hyengineering Physics - | 4.10 wwikT ve have On substituting x = @ , w. (12) . h Number of oscillators per unit volume in the wavelengt range A and A+d)h is given by 8n dh -. (18) 3 The energy density of radiation between wavelengths AandA+dd is given by Number of oscillators E ay =| Per unit volume Average energy a” ~ | in wavelength range per oscillator A and 2 + da 8ndd Ay (14) E, dA = —— x —_— a » if give 8n da Acld eye a) +» (15) - (16)quantum Physics This eqn (16) ,, eal epres of wavelength. Sents Planck's radiation law in terms Note: Planck’s law is also @. ae x i gubstituting A = Pressed in terms of frequency by C/V and d), = 2 ey in eqn (15) Then, we have = region which is confirmed by deduction of Wien’s law and Rayleigh’s Jean law from Planck's law. 4) DEDUCTION OF WIEN’S DISPLACEMENT LAW FROM PLANCk’s LAW We know that Wien’s law holds good only at shorter wavelength. Therefore, when 4 is very small, v is very large, hence Ay AVIAT i >> land e is large when compared to 1. Thus, ‘1’ is neglected in the denominator of eqn (16) Av/AT hv/RT je., @ -1=e : Hence, the eqn (16) reduces to 8 he . (18) = 75 Aviat e This eqn (18) represents Wien’s displacement law. Thus, Planck’s law reduces to Wien’s law at shorter wavelengths.4.12 Engineering Physics ~ | : "= LAW 45) DeDuction OF RAYLEIGH - JEAN'S | FROM PLANCK'S LAW d only at We know that Rayleigh - Jean's law holds oy is very longer Wavelength. Therefore, when ) is very large, h small, and on <<1 hv/kT ly 1+ op (by using exponential series and neglecting higher orders) Now, eqn (16) reduces to & " oo a = = —i <19° " ~ ut w (19) This eqn (19) represents Rayleigh - Jeans law. Hence, Planck’s law reduces to Rayleigh - Jean’s law at longer wavelength. Thus, Planck’s law reduces to Wien’s law of radiation at smaller wavelengths and to Rayleigh - Jeans law at longer wavelengths.MATTER WAVES Wave Nature of Particle The light radiation behaves like a wave in interference and diffraction experiments. The same light radiation behaves like a particle in photoelectric effect and Compton effect. Thus, the light radiation has dual (two) nature ie., wave nature and particle nature. The idea of wave nature of the particle was put forward from the observation of dual nature of light radiation. de - Broglie’s Hypothesis Louis de-Broglie proposed a very bold and novel suggestion that like light radiation, matter or material particle also posseses dual (two) characteristics i.e, particle -like and wave - like since nature loves symmetry. The moving particles of matter such as electrons, protons, neutrons, atoms or molecules exhibit the wave nature in addition to particle nature. According to de - Broglie hypothesis, a moving particle is - always associated with waves. (Fig. 4.7) | a4 (a) Particle nature Fig. 4.7 Particle and wave nature 0! (b) Wave nature f matter particlePROPERTIES OF MATTER WAVES I If the mass of the particle is smaller, then the wavelength associated with that particle is longer. . If the velocity of the particle is small, then the wavelength associated with that particle is longer. If v=0, then =~, ie, the wave becomes indeterminate and if v=~, then 4=0. This indicates that de - Broglie waves are generated by the motion of particles. These waves do not depend on the charge of the particles. This shows that these waves are not electromagnetic waves. The velocity of de - Broglie’s waves is not constant since it depends on the velocity of the material particle.Engineering Phy (er TION 4.11] SCHROEDINGER WAVE EQUA describes the way, . uation Schroedinger wave ed 1 form. It is the basic nature of a particle in mathematical equation of motion for matter waves. then there should be has wave properties, If the particle has 1 of that some sort of wave equation to describe the behaviow: particle. Schroedinger connected the expression of de-Broglie's wavelength with the classical wave equation for a moving particle. He obtained a new wave equation. This wave equation is known as Schroedinger wave equation. Forms of Schroedinger wave equations There are two forms of Schroedinger wave equations. They are (a) Time independent wave equation (b) Time dependent wave equation 23 SCHROEDINGER TIME INDEPENDENT WAVE EQUATION (Derivation) Consider a wave associated with a moving particle. Let x, y, 2 be the coordinates of the particle and y wave function for de - Broglie’s waves at any given instant of time ¢, (Fig 4.10)Quantum Physics 4.35 Fig 4.10 The classical differential equation for wave motion is given by dy , fy wy _ ay 52 +e t+ tao ) ax dy dz v ot Here, v is wave velocity. The eqn (1) is written as 2 2 ldy Vv=s—a we (2) vat 2 2 2 a Oc : where V = i + — z + —z_ is a Laplacian’s operator. Ix dy Iz The solution of eqn (2) gives y as a periodic variations in terms of time ¢, iwt WO, 2, 0) =W, (GY ze yy ena . (8) oO Here, wy, (x,y,z) is a function of x,y,z only, which is the amplitude at the point considered. w is angular velocity of the Wave,Engineering Physicg i ae i to t, we get Differentiating the eqn (3) with respect oy _ : fied 2 = " T & ats & « ca Q le ul “i 2 ° é ay 2 w (4) -iwt [v@=-1 vewe | Substituting eqn (4) in eqn (2), we have 2 vw+2 yo - (5) Vv We know that angular frequency « = 2ny = ats Here, v is the frequency [= v= i] @ Qn re w (6) Squaring the eqn (6) on both sides, we get 2 _ an wn (1) 2 2Quantum Physics 4.37 Substituting eqn (7) in eqn (5), we have 2 4n a (8) oe h on substituting, A = i in eqn (8), we get an Vy +—yv=0 22 mv v anim" 9 wt a. a» (9) If E is total energy of the particle, V is potential energy and 5m is kinetic energy, then Total energy = Potential energy + Kinetic energy 12 E = Vtgm 2(E - V) = mv 2 mv = 2(E-V) Multiplying by m on both sides, we have — ie my? = 2m(E-V) Substituting eqn (10) in eqn (9), we get 2 4 Wy + oy x amE-V)v = 0ineering Phys; 4.38 i Sig The eqn (11) is known as Schroedinger tin, independent wave equation. h. Let us now introduce # = 5, eqn (11), P - an ee » (12) “-73o° 42 20 4n where i is a reduced Planck’s constant The eqn (11) is modified by substituting 7, vy +7; G-Yy=0 A 8n° Vy + are =0 axon 2 2m Vv t 77 GY 0 wa (13) dn On substituting eqn (12) in eqn (13), Schroedinger time-independent wave equation is written as w. (14) In eqn (14), there is no term representing time. That is why it is called as time independent equation. Note:juantum Physics Qi y' 4.39 Special case If we consider one-dimens; . ional motion ie., particle moving along only X - direction, then Schroedinger time independent equation (14) reduces to + (15) (413) SCHROEDINGER TIME DEPENDENT WAVE EQUATION Schroedinger time dependent wave equation is derived from Schroedinger time independent wave equation. The solution of classical differential equation of wave motion is given by iwt W195 2,8) = Woy, ze » Differentiating eqn (1) with respect to time t, we get oy . -iwt w» (2) Fy = EWE oy : -iot a ay = EAM) Wye (2° @ = 2m) a = -2nivy w (3) ree) ® _ _oniBy oBa hove; at h h ay EYE a-+] ae REY 2n |1 Engineering Ph Sicg 4.40 (4 ov y ot ‘ have Multiplying i on both sides 10 eqn (4), we (E __fl=lu i% . ninil | ‘(a jw _ zy te ixiel =~] oO » 6, Meg ) hae iM Schroedinger time independent wave equation is given by .. (6) Vy + 2m @E-v)y =0 h 2 2m =0 Vy t+ pew = On substituting Ey from eqn (5) in eqn (6), we get 2 2m|.. oy 2m as =0 Vt ud at vy «7 . (8)Quantum Physics 4.41 2 2m he 2 where H -( 37 V+ v| is Hamiltonian operator 10 E-ih az 8 energy operator. The eqn (8) is known as Schroedinger time dependent Ff im Note: In eqn (7), there is the term fe That is why it is called time dependent wave equation. representing time. oy ot PHYSICAL SIGNIFICANCE OF WAVE FUNCTION wy The variable quantity which describes de-Broglie wave is called wave function w. It connects the particle nature and its associated wave nature statistically. The wave function associated with a moving particle at a particular instant of time and at a particular point in space is related to the probability of finding the particle at that instant and at that point. . The probability 0 corresponds to the certainty of not finding the particle and probability 1 corresponds to certainty of finding the particle. ie., Sif yv ydt=1, if particle is present. = 0, if particle is not present. where y — complex conjugate of w rticle at a particular ili f finding a pa : . The probability of 1 ig eeeteare function region must be real and positive, y is in general a complex quantity.Engineering Physics | 4.42 ) BOX (4.15) PARTICLE IN A ONE-DIMENSIONAL en two rigiq ing betwet Consider a particle of mass ™ moving : walls of a box at x=0 and x=a along x-axis. This particle is bouning back and forth inane of the box. The potential energy (V) of the aa fa box is constant. It is taken as zero for simplicity (fig. 4.4). The walls are infinitely high. The potential energy _. the particle is infinite outside the walls. Thus, the potential function is given by 0 for 0O< x x2a This potential function is known as square well potential. (fig. 4.11) Fig. 4.11 Particle in a boxQuantum Physics eae The particle can not come out of the box. Also, it can not exist on the walls of the box. So, its wave function w is 0 for x < 0 and x > a. Now, task is to find the value y within the box i.e., between x=0 and x=a. Schroedinger’s wave equation in one-dimension is given by 2 dw 2m pao Pa OM =0 dx - (1) Since V=0 between the walls, the eqn (1) reduces to dy | ImE mE GW , ImE (2) de oe 2mE put z = KR in eqn (2), we get 2 d . w (3) oY ey =0 dx The general solution of eqn (3) is given by w(x) = Asinkx + Beoskx ws (4) Here, A and B are two unknown constants. The values of the constants A and B are determined by applying the boundary conditions. Boundary condition (i) y = 0 atx =0 Applying this condition to eqn (4), we have : “.' sind = 0 =Asin0+Bcos0 [ cos 0 nou a = 0 =0+Bxl Hence, B= 0Engineering Physicg _ 4.44 ii =a Boundary condition (ii) y = 9 at * Applying this condition to eqn (4), we have 0 = Asinka + 0 [. B=o) Asin ka = 0 It is found that either A=0 or sinka = 0 A can not be ‘0’ since already one of the constants B is 0. If A is also 0, then the wave function is zero even in between walls of the box. Hence, A should not be zero. sinka = 0 sin ka is ‘0’ when ka takes the value of nm je., ka = nn where n is positive integer 1, 2, 3... p= tm = (5) - a On squaring eqn (5), we have ae .. (6) 2 nu k= a We know that 4° = 2% _ 2mE [8-8] h h Qn 4n 2 (2mx4n )E k= z 2 2 8nmE 1G) k= Ah Equating eqn (6) and eqn (7), we haveQuantum Physics 4.45 2 2 nw 80 mE . nw (8) + (9) Here n=1, 2,3... For each value of n, there is an energy level. The particle in a box cannot possess any arbitrary amount of energy. It can only have discrete energy values specified by eqn (8). In otherwords, its energy is quantised. Each value of E, is known as eigen value and the corresponding y,, is called as eigen function. Normalisation of wave function The constant A is determined by normalisation of wave function as follows. Probability density is given by y yw . Ax We know that y, (x) = Asin — nnux * _ N1x . Ww Wwe Asin—— x Asin—— Ps yv when the wave function is real (not complex)vs NV The second term of the integral becomes 2 ov" Engineering Ph, Rigg we g{ 2% .. + _ gsi al Qy jg some where inside tig te particle inside the i It is certain that the par” box. Thus, the probability of findin8 of length a is given by f vydee! Gy 0 . . we have Substituting wy from eqn (10) in eqn a, a J A’ gin? 22 ay = 1 a 0 _ 2nnx 2 J _ 2 —] dx =1|*.*sin'@ = SS 2 0 @ a g nan ~ ax / ad 1 cos 22 -1 oe aad 3 0 \ 0 \ a 2ntec alia 1{s— A [3] 2) Onn ae 0 —_ e 0 zero at both limitsQuantum Physics 4.47 2 Thus, os 22 2 a +» (12) On substituting eqn (12) in eqn (9), we have The eigen function (y,) belongs to eigen values E, and it is expressed as ++ (18) This expression (13) is known as normalised eigen function. The energy E,, and normalised wave functions W,, are shown in fig. 4.12. Fig. 4.12 Energy levels and wave functions Special cases From eqns (8) and (13), the following cases can be taken and they explain the motion of electron in one dimensional box.INeerin ae Ph Sigg Case (i) : For n=! 2 h Ey=> 2 “l gma c TH sin =| vy, = Va H V, @) is maximum at exactly middle of the bo, a lence, y, (x shown in fig. 4,12. Case (ii) : For n=2 2 h g, =) = 8ma Hence, y, (x) is maximum at quarter distance from either sides of the box as shown in 4,12. 4/2. (3nx Wo (x) = z sl . Hence, Wy () is maximum at exactly middle and one-sixth distance from either sides of the box as shown in fig. 4.12.