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PRINCIPLES OF QUANTUM MECHANICS, , Engineering physics, , Introduction to Quantum mechanics:, Classical physics is basically based on three Newton’s laws of motion. According to, classical mechanics, the equation of motions explain the behavior of the macroscopic, particle systems. But the classical mechanic concepts cannot explain the atomic, phenomenon, stability of an atom, spectrum of hydrogen, black body of radiation,, photo electric effect, Compton Effect and Raman Effect etc., , Black body Radiation:,  A black body is body which absorbs radiations of all wavelengths incident upon it., It neither reflects not transmit any of the incident radiation and hence it appears, as black body.,  When black body is heated, it emits radiation which is known as black body, radiation.,  Ferry described the black body which consists of hallow thick walled sphere, painted internally with black lamp and provided a circular small opening to enter, the radiation.,  When any radiation enters opening, it suffers, multiple reflections inside the sphere and is, finally absorbed.,  When the walls of the sphere is heated to a, temperature T, the radiation emitted from, the sphere through the opening. This is called, as black body radiation., , Spectral Distribution:, Scientists were interested to know how the energy is distributed among the different, wavelengths in the spectrum of black body., But among all, Lummer and Pringsheim described the spectrum of the black body and, gave important findings by plotting a graph., , Vignana Bharathi Institute of Technology

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PRINCIPLES OF QUANTUM MECHANICS, , Engineering physics, ,  The energy of the black body is not uniformly distributed in the radiation, spectrum.,  At a given temperature, the intensity of radiation increases with increase in the, wavelength and becomes maximum at a particular wavelength ( m ). With further, increase in the wavelength, the intensity of radiation decreases.,  The wavelength corresponding to the maximum energy represented by the peak, of the curve shifts towards shorter wavelengths as the temperature increases., From these findings it can be easily explain the black body ration., , Wien’s Law:, Wien observed the results of Lummer and Pringsheim and derived the expression for, the energy of the black bod radiation., He explained that the wavelengths corresponding to the maximum energy ( m ), decreases as temperature increases., Therefore mT  Constant ---------------- (1), and, , E  5  Constant ---------------- (2), , This relation is known as Wien’s displacement law., Wien’s formula works well only for shorter wavelengths. There are considerable, deviations at long wavelengths and at high temperatures., , Vignana Bharathi Institute of Technology

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PRINCIPLES OF QUANTUM MECHANICS, , Engineering physics, , Rayleigh Jeans Law:, On the basis of the classical physics, Rayleigh and Jeans derived the expression for, energy density of a hot body., E d  , , 8 kT, d  --------------- (3), 4, , They observed that energy density increases with frequency and becomes infinite., Rayleigh-Jeans formula agreed with long wavelengths but not works for shorter, wavelengths., None of these two laws could explain the entire black body. The fundamental, assumptions of classical theory were fault. In 1901 Max Plank proposed a new, hypothesis known as theory of quanta., , Plank’s Radiation Law:, Max Plank explained the energy distribution in black body radiation. He derived the, radiation law using some assumptions., 1. A black body contains number of simple harmonic oscillators which can vibrate, with all possible frequencies., 2. The energy emitted from the oscillators is taken as discrete rather than, continuous., 3. The frequency of radiation emitted by an oscillator is the same as the frequency, of its vibration., 4. An oscillator cannot emit energy in a continuous manner, it can emit energy in, the multiples of a small unit called (photon). If an oscillator is vibrating with a, frequency (  ), it can only radiate in quantas of magnitudes h , i.e. the oscillator, can have only discrete energy values En, given by, En  nh  n Where, ,   h, , 5. Here n is an integer and h is plank’s constant ( h  6.625  1034 Joules  s ec ), , 6. The oscillator can emit or absorb radiation energy in packets of h values, ( 0, h , 2h ,3h , 4h ,5h....nh ), , Vignana Bharathi Institute of Technology

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Engineering physics, , PRINCIPLES OF QUANTUM MECHANICS, , If N be the total number of the Plank’s oscillators, and E be the their total energy, then, the average energy per Plank’s oscillator,  , ,  given by, , E, ---------------- (1), N, , Let there be N 0 , N1 , N 2 , N3 , N 4 ,......N r etc oscillators having energy, , 0,  , 2 ,3 , 4 ......r etc, N  N0  N1  N 2  N3  N 4 ,......  N r, , -------------- (2), , And E  0   N1  2 N 2  3 N3  4 N 4 ,......  r N r, , -------------- (3), , According to the Maxwell’s distribution formula, the number of oscillators having, ,  r , energy r is given by N r  N 0 exp ,  --------------- (4),  kT , Where k is Boltzmann constant., Substituting the values of N 0 , N1 , N 2 , N3 , N 4 ,......N r from (4) in equation (2) we get,  , N  N 0  N 0 exp ,  kT, ,  , N  N 0 1  exp ,  kT, , , ,  2,   N 0 exp , ,  kT, , ,  2,   exp , ,  kT, , ,  3,   N 0 exp , ,  kT, , ,  3,   exp , ,  kT, , ,  r,   .......  exp , ,  KT, , 2, 3, 4, n, We know that 1  x  x  x  x  ......  x , , N , , N0, ,    , 1  exp  kT  , , , , , ,  r ,   .......  N 0 exp , , ,  KT , , , , , 1, 1 x, , -------------- (5), , The total energy given E by, ,   ,  2 ,  3 ,  r , E  ( N 0  0)   N 0 exp ,   2 N 0 exp ,   3 N 0 exp ,   .......  r N 0 exp , ,  kT ,  kT ,  kT ,  kT ,    ,   ,  2 ,  3 ,  (r  1)  , E   N 0 exp ,  1  2 exp ,   3exp ,   4 exp ,   .......  r exp , ,  kT  ,  kT ,  kT ,  kT ,  kT  , 2, 3, r 1, We know that 1  2 x  3x  4 x  ......  rx , ,  ,  E   N 0 exp ,  kT, , 1, , ,  ,  , 1, , exp, , ,  kT, , , 1, (1  x) 2, , , , , , 2, , --------------- (6), , Vignana Bharathi Institute of Technology

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Engineering physics, , PRINCIPLES OF QUANTUM MECHANICS, , Now the average energy of oscillator is given by,    1  exp    ,  N 0 exp , , , , E, kT  ,  kT  , ,   , , 2, N , N0,    , 1, , exp, , , ,  kT  , , ,   ,  exp , , kT , ,  , ,    , 1  exp  kT  , , , ,   ,   , ,  exp ,  exp , ,   exp kT,  kT  ,  kT ,  , , , exp , 1, 1, kT, 1 , ,  exp , , kT , , , , , , , , ,  , , , exp , 1, kT, ,  , , h, ---------------- (7), exp h, 1, kT, , , , , , , , , , , , , , , , This is the expression for the average energy of oscillator in the black body., From the Maxwell distribution law, the number of oscillators per unit volume in, frequency range  and   d is given by, , 8 2, N  3 d --------------- (8), c, We know that E  N  , , ,  8 2,  , h, , Therefore E d   3 d  , ,  c,   exp h kT  1 , , , , , , E d , , , , 8 h 3, 1, , d, h, , c3, exp, 1, kT, , , , , , ---------- (9), , This is known as Plank’s law of radiation., , Vignana Bharathi Institute of Technology

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Engineering physics, , PRINCIPLES OF QUANTUM MECHANICS, , It can also expressed in terms of wavelength as:, , , , c, c, and d  2 d , , , 8 h  c ,   , c3   , 3, ,  E d  , , E d  , , 8 hc, , 5, , 1,  hc, exp ,   kT, , 1,  hc, exp ,   kT, , ,  1, , ,  c, ,  2 d ,  , ,  1, , , d, , ---------- (10), , This formula agrees well with the experimental curves throughout the entire range of, wavelengths., Derivation of Wien’s Formula from Plank’s Law:, We know that the plank’s law is, E d  , , 8 hc, , 5, , 1, d  --------------------- (1),  hc , exp ,  1,   kT , , Let C1  8 hc & C2 , ,  E d  , , C1, , 5, ,  E  5  C1 , , 1, C, exp  2,  T, , hc, k, ,  1, , , 1, ------------- (2),  C2 , exp ,  1,  T , , Which is a good agreement with Wien’s Law and also known as Wien’s Displacement, law., Derivation of Rayleigh Jeans Formula from Plank’s Law:, We know that, ,  hc, exp ,   kT, , hc, 1 hc, 1 hc, 1 hc, , , , ,  .....,   1,  kT 2!  kT 3!  kT 4!  kT, , , T is very high value we can neglect the higher order terms, ,  hc,  exp ,   kT, , hc, ,   1,  kT, , , Vignana Bharathi Institute of Technology

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Engineering physics, , PRINCIPLES OF QUANTUM MECHANICS, , 8 hc, , Therefore from equation (1) we get E d    5 , , 1, d, hc, 1, 1,  kT, , 8 hc, 1, , d, 5, hc, ,  kT, 8 hc  kT, E d   5 , d, , hc, ,  E d  , , E d  , , 8 kT, d  --------------------- (3), 4, , Which is Rayleigh jean law and is applicable for longer wavelength., , The energy distribution curve derived by Plank theoretically represents the, experimental results accurately, i.e. it explain entire black body radiation., , Photoelectric effect:, The emission of electrons from a metal plate when illuminated by light or any other, radiation of suitable wavelength or frequency is called photoelectric emission. The, emitted electrons are called photoelectrons and the phenomenon is called photoelectric, effect.,  This phenomenon was discovered by Hertz when he allowed ultraviolet light to, fall on zinc plate.,  This phenomenon was verified by Hallwaches, Lenard, J.J.Thomson and Millikan., , Vignana Bharathi Institute of Technology

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PRINCIPLES OF QUANTUM MECHANICS, , Engineering physics, ,  Millikan discovered that alkali metals like Na, K, Rb, and Cs eject electrons when, visible light falls on them.,  He investigated effect with a number of alkali metal over a wide range of light, frequencies., Experimental Study of Photo-electric effect:,  A simple experimental arrangement to study the photoelectric effect is shown, in fig.,  The apparatus consists of two photo sensitive surfaces A and B. enclosed in, an evacuated quartz tube.,  The plate A is connected to the negative terminal of a battery while the plate, B is connected to positive terminal of the battery through a galvanometer G.,  In the absence of any light there is no emission of the electrons and hence no, current flow through the galvanometer.,  When a monochromatic light is allowed to fall on the plate A, current starts, flowing in the circuit which is indicated by galvanometer. This current is, known as photo current., The number of photo electrons emitted and their kinetic energy depends, upon the following factors:, 1. The potential differential between the two plates., 2. The intensity of the incident radiation., 3. The frequency of the incident radiation., 4. Type of the photo metal used., , Characteristics of the photo electrons:, The effect of the potential difference (V), For a given photo metal A, and at fixed frequency and, intensity of the incident radiation, the effect of the, potential difference (V) on the photo current (I) is, shown in fig., , Vignana Bharathi Institute of Technology

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PRINCIPLES OF QUANTUM MECHANICS, , Engineering physics, ,  If the potential difference (V) of plate B is increased photo electric current (I) is, also increased and reached a maximum current known as saturation current.,  Further increasing in the Voltage hardly produces any appreciable increase in, current.,  If the potential kept zero, the current still flows in the same direction.,  If the potential of the plate made negative, the photocurrent does not, immediately drop to zero due to their electromagnetic force.,  If this negative or retarding potential further increased the photo current, decreases and finally becomes zero at a particular potential.,  The negative potential difference at which the photo electric current becomes, zero is called cutoff current or stopping potential (V0)., Effect of intensity of incident radiation:, For given photo metal and fixed frequency of, incident radiation, the effect of the intensity of, radiation on photo current is shown in fig.,  If the intensity of the incident radiation is, increased from I1 to I3 and the experiment, is repeated, then the photo electric, current increases in the same ratio for all potentials.,  Saturated current is increased with increase in the intensity.,  But in all cases the stopping potential is same. i.e. the stopping potential is, independent of the intensity of the incident radiation and the saturation current, is proportional to the intensity of incident radiation., The effect of the frequency of incident radiation:, The effect of the varying frequency of the incident, radiation while keeping the same metal and fixed, intensity of incident radiation is shown in fig., Stopping potentials are measured for different for, different frequencies., , Vignana Bharathi Institute of Technology

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PRINCIPLES OF QUANTUM MECHANICS, , Engineering physics, , The graph shows that the frequency 0 the stopping potential is zero. Then the, frequency is known as threshold frequency., Hence it can be defined as the minimum frequency ( 0 ) of the incident radiation which, can cause photoelectric emission, i.e., this frequency is just able to liberate electrons, without giving them additional energy., , The effect of Photo metal:,  The variation of stopping potential with, frequency of incident radiation is shown in, the fig.,  When the stopping potential is measure for different frequencies of different, metal surfaces, it is clear that the lines have the same slope but their interactions, with frequencies are different.,  Thus the threshold frequency is a function of photo metal., , Fundamental laws of photo-electric emission:, 1. The number of electrons emitted per second, i.e., photo electric current is, proportional to the intensity of the incident light., 2. If the frequency is less than the threshold frequency no electrons can be emitted, from the metal surface however intense the incident radiation may be. The, wavelength corresponding to the threshold frequency is called as threshold, wavelength., 3. The maximum velocity or the kinetic energy of the photoelectrons depends on, the frequency of incident radiation but not on the intensity of radiation., 4. The rate at which the electron emission from the photo metal is independent of, its temperature., 5. Electron emission from a photosensitive surface is almost instantaneous and, emission continuous as long as the frequency of incident radiation is greater than, , Vignana Bharathi Institute of Technology

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PRINCIPLES OF QUANTUM MECHANICS, , Engineering physics, , the threshold frequency. The time lag between the incident radiation and the, emission of electrons is less than 10-8 seconds., 6. For a given metal surface, stopping potential is directly proportional to the, frequency of the incident radiation but not in the intensity of the incident, radiation., , Einstein’s Photoelectric Equation:, On the basis of quantum theory, Einstein derived an equation for the photo electric, effect known as Einstein photo electric equation., Einstein assumed that,, , , Light consists of photons or quanta of energy, energy of each photon is h ., ,  Each incident photon colloids with an electron inside an atom and give its all, energy to the electron.,  The part of its energy used by electron to come out of the metal surface and, remaining part of its energy is used to giving kinetic energy to the emitted, electron.,  The part of the energy which is used to eject the electron from the metal surface, is known as photoelectric work function of the metal denoted by W0., Thus, , 1, h  W0  mv 2 ------------------- (1), 2, , Where v is velocity of the emitted electron, this equation is known as Einstein equation, of the photo electric equation., If the total incident energy is utilized to eject the electron from the metal surface then, the kinetic energy of the emitted electron will be zero., Then W0  h0, , ------------------ (2), , Where 0 is called as threshold frequency and it can be defined as the minimum, frequency which can cause photo electric emission., Substituting the value of work function in equation (1), we get, , Vignana Bharathi Institute of Technology

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PRINCIPLES OF QUANTUM MECHANICS, , Engineering physics, , 1, h  h0  mv 2, 2, 1,  mv 2  h  h0, 2, 1,  mv 2  h(  0 ) ---------- (3), 2, This is another form of Einstein’s Photo-electric equation., This equation predicts all the experimental results., 1. For a particular metal surface work function is constant and hence, , 1, KE  mv2  h, 2,  v2  , Thus the increase in the frequency of incident radiation causes increase in the velocity, of photoelectric electrons provided intensity of incident radiation is constant., 2. An increase in the intensity of incident radiation is equivalent to an increase in the, number of photons fall on the metal surface. At a frequency of incident radiation is, greater than the threshold frequency, then number of emitted electrons will, increase. Hence the intensity of emitted electrons is directly proportional to the, intensity of incident radiation., 3. If V0 is the stopping potential then the kinetic energy of the emitted electron is, equal to the potential energy of the electron., We know that, Therefore, , 1 2, mv  h  h0, 2, ,  eV0  h  h0,  V0 , , h h0, , ------------------ (4), e, e, , As h, e and 0 are constants for a given photo metal, surface a graph between stopping potential and frequency, shows a straight line with the slope h, , e, , . This is same as, , experimental results., , Vignana Bharathi Institute of Technology

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PRINCIPLES OF QUANTUM MECHANICS, , Engineering physics, , Compton Effect:, Arthur H. Compton investigated the on the scattering of X-rays by materials of low, atomic numbers., Compton effect conforms the particle nature of electromagnetic radiation by showing, that photons carry energy and momentum like any material particle., Compton stated that when a beam of monochromatic X-rays is scattered by a, material, the scattered radiation consists of two components of X-rays. One of the, components having longer wavelength than that of the incident X-rays and another, component is having same wavelength as that of incident beam., , Experimental arrangement:, , The Schematic diagram of apparatus for Compton effect is shown in fig.,  A monochromatic X-rays of wavelength    from the molybdenum, , target is, , collimated by passing it through the slits L1 and L2.,  The collimated beam is then incident on the graphite block. The graphite block, scattered the X-rays in different directions.,  The scattered X-rays are detect by the Bragg’s Spectrometer. The scattered angle, ,   and the intensities are can be determined by Bragg’s spectrometer.,  Graphs are plotted for variation of intensity verses wavelength of scattered Xrays., , Vignana Bharathi Institute of Technology

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Engineering physics, , PRINCIPLES OF QUANTUM MECHANICS, ,  The graph is consist of only two peaks which shows that two wavelengths of Xrays are scattered.,  The X-rays whose wavelength is changed is known as modified component, and, whose wavelength is not changed is known as unmodified component.,  The difference in the wavelengths between the modified and unmodified X-rays is, known as Compton shift. It is denoted as      ,  The Compton shift is independent of the wavelengths of both modified and, incident beams. It is depends on the scattering angle   ., Compton assumed that..., , , X-rays are quantized and regarded as photons. A photon carries energy h, ,  Electron in the atom of target material considered as at rest initially.,  When scattering occurred, there is a simple elastic collision between a photon, and an electron.,  As a result of collision, the incident photon loses a part of its energy and, scattered with an angle   , and the electron gains kinetic energy and recoils in a, direction   of incident X-rays., Derivation of Compton shift:      , Let a beam of X-rays of energy h and momentum, , h, collide with an electron at rest., c, , As the collision is an elastic collision, the energy and momentum of the system is, conserved., , Vignana Bharathi Institute of Technology

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Engineering physics, , PRINCIPLES OF QUANTUM MECHANICS, , Hence the energy and momentum of both photon (X-rays) and electron is given as, Before collision, Photon, , Electron, , After collision, , Energy, , h, , h , , Momentum, , h, c, , h , c, , Energy, , mo c 2, , mc 2, , Momentum, , 0, , mv, , Conservation of energy:, According to the principle of conservation of energy,, Total energy of the system before collision = Total energy of the system after collision, , h  mo c 2  h   mc 2, ,  h  mo c 2  h   mc 2, , --------- (1), , Conservation of momentum:, According to the principle of conservation of momentum,, Total momentum of the system before collision = Total momentum of the system, after collision, Therefore, Horizontal component of momentum is, , h, h , 0, cos  mv cos , c, c, h h , , cos  mv cos  ------------- (2), c, c, Squaring on both sides, , h 2 2 h 2 2, h 2 , 2,  2 cos   2 2 cos  m 2v 2 cos 2 , 2, c, c, c, Vertical component of momentum is, , 00, , h , sin   mv sin , c, , h , sin   mv sin , c, , Vignana Bharathi Institute of Technology

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PRINCIPLES OF QUANTUM MECHANICS, , Engineering physics, , Squaring both sides, , h 2 2 2, sin   m 2v 2 sin 2  -------------- (3), 2, c, Adding these two equations (2) & (3), we get, , h 2 2 h 2 2, h 2 , h 2 2 2, 2,  2 cos   2 2 cos  2 sin   m 2v 2 cos 2   m 2v 2 sin 2 , 2, c, c, c, c, , h 2 2 h 2 2, h 2 , 2, 2,  2  cos   sin    2 2 cos  m 2v 2  cos 2   sin 2  , 2, c, c, c, h 2 2 h 2 2, h 2 , , , 2, cos  m 2v 2, 2, 2, 2, c, c, c, h 2 2  h 2 2  2h 2  cos  m 2v 2c 2 ------------ (4), Squaring the equation (1) we get, , h 2 2  m 2 oc 4  h 2 2  2h moc 2  2h h   2h moc 2  m 2c 4 ------------- (5), (5)  (4), h 2  2  m 2 o c 4  h 2  2  2h moc 2  2h 2   2h moc 2  h 2  2  h 2  2  2h 2  cos,  m 2c 4  m 2v 2c 2, , m 2 o c 4  2hmoc 2       2h 2  1  cos   m 2c 2  c 2  v 2  ------------ (6), We know that the relation between m and mo is given as, , m, , mo, v2, 1 2, c, , ,  m2, Squaring on both sides m 2   o 2,  1  v, c2, , , , , mo 2c 2, 2, , m, , , c2  v2, , , ,  m2  c 2  v 2   mo 2c 2 ------------- (7), Substitute this value in equation (6) we get, , Vignana Bharathi Institute of Technology

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PRINCIPLES OF QUANTUM MECHANICS, , Engineering physics, , m 2 o c 4  2hmo c 2       2h 2  1  cos   mo 2c 2c 2, , m o2c 4  2hmo c 2       2h 2  1  cos   mo 2 c 4, 2hmo c 2       2h 2  1  cos , mo c 2, ,       h 1  cos, , , , , ,  1 1,  mo c 2     h 1  cos ,    , ,    ,  moc 2     h 1  cos ,  c c,  moc        h 1  cos , ,       , , c, , , , ,  , , , h, 1  cos , mo c, ,          , , h, 1  cos , mo c, , -------------- (8), , This equation is known as Compton shift., From this equation it is clear that,, 1. The Compton shift is independent of the wavelengths of both incident and, modified X-rays and It is solely depends on the scattering angle., , , , 2. As  values varies from 0 to 180, o, , o, ,  , cos values varies from (1 to  1) and, , , , hence Compton shift is varies from  0 to, , , , 2h , , moc , , Vignana Bharathi Institute of Technology

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Engineering physics, , PRINCIPLES OF QUANTUM MECHANICS, , Wave and particle nature:, To understand the concept of dual nature, one should have the knowledge of, characteristics of particles and waves., It easy to understand the concept of particle, since it is a physical quantity and it can be, seen. A particle has a definite mass and occupies a particular space. The characteristics, of the particle are, 1. Mass (m) 2. Velocity (v), , 3. Momentum (p), , 4.Energy (E), , A wave can be simply defined as spreading out of the disturbance in a medium in all the, direction uniformly. It is not possible to say the wave is present here and there. The, characteristics of a wave are, 1. Wavelength (  ) 2. Frequency (  ), , 3. Amplitudes (A), , 4. Phase (  ), , 6. Intensity (I), , 5. Wave Velocity (  ), , In experiments like photoelectric effect and Compton effect exhibits the particle nature,, where as the in the experiments like interference and diffraction, it exhibits wave, nature., Hence we concluded that radiation is exhibiting dual nature. But it is not possible to, exhibit both the wave and particle nature at the same time., , De-Broglie Hypothesis: 1. De-Broglie in 1924 extended the wave-particle duality to material particles like, electrons, protons, and neutrons can behave as wave. According to his hypothesis,, a moving particle is associated with a wave with a certain wavelength is known as, De Broglie wave or matter wave., 2. His hypothesis is based on the fact that ‘nature loves symmetry’. That is if radiation, exhibits dual nature of wave and particle; matter will also exhibit the same dual, nature., 3. According to de-Brogile a moving particle, whatever its nature, has wave properties, associated with it. He proposed that the wavelength  associated with any moving, particle of momentum p is given by, , Vignana Bharathi Institute of Technology

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PRINCIPLES OF QUANTUM MECHANICS, , , , h, p, , , , h, mv, , Engineering physics, , Such waves associated with the matter particles are called matter waves or de-Broglie, waves., 4. De-Broglie indicated that an electron in a Bohr orbit moving around the atom's, nucleus would possess standing waves. De Broglie used his matter-wave hypothesis, to explain quantization of atomic orbital., , 5. The n full wave lengths of a de Broglie electron wave fit around the circumference of, the electron’s circular orbit. That is n  2 r, , Expression for de-Broglie wavelength:According to the Planck’s and Einstein theories the energy of a photon whose frequency, ,  can be expressed as, E  h ------------------- (1), Where h is the Planck’s constant., According to Einstein’s mass energy relation, ,   m c 2 ------------------- (2), From eq (1) and (2), h  m c, , h, , 2, , c,  mc2, , , Vignana Bharathi Institute of Technology

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PRINCIPLES OF QUANTUM MECHANICS, , , , , , Engineering physics, , h, m c, , For electron we put v in place of c, , ------------------------------ (3), This is called de-Broglie’s equation., The wave length of the matter wave is inversely propositional to its momentum., , Wave length of the matter wave in terms of potential V:(Wave length of charged partical), The energy of the electron in terms of potential can expressed as, , E  eV ---------------------------------- (4), The kinetic energy of an electron is, , E, , 1 2, mv --------------------------------- (5), 2, , From equation (4) and (5), , 1, eV  mv 2, 2, Multiplying by ‘m’ in either sides of the equation, , meV , , 1 2 2, mv, 2, , P  2meV, Now from equation (3), , , , h, 2meV, , Substituting the value of h, m and e, the wavelength associate with electron, , , , 6.625  1034, 2  9.1  1031  1.6  1019 V, , Vignana Bharathi Institute of Technology

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PRINCIPLES OF QUANTUM MECHANICS, , , , Engineering physics, , 12.27  1010, m, V, , Wave length of the matter wave in terms of energy E:If E is the kinetic energy of the electron then, , E, , 1 2, m2v 2, mv , 2, 2m, , p2, E,  p  2mE, 2m, Now from de Broglie hypothesis the wave length of the matter wave in terms of energy, is  , , h, 2mE, , Properties of matter waves:1. The wave length of the matter wave is inversely proportional to the mass of the, particle. The larger the mass of the particle, the shorter will be the wave length, vice versa., 2. The wave length of the matter waves is is inversely proportional to the velocity of, the particle., 3. The matter waves are produced whenever the matter particle (charged or, uncharged) is in motion where as the electromagnetic waves are produced, whenever charged particles are in motion. This property shows that the matter, waves are not electromagnetic waves., 4. The velocity of the matter wave is not constant. The velocity of the matter wave, depends on the motion of a material particle. The velocity of the electromagnetic, radiation is constant., 5. The wave nature of matter introduces an uncertainty in the location of the, position of the particle., , Vignana Bharathi Institute of Technology

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PRINCIPLES OF QUANTUM MECHANICS, , Engineering physics, , 6. The velocity of the matter wave is greater than the velocity of light. Equating the, Einstein’s equation and Planck’s equation we, get, , E  hv, , And, , E  mc 2, , mc 2, hv  mc  v , h, 2, , The wave velocity is w  frequency  wavelength, w, , Substituting  , , w, , mc 2, , h, , h, we get, mv, , mc 2 h c 2, c2, , ,  w, h mv v, v, , As the particle velocity v cannot exceed velocity of light c, w is greater than the, velocity of light c ., , Davisson and Germer experiment:The first practical evidence for the matter waves was given by C. J. Davisson and L.H, Experimental arrangement: .Germer in 1927. This was the first experimental support for De-Broglie’s hypothesis., , Vignana Bharathi Institute of Technology

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PRINCIPLES OF QUANTUM MECHANICS, , Engineering physics, , 1. The experimental arrangement is shown in fig. it consists of three parts; they are, electron gun, target set up and circular scale arrangement. The whole, experiment is kept in vacuum., 2. The electron gun produces a fine beam of electrons of a required velocity. It, consists of filament (F), low tension battery (LTB), high tension battery (HTB) and, pin holes provided in the cylinder (C)., 3. When tungsten filament ‘F’ is heated by low tension batteries (LTB) then, electrons are produced. These electrons are accelerated to a required velocity by, applying sufficient potential through the high tension battery (HTB), across the, cylinder ‘C’., 4. The accelerated electrons are collimated into a fine beam of pencil by passing, them through a system of pin holes provided in the cylinder ‘C’., 5. The target set up help to get diffraction pattern. The target is typically nickel, crystal. The fast moving beam of electrons from electron gun is made to incident, on the nickel target, which can be rotated about an axis perpendicular to the, plane of the diagram., 6. The electrons are reflected in all possible directions by atoms at lattice points in, the surface planes, which acts as a diffraction grating., 7. In the circular scale arrangement, an electron collector is fixed to a circular scale, which can collect the electrons and can move along the circular scale. The, electron collector is connected to a sensitive galvanometer to measure the, intensity of electron beam entering the collector., , Calculation of the wavelength associated with electrons:When a potential of 54v is applied the first order diffraction maximum is observed at, angle of 500 between incident and reflected rays., It can be observed in the plot of variation of number of scattering electrons with the, angle of diffraction as 650., The inter planar spacing (d) of nickel crystal is 0.091nm, which is measured by the x-ray, diffraction method., , Vignana Bharathi Institute of Technology

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PRINCIPLES OF QUANTUM MECHANICS, , Applying Bragg’s law, We know that, , Engineering physics, , 2d sin   n, ,   650, d  0.091nm, n 1, , Now from the Bragg’s law equation, we get,   1.648A0, , The de-Brogile wavelength associated with the electron, when a potential difference of, 54v is applied., According to de-brogile wave length, , , 12.26 12.26, ,  1.66 A0, V, 54, , The wavelength of the electron beam calculated from Bragg’s law and de-Brogile’s, equation are in good agreement. Hence the wave nature of the particle is proved, experimentally., The drawback of this experiment is that whether the diffraction pattern formed is due, to electrons (or) electromagnetic radiation generated by fast moving electrons are not, known., , Heisenberg’s uncertainty principle:In 1927 Heisenberg proposed a very interesting principle known as uncertainty principle, as a consequence of the dual nature of the matter., Statement:It is impossible to determine precisely and simultaneously the value of pair of, physical variables which describe the motion of an atomic system, , Vignana Bharathi Institute of Technology

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PRINCIPLES OF QUANTUM MECHANICS, , Engineering physics, , 1. According to “uncertainty principle” if we obtained a perfect knowledge of the, position of an electron we have no way of knowing its momentum, and the viseverse., 2. If particle is moving based on classical mechanics, at any instant we can find its, position and momentum. In wave mechanics, we regard a moving particle as a wave, group. The particle that corresponds to this wave group may be located anywhere, within the group at any given time., 3. In the middle of the group, the probability of finding the particle is more but the, probability of finding the particle at any other point inside the wave group is not, zero., 4. Narrow the wave group higher will be the accuracy of locating the particle. At the, same time, one cannot define the wavelength  of the wave accurately when the, , , , h , , , wave group is narrower. Since measurement of particle’s momentum   ,  also, mv, becomes less accurate., 5. On the other hand when we consider a wide wave group, wavelength  can be well, defined and hence measurement of momentum becomes more accurate, at the, same time, since the width of the wave group is large, locating the position of the, particle becomes less accurate. Thus we have uncertainty principle., 6. If x and p are the uncertain in the position and momentum measurements then,, according to uncertainty principle, x p , , h, 4, , Another form of the uncertainty is in energy and time. If the energy is emitted in the, form of electromagnetic waves, we cannot measure the frequency v of the waves, accurately in the limited time available., Since, , v , , 1, t, , Hence the corresponding uncertainty in energy E is,  E  h v, , Vignana Bharathi Institute of Technology

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PRINCIPLES OF QUANTUM MECHANICS, , And so, , E , , Or, , Engineering physics, , h, t, , E  t  h, , A more precise calculation based on the nature of wave group modifies this result to, E t , , h, 4, , This gives uncertainty in the measurement of an energy and time of a process., , Schrödinger wave equation:In 1926 Schrödinger presented wave equation by using de Broglie ideas of matter, waves. The Schrödinger‘s equation is the fundamental equation of quantum mechanics., It is the differential equation for the de-broglie waves associated with particles and, describes the motion of the particles., Schrödinger introduced a mathematical function  ( x, y, z , t ) which is the variable, quantity associated with the moving particle, and is a complex.  is called “wave, function” as is characterizes the waves associated with the particle., If a particle of mass ‘m’ moving with a velocity ‘v’ is associated with group of waves. Let, us consider a simple form progressing wave traveling in positive x-direction as time t is, ,    0 sin ( t  k x) ------------------ (1), ,    (x,t) and, , Where, ,  0 is amplitude., , Differentiating  partially with respect to x twice,, , 2, k, x2, , 2, ,  0 sin ( t  k x), ,   k 2, , , ,  2, k, x2, , 2, ,   0 -------------------- (2), , Where k is wave number and is given as k , , 2, , , Equation (2) is the differential form of the classical wave equation. Now we incorporate, de Broglie wavelength  , , h, and k into equation (2)., mv, , Vignana Bharathi Institute of Technology

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PRINCIPLES OF QUANTUM MECHANICS, , Engineering physics, , Then we obtain,  2  4 2 m, , x2, h2, , 2, , v, , 2, ,   0 ----------------------------- (3), , The total energy E of the particle is sum of its kinetic energy K and potential energy V,, i.e. E  K  V ------------------------------------ (4), 1, 2, , 2, and K  mv -------------------------------------- (5), , m, , 2, , v 2  2 m ( E  V ) ----------------------------------- (6), , Substituting equation (6) in equation (3),  2  8, , x2, , 2, , m(E  V ),  0, h2, , In quantum mechanics, the value, , h, h, occurs most frequently. Hence we denote h , ., 2, 2, , Using this notation, we have, 2, 2 m ( E  V), ,   0 ------------------------------ (7), 2, x, h2, , Where   (x,t) .,  2   2   2  2 m ( E  V), , , ,   0 ------------------------- (8), x2, y2, z2, h2, , Using Laplacian operator,, 2, , 2, 2, 2, -------------------------------- (9), , , x2 y2 z2, , The equation (9) can be written as, , , , 2, ,  , , 2 m ( E  V),  0, h2, , This is Schrödinger time independent wave equation., , Vignana Bharathi Institute of Technology

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PRINCIPLES OF QUANTUM MECHANICS, , Engineering physics, , Born interpretation on  (OR), Physical significance of the wave function ( ): 1. The probability that a particle will be found at a given place in space at a given, instant of time is characterized by the function   x, y, z , t  .It is called the wave, function. This function can be either real or complex., 2. A satisfactory interpretation of the wave function  associated with a moving, particle was given by Born in 1926. He postulated that the square of the magnitude, of the wave function , , 2, , (or)   * evaluated at a particular point at any instant, , represents the probability of finding the particle at that point., , 3. , , 2, , is called the ‘probability density’ and  is the ‘probability amplitude’. According, , to this interpretation, the probability of finding the particle within an element of, 2, volume d (dxdydz ) is  d , since the particle is certainly somewhere in space. So, , the integral of  d over all space must be unity, that is, 2, ,  , , 2, , d  1, , The wave function that obeys this equation is said to be ‘normalized’., 4. There are certain limitations for  :(1)  must be finite every where: – if for instance  is infinite for a particular point,, the same would be true for the wave function * . It would mean an infinitely large, probability of finding the particle at that point. This would violate the uncertainty, principle. There fore  must have a finite or zero value at any point., (2)  must be single-valued : - If  has more than one value at any point, it would, mean more than one value of probability of finding the particle at that point which is, obviously ridiculous., (3)  must be continuous and have a continuous first derivative every where, , Vignana Bharathi Institute of Technology

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PRINCIPLES OF QUANTUM MECHANICS, , Engineering physics, , d 2, It necessary for the Schrödinger equation is that, must be finite every where. This, dx 2, , can be so only if, , d, d, has no discontinuity at any boundary. So, is a continuous, dx, dx, , function then  is also continuous across the boundary., , Particle in one dimensional potential box:Consider a particle moving inside a box along the X-direction. The particle is bouncing, back and forth between the walls of the box. The box has potential barriers at X=0 and, X=L i.e. the box is supposed to have walls of infinite height at X = 0 and X = L. the particle, has mass m and its position x at any instant is given by 0 < X< L., The potential energy V of the particle is infinite on both sides of the box. The potential, energy V of the particle can be assumed to be zero between X = 0 and X = L., In terms of the boundary conditions imposed, by the problem, the potential function is, V  0 For 0 < X < L, V   For X  0, V   For X  L, , The particle cannot exist outside the box and, so its wave function  is 0 for X  0 and X  L . Our task is to find what  is within the, box i.e. in between x = 0 and x = L. Within the box, the Schrödinger’s equation becomes, d 2  8 2 m, , E   0 ----------------------- (1), dx2, h2, 8 2 mE,  k 2 , the equation becomes, Putting, 2, h, d 2, k, dx2, , 2, ,   0 ------------------------------ (2), , The general solution of equation (2) is,   A sin k x + B cos k x -------------------- (3), , The boundary conditions can be used to evaluate the constants A and B in Eq (3)., , Vignana Bharathi Institute of Technology

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Engineering physics, , PRINCIPLES OF QUANTUM MECHANICS,   0 at x  0 and hence B  0,   0 at x  L . Hence 0  A sin k L, , Since A  0 , kL= n where n is an integer or k , Thus, , , , n, , n, L, , n x, ----------------------------- (4), L, , ( x )  A sin, , The energy of the particle, En , , k 2h2, h2n2 2, , ----------------------------- (5), 8 2 m L 2 8 2 m, , E, , n, , n2h2, , ---------------------------------- (6), 8mL 2, , For each value of n, there is an energy level and the corresponding wave function is, given by Equation (4)., Each value of En is called an Eigen value and corresponding  n is called wave function., Thus inside the box the particle can only have discrete energy values specified by, equation (6). Note that particle cannot have zero energy., The value of ‘A’ in equation (4) can be obtained by applying normalization condition., Since the particle is inside the box of length L, the probability that the particle is found, inside the box is unity., , , , , L, , 2, , A, , , , 0, , , , L, 0, , *, , sin, , 2, , n  x, ,  1,  L , , 2n x, ,  1  cos, L dx=1, , 0 , 2, , , , , , , , A, , , , A2, , L, 1, 2, , , , A, , 2, L, , 2, ,  d x 1, , L, , The normalized wave functions of the particle = , , n, , , , 2, n x, sin, L, L, , Vignana Bharathi Institute of Technology

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PRINCIPLES OF QUANTUM MECHANICS, , Engineering physics, , The normalized wave functions 1 ,  2 and  3 and corresponding probability density, functions  n are plotted in fig., 2, , Vignana Bharathi Institute of Technology