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NPTEL – Electrical & Electronics Engineering – Semiconductor Nanodevices Wave Particle Duality and Heisenberg Principle, Schrodinger Wave Equation, Fermi-Dirac and Bose-Einstein Distributions R. John Bosco Balaguru Professor School of Electrical & Electronics Engineering SASTRA University B. G. Jeyaprakash Assistant Professor School of Electrical & Electronics Engineering SASTRA University Joint Initiative of IITs and IISc – Funded by MHRD Page 1 of 17 NPTEL – Electrical & Electronics Engineering – Semiconductor Nanodevices Table of Content 1 INTRODUCTION…………………………………………………………………. 3 1.1 THE FAILURE OF CLASSICAL MECHANICS ………………………………………………….3 1.2 ORIGIN OF MATTER WAVES …………………………………………………………………...3 2 WAVE PARTICLE DUALITY ………………………………………………….5 2.1 DUALITY RELATION ………………………………………………………….............................5 3THE PRINCIPLE OF UNCERTAINITY……………............................................7 3.1 CONCEPT ……………………………………………………………………………………………7 3.2 STATEMENT ………………………………………………………………………………………..7 4 SCHRÖDINGER WAVE EQUATION ………………………………………….8 4.1 WHY SCHRÖDINGER’S EQUATION FOR NANOSTRUCTURED SYSTEMS? ………………..8 4.2 SCHRÖDINGER’S EQUATION …………………………………………………………………….9 5 FERMI-DIRAC AND BOSE-EINSTEIN DISTRIBUTIONS …………………10 5.1 NEED FOR STATISTICS …………………………………………………………………………..10 5.2 BOSE – EINSTEIN STATISTICS ………………………………………………………………….10 5.3 CONCEPTS …………………………………………………………………………………………11 5.4 INFERENCES ………………………………………………………………………………………12 5.5 FERMI – DIRAC DISTRIBUTION ………………………………………………………………...12 6 SOLVED PROBLEMS …………………………………………………………….14 7 REFERENCES ……………………………………………………………………..17 Joint Initiative of IITs and IISc – Funded by MHRD Page 2 of 17 NPTEL – Electrical & Electronics Engineering – Semiconductor Nanodevices 1 Introduction 1.1 The failure of classical mechanics Bohr’s theory of atoms proposed in 1913 was able to account for the basic laws of the theory of thermal radiation and spectroscopy, yet the explanation about the obtained spectra was not sufficient. Classical physics was able to calculate the brightness of spectra but could not account for their origin. Quantum physicson the other hand was able to explain the essence of spectra but could not calculate the brightness of the spectra. Bohr concluded that both the theories should be harnessed to extend the areas where they more or less coincide. Bohr rejected the classical views on electron motion and introduced the concept of electron orbiting the nucleus the same way that the earth moves round the sun. His theory gave the correct explanation for the photon‘s origin in atom but its process remained unanswered and did not follow any of its postulate. Bohr’s theory was a tremendous leap (step forward) in understanding atomic world yet its limitations had to be broken down with the aid of quantum tool, which emerged very soon in the form of dual nature concept. 1.2 Origin of matter waves In 1929, a remarkable article by a then unknown French physicist, Louis de Broglie, appeared in the September issue of English “Philosophical Magazine”, in which the author described the possible existence of matter waves. That was the era, where the electromagnetic waves and sound waves were known, which are quite material in the sense that they can be perceived by our sensory organs or recording instruments. The de Broglie waves on the other hand were not perceivable. To elucidate on the significance of the discovery of the matter waves, let us try to understand the difference between the matter waves and other types of waves. Last century physicists discoveredthe sound (mechanical) and light (electromagnetic) waves. Sound waves need medium like air, water and matter generally for its propagation i.e., on the moon, spacecraft will start up in absolute silence. But the light waves on the other hand need no medium i.e., on the moon, astronauts will watch dazzling fire eject from the bottom of their space rocket in complete silence. Thus, in vacuum one can see and cannot hear.The matter waves proposed by de Broglie are unorthodox and paradoxical and do not resemble either mechanical or electromagnetic waves. Anyway, what are these matter waves? De Broglie suggested that these waves are generated due to the motion of anybody like a planet, a stone, a particle of dust or an electron. We usually device instruments to detect waves outside the nature’s window of human detectable range; human eye (0.4 to 0.7 microns) and ear (20-20kHz). But then, why can’t we see or detect de Broglie waves? Joint Initiative of IITs and IISc – Funded by MHRD Page 3 of 17 NPTEL – Electrical & Electronics Engineering – Semiconductor Nanodevices For example, radio receivers pick up radio waves alone; scintillation counters detect gamma rays and so on. Therefore, matter waves should also be detectable by an appropriate detector. However, to understand why they remain obscure, we should consider the wavelength of the matter waves proposed by de Broglie, λ= h mv Let us consider three objects of different scale in mass and dimensions; (a)the planet(earth), (b)human body and (c)an electron, to understand the secret of why matter waves are imperceptible. a) For planet earth: The mass of the earth is 6x1027 and its velocity of orbital motion (around the sun is 3x106 cm/sec approx). Then its de Broglie wavelength is λ= 6.6 x10 −27 h = = 36 x10 −61 cm 27 6 mv 6 x10 x3x10 This value is extremely small and no existing instrument can record something that small. b) Let us calculate the wavelength of a human, whose weight is 50x103gm, moving with a speed of 85 cm per sec. From de Broglie’s wavelength formula, λ= 6.6 x10 −27 = 1.38 x10 −33 cm 3 56 x10 x85 Even this is too small to be ever detected by the present instruments. c) Now let us consider the matter waves for electron. It has a mass of 10-27 gm. The electron will acquire a velocity of 6.1x107 cm/sec when it is placed between an electric field of potential difference one volt then the wavelength of the matter wave is, λ= 6.6 x10 −27 = 10 −7 cm 7 − 27 6 x10 x10 This corresponds to the wavelength of x-rays and is detectable with the principle of diffraction. Hence the presence of matter waves at the nano scale dimensional particle is traceable and so the presence of nanoparticle could be analyzed in terms of de Broglie wavelength. The detection of matter waves confirms the presence of moving particle say electron which ultimately decides the conductivity in nano devices. Hence de Broglie concept got its significance at the nano dimensions. Joint Initiative of IITs and IISc – Funded by MHRD Page 4 of 17 NPTEL – Electrical & Electronics Engineering – Semiconductor Nanodevices With this brief introduction, let us start the discussion on the first module of our course. 2 Wave particle duality The realization of matter at the finest level leads to a new branch of physics – quantum physics – the ultimate conceptual basis to study and implement Nanoscience. It is the field of optical science which made the scientiststhinksaway from corpuscular theory to explain few phenomenons in optics such as interference. The concept of dual nature of particles came into existence to satisfy the above mentioned properties as it was first proposed by Louis de Broglie in 1923 A.D. The presence of matter waves was first experimentally verified by C.J. Davisson and L.H.Germer at the Bell Telephone Laboratories. Later many confirmatory experiments were carried out (like e/m), which suggested particle nature of matter. 2.1 Duality relation The relation between energy E and frequency υ of a photon as given by Einstein is E = hν (1) Assuming particle nature of photons, then the energy E of photon as obtained by Compton effect is (2) E 2 = p 2 c 2 + m02 c 4 , which yields E = pc (3) Eqn. 3, is obtained by considering the rest mass of the photon. Thus, comparing Eqn. 1 and 3, we obtain p = hν / c or, p = h/λ This gives the relation between wave and particle nature of photons. Hence Louis de Broglie proposed that, all moving particles have wave nature and the wavelength is given by, λ = h / p = h / mv From photons this duality nature was extended to sub atomic particles such as electrons and protons which follow the quantum mechanical laws for its behaviour. These particles (matter) exhibitboth particle and wavebehaviorssimultaneously. Joint Initiative of IITs and IISc – Funded by MHRD Page 5 of 17 NPTEL – Electrical & Electronics Engineering – Semiconductor Nanodevices Thus, de Broglie related the particle and wave natures of matter by providing a relation between the particle properties; mass and velocity and its wave property viz. wavelength. From the example discussed about the de Broglie wavelength associated with planet earth, human being and an electron, it can be seen that de Broglie wavelength reduces with the mass. The wavelength is length in space over which there is probability of finding the particle at a given instant. Therefore, we can say that with the increase of mass, the matter prefers particle nature. Louis de Broglie’s review: If it is possible for the radiation to have the dual nature, then it should also be possible for particles like electron to exhibit wave properties under suitable conditions. In support to his view, he quoted three main points * Nature is symmetrical * There is a close parallelism between mechanics and geometrical optics * The stable orbits for electron as proposed by Bohr. Different forms of de Broglie wavelength: h h = p mv h Relating wavelength and kinetic energy, λ = 2m E k Relating wavelength and momentum λ = (i) (ii) (iii) (iv) since p = 2mEk i.e., for non-relativistic cases The kinetic energy of a charged particle carrying q charges is given by E k =qV where V is the accelerating potential then, h λ= 2mqV The wavelength of the material particle in thermal equilibrium, is obtained by rewriting the kinetic energy of the particle as E k = kT( k- Boltzmann’s constant and T – equilibrium temperature) is, h h = λ= 2mEk 2mkT Thus, the particles (matter) with velocity v have the wave nature associated with it. This is resulted in wave mechanics – an insight tool, which helpsin understanding and predicting the particle nature of matter at the atomic scale as well as in its higher dimension say nanodimension. The evidence for matter waves was provided by several experiments. Few to be mentioned are • Davisson and Germer’s electron diffraction experiments Joint Initiative of IITs and IISc – Funded by MHRD Page 6 of 17 NPTEL – Electrical & Electronics Engineering – Semiconductor Nanodevices • G.P. Thomson’s experiment • Double silit interference pattern with electron • Straight edge diffraction pattern with electron • Braff reflection of Helium and Neutron beams 3 The principle of uncertainty Werner Karl Heisenberg Though classically, the determination of position and velocity of a moving particle is possible, its satisfactory description fails at atomic dimensions The uncertainty principle proposed by a German physicist Werner Heisenberg in 1927 is a consequence of dual nature of matter i.e. wave nature of a particle. 3.1 Concept There exists a fundamental limit in the accuracy of measuring variables such as position, energy, momentum and angular momentum which describes the behavior of microphysical system. 3.2 Statement Heisenberg principle can be stated as “A particular pair of physical entities can determined precisely and simultaneously only with a minimum tolerance of the order of ħ” ∆p ∆x ≥ ħ ∆E ∆t ≥ ħ ∆J ∆θ≥ ħ where, p – momentum, x- position, E- energy, t- time, J- angular momentum, and θ angular position. Joint Initiative of IITs and IISc – Funded by MHRD Page 7 of 17 NPTEL – Electrical & Electronics Engineering – Semiconductor Nanodevices Thus, uncertainty principle is the valley point or the limit of classical dynamics, beyond which the quantum nature (duality nature) has to be employed to the particles when the dimension of the system shrinks (from macro to atomic). Few notable applications of uncertainty principles are • • • • higher strength of nuclear force absence of electron in the nucleus stability of the atom determination of binding energy of the hydrogen atom and radius where the results matches the Bohr’s theory. 4 Schrödinger wave equation 4.1 Why Schrödinger’s equation for nanostructured systems? It is a partial differential equation which describes the physical status of the quantum system in a time bounded situation (also in a time independent situation). Austrian Physicist Erwin Schrödinger in 1924 formulated this equation of motion and published in 1925. Analogous to Newton’s equation of motion in classical dynamics, Schrödinger equation helps us to understand the dynamics of nano world, though it could be used in studying the whole universe. The terms that the Schrödinger equation involves are nothing but the energy terms and these are conserved i.e., T.E.=P.E+K.E Of course the energy is quantized. In other words, an electron in an atom can only possess discrete values, which explains sharps lines observed in atomic spectroscopy. The Schrödinger equation operates on the wave function, ψ of a particle. Tunneling effect is central idea in developing single electron transistors (SET) and the next generation tunneling semiconductor components. The schrodinger equation can be used to understand the concept of quantum tunneling and also to predict the probability of tunneling. So, Wave function means… In quantum mechanics, the description of a system-particle is given in terms of wave function. The wave function has all relevant information about the system say its Joint Initiative of IITs and IISc – Funded by MHRD Page 8 of 17 NPTEL – Electrical & Electronics Engineering – Semiconductor Nanodevices position, momentum, energy and so on. It is nothing but the representation of a matter wave. A particular system is said to be real if it could be represented by a wave function. The wave function is represented by the Greek letter ψ and is defined as ψ(x,y,z,t) = ψ(x,y,z) e-iwt. Although, it is complex in nature due to the imaginary part, the probability function, which is given by its square, is real.The probability of finding a particle in a volume dV is mathematically represented as ∫∫∫ψ 2 dV If this probability function is equated to the maximum probability of one, i.e., if ∫∫∫ψ 2 dV =1, the function is said to normalized and the existence of the particle is made sure within the volume dV. The wave function is orthogonal i.e. product of two wave functions, ψ i (x) and ψ j (x) vanishes integrated in space. i.e. ∫ψ i ( x)ψ *j ( x) = 0 , i≠j whereψ j* (x) is a complex conjugate function. This orthogonalityrelation implies that each particle is unique. 4.2 Schrödinger’s equation With the aid of the wave functionψ, Schrödinger proposed a wave equation, Ervin Schrödinger which includes the kinetic and potential energy of the system and the position dependent potential as, Joint Initiative of IITs and IISc – Funded by MHRD Page 9 of 17 NPTEL – Electrical & Electronics Engineering – Semiconductor Nanodevices ∆2ψ + 2m ( E − V )ψ = 0 h2 where m – mass of the particle, total energy E and potential energy V. This equation describes the behavior of any real microscopic system. For a free particle, this equation can be written as ∆2ψ + 2m ( E )ψ = 0 h2 Where V=0, since a free particle is independent of position. 5 Fermi-Dirac and Bose-Einstein Distributions 5.1 Need for statistics Schrödinger’s equation is an important tool to describe a particle along with its energy and analyze its complete dynamic behaviour. When we extend the size of the system from one particle to many particles, the Schrödinger equation will be a linear combination of wave function of all the particles and its solution is a tiresome work. To overcome this problem we study a system of particles in terms of statistics. For example if we consider electron cloud in a conductor, all electrons have their common basic character and difference will be the energy it possess and is classified based on Pauli’s principle. They are termed as indistinguishable. Based on the parameters like charge, spin and so on, the particles are grouped as bosons, fermions and so on. The two statistics which treats the particles quantum mechanically arei) Bose – Einstein and ii) Maxwell – Boltzmann. These will be discussed here briefly. 5.2 Bose – Einstein Statistics While discussing the theory of radiation and ultraviolet catastrophe,SatyendraNath Bose observed that, the result does not match with the contemporary theory (Maxwell Boltzmann statistics). During his lecture at the University of Dhaka, he unexpectedly made an error and predicted that “the probability of getting head while tossing a coin is one third”. This interestingly matched with the experimental results. Thus, he emphasized that the Maxwell – Boltzmann’s theory no longer holds good for particles at the microscopic level where the quantum effects has to be considered. His theory was initially not approved by the Physics journals. Then he communicated his paper to Einstein and got immediate approval from him. Joint Initiative of IITs and IISc – Funded by MHRD Page 10 of 17 NPTEL – Electrical & Electronics Engineering – Semiconductor Nanodevices SatyendraNath Bose Albert Einstein As a collaborative work with Einstein, he established his statistic under the banner, “Bose-Einstein statistics”. They extended this idea to atoms and proposed ‘BoseEinstein condensation’. 5.3 Concepts A system which has particles of identical and indistinguishable in nature is not restricted by Pauli’s exclusion principle. These particles are termed as bosons and are represented by symmetric wave function. The spin of bosons is either zero or one (eg. Mesons, Helium nuclei etc.,). Suppose, if a system has n i number of particles arranged in the g i (where g i refers to the degeneracy of each energy level) quantum states then the total possible ways of distributing is (n i +g i -1)! Then, the possible number of distinct arrangement is (ni + g i − 1)! n!( g i − 1)! This could be extended for many number of available energy states and it is summed as P= ∏ i (ni + g i − 1)! ni !( g i − 1)! The maximum value of P is obtained by finding its first order differential maxima i.e., d (ln P)=0 By applying Stirling’s approximation and solving we get the expression for the number of particles occupying the n i state is given as, Joint Initiative of IITs and IISc – Funded by MHRD Page 11 of 17 NPTEL – Electrical & Electronics Engineering – Semiconductor Nanodevices ni = g (e i (α + βEi ) ) −1 Where β=1/kT 5.4 Inferences • Particles which are indistinguishable do not obey Pauli’s exclusion principle • There is no upper limit to the number of particles occupying the same quantum state • At absolute zero, the energy of the particles is zero. • At higher temperature, Bose- Einstein distribution approaches Maxwell – Boltzmann distribution • This statistics is applicable to indistinguishable and symmetrical particles like photons. 5.5 Fermi – Dirac Distribution Enric Fermi Paul Adrien Maurice Dirac Enrico Fermi and Paul Dirac mathematically defined the statistics independently, even though Fermi proposed it earlier than Dirac. Fermi-Dirac distribution assumes that, the particles are identical and indistinguishable yet obeys Pauli’s exclusion principle. Their spin is half integral which leads the intrinsic probability of g i to be 2. These particles are said to posses’ antisymmetry wave function. Joint Initiative of IITs and IISc – Funded by MHRD Page 12 of 17 NPTEL – Electrical & Electronics Engineering – Semiconductor Nanodevices Let there be n i number of particles in a system to be placed in E i energy level has the available degeneracy state is g i . The different ways of arranging the n i particles, = g i (g i -1) (g i -2) …g i -(n i -1) = gi! ( g i − ni )! i.e., first particle has g i position to occupy, and then second particle has (g i -1) position to occupy and so on. Thus by extending it to the more number of energy levels E1, E2, E3… The maximum probability distribution is P=∏ i g i! ni ( g i − ni )! The maxima is obtained by taking d(ln P) =0. Also for most probable distribution ∑ ni = total particles in the system and ∑ ni Ei = U , the total energy of the system. On i i solving mathematically the number of particles in the ith state is ni = g [e i (α + βEi ) + 1] whereβ=1/kT. The other methods of deriving Fermi – Dirac distribution are 1. using canonical distribution 2. using Lagrange multipliers Points to be noted: • In the classical limits F –D and B- E statistics reduce to M – B statistics. • Though the inter-molecular forces are neglected, the individual particles are not independent as it requires symmetry wave function. • The quantum statistics are applicable to systems like metals, liquid helium etc., Joint Initiative of IITs and IISc – Funded by MHRD Page 13 of 17 NPTEL – Electrical & Electronics Engineering – Semiconductor Nanodevices 6 Solved problems 1. A bird of mass 250 g flies with a velocity of 75 m/sec for a direction of 10 min. Calculate the de Broglie wavelength associated with the bird. Reason out why the bird does not explicitly show the wave nature. Answer: The de Broglie wavelength λ is given by, h λ= mv λ= 6.62 x 10 −34 (250 x 10 −3 x 75) = 3.53 x 10 −35 m Reason: The length of the wave is much smaller than the dimensions of the bird. Hence it wave nature is not explicit. 2. Given that the mass of the neutron as 1.674 x 10-27 kg of the Planck’s constant h=6.60 x 10-34 J sec. Calculate the de Broglie wavelength for the same of energy 20 eV. Answer: The de Broglie wavelength in terms of energy is given by, λ= λ= h = mv h 2mE ; E = 20eV = 20 x 1.6 x 10 −19 J 6.62 x 10 −34 (2 x 1.67 x 10 − 27 x 20 x 1.6 x 10 −19 ) 1 2 = 6.19 x 10 −12 m Joint Initiative of IITs and IISc – Funded by MHRD Page 14 of 17 NPTEL – Electrical & Electronics Engineering – Semiconductor Nanodevices 3. Calculate the wavelength of thermal neutrons at 21°C with the Energy = kT. Answer: The de Broglie wavelength is λ= h = mv h = h 2mE 2mkT −34 h = 6.62 x 10 J . sec ; where E = kT m = 1.67 x 10 − 27 kg k = 1.38 x 10 − 23 J K T = 27 C = (273 + 21) = 294 K ο λ= 6.62 x 10 −34 (2 x 1.67 x 10 − 27 x 1.38 x 10 − 23 x 294) 1 2 = 1.7 A ° 4. Calculate the momentum of an e- possessing de Broglie wavelength6.6x10-11m. Answer: The de Broglie wavelength λ h λ= p (or ) p= 6.62 x 10 −34 6.6 x 10 −11 = 1 x 10 − 23 kg.m sec 5. A bullet of mass 18 gm takes 1 sec to reach its target board. Find the probability of missing the target. Consider that the shots are fired accurately. Answer: According to uncertainty principle, Joint Initiative of IITs and IISc – Funded by MHRD Page 15 of 17 NPTEL – Electrical & Electronics Engineering – Semiconductor Nanodevices ∆E. ∆t = ; E − energy, t − time & ∆y.∆p = ; y − position, p − momentum also momentum = ∆p = 2m(∆E ) Therefore Uncertainity in position ∆y = = ∆p = 2m(∆E ) 2 m( t ) = t = 2m ht 4πm Given that t = 1sec , m = 18 gm , h = 6.625 x 10 − 27 The chance of mis sin g is = 6.625 x 10 − 27 = 5.41 x 10 −15 m 4 (3.14 x 18) 6. If suppose a proton move as a speed of 2x106 m/s and if we could measure its speed with an accuracy of 60%. What is the uncertainty in determining the position of the same? Answer: The momentum of the proton p = mv = mass of proton * speed = 1.6276 x 10 − 27 x 2 x 106 = 3.3452 x 10 − 21 kg. m s The uncertainity in det er min ing its speed 60 60 ∆p = p x = 3.3452 x 10 − 21 x = 2.00712 x 10 − 21 100 100 By Heisenberg ' s priniciple the min imum uncerta int y is det er min ing its position is, 6.623 x 10 −34 ∆x = = = 0.5259 x 10 −13 m − 21 ∆Px 2π x 2.00712 x 10 Joint Initiative of IITs and IISc – Funded by MHRD Page 16 of 17 NPTEL – Electrical & Electronics Engineering – Semiconductor Nanodevices 7 References 1. Richard Liboff, “Introductory Quantum Mechanics”, 4th edition, Addison Wesley, 2002. 2. David J Griffith, “Introduction to Quantum Mechanics”, 2nd edition, Benjamin Cummings, 2004. 3. EliahuZaarur, PhinikReuven, “Schaum's Outline of Quantum Mechanics”, McGraw-Hill, 1998. 4. Stephen Gasiorowicz, “Quantum Physics”, 3rd edition, Wiley Publishing, 2003. Joint Initiative of IITs and IISc – Funded by MHRD Page 17 of 17