2.2 WAVE AND PARTICLE DUALITY OF RADIATION

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1 Quantum Mecanics.1 INTRODUCTION Te motion of particles wic can be observed directly or troug microscope can be explained by classical mecanics. But wen te penomena like potoelectric effect, X-rays, ultraviolet catastrope, superconductivity were discovered, classical pysics failed to explain suc penomena. Te microworld of atoms obeys different laws. Te new laws applicable for microparticles constitute quantum mecanics. Te revision of classical concepts began wit te seminal ypotesis of Planck and many distinguised pysicists suc as Einstein, Bor, de Broglie, Scrödinger, Born, Pauli, Heisenberg, Dirac and oters contributed to te development of quantum mecanics. Quantum Mecanics is a branc of pysics dealing wit te beaviour of matter and energy on te microscope scale of atoms and subatomic particles. Quantum mecanics is fundamental to our understanding of all te fundamental forces of nature except gravity. It provides te foundation to several brances of pysics, including Electromagnetism, Particle Pysics, Condensed Matter Pysics, and some parts of Cosmology. Quantum Mecanics is essential to understand te teory of cemical bonding (and ence, te entire subject of cemistry), structural biology, and tecnologies suc as electronics, information tecnology (IT), and nanotecnology. Hundreds of experiments and commendable work in applied sciences ave proved quantum mecanics a successful and practical science.. WAVE AND PARTICLE DUALITY OF RADIATION Te concept of a particle is easy to grasp. It as mass, it is located at some definite point, it can move from one place to anoter, it gives energy wen slowed down or stopped. Tus, te particle is specified by: (i) mass, m; (ii) velocity, v; (iii) momentum, p; and (iv) energy, E. Te concept of a wave is a bit more difficult tan tat of a particle. A wave is spread out over a relatively large region of space, it cannot be said to be located just ere and tere, it is ard to tink of mass being associated wit a wave. Actually, a wave is noting but a spread out disturbance. A wave is specified by its: (i) frequency, (ii) wavelengt, (iii) pase or wave velocity, (iv) amplitude, and (v) intensity. Capter_.indd 4 1/4/17 4:8:1 PM

2 44 Applied Pysics waves. His suggestion was based on te fact: if radiation like ligt can act like wave sometime and like a particle at oter times, ten te material particles (e.g., electron, neutron, etc.) sould also act as wave at some oter times. According to de Broglie s ypotesis, a moving particle is associated wit a wave wic is known as de Broglie wave. Te wavelengt of te matter wave is given by l mv p were m is te mass of te material particle, v its velocity and p is its momentum. A particle A wave Fig..1 Diagrammatic representation of a particle and a wave. A particle is localized at a point in space wereas a wave spreads over a large volume. de Broglie wavelengt. Te expression of te wavelengt associated wit a material particle can be derived on te analogy of radiation as follows: Considering te Planck s teory of radiation, te energy of a poton (quantum ) is given by E g c... (1) l were c is te velocity of ligt in vacuum and l is its wavelengt. According to Einstein energy mass relation E mc... () From equations (1) and (), we get mc c c or l l mc or l... (3) mc were mc p (momentum associated wit poton). If we consider te case of a material particle of mass m and moving wit a velocity v, i.e., momentum mv, ten te wavelengt associated wit tis particle is given by l... (4) mv p If E is te kinetic energy of te material particle, ten E 1 mv or p me 1 mv p m m \ de Broglie wavelengt l m E... (5) Capter_.indd 44 11/4/17 11:48:11 AM

3 Quantum Mecanics 45 de Broglie s wavelengt associated wit electrons. Let us consider te case of an electron at rest mass m, and carge e wic is accelerated by a potential V volts from rest to velocity v, ten 1 m ev v ev or v m or l If V 1 volts, ten l m mv m ev ev m ( V ) 1. 6 V Å l 1.6 Å.... (6) Equation (6) sows tat te wavelengt associated wit an electron accelerated to 1 volts is 1.6 Å. Wave Velocity of de Broglie Wave Te wave velocity u is given by te following expression: u g l were g is te frequency of matter waves. According to Planck s teory, E g and from Einstein s relation, E mc \ g mc or g mc For matter waves, l mv Substituting tese values, we get (1) () u mc c (3) mv v were v is te speed of material particle wic is less tan te speed of ligt c. Te wave velocity can also be expressed in terms of wavelengt, we know tat 1 g E mv ev È 1 mv ev Í Î (4) Capter_.indd 45 1/4/17 4:8: PM

4 46 Applied Pysics Multiplying and dividing te rigt-and side of equation (4) by / m, we ave g mev 1 were l m m l \ u g l 1 l m l ml mev... (5) Equations (3) and (5) are te different forms of te wave velocity of de Broglie wave..4 EXPERIMENTAL VERIFICATION OF DE BROGLIE S THEORY OF MATTER WAVES: DIFFRACTION OF MATERIAL PARTICLES (i) Davisson and Germer Experiment on Electron Diffraction If a material particle as a wave caracter, it is expected to sow penomena like interference and diffraction. In 197, Davisson and Germer demonstrated tat a beam of electrons does suffer diffraction. Teir apparatus is sown in Fig... Electrons from a eated filament are accelerated troug a variable potential V and emerge from te electron gun G. Tis electron beam falls normally on a nickel crystal C. Te electrons are diffracted from te crystal in all directions. Te intensity of te diffracted beam in different directions is measured by a Faraday cylinder F (connected to a galvanometer), wic can be moved along a circular scale S. Te crystal can be turned about an axis parallel to te incident beam. Tus, any azimut of te crystal can be presented to te plane defined by te incident beam and te beam entering te Faraday cylinder. First of all, te accelerating potential V is given a low value, and te crystal is turned at any arbitrary azimut. Te Faraday cylinder is moved to various positions g Fig.. Davisson Germer experiment setup. Capter_.indd 46 1/4/17 4:8:1 PM

5 Quantum Mecanics 47 on te scale S and te galvanometer current at eac position is noted. Te current, wic is a measure of te intensity of te diffracted beam, is plotted against te angle φ between te incident beam, and te beam entering te cylinder. Te observations are repeated for different accelerating potentials and te corresponding curves are drawn as sown in Fig..3. f f 5 44 volts 48 volts 54 volts 6 volts Fig..3 Variation of intensity wit angle f It is seen tat a bump begins to appear in te curve for 44-volt electrons. Wit increasing potential, te bump moves upwards and becomes most prominent in te curve for 54 volt electrons at φ 5. At iger potentials te bump gradually disappears. Te bump in its most prominent state verifies te existence of electron waves. For, according to de Broglie, te wavelengt associated wit an electron accelerated troug V volts is 1 6 l. V Å, Hence, te wavelengt associated wit a 54-volt electron is l 1. 6 (54) 1.67 Å. Now, it is known from X-ray analysis tat te nickel crystal acts as a plane diffraction grating wit grating space d.91 Å (Fig..4). According to experiment, we ave a diffracted electron beam at φ 5. Tis arises from wave-like diffraction from te family of Bragg atomic planes. Te corresponding angle of Ê incidence relative to te family of Bragg planes is θ 65 Ë Á using te Bragg s equation (taking te order n 1), we get l d sin q ˆ. Hence, (.91 Å) sin Å. Tis being in excellent agreement wit te wavelengt computed from de Broglie ypotesis, sows tat electrons are wave-like in some circumstances. Oter fundamental particles like neutrons also sow wave-like properties. Te Davisson- Germer experiment tus provides direct verification of de Broglie ypotesis of te wave nature of moving particles. Capter_.indd 47 1/4/17 4:8:3 PM

6 Quantum Mecanics 49 Te diffraction patterns produced by electron beams were strikingly similar to te X-ray diffraction patterns obtained from powder samples. Tus, te experiment of G.P. Tomson and Kikuci provided irrefutable proof to te existence of de Broglie waves..5 PROPERTIES OF MATTER WAVES Following are te properties of matter waves: (1) Ligter is te particle, greater is te wavelengt associated wit it. () Smaller is te velocity of te particle, greater is te wavelengt associated wit it. (3) Wen v ten l, i.e., wave becomes indeterminate and if v ten l. Tis sows tat matter waves are generated by te motion of particles. Tese waves are produced weter te particles are carged particles or tey are uncarged (l is independent of carge). Tis fact reveals tat mv tese waves are not electromagnetic waves but tey are a new kind of waves (electromagnetic waves are produced only by motion of carged particles). (4) Te velocity of matter waves depends on te velocity of material particle, i.e., it is not a constant wile te velocity of electromagnetic wave is constant. (5) Te velocity of matter waves is greater tan te velocity of ligt. Tis can be proved as under: We know tat E g and and E mc \ g mc or v mc Te wave velocity (v p ) is given by v p g l mc \ l mv mv v p c v As particle velocity v cannot exceed c (velocity of ligt), ence v p is greater tan te velocity of ligt. (6) Te wave and particle aspects of moving bodies can never appear togeter in te same experiment. Wen we can say is tat waves ave particle-like properties and particles ave wave-like properties and te concepts are inseparably linked. Matter wave representation is only a symbolic representation. (7) Te wave nature of matter introduces an uncertainty in te location of te position of te particle because a wave cannot be said exactly at tis point or exactly at tat point. However, were te wave is large (strong), tere is a good cance of finding te particle wile, were te wave is small (weak) tere is less cance of finding te particle. Capter_.indd 49 1/4/17 4:8:5 PM

7 5 Applied Pysics Wavelengt of Macroscopic Bodies For an electron aving an energy 1 ev, te de Brogile wavelengt is 1.33 Å wic is more tan te size of te electron. Wereas te wavelengt of macroscopic bodies is insignificant in comparison to te size of te bodies temselves even at very low velocities, e.g., if we consider a cricket ball of mass 5 gm flying wit velocity km/r, its wavelengt comes to Js l 5. kg 13.9 m/s m 1 4 Å wic is insignificant in comparison to te size of te ball..6 WAVE PACKET, PHASE VELOCITY AND GROUP VELOCITY Wave velocity is also called pase velocity. Wave motion is a form of disturbance wic travels troug a medium due to repeated periodic motion of te particles of te medium about teir mean positions, te motion being anded over from one particle to te next. Te individual oscillators wic make up te medium only execute simple armonic motion about teir mean positions and do not temselves travel troug te medium wit te wave. Every particle begins its vibration a little later tan its predecessor and tere is a progressive cange of pase as we travel from one particle to te next. It is te pase relationsip of tese particles tat we observe as a wave and te velocity wit wic te plane of equal pase travels troug te medium is known as pase velocity or wave velocity. Let us assume tat a particle like an electron can be described matematically by a sine wave y A sin (wt kx). Tis wave as no beginning and no end. It is of infinite extent and completely nonlocalised. But particle (electron) is confined to a very small volume. Terefore, a monofrequency wave cannot represent a particle. It implies tat te de Broglie waves are not armonic waves but could be a combination of several waves. A superposition of several waves of sligtly different frequencies gives rise to a wave packet. Suc a wave packet possesses bot wave and particle properties. Te regular separation between successive maxima in a wave packet is te caracteristic of a wave and at te same time it as a particle-like localization in space. A wave packet can be described in terms of a superposition of individual armonic waves of sligtly different frequencies centred on a frequency, v. Let a superposition of two waves be y 1 A sin (k 1 x w 1 t) y A sin (k x w t) \ y y 1 + y A sin (k 1 x w 1 t) + A sin (k x w t) A sin È1 1 È Í ( k1+ k) x- ( w1+ w ) t Î cos 1 1 Í ( k1+ k) x- ( w1+ w) t Î Capter_.indd 5 1/4/17 4:8:6 PM

8 5 Applied Pysics te group and pase velocities are te same, i.e., in case of electromagnetic waves in vacuum. Relation between Particle Velocity and Group Velocity Consider a particle of rest mass m and moving wit velocity v. Let w be te angular frequency and k be te wave number of de Broglie waves associated wit a particle. Ê w pg p Á Ë k p l mc p mv ˆ p pm v 1- c m 1- v c. v c wave velocity v p w k c v As v < c, te pase velocity of te associated wave is always greater tan c, te velocity of ligt. Te group velocity v g d w dw/ dv. dk dk/ dv -3 / dw pmc v v p Ê ˆ Ê ˆ 1 - dv Ë Á Á Ë c c Ê Á1- Ë dk dv dk dv \ v g pm Ê v - Á1 Ë c 1 ˆ mv Ê - ˆ Ê p È v ˆ + Í 1 Ë Á Á1- Í Ë c Î m Ê v ˆ È p v v - Á Ë c ÎÍ c c pm Ê v ˆ 1- Á Ë c mv v c ˆ - / -3 / -3 / - 3 / dw / dv pmv dk / dv [ 1- v / c ] 3 / 3 / ( 1- v / c ) pm v 3 / Ê - vˆ. Ë Á c \ v g v Tus, te de Broglie wave group travels wit te same velocity as tat of te particle. Capter_.indd 5 1/4/17 4:8:34 PM

9 .7 WAVE FUNCTION Quantum Mecanics 53 Waves represent te propagation of a disturbance in a medium. Ligt waves are represented by electromagnetic field variations and sound waves are represented by pressure variations. Te de Broglie wave associated wit a moving particle cannot be specified in a similar manner, since electrons ave wave properties, it may be assumed tat a quantity y represents a de Broglie wave just as te electric vector represents a ligt wave. Te quantity y is called te wave function. y (x, y, z, t) is function of space and time coordinates and it represents position of particle at some time t. However, it is not possible to locate a particle precisely at position x, y, z, tere is only a probability of te particle being at te specific point (x, y, z). y is usually a complex quantity..8 PHYSICAL INTERPRETATION OF WAVE FUNCTION Te first and te simple interpretation of y was given by Scrödinger imself in terms of carge density. We know tat in any electromagnetic wave system if A is te amplitude of te wave, ten te energy density, i.e., energy per unit volume is equal to A, so tat te number of potons per unit volume, i.e., poton density is equal to A /g or te poton density is proportional to A as g is constant. If y is te amplitude of matter waves at any point in space, ten te particle density at tat point may be taken as proportional to y. Tus, y is a measure of particle density. Wen tis is multiplied by te carge of te particle, te carge density is obtained. In tis way, y is a measure of carge density. It is observed tat in some cases, y is appreciably different from zero witin some finite region known as wave packet. It is natural to ask, were is te particle in relation to wave packet? To explain it, Max Born suggested a new idea about te pysical significance of y wic is generally accepted nowadays. According to Max Born yy* y gives te probability of finding te particle in te state y, i.e., y is a measure of probability density. Te probability of finding a particle in volume dt dx dy dz is given by y dx dy dz. For te total probability of finding te particle somewere is, of course, unit, i.e., te particle is certainly to be found somewere in space. ÚÚÚ y dx dy dz 1 y satisfying above requirement is said to be normalized. Normalization Condition If at all te particle exists, te particle is certainly somewere in te universe terefore, te probability of finding te particle somewere in te universe must be unity. Since te probability of its being located in an elemental volume is proportional to y dx dy dz, it is convenient to coose te constant of proportionality suc tat te sum of te probabilities over all values of x, y, z must be unity. Tus, Capter_.indd 53 1/4/17 4:8:35 PM

10 54 Applied Pysics ÚÚÚ - yy * dx dy dz 1 Tis is called te normalization condition. A wave function satisfying te above condition is said to be normalized. Requirement of an Acceptable Wave Function Besides being normalizable, an acceptable wave function must fulfil te following requirements: 1. y must be finite everywere. If, for instance Y is infinite, it would mean an infinitely large probability of finding te particle at tat point. Tis would violate te uncertainty principle \ y must ave a finite or zero value at any point.. y must be single valued. If y as more tan one value at any point, it would mean more tan one value of probability of finding te particle at tat point wic is obviously ridiculous. 3. It must be continuous and ave a continuous first derivative everywere. Tis is necessary from Scrödinger equation itself wic sows tat d y/ dx must be finite everywere. Tis can be so only if d Y/dx as no discontinuity at any boundary. Furtermore, te existence of d y/dx as a continuous function implies y too is continuous across te boundary. Tese requirements wic must be fulfilled by an acceptable wave function carry great significance wen te Scrödinger s steady state equation for a given system is solved to obtain a wave function wic fulfils tese requirements, ten we find tat te equation can be solved only for te value of energy of te system. Tus, energy quantization appears in wave mecanics as a natural feature of te solution of te wave function. Te values of energy for wic Scrödinger equation can be solved are called eigenvalues and te corresponding (acceptable) wave functions are called eigenfunctions..9 HEISENBERG S UNCERTAINTY PRINCIPLE A monocromatic wave is infinite in extent so instead of associating a single monocromatic de Broglie wave wit a moving particle, we associate a wave packet consisting of a group of waves of nearly equal amplitude centred around de Broglie l /p of te particle. Te wave packet as te dimensions of te localized particle and travels wit te same velocity as te particle. Te association of a group of waves wit a moving particle tat te position of te particle at any instant of time cannot be specified wit any desired degree of accuracy. Te particle can be somewere witin te group of waves, i.e., witin a small region Dx of space (Dx is linear spread of te wave packet) Fig..7. Te probability of finding te particle is maximum at te centre of te wave packet and falls to zero at its ends. Terefore, tere is an uncertainty Dx in te position of te particle. A wave packet is formed by waves aving a range of wavelengt. Capter_.indd 54 15/4/17 3:1:5 PM

11 56 Applied Pysics.1 ELECTRON DIFFRACTION EXPERIMENT Let us consider a stream of electrons moving along x direction passes troug a narrow slit of widt d ( Dy). If Dy is comparable to te wavelengt of te electron beam, ten te electrons are diffracted. According to single slit diffraction pattern a central maxima and two secondary minima are formed as sown in Fig..4. According to diffraction teory, d sin q l Fig..8 Electron diffraction experiment. or l Dy sin q (1) Q d Dy uncertainty in position of electron before being diffracted. Before diffraction a slit electron as momentum p x only along x direction and zero in y direction. Terefore, uncertainty in y component of momentum of Dp y. Because of te diffraction effect at te slit, te particle acquires a small component of momentum p y in y direction. Te original momentum of te particle in te X direction p x decreases so tat te resultant momentum p remains constant. Te original momentum of te particle in te y direction was accurately known to be zero. Terefore, Dp y is te uncertainty introduced in te y component of te momentum. \ Dp y is te uncertainty produced in y component of momentum. Particles tat strike te screen at a point A, te first minima must ave left te slit at an angle q, given by tan q q Dp y px Equating (1) and () () Dpy p l Dy x \ Dy. Dp y l. p x \ Dy. Dp y Q l p x wic is te uncertainty principle, i.e., if we try to improve te accuracy in y, we ave to reduce to Dy using a finer slit wic results in turn in a wider pattern. It leads to a larger Dp y. Capter_.indd 56 11/4/17 1:57:59 AM

12 Quantum Mecanics g-ray MICROSCOPE EXPERIMENT Consider an experiment for measurement of te position of an electron at rest using microscope. Since te size of electron is very small, ligt wave cannot be used for measurement as it would be 1 4 to 1 5 times smaller tan te size of electrons. To illuminate an electron g rays can be used. To determine te location of te electron, one poton must bounce off te electron, and pass troug te microscope into our eyes. A g-poton carries a very large momentum. Wen te poton strikes te electron, part of te momentum and energy are transferred to te electron due to te Compton scattering. Tus, wen te scattered poton is registered by te microscope, only te earlier position of te electron can be deduced but te momentum of te electron is altered. Incident poton Before collision Scattered poton After collision Electron Recoiling electron (a) (b) Fig..9 An experiment were an attempt is made to locate electron by illuminating it wit g-rays. Te g-rays cause recoil of electron tus frustrating our efforts to know te electron position. Let te incoming poton momentum be /l. Te uncertainty in te electron momentum after te scattering be Dp. Tis Dp can ave maximum values as te momentum of incident poton, i.e., /l \ Dp l Te position of te electron can be determined witin one wavelengt of poton. Te uncertainty in position is Dx l. \ Dx.Dp l. l wic is te uncertainty principle..1 APPLICATIONS OF UNCERTAINTY PRINCIPLE (i) Non-existence of electrons and existence of protons and neutrons in te nucleus: Te radius of nucleus of any atom is of te order of 1 14 m. If an electron is confined inside te nucleus, ten te uncertainty in te position Dx of te electron is equal to te diameter of te nucleus, i.e., Dx 1 14 m. Capter_.indd 57 11/4/17 1:58: AM

13 Quantum Mecanics 59 (ii) Binding energy of an electron in atom: In an atom, te electron is under te influence of electrostatic potential of positively carged nucleus. It is confined to te linear dimensions equal to te diameter of electronic orbit. Te uncertainty in te position Dx of an electron is of te order of R were R is te radius of te orbit and te corresponding uncertainty in te momentum component Dp x is given by Dp x > p.dx Dp x > p. R wic sows te momentum of electron in an orbit is at least p ~ Dp x ~, R ª 1 p. 1 m R K.E. of electron p 1 Ê m Ë Á ˆ 4 R p m 3p mr Te potential energy of an electron in te field of nucleus wit atomic number z is given by -ze V 4pŒ R Te total energy K.E + P.E. ze - 3p mr 4p Œ R ( / p ) ze - 8mR 4p Œ R È ( ) z( ) ev - ÎÍ R R were, p È E - z ev ÎÍ R R Taking R 1 1 m, we ave E (1 15z)eV Now, te binding energies of te outermost electrons in H and He are 13.6 ev and 4.6 ev respectively. So, te value of binding energy derived on te basis of uncertainty principle is acceptable as tese are comparable in magnitudes. (iii) Finite widt of spectral lines: From Heisenberg s principle of energy and time relation, we ave DE. Dt > ħ Since te lifetime of electron in an excited orbit is finite (1 8 sec), so te energy ħ levels of an atom given by DE must ave a finite widt wic means tat Dt te excited levels must ave a finite energy spread, i.e., radiation given out wen Capter_.indd 59 11/4/17 1:58:9 AM

14 6 Applied Pysics an electron jumps must be truly monocromatic, i.e., te spectral lines can never be sarp but must ave a natural spectral widt. (iv) Strengt of nuclear force: If we assume te nuclear radius is of te order r cm. From uncertainty principle, momentum will be of te order p ħ \ K.E. will be of te order r K.E. p ħ 1 MeV m mr were M corresponds to te mass of a nucleus (poton or neutron). Since te nucleus is bound, so tat B.E. sould be greater tan K.E. wit ve sign so te B.E. of a nucleus is of te order of 1 MeV..13 ONE-DIMENSIONAL TIME-DEPENDENT SCHRÖDINGER EQUATION In 196, Erwin Scrödinger formulated te wave equation for matter waves wic is known as Scrödinger s equation. It plays te same role in quantum mecanics as Newton s second law does in classical mecanics. Te motion of an atomic particle can be determined using Scrödinger s wave equation. Let us consider a microparticle. Let y be te wave function associated wit te motion of tis microparticle y function represents te wavefield of te particle. It is smaller tan E & B used to describe te electromagnetic waves and to transverse displacement for waves on a string. For one-dimensional case, te classical wave equation as te following form x x 1 (1) x v t A solution of te above equation is te familiar plane wave ( ) x( x t) Ae ikx -w t 1 () were w vk and v is pase velocity. For microparticle, w and k may be replaced wit E & P using Einstein & de Broglie relations. w E p E E w pv Ô ħ (3) & p k Ô p p P P k l Ô l ħ Substituting equation (3) in () & replacing x (x, t) wit wave function y (x, t) Differentiating wit respect to t y (x, t) Ae i(et px)/ħ (4) y - i E y t ħ (5) Capter_.indd 6 11/4/17 1:58:1 AM

15 6 Applied Pysics Te rigt-and side of equation (1) is a function of t only and te left-and side a function of x only. Terefore, equation (1) must be valid for any x and t, it can be so only if te two sides of equation (1) are equal to a constant. Setting tis constant equal to energy E, we get - + ħ 1 y( x) V( x) E m y( x) x \ -ħ y + Vy Ey (1) m x Tis is time independent Scrödinger equation. In tree dimensions, te time independent Scrödinger equation, is written as were, + + x y z -ħ y+ V() r y Ey (13) m Equation (1) is frequently written in te form Hy Ey. were H is Hamiltonian operator ħ H - + V (14) m.15 MOTION OF A FREE PARTICLE Consider an electron moving freely in space along te positive x direction and not acted upon by any force. Since no force is acting on te electron and its potential energy is zero. Scrödinger s time independent equation is, Q - m d ħ dx d y + Vy Ey (1) dx y 8p + m ħ p & V If K 8 p me te above equation reduces to Ey () (3) d y + K y (4) dx Capter_.indd 6 1/4/17 4:8:59 PM

16 Te general solution to te above equation is Quantum Mecanics 63 y(x) Ae ikx + Be ikx (5) were A and B are constants. As it is assumed tat te waves propagating only in te positive x direction, we can write y (x, t) Ae ikx e iwt Since te electron is moving freely, tere are no boundary conditions and ence no restriction on K. All values of te energy are allowed. Te allowed values of energy form a continuum and are given by E 8p m K A freely moving electron terefore possesses a continuous energy spectrum as sown in Fig..1. me P K p ħ ħ l Te K vector describes te wave properties of te electrons. As E a K te grap between E & K is a parabola as sown in Fig..1. Te momentum is well defined in tis case. Terefore, according to te uncertainty principle, it is difficult to assign a position to te electron, i.e., te electron position is indeterminate. E K K Fig..1 Parabolic relationsip between E & K in case of free electron..16 PARTICLE TRAPPED IN ONE-DIMENSIONAL INFINITE POTENTIAL WELL Consider te motions of electrons in one-dimensional deep potential well bounded by ig potential barriers. Electrons can propagate along X-axis and can get reflected from te walls at x and x L as sown in Fig..11 and tus it can propagate bot in positive and negative x directions witin te well te potential energy is zero and at te boundaries, i.e., x and x L, potential is very ig almost. Terefore, te probability of finding te electron outside te well must be zero, i.e., y at x < and x > L. Te Scrödinger equation is y+ 8 p m ( E - V) y Capter_.indd 63 1/4/17 4:9:1 PM

17 Quantum Mecanics 65 \ L np (8) K Tis equation (6) implies tat te wave equation as solutions only wen te electron wavelengt is restricted to discrete values suc tat only a wole number of alf wavelengt formed over te lengt L, i.e., te electron form standing wave pattern witin te potential well. Substitute (7) in () K n p L 8p me \ E n. n (n 1,, 3,.) (9) 8ml Te above equation sows only tose values of energy are wic are possible for n to be an integer. 9 So E 1, E E, 3... etc. 8mL ml 8mL are allowed energy states. It is sown in Fig..1. Energy of particle in a box can take only discrete values, i.e., it is quantized. Te value of energy E 1 for n 1 is called zero point energy, wic signifies tat tere must be some movement of particles (atoms, molecules, etc.) at te absolute zero temperature. n 4 E 4 ml n 3 E 3 9 8mL n n 1 E E 1 ml 8mL Fig..1 Energy level diagram for an electron confined to a one-dimensional potential well. Note tat te electron cannot ave zero energy. Te lowest allowed energy is E 1. Te wave functions corresponding to te above allowed discrete energy levels can be obtained as follows: y Ai sin Kx Applying normalized condition. L Ú y( x) dx 1we get, ia sin Kxdx 1 L Ú Capter_.indd 65 15/4/17 3:5:6 PM

18 66 Applied Pysics 1 coskx sin Kx - L Ê 1 cos Kx ˆ Ú ia - dx 1 Ë Á \ ia ia ia ia Ï L ÔÊ -sin Kx ˆ 1 x Ë Á + Ê Ë Á ˆ Ì ÓÔ L Ô Ô ÏÊ -sin KL ˆ L Ì - 1 Ë Á + Ó Ï -sin np L 1 K n p Ì + Ó L Ê Ë Á ia Lˆ 1 L Inserting te value of ia into equation, we get y(x) n L sin p L x y(x) n L sin p L x Analysis of te above result sows 1. Wile y n may be negative, y n is always positive y n gives te probability of finding te electron at certain place witin te well. Wave function and y n can be plotted as sown in Fig..13. For n 1, te probability of finding a particle is largest in te middle of te box for most of time. Fig..13 Te allowed wave functions and te corresponding probability distributions for an electron trapped in a one-dimensional potential well. Capter_.indd 66 1/4/17 4:9:1 PM

19 68 Applied Pysics 5. Uncertainty principle, Dx. Dp ħ DE. Dt ħ DL. Dq ħ 6. E n n 8ma 7. y(x) a n x sin p a QUESTIONS WITH ANSWERS 1. Wat are te assumptions of Planck s quantum teory? Te assumptions of Planck s quantum teory are: Atoms in te black body are assumed to be simple armonic oscillators. Te armonic oscillators do not emit energy continuously. Energy is emitted in te form of quanta of magnitude v.. State de Broglie s ypotesis. According to de Broglie s ypotesis, a particle moving wit a velocity v and mass m as a wave associated wit it. Te wavelengt of tis wave is given by l mv p 3. Wat is te pysical significance of te wave function? Te wave function y(x, y, z, t) signifies te probability of finding a particle in space at any given instant of time. It relates statistically te moving particle and its matter wave. It is a complex quantity. y (x, y, z) gives te probability density of te particle, wic is a real quantity. 4. Wic experiments were carried out to verify uncertainty principle? Electron diffraction experiment and g-ray microscope experiment were used to verify uncertainty principle. 5. Wat is a wave function? A variable quantity wic caracterizes de Broglie wave is known as wave function and is denoted by te symbol y. 6. Mention pysical significance of te wave function. It relates te particle and wave nature of matter statistically. It is a complex quantity and ence cannot be measured. It must be well beaved. Tat is, single valued and continuous everywere. If te particle is certainly to be found somewere in space, ten te probability value is equal to one. Capter_.indd 68 11/4/17 1:58:15 AM

20 ÚÚÚ i.e., P dx dy dz 1 v y Quantum Mecanics 69 A wave function satisfying te above relation is called a normalized wave function. 7. Write te Scrödinger time independent and dependent wave equation and explain te terms. Scrödinger s time independent wave equation is y m + E V y ħ ( - ) Scrödinger s time dependent wave equation is -ħ y y+ Vy iħ m t were y is te wave function and Y is a function of Cartesian coordinates. 8. Wat are Hamiltonian and energy operators? Te Scrödinger time independent wave equation is -ħ y y+ Vy iħ m t Te above equation can be written as or Hy Ey Ê -ħ ˆ y Á + V y iħ Ë m t Ê -ħ ˆ were H Á + V Ë m is called te Hamiltonian operator and E iħ is called te energy operator. t 9. Wat is zero point energy? Te possible energies of a particle in a box of lengt a is given by n E 8ma [were n 1,, 3,.] If n 1 ten E 8ma Tis is te energy of te ground state of te particle. Since te particle in a box cannot be at rest, its minimum energy is positive and is often called te zero point energy. 1. Wat are eigenvalues and eigenfunctions? Te allowed values of energy for different values of n are given by Capter_.indd 69 11/4/17 1:58: AM

21 Quantum Mecanics 71 E ml ( ) ( 1 ).6 1 Joules ev E.164 ev. Compute de Broglie wavelengt of 1 11 KeV neutrons (m n kg). Solution: Given data: E 1 11 KeV ev E J J Formula: l me l m 3. An electron microscope uses 1.5 Ke V electrons. Find its ultimate resolving power on te assumption tat tis is equal to te wavelengt of te electron, given tat e e.s.u., m gm erg-sec. Solution: Given data: E 1.5 KeV ev J m gm J sec. Formula: l me l m.348 A.U. So, te resolving power of microscope is.348 Å. 4. Wat is de Broglie wavelengt of an electron wic as been accelerated from rest troug a potential difference of 1 V? Solution: Given data: V 1 V. Capter_.indd 71 1/4/17 4:9:4 PM

22 7 Applied Pysics Formula: l 1. 6 V Å Å 5. Estimate te amount of accelerating voltage to wic electrons are to be subjected in order to associate tem wit de Broglie wavelengt of.5 Å. Given tat m gm Solution: Given data: l.5 Å cm m gm J. sec. 1 6 Formula: y. V A.U. \ V ( 1. 6 ) l ( 1. 6 ) ( 5. ) 61. volts 6. Calculate de Broglie wavelengt associated wit a proton moving wit a velocity equal to 1 t of te velocity of ligt. Solution: Given data: v p m/sec Formula: l m p kg mv m. 7. Compute te de Broglie wavelengt of a proton wose kinetic energy is equal to te rest energy of an electron. Mass of a proton is 1836 times tat of electron. Solution: Given data: m p 1836 m e kg E m c (3 1 8 ) J Capter_.indd 7 1/4/17 4:9:7 PM

23 74 Applied Pysics \ p - l E J ev ev 11. Calculate te de Broglie wavelengt of an a particle accelerated troug V. m a kg. Solution: Given data: m a kg V V e a e Formula: E ev l me a a l m ev a a m 1. A beam of 1 kv electrons is passed troug a tin metallic seet wose interplanar spacing is.55 Å. Calculate te angle of deviation of te first order diffraction maximum. Solution: Given data: V 1 kv V d.55 Å m n 1 Formula: 1 6 l. Å and d sin q nl V l Å Ê nl ˆ q sin 1 Ë Á d sin -1 Ê Á - Ë sin 1 (.11145) An electron and a 15 gm baseball are travelling at m/s measured to an accuracy of.65%. Calculate and compare uncertainty in position of eac of te bodies. Solution: Given data; v e m/s, -1 1 ˆ Capter_.indd 74 1/4/17 4:9:34 PM

24 accuracy.65% m 15 gm.15 kg v m m/s v e m/s 1 ħ Formula: Dx mdv p mdv (i) Dv e v e.65% m/s Dx e m - 31 ( )(. 143) (ii) Dv m.143 m/s ħ Dx m m mdv m Quantum Mecanics 75 Te results sow tat te uncertainty in position of te electron is very large compared to its dimensions ( 1 15 m) wereas te uncertainty in position of te baseball is nearly as small as zero. 14. An electron as a speed of 5 m/s correct up to.1% wit wat minimum accuracy can you locate te position of tis electron. Solution: Given data: v e 5 m/s Dv e accuracy.1% m/s Dx? 1 Formula: Dp mdv m. Dv kg m/s Te uncertainty in position is given by Dx pd p mm 15. An electron as a speed of 6 m/s wit an accuracy of.5%. Calculate te uncertainty wit wic we can locate te position of te electron. Solution: Given data: ħ J m kg v 6 m/s Dv m/s Capter_.indd 75 1/4/17 4:9:38 PM

25 Quantum Mecanics In jumping from an excited state to a stationary state, an atom takes 1 8 sec. Wat is te uncertainty in te energy of te emitted radiation? Solution: Given data: Dt 1 8 sec Formula: DE. Dt p Te excited atom gives up its excess energy by emitting one or more potons of caracteristic frequency. Te average time gap between te excitation of an atom and te time it radiates energy is 1 8 sec. So, te poton energy is uncertain by an amount given by DE p D t p J and te frequency of te ligt is uncertain by an amount. Dv DE Hz and tis limit cannot be reduced.. Using uncertainty relation, calculate te time required for te atomic system to retain rotation energy for a line of wavelengt 6 Å and widt 1 4 Å. Solution: Given data: dl 1 4 Å 1 14 m l 6 Å m Formula: E c l \ DE c l Dl and DE. Dt p 1 \ Dt p DE ħ l l p ħcdl pc. Dl -7 ( 6 1 ) sec. 1. If te uncertainty in te location of a particle is equal to its de Broglie wavelengt, wat is te uncertainty wit velocity? Solution: Given data: Dx l Formula Dx Dp Dx. mdv Dv v ml m / mv \ Dv v Capter_.indd 77 1/4/17 4:9:47 PM

26 78 Applied Pysics. A nucleon is confined to a nucleus of diameter m. Calculate te minimum uncertainty in te momentum of te nucleon. Also calculate te minimum kinetic energy of te nucleon. Solution: Given data: (Dx) max m Formula: (Dp) min (Dx) max ħ (Dp) min -4 ( Dx) max kgm sec. Now p cannot be less tan (DP) min p min (DP) min \ E P ( ) - m ev Find te energy of an electron moving in one dimension in an infinitely ig potential box of widt 1 Å, given mass of te electron kg m and Js. Solution: Given data: a 1 Å 1 1 m m kg Js Formula: E 8ma - ( ) ` E ( 1 ) J J J ev ev 4. An electron is bound by a potential wic closely approaces an infinite square well of widt m. Calculate te lowest tree permissible quantum energies te electron can ave. Solution: Given data: a m Formula: E n n 8ma -19 E 1 ( ) J (. 5 1 ) Capter_.indd 78 1/4/17 4:9:51 PM

27 8 Applied Pysics \ E Furter, or ( ) -7-1 ( )( 3 1 ) joule d sin q n l sin q n -1 l 1 ( 3 1 ) d ( ) q sin 1 (.4936) A ydrogen atom is.53 Å in radius. Use uncertainty principle to estimate te minimum energy an electron can ave in tis atom. Solution: Given x max.53 Å Heisenberg s uncertainty principle DD x p p (D x) max(dp) min p and (K.E.) min p min ( p ) min m D [ pmin Dpmin ] m. ( Dp) min 1 Dx p (i) kg m/sec -4 and (K.E.) min ( Dp ) min ( ) -31 m J 8. Te speed of an electron is measured to be m/sec to an accuracy of.3%. Find te uncertainty in determining te position of tis electron. Solution: Given v m/sec Formula used is DD x p p Dv v m/sec 1 1 and Dp mdv kg m/sec Dx 314. Dp m -31 Capter_.indd 8 1/4/17 4:3: PM

28 Quantum Mecanics An electron as speed of m/sec wit an accuracy of.1%. Calculate te uncertainty in position of an electron. Given mass of an electron as kg and Planck s constant as J sec. Solution: Given v m/sec and Dv m/sec m/sec. Formula used is 1 DD x p or Dx p p Dp Dp mdv Dx 1 Dp p Dx m 3. If an excited state of ydrogen atom as a lifetime of sec, wat is te minimum error wit wic te energy of tis state can be measured? Given J sec. Solution: Given t sec. Formula used is DEDt p DE p Dt J DE J 31. Find te energy of an electron moving in one dimension in an infinitely ig potential box of widt 1. Å. Given m kg and J sec. Solution: Given l m, m kg and J sec. Formula used is n E n 8mL n - ( ) ( 1. 1 ) For n 1, and for n, n J E J Capter_.indd 81 1/4/17 4:3:3 PM

29 or Dp D(mv) or mdv mv p Dv v p 1 p mv p D x p p Quantum Mecanics Te position and momentum of.5 KeV electron are simultaneously determined. If its position is located witin. nm, wat is te percentage uncertainty in its momentum? Solution: Given E J and x. 1 9 m. Now DxDp p and momentum p me so p or p kg m/sec or Dp 1 1 p D x - p. 1 9 Dp Percentage uncertainty in momentum Dp p kg m/sec % EXERCISE 1. Explain de Broglie s concept of matter waves.. Write an expression for te wavelengt of a particle. 3. Wat do you understand by a wave packet? Wat is te relationsip between pase velocity and group velocity? 4. Distinguis between pase velocity and group velocity. Sow tat te de Broglie wave group associated wit a moving particle travels wit te same velocity as tat of te particle. 5. Can a wave given by an equation y A sin(wt kx) represent a particle? Explain te concept of a wave packet? How does tis concept lead to Heisenberg s uncertainty principle. Capter_.indd 83 1/4/17 4:3:1 PM

30 Quantum Mecanics Compute te energy difference between te ground state and te first excited state for an electron in a one-dimensional rigid box of lengt 1 8 cm. Given m kg and J sec. [Ans: 114 ev] 6. Calculate te value of lowest energy of an electron in one dimensional force free region of lengt 4 Å. [Ans: J] 7. Te lowest energy possible for a certain particle entrapped in a box is 4 ev. Wat are te next tree iger energies te particle can ave? [Ans: 16 ev, 36 ev and 64 ev] 8. Find te energy levels of an electron in a box 1 nm wide. Mass of electron is kg. Also find te energy levels of 1 gm marble in a box 1 cm wide. [Ans: 6. 1 J, J and J; and for marble J, J and J] Capter_.indd 85 11/4/17 1:58:3 AM