Module 3: Quantum Mechanics PDF

# Module 3: Quantum Mechanics ## Black Body The term black body was introduced by Kirchhoff in 1860. A black body is a perfect absorber and radiator. A perfect black body does not exist in nature. Fery has devised a black body which approximates very closely to the properties of a perfect black...

# Module 3: Quantum Mechanics ## Black Body The term black body was introduced by Kirchhoff in 1860. A black body is a perfect absorber and radiator. A perfect black body does not exist in nature. Fery has devised a black body which approximates very closely to the properties of a perfect black body. - (a) Black body as an absorber - (b) Black body as an emitter Fery's black body consists of a metallic cavity in the form of a double walled hollow copper sphere with a lamp black coating from inside and nickel polished from outside. The spherical enclosure has a narrow conical hole O. The inner space between the walls of the spherical enclosure is evacuated to prevent loss of heat energy by conduction and convection. To avoid any chance of radiation getting out, an inward conical projection P is made. At each radiation falling on the hole enter the enclosure and suffers multiple reflections at the interior surface before it escaped, in this way it behaves like a perfect absorber. When the spherical enclosure is heated at constant temperature, the radiation coming out, called black body radiation, the device behaves like a perfect radiator. ## Black Body Radiation Spectrum During the years 1893 to 1897, Lummer and Pringsheim carried out a series of experiments on the energy distribution in the spectrum of black body radiation. In the experiment, radiation was produced by electrically heating a hollow enclosure with a small opening, which acts as a black body. The temperature of the black body was measured with the help of a thermocouple. The experimental arrangement is shown in the figure below: The radiation from the black body O falls on a narrow slit S by means of a concave mirror M1. The slit S is in the focal plane of another concave mirror M2 which reflects the radiation was dispersed into a spectrum by a fluorspar prism P (it avoids absorption of radiation) placed on a turn table of a spectrometer. The spectrum was focussed onto a Lummer-Kurlbaum linear bolometer (radiation detector) by means of another concave mirror M3. On slightly rotating the mirror M3 about a vertical axis, the radiation of different wavelengths fall on the bolometer one after the other, which is connected to a sensitive galvanometer G, which gives deflection in proportion to the intensity of each line. of the spectrum. This experiment was repeated by for different emission temperatures of the black body. Their results are represented graphically, where the amount of energy emitted in a narrow wavelength is plotted against wavelength of each range. From the curve, it is seen that: 1. At any one temperature the energy is not uniformly distributed, the intensity of radiation initially increases with wavelength, reaches a maximum value and finally decreases with the further increase of wavelength. 2. The total energy of radiation at a given temperature is represented by the area under the curve corresponding to that temperature. 3. It is found that the area under the curve is directly proportional to the fourth power of the absolute temperature of the black body, i.e. E α T^4. Stefan's Law is verified. 4. It is also observed that the wavelength corresponding to the maximum energy λm, shifts to shorter wavelength side as the temperature of the body increases. Therefore, λm*T = constant, Wein's displacement law is verified. As Wein's formula for energy desbubution is in good agreement with experimental curves for short wavelength only. Rayleigh Jeans formula could be made to fit the curves in the region of long wavelength. Therefore, none of these theoriticised formulae could not explain radiation curve. ## Planck’s Quantum Hypothesis In 1900, Max Planck proposed the radiation law by using following assumptions and hypothesis: 1. The cavity of an experimental black body also contains electrical linear oscillators of molecular dimensions which can vibrate with all possible frequencies. The frequency of radiation emitted by an oscillator is the same as the frequency of its vibration. 2. The linear oscillator cannot emit energy in a continuous manner, but in the multiple of a small und called quanta (photon). If an oscillator is vibrating with a frequency v. it can radiate in quanta of magnitude hv. i e. the linear oscillator can have only discrete energy values En given by > En=nhv. Where n=0,1,2,3,4,--- > n→ integer > h→ planck constant. 3. The oscillators do not emit or absorb radiation energy continuously but only in a certain multiples of packets of hv. This implies that the exchange of energy between radiation and matter are limited to discrete set of values 0, hv, 2hv, 3hv,---nhv. ## Average Energy Of Planck’s Oscillator If N be the total number of oscillators & E their respective total energies > E = ΣE/N Let No, N1, N2, Nr, -- be the number of oscillators having energies 0, hv, 2hv, --Kythu The no. of oscillators having rhy will be. > Nr = No e-rholky. And the total number of oscillators is > N= No +N, N₂+---- > = Not Noe-holkt +Noe-zhulrst--. > = No [ite-holkt + e-aвиткт темникт+--] > N = No (1-ehulkt) > N= No/(1-e-hulkt) Total energy of all oscillators > E = No XO +N, Xhu+N2x2hv+---NrXrhv+-- > = hv No e-hulkt tahu Noe-shulkT+ > E = Noe-hulkt hv [I+ 2e-holkT+3e2holkT+] > = No e-holkt (etholky2] (1+2+3x²+-5] > E= No e-hr IkrUIKT) 2 Substituting the values of N and E from eq'in (2) & (4) in Equation (1). > E = (e-hylkt hv) / (1-e-hvikt) = (hv) / (eholkt-1) This is the expression for the average energy of a Planck's Oscillator. ## Planck’s Radiation Formula The number of oscillators per unit volume lying in the frequency range v and v+dv, as obtained by Rayleigh & Jean's is 8πv^2dv/c^3, c → velocity of light. Therefore, the energy density of radiation Uv, in the frequency range v and v+dv is > Uodv = (No of oscillator/volume) x average energy > = 8πv^2dv/c^3 * (hv) / (eholkt-1) > Uvdv = 8πh v^3dv / c^3 (ehulkt-1) Planck's radiation formula in terms of frequency The above formulae explains the entire curve of black body radiation, It can be verified as below: At high frequencies hv>>KT & e holkt→0, which mean that UD/vdvo → No more ultra catastrophe. At low frequencies, where Rayleigh-Jeans formula is good approximation to the data. hv << KT & hv/kTE! > ex = 1+x++--; If f x us small ex≈ ex~1+x. 1the > for holky CCI > hv & kt. > ehulkt -I > ≈ KT > 1+hu-1 / KT > by > At low frequencies Planck's formula becomes > Uvdv & v^3 (KT) do ~ 8AKT v^2dv ## Wave Particle Duality A particle means an object with a definite position in space which cannot be simultaneously occupied by another particle and identifiable by their distinct properties such as mass, momentum, Kinetic energy, spin and electric charge. On the other hand, a wave means a periodically repeated pattern in space which is specified by its wavelength, frequency, amplitude of the disturbances, intensity, Energy and momentim. A wave is spread out or delocalised in space. Two or more waves can coexist in the same region and even superimpose to form resultant wave. ## De-Broglie's Concept of Matter Waves  Upto 1924 Physicists were of the view that only light or Electomagnetic radiation possesses dual character, sometimes they behane as a wave and at the other as a particle. The matter was considered Exclusively corpuscular in nature. In 1924, Louis de- Broglie proposing that wave-particle duality may not be a monopoly of radition, but a universal character of nature. He suggested a new idea of the Existance of matter waves on theoretical ground. He suggested that matter also possess dual Character as light. Like Electromagnetic radiation material particles such as Electrons, protons, and neutrons also show wave - like character. According to de-Broglie concept, a moving particle always has a wave associated with it and the motion of the particle is guided by that wave in similar manner as photon is controlled by a wave. According to de-Broglie, the wavelength of a wave associated with moving particle depend upon the mass of the particle and its velocity. The wavelength of matter wave is given by > λ =h/mv = h/p h is planck's constant and p the momentem of material particle. ## De - Broglie wavelength of photon A light wave of frequency v is associated with a photon of energy E, given by Plank's relation as > E = hv According to Einstein's theory of relativity > E=mc^2 Comparing above two equaticons, we get > hν = mc^2 for photon If p is the momentum of photon, then p = mc. > hν = pc > hν = bλλ (:: c= λν) > λ = h/p This is the de-Broglie wavelength for photons. de-Broglie furthur suggested that the same Expression also be used for the wavelength of material particles or matter waves. > λ = h/mv if E is the kinetic energy of the material particle then, > E = (1/2)mν^2 = p^2/2m > p = √2mE Therefore, The de-Broglie wavelength λ associated with a particle of mass m and kinetic Energy E may also be written as > λ = h/√2mE If a charged particle carrying charge q is accelerated through a potential difference of Violts, then the wavelength of the wave associated with this moving charged particle is obtained as follows: The Kinetic energy of the charged particle E = qV De - Broglie wavelength > λ = h/√2mqV For a material particles, like neutrons, which possess Maxwellian distribution of velocities at absolute temp T in thermal Equilibriom, the ang. kinetic energy is given by > E = 3KT K is Boltzmann's Constant. The wavelength of a wave associated with such a particle is given by > λ = h/√3mKT Equation (2) to (5) represent de-Broglie's wavelength of a wave associated with moving particles ## De-Broglie wavelength of Electron wave Suppose a stream of Electrons of rest mass mo and charge e be accelerated from rest to a potential difference V. The Rinetic Energy or velocity of these electron (in non-relativistic case) is given by > (1/2)mo*ν^2 = eV >>> ν = √2eV/mo The wavelength of de-Broglie wave associated with electron is given by > λ = h/mv = h/ √2meV (m=mo for nonrelativistic case) Substituting value of ν from eq (1) > λ = h/(√2meV * √mo) = h/√2meV This is the requised expression for wavelength of de-Broglie wave associated with an Electron in the non-relativistic cases. substituting value of h = 6.6 * 10^-34 J-sec. , mo = 9.1 * 10^-31 kg and e = 1.6*10^-19 Coulomb we get > λ = 6.625*10^-34 / √2*91*10-31 * 1.6*10^-19 V = 12.29 Ao ## Properties of matter waves - Matter waves are associated with only moving material particle. - Particle nature and wave nature never appear simultaneously. - Matter waves moves with the speed of moving material particle. - They are generated from charged as well as neutral particles. ## Difference between matter waves & EM waves - E.M waves travel with speed of light, matter wave travel with ve less than the velocity of light. - E.M waves travel without medium but matter wave required medium. - EM waves generated from charge particle but matter waves generated from charged and material particle. ## Wave function The quantity whose Variations builds up matter waves is called wave function (Ψ). If a particle is moving there should be some wave equation to describe the motion of particle. Schrödinger derive the equation for this which is known as fundamental Equation for wave mechanics. ## Born interpretation wavefunction Ψ The quantity whose Variation builds up matter waves is called wavefunction (Ψ). The value of wave function associated with a moving particle at particular point (x, y, z) in space at the time 't', introduced by Schrödinger asg > where Ψ(x,t) = Ψ0(r) sinwt > 42 => Particle density > As in EM wave Ψ = A sin ut > where A^2 Intensity of wave > ie A^2(nv); n is number of photon per unit Volume. > A represent photon density. But this Explanation fails to explain the flight of a single material particle where 42 changes but particle density remains the same. To overcome this discrepancy Max Born, proposed a new idea. According to him; 4^2 does not measure the particle density at any point but the |Ψ|^2 is called probability density and Ψ is probability amplitude. The probability of finding the particle at a given point is proportional to |Ψ|^2 at that point and probability of finding the particle within an element of volume d5= dx dy dz is |Ψ|^2d5. Since the particle is somewhere in space the integral over the whole space must be unity, that is > ∫∞ -∞ |Ψ|^2 d5 = 1 A wavefunction Ψ satisfying the above relation is Called normalised wave function. Every acceptable wavefunction must be normalisable. An acceptable state function Ψ must fully the following Conditions; - Normalised wave function must be single-Valued: > If state function Ψ has more than one value at any point, it would mean an infinitely large probability of finding the particle at that point. - It must be finite everywhere: > If it is infinite for a particular point, it would to mean an infinitely large probability of finding the particle at that point. - Ψ must be continuous throughout the entire space and have a continuous first derivative: > This condition arise from Schrödinger equation, which shows that the second derivative d^2Ψ/dx^2 must be finite every where. ## Schrödinger Time-Independent wave equation Schrödinger Time-independent wave Equation is a second order differential Equation which is applicable when force acting upon the particle does not depend upon time but vary with the position of a particle. General wave Equation, for a particle, > ∂^2Ψ/∂x^2 = (1/ν^2)* ∂^2Ψ/∂t^2 Let the solution of eq(1) can be > Ψ(x,t) = Ψ0 e^i(Rx-wt) Differentiating Eq (2) wirit time twice > ∂Ψ/∂t = iω Ψ0 e^i(Rx-wt) > ∂^2Ψ/∂t^2 = -iω^2 Ψ0 e^i(Rx-wt) But value of (3) in equation (1) > ∂^2Ψ/∂x^2 + (ω^2/ν^2) Ψ = 0 where ν is the velocity of wave. We know that ω^2 = (2πν)^2 = (2π/λ) = ω^2 = (2π/h) = 2π * ν = 2πν/ λ > ν = (ω λ )/2π = (ω λ )/2π = (1/4π^2) * (ωλ/h)^2 = w^2/4π^2ν^2 If E and V are the total and botential Energies of particle respectively, then its kinetic Energy T is given by > T = E-V In this case, the potential Energy is independent of time. > T^2 = (p^2)/2m > P^2 = 2m (E-V) > P^2 = 4π^2 * 2m (E-V) > P^2 = 8π^2 * m (E-V) > ω^2 = (8π^2 * m (E-V))/h^2 > ω^2 = (2m (E-V))/h^2 But in eq (4) > ∂^2Ψ/∂x^2 + 2m(E-V)/h^2 Ψ = 0 This is Schrödinger's time-independent (Steady State) wave equation for a particle waves. For a foree particle V=0, put it eq (4), we get Schrödinger time independent equation for a free particle as > ∂^2Ψ/∂x^2 + 2mE/h^2 Ψ = 0 ## Schrödinger Time-dependent wave equation Let general wave Equation be > ∂^2Ψ/∂x^2 = (1/ν^2)* ∂^2Ψ/∂t^2 Let the solution of eq(1) > Ψ(x,t) = Ψ0 e^(Rx-wt) Differentiate equation (2) wrt time > ∂Ψ/∂t = -iω Ψ0 e^(Rx-wt) > ∂^2Ψ/∂t^2 = - ίω Ψ > ∂^2Ψ/∂t^2 = > (E=ħw) From Schrödinger time independent wave equation- > ∂^2Ψ/∂x^2 + 2m (E-V) Ψ = 0 > ∂^2Ψ/∂x^2 = -2m (E-V) Ψ > -2m ∂^2Ψ/∂x^2 = 2m (E-V) Ψ > -h^2 * ∂^2Ψ/2m ∂x^2 = (E-V) Ψ > -h^2 * ∂^2Ψ/2m ∂x^2 + VΨ = EΨ > -h^2 * ∂^2Ψ/2m ∂x^2 + VΨ = iħ * ∂Ψ/∂t This is Expression for Schrödinger time-dependent equation. ## Particle in one - Dimensional Bon or infinite square well (Infinite Potential well) Let us consider a case in which a particle is trapped in a rectangular one dimensional potential box. Particte is bound to move along the x-axis between x = o to x=L. Suppose that the potential energy of the particle is zero inside the box, but becomes infinite at the walls and outside, that is > V= o for 0<x<L > V= ∞ for O<x and x>L Under these conditions, particle is said to move in an infinitely deep botential wall The Schrodinger equation for the particle within the box (V=0) is > d^2Ψ/d(x)^2 + 2ME/h^2 Ψ = 0 Let > 2ME/h^2 = k^2 > d^2Ψ/d(x)^2 + K^2Ψ = 0 The general solution of differential Equation (3) is > Ψ(x) = AsinKx + B CosKx Where A & B are Constants The value of A & B are determined by applying boundary conditions. Abblying the boundary Condition, Ψ= 0 at x = 0 in eq(4) we get > 0 = Asin0 + B Cos 0 or B = 0 By applying second boundary condition, that is Ψ= 0 at x=L to Equation (4) , we get > AsinKL = 0 > K = nπ/L or KL = nπT > K= nπ/L where n = 1, 2, 3 ---- n=0 is not possible because it gives Ψ = 0 Everywhere, it means that particle is not Existing in the box. Substituting the value of K from Eq (6) in eq(2) we get > n^2 * h^2 / 2m = 2m E / h^2 > En = n^2 * h^2 / 8mL^2 where n=1,2,3---- It means particle can not have an arbitrary Energy, but Can have only certain discrete Energy for n=1,2,3----. Each permitted Energy is called Eigen Value of the particle. The wave function Ψ corresponding to each eigen value are called Eigen functions. Jo To find the value of eigenfunction of the particle, but B= 0 and K = nπ/L, we find that permitted Solutions of Schrödinger's wave equation (1) are > Ψn (x) = A sin (nπx/L) To find the value of A, we apply normalisation condition > ∫∞ -∞ |Ψn(x)|^2 dx = 1 > ∫∞ -∞ |A sin (nπx/L)|^2 dx = 1 > A^2 * ∫∞ -∞ (1-cos2nπx/L) dx = 1 > (A^2/2) * [x- (sin2nπx/L * 2nπ/L)]∞ -∞ =1 > (A^2/2) * (L) = 1 > A = √2/L Normalised wavefimetion is > Ψn= √2/L sin nπx/L ## Compton Effect In 1921 A. H. Compton, While studying monochromatic X-rays scattered by Carbon atoms I found that when a monochromatic beam of high energy photon is scattered by a target rich in electrons, the scattered beam contain photon not only of the same wavelength as that of incident photon but also the photon of longer wavelength, (or shorter frequency). Compton derived an expression for the change in wavelength of scattered photon considering the elastic collision between the incident photon and free electron of the scattering material and by applying the law of conservation of energy and momentum. det a photon of energy hv and momentum hv/c collides with a free electron at rest and give up a fraction of its energy to the free electron and scattered. Let the scattered photon is emitted at an angle θ with the energy hv' and momentum hv'c and the election recoils at an angle φ with momentum mr. Now according to the princable of conservation of energy > Energy of the system before collision to is equal to Energy of the system after collisen >> hv + moc^2 = hu' + mc^2 Applying principle of conservation of momentum. > Momentum before collision = Momentum after collision In a direction along the direction of incidence > hv/c + 0 = hu'cosθ/c + mv cos φ In a direction perpendicular to the direction of incidence > 0+ 0 = hu'sing/c - mv sing Rearranging equations (2) & (3), we have > mv cos φ = h/c (ν - ν'cosθ) > mv sin φ = h/c ν'sinθ Squaring and adding equation (4) & (5) we get > m^2*v^2= [h^2/c^2] (ν^2 + ν'^2*cos^2θ - 2νν'cosθ) + ν^2 * sin^2θ > m^2*v^2 = h^2/c^2 (ν^2 + ν'^2 - 2νν'cosθ). > p^2*c^2 = h^2(ν^2 + ν'^2 - 2hνν'cosθ). > p^2*c^2 = h^2 ν^2 + h^2 ν'^2 - 2h^2 νν'cosφ - 2pmν Now from Equation (1), using E=mc^2 = √p^2*c^2 + m^2*c^4 > hv + moc^2 = hu' + √p^2*c^2 + m^2*c^4 > (hv-hu') +moc^2 = √p^2*c^2 + m^2*c^4 Sequaring above equation > [(hv-hu') +moc^2]^2 = p^2*c^2 + m^2*c^4 > ⇒ (hv-hv1)^2 + m^2 c^4 + 2(hv-hu) moc^2= p^2*c^2 + m^2*c^4 > ⇒ h^2 v^2 + h^2 v'^2 -2hv'v'moc^2 + 2h(ν-ν') moc^2= p^2*c^2 From (6) & (7) > ⇒ h^2 ν^2 + h^2 ν'^2 -2h^2 νν'cosθ = h^2 ν^2 + h^2 ν'^2 - 2h^2 νν' > ⇒ -2h^2* νν' (1-cosΘ) = 2K (ν-ν') moc^2 > ⇒ ν - ν' = (hνν' (1-cosΘ)) / moc^2 > 2h (ν-ν') moc^2 = 2h^2 νν' (1-cosΘ) >> ν-ν' = h/moc^2 (1-cosΘ) >> 1/ν' - 1/ν = h/(moc^2) (1-cosΘ) 1. At the values h, mo and c^2 are positive and maximum value of cosΘ is 1, the equation (8) shows that ν> ν', that is the frequency of incident photon (or radiation) is always greater than the frequency of the scattered photon. 2. Equation (8) can be written as, > (1 / ν') = (1 / ν) + (h/(moc^2) (1-cosΘ) > Δλ = λ' - λ = h/moc (1-cosΘ) > Δλ = 2h/(moc) * sin^2(θ/2) 3. This is required expression of compton shift Δλ. Δλ depends on angle of scattering & is independent of the wavelength of incident photon. 4. λ' > λ 5. when θ=0, Δλ=0 ,compton Shift is zero ie, no scattering occurs along the direction of incident radiation. 6. When θ=π/2 , Δλ = h/(moc)= 6.62*10^-34/(9.1*10^-31 * 3*10^8) = 2.42*10^-12 = 0.0242 A° 0.0242 A° compton wavelength (constant) 7. When θ=π, Δλ = 2x 0.0242 = 0.048JZA°. 8. As angle of scattering θ varies from 0 to 180°, the wavelength of scattered photon varied from λ to λ + 2h/(mol) 9. Direction of Recoiled Compton Election - tan φ = (ν'sinθ)/(ν-ν'cosθ) - or tan φ = (λ sinθ)/(λ' - λcosθ) Substituting the value of ν' from eq ⑧ after rearranging as ν' = ν/(1+ 2asin^2(θ/2)), where a = hv/ moc^2 - tan φ = [(cot θ/2 * 1+2asin^2(θ/2))] 10. Kinetic Energy of recoiled electrom: - K.E. of the recoiled electrom = hv - hu' Subslituting the ν' from equation C - K.E. of the recoiled electron = hv - hν(✓(1+2asin^2(θ/2))/ (1+2asin^2(θ/2)) = hv - ( 2asin^2(θ/2)/(1+2asin^2(θ/z)) Maximum KE of the recoil electron. > Sin θ/2 =1 or θ/2 =π/2 > .. θ = π Maximum KE = 2hν^2/(moc^2 (1+2hv/moc^2)) 11. Unmodified Radiation: → > The scattered radiation with unchanged frequency called unmodified radiation or coherent scattering 12. Modified Radiation: →. > The scattered radiation with changed frequency called modified radiation or incoherent scattering.