DOCSLIB.ORG

Spectroscopy and the “Particle in a Box”

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Spectroscopy and the “Particle in a Box” Spectroscopy and The “Particle in a Box” DEPARTMENT OF CHEMISTRY UNIVERSITY OF KANSAS Spectroscopy and the “Particle in a Box” Introduction The majority of colors that we see result from transitions between electronic states that occur as a result of selective photon absorption. For a molecule to absorb a photon, the energy of the impinging photon must match the energy difference between the initial state and some excited state of the molecule. We can describe this concept using the equation Ephoton = hυ = ΔEmolecule = Eupper state - Elower state (1) in which E represents the energy of the photon or molecule being studied, h is Planck's constant, and υ is the frequency. To predict the color of a specific molecule from fundamental physical chemistry principles, one must know the array of possible molecular energy levels (quantized rotational, vibrational, and electronic energy levels). Molecules of a colored object absorb visible- light photons when they are excited from their lowest-energy electronic state (called the “ground state”) to a higher-energy electronic state (called an “excited state”). In principle, the various electronic states of an atom or molecule may be calculated quantum- mechanically. In fact, quantum mechanics can be used to predict the allowed set of energy levels for an atom or molecule. For larger molecules, determining these energy levels requires making approximations and it can be computationally intensive (For simpler molecules, free software is available that would allow you to carry out such calculations on your home computer). With a carefully chosen set of molecules, however, we can study some of the principles of quantum mechanics in the general chemistry laboratory. Some ultraviolet (UV) light and visible light-absorbing molecules are members of a special group for which the simple "particle in a box" quantum-mechanical model applies nicely. This model can be used to predict the energy levels of electrons responsible for UV or visible wavelength transitions — if we are willing to make some assumptions. Imagine that a particle of mass m (in this case, the particle is an electron) travels in one dimension (x) between two walls separated by a distance L. Then assume that the potential energy between 1 these walls (i.e., from 0 ≤ x ≤ L) is constant, while the potential energy jumps to infinity at the walls. This assumption allows us to draw a simple potential energy diagram like that shown on page one. Solving the Schrödinger equation for this simple one-dimensional particle in a box system yields the following allowed energies: 2 2 n h En = 2 n = 1,2,3,... 8mL where h is Planck's constant, m is the mass of the particle, and L is the length of the one- dimensional box. Note that the different allowed energies are labeled by the quantum number n, which can only take on integer values. If the energy of a “particle in a box” is measured, these are the only results that will be found — no other energies are possible results according to quantum mechanics. These energies can be qualitatively understood by considering the wave-particle duality in quantum mechanics, wherein objects we normally think of as particles in some ways behave as waves and vice-versa. This means that an electron in a box must be described as a wave in quantum mechanics with wavelength λ=h/p. This “wave” is related to the probability of finding the electron — in fact the electron probability distribution of positions is the square of the wavefunction that describes the particle — and hence the wave must go to zero outside of the box where the potential energy is infinitely high. That is, the wave describing the electron must fit neatly in the box so that it goes to zero at the edges. This is clearly the case when the length of the box, L, is equal to λ/2, λ, 3λ/2, 2λ, 5λ/2, .... Combining this with the expression for λ above and using E=mv2/2 gives the allowed energies. Note that this somewhat hand-waving explanation also shows how the energy levels and their spacings depend on L and m. As L becomes very large the energies get closer and closer together, eventually becoming continuous (no longer “quantum”); this is due to L being much larger than wavelength λ for a particle with a typical energy. Similarly, increasing the mass has the same qualitative effect as making the box larger, which is why you (a “particle”) do not notice quantum effects when you sit in a room (a “box”), even though your motion is fundamentally described by quantum mechanics. Stated another way, discrete energy level spacing is observed for very low- mass particles confined to small quarters (in this case, an electron within an atom or molecule that gives a small value for mL2). For the molecules considered in this experiment, the electronic energy level spacing corresponds to the energy of a visible photon. Specifically, there must be high-energy valence electrons capable of traveling "freely" over the length of the molecule, L. These "free" electrons behave approximately like the particles in a one-dimensional box. 2 To understand the electronic structures of the three compounds you will study in this experiment, begin by considering the bonding between the carbon atoms in a very simple organic molecule, ethene (C2H4). A line-bond structure for ethene is shown above, together with a more detailed orbital cartoon. Each carbon atom utilizes sp2 hybrid orbitals to overlap with the 1s orbitals from two hydrogen atoms, while the remaining sp2 hybrid orbital overlaps with an sp2 hybrid orbital from the adjacent carbon atom. This "end-on" overlap is called σ-bonding (“sigma bonding”). This experiment is particularly concerned with a second type of bonding in which the p-orbitals on adjacent carbon atoms overlap. For purposes of electron bookkeeping, each p-orbital can be assumed to contain one electron. The overlap between p-orbitals on the adjacent carbons in ethene is "side-on" and termed a π-bond (“pi bond”). Hence, the double bond between the ethene carbon atoms actually consists of two distinct components, a σ-bond and a π-bond. More complex organic molecules containing alternating single and double bonds are said to be conjugated. The compounds used in this experiment are indeed conjugated, but the chemical structure of butadiene (shown below) is a useful, simpler example of a conjugated molecule. An examination of the π-bonding in butadiene shows that a p-orbital in each carbon atom overlaps with the p-orbital(s) of its neighbor(s). Again, each p-orbital is formally said to contain 1 electron. More extensively conjugated molecules called polyenes have formal structures with alternating single and double bonds along the carbon atom chain. The electrons in the π-bonds of polyenes can be considered to be delocalized over all the atoms of the conjugated chain, and thus can be thought of as moving somewhat freely along the length of the chain. Viewing each electron as a particle in a box is then a fairly crude, but physically reasonable, model of electrons moving along a chain of carbon atoms. For many compounds, it is remarkably successful in modeling the behavior of the quantum mechanical particle in a box. In this experiment, you will measure the light absorption properties of a carefully chosen set of organic molecules and relate the absorption spectra to the particle in a box quantum mechanical model for the electrons. 3 Pre-lab Review the instructions for the operation of the OceanOptics Spectrophotometer. You should have used this instrument in a previous lab and it will be used during this and future laboratory exercises. Thus, the operation of this instrument should be familiar to you. This pre-lab assignment is due at the beginning of lab. You will not be allowed to start the experiment until this assignment has been completed and submitted to your TA. List all of the chemicals you will use for this week's experiment. For each chemical, list specific safety precaution(s) that must be followed. To find specific safety information, please obtain a Materials Safety Data Sheet (MSDS) on the chemical of interest. MSDSs can be found through an internet search (e.g., google) or from the following website: www.hazard.com Read the MSDS and find specific safety concerns for each chemical. Be sure to include the route(s) of entry and the possible acute and chronic effects of exposure, if given. Using your own words, complete the OBJECTIVE and PROCEDURE sections in your lab notebook. (See the Maintaining Your Laboratory Notebook link on the lab website to learn what these lab notebook sections entail.) Also, please write out answers to the following questions: 1) The particle in the box energy, En, involves Planck’s constant, h, and the particle mass, m. What particle is involved in the transitions you will measure in this experiment? What are the values of h and m? 2) According to the assumptions of the particle-in-a-box experiment, how many freely moving π- electrons are there between the phenyl rings in each of the three organic compounds you will be studying? 3) Using the particle in the box equation and a wavelength of 400 nm, what is the length of the box of 1,4-diphenyl-1,3-butadiene? 4) The particle in a box theory assumes that potential energy is constant along the entire conjugated carbon-carbon chain. Are there any flaws in this assumption? Procedure Safety: Goggles must be worn at all times. Organic solvents (e.g., cyclohexane) should be collected in a separate container as waste.
Load more

Recommended publications
  • A Study of Fractional Schrödinger Equation-Composed Via Jumarie Fractional Derivative
    A Study of Fractional Schrödinger Equation-composed via Jumarie fractional derivative Joydip Banerjee1, Uttam Ghosh2a , Susmita Sarkar2b and Shantanu Das3 Uttar Buincha Kajal Hari Primary school, Fulia, Nadia, West Bengal, India email- [email protected] 2Department of Applied Mathematics, University of Calcutta, Kolkata, India; 2aemail : [email protected] 2b email : [email protected] 3 Reactor Control Division BARC Mumbai India email : [email protected] Abstract One of the motivations for using fractional calculus in physical systems is due to fact that many times, in the space and time variables we are dealing which exhibit coarse-grained phenomena, meaning that infinitesimal quantities cannot be placed arbitrarily to zero-rather they are non-zero with a minimum length. Especially when we are dealing in microscopic to mesoscopic level of systems. Meaning if we denote x the point in space andt as point in time; then the differentials dx (and dt ) cannot be taken to limit zero, rather it has spread. A way to take this into account is to use infinitesimal quantities as ()Δx α (and ()Δt α ) with 01<α <, which for very-very small Δx (and Δt ); that is trending towards zero, these ‘fractional’ differentials are greater that Δx (and Δt ). That is()Δx α >Δx . This way defining the differentials-or rather fractional differentials makes us to use fractional derivatives in the study of dynamic systems. In fractional calculus the fractional order trigonometric functions play important role. The Mittag-Leffler function which plays important role in the field of fractional calculus; and the fractional order trigonometric functions are defined using this Mittag-Leffler function.
    [Show full text]
  • Lecture 16 Matter Waves & Wave Functions
    LECTURE 16 MATTER WAVES & WAVE FUNCTIONS Instructor: Kazumi Tolich Lecture 16 2 ¨ Reading chapter 34-5 to 34-8 & 34-10 ¤ Electrons and matter waves n The de Broglie hypothesis n Electron interference and diffraction ¤ Wave-particle duality ¤ Standing waves and energy quantization ¤ Wave function ¤ Uncertainty principle ¤ A particle in a box n Standing wave functions de Broglie hypothesis 3 ¨ In 1924 Louis de Broglie hypothesized: ¤ Since light exhibits particle-like properties and act as a photon, particles could exhibit wave-like properties and have a definite wavelength. ¨ The wavelength and frequency of matter: # & � = , and � = $ # ¤ For macroscopic objects, de Broglie wavelength is too small to be observed. Example 1 4 ¨ One of the smallest composite microscopic particles we could imagine using in an experiment would be a particle of smoke or soot. These are about 1 µm in diameter, barely at the resolution limit of most microscopes. A particle of this size with the density of carbon has a mass of about 10-18 kg. What is the de Broglie wavelength for such a particle, if it is moving slowly at 1 mm/s? Diffraction of matter 5 ¨ In 1927, C. J. Davisson and L. H. Germer first observed the diffraction of electron waves using electrons scattered from a particular nickel crystal. ¨ G. P. Thomson (son of J. J. Thomson who discovered electrons) showed electron diffraction when the electrons pass through a thin metal foils. ¨ Diffraction has been seen for neutrons, hydrogen atoms, alpha particles, and complicated molecules. ¨ In all cases, the measured λ matched de Broglie’s prediction. X-ray diffraction electron diffraction neutron diffraction Interference of matter 6 ¨ If the wavelengths are made long enough (by using very slow moving particles), interference patters of particles can be observed.
    [Show full text]
  • Dirac Particle in a Square Well and in a Box
    Dirac particle in a square well and in a box A. D. Alhaidari Saudi Center for Theoretical Physics, Dhahran, Saudi Arabia We obtain an exact solution of the 1D Dirac equation for a square well potential of depth greater then twice the particle’s mass. The energy spectrum formula in the Klein zone is surprisingly very simple and independent of the depth of the well. This implies that the same solution is also valid for the potential box (infinitely deep well). In the nonrelativistic limit, the well-known energy spectrum of a particle in a box is obtained. We also provide in tabular form the elements of the complete solution space of the problem for all energies. PACS numbers: 03.65.Pm, 03.65.Ge, 03.65.Nk Keywords: Dirac equation, square well, potential box, energy spectrum, Klein paradox Aside from the mathematically “trivial” free case, particle in a box is usually the first problem that an undergraduate student of nonrelativistic quantum mechanics is asked to solve. Normally, he goes further into obtaining the bound states solution of a particle in a finite square well. It is much later that he works out more involved exercises like the 3D Coulomb problem [1]. Unfortunately, in relativistic quantum mechanics the story is reversed. One can hardly find a textbook on relativistic quantum mechanics where the 1D problem of a particle in a potential box is solved before the relativistic Hydrogen atom is [2]. The difficulty is that if this is to be done from the start, then one is forced to get into subtle issues like the Klein paradox, electron-positron pair production, stability of the vacuum, appropriate boundary conditions, …etc.
    [Show full text]
  • Analytic Solution of the Schrцdinger Equation: Particle in A
    Analytic solution of the Schrödinger equation: Particle in a box Notes on Quantum Mechanics http://quantum.bu.edu/notes/QuantumMechanics/ParticleInABox.pdf Last updated Tuesday, September 26, 2006 16:04:15-05:00 Copyright © 2004 Dan Dill ([email protected]) Department of Chemistry, Boston University, Boston MA 02215 One way to work with the curvature form of the Schrödinger equation, curvature of y at x ∂-kinetic energy at x μyat x, is try to guess what wavefunctions, y, satisfy it. For systems in which the kinetic energy changes with position in complicated ways, systematic implementation of this so-called analytic approach typically requires sophisticated mathematical machinery and so it not as easy to understand as the visual approach we will use. However, there is one example of the analytic approach that is very easy to understand, namely the case where a particle of mass m is confined in a one-dimensional region of width L; in this region it moves freely but it is not able to move outside this region. Such a system is called a particle in a box. It is one of the most important example quantum systems in chemistry, because it helps us develop intuition about the behavior on electrons confined in molecules. To use the analytical method, we begin by writing the Schrödinger equation more precisely as 2 m curvature of y at x =-ÅÅÅÅÅÅÅÅÅÅÅÅ μ kinetic energy at x μyat x. Ñ2 The reason the particle is able to move freely in the specified region, 0 § x § L (inside the "box"), is that its potential energy there is zero.
    [Show full text]
  • Schrödinger Equation - Particle in a 1-D Box
    Particle in a Box Outline - Review: Schrödinger Equation - Particle in a 1-D Box . Eigenenergies . Eigenstates . Probability densities 1 TRUE / FALSE The Schrodinger equation is given above. 1. The wavefunction Ψ can be complex, so we should remember to take the Real part of Ψ. 2. Time-harmonic solutions to Schrodinger equation are of the form: 3. Ψ(x,t) is a measurable quantity and represents the probability distribution of finding the particle. 2 Schrodinger: A Wave Equation for Electrons Schrodinger guessed that there was some wave-like quantity that could be related to energy and momentum … wavefunction 3 Schrodinger: A Wave Equation for Electrons (free-particle) (free-particle) ..The Free-Particle Schrodinger Wave Equation ! Erwin Schrödinger (1887–1961) 4 Image in the Public Domain Schrodinger Equation and Energy Conservation ... The Schrodinger Wave Equation ! Total E term K.E. term P.E . t e rm ... In physics notation and in 3-D this is how it looks: Electron Maximum height Potential and zero speed Energy Zero speed start Incoming Electron Fastest Battery 5 Time-Dependent Schrodinger Wave Equation To t a l E K.E. term P.E . t e rm PHYSICS term NOTATION Time-Independent Schrodinger Wave Equation 6 Particle in a Box e- 0.1 nm The particle the box is bound within certain regions of space. If bound, can the particle still be described as a wave ? YES … as a standing wave (wave that does not change its with time) 7 A point mass m constrained to move on an infinitely-thin, frictionless wire which is strung tightly between two impenetrable walls a distance L apart m 0 L WE WILL HAVE MULTIPLE SOLUTIONS FOR , SO WE INTRODUCE LABEL IS CONTINUOUS 8 WE WILL HAVE e- MULTIPLE SOLUTIONS FOR , SO WE INTRODUCE LABEL n L REWRITE AS: WHERE GENERAL SOLUTION: OR 9 USE BOUNDARY CONDITIONS TO DETERMINE COEFFICIENTS A and B e- since NORMALIZE THE INTEGRAL OF PROBABILITY TO 1 L 10 EIGENENERGIES for EIGENSTATES for PROBABILITY 1-D BOX 1-D BOX DENSITIES 11 Today’s Culture Moment Max Planck • Planck was a gifted musician.
    [Show full text]
  • “Particle-In-A-Box” Problems in Quantum Mechanics
    2-DIMENSIONAL “PARTICLE-IN-A-BOX” PROBLEMS IN QUANTUM MECHANICS Part I: Propagator & eigenfunctions by the method of images Nicholas Wheeler, Reed College Physics Department July 1997 Introduction. All strings make the same music, and support the same physics, because all strings have the same shape. The vibrational physics of a drumhead is, on the other hand, shape-dependent—whence Mark Kac’famous question: “Can one hear the shape of a drum?”1 Relatedly, all instances of the quantum mechanical “particle-in-a-box” problem are, in the one-dimensional case, scale- equivalent to one another, but higher-dimensional instances of the same problem are (in general) inequivalent. Though the one-dimensional box-problem yields quite readily to exact closed-form analysis—the simplest (but only the simplest!) aspects of which can be found in every quantum text—the higher-dimensional problem (N ≥ 2) is in most of its aspects analytically intractable except in certain special cases. In previous work2 I have identified a class of cases that yield to analysis by what might be called the “quantum mechanical method of images.” I undertake the present review of the essentials of that work partly to make it more readily available to students of quantum mechanics, but mainly to be responsive to an expressed need of my colleague Oz Bonfim, who speculates that some of my results may be relevent to his own work in connection with the so-called “Bohm interpretation of quantum mechanics.”3 I have also an ulterior motivation, 1 Amer. Math. Monthly 73, 1 (1966). This classic paper is reprinted in Mark Kac: Selected Papers on Probability, Number Theory & Statistical Physics () 2 feynman formalism for polygonal domains (–); circular boxes & rigid wavepackets (–).
    [Show full text]
  • Quantum Mechanics (Surprising !!!) the Analysis of Electronic Flow)
    2/6/2019 INE: Revision INTRODUCTION TO NANOELECTRONICS Week-6, Lecture-2 Sneh Saurabh 6th February, 2019 Introduction to Nanoelectronics: S. Saurabh Overview 2 A free particle: Significance in Solid State Physics Seminal concept in solid state physics Impact on Analysis . In crystalline solids electrons behave as if . Reduces the complicated problem of they are in vacuum, but with an effective electron waves in a crystalline solid to that mass different from their natural mass of waves in vacuum (Dramatically eases Quantum Mechanics (Surprising !!!) the analysis of electronic flow) . Energy-momentum relation 푝 . Valid for several materials such as Copper (metal) and Silicon (semiconductor) in wide 퐸 푝 = 퐸 + ∗ 2푚 range of energies . Energy-wavenumber (퐸 − 푘) relation (using . Parabolic 퐸 − 푘 푟푒푙푎푡푖표푛푠ℎ푖푝 푝 = ℏ푘) A Free particle ℏퟐ풌ퟐ (퐷푖푠푝푒푟푠푖표푛 푟푒푙푎푡푖표푛푠ℎ푖푝) is not valid for 퐸 푘 = 퐸 + some materials such as graphene 2푚∗ Introduction to Nanoelectronics: S. Saurabh Quantum Mechanics 3 Introduction to Nanoelectronics: S. Saurabh Quantum Mechanics 4 1 2/6/2019 A free particle + particle in a box ! Wave function of an electron in a confined nanoelectronic device can be treated as combined effect of (a free particle + a particle in a box) Quantum Mechanics Bulk Quantum Quantum Quantum (3D) Well (2D) Wire (1D) Dots (0D) 3-D Schrödinger Equation Introduction to Nanoelectronics: S. Saurabh Quantum Mechanics 5 Introduction to Nanoelectronics: S. Saurabh Quantum Mechanics 6 Schrödinger Equation: 3-D Schrödinger Equation: 3-D Separable solution (1) Three-dimensional time-independent Schrödinger equation: . Applying quantum mechanics to nanostructures (which are naturally 3-D objects) requires 퐻휓(푥, 푦, 푧)= 퐸휓(푥, 푦, 푧) extending Schrödinger Equation to 3-D where: ℏ 휕 휕 휕 .
    [Show full text]
  • Optical Physics of Quantum Wells
    Optical Physics of Quantum Wells David A. B. Miller Rm. 4B-401, AT&T Bell Laboratories Holmdel, NJ07733-3030 USA 1 Introduction Quantum wells are thin layered semiconductor structures in which we can observe and control many quantum mechanical effects. They derive most of their special properties from the quantum confinement of charge carriers (electrons and "holes") in thin layers (e.g 40 atomic layers thick) of one semiconductor "well" material sandwiched between other semiconductor "barrier" layers. They can be made to a high degree of precision by modern epitaxial crystal growth techniques. Many of the physical effects in quantum well structures can be seen at room temperature and can be exploited in real devices. From a scientific point of view, they are also an interesting "laboratory" in which we can explore various quantum mechanical effects, many of which cannot easily be investigated in the usual laboratory setting. For example, we can work with "excitons" as a close quantum mechanical analog for atoms, confining them in distances smaller than their natural size, and applying effectively gigantic electric fields to them, both classes of experiments that are difficult to perform on atoms themselves. We can also carefully tailor "coupled" quantum wells to show quantum mechanical beating phenomena that we can measure and control to a degree that is difficult with molecules. In this article, we will introduce quantum wells, and will concentrate on some of the physical effects that are seen in optical experiments. Quantum wells also have many interesting properties for electrical transport, though we will not discuss those here.
    [Show full text]
  • Quantum Mechanics Part 2
    Quantum Mechanics Part 2 Ch 40 The Schoedinger Eqn. Making sense of The Schoedinger Equation There are physics laws which are encoded in equations e.g. F=ma; Einitial=Efinal etc Is there a rule for ψ ????? Schoedinger ~ 1924 dx2ψ () 2 m =−[()]()EUx − ψ x dx22= Why do we believe it Because it works! What is the program? Justifying it This isn’t the real reason we believe it real reason is because of experiment Making models of real life and solving Periodic Table of the elements Molecular binding (chemistry) tunneling (decay) transistors Zero point energy of the vacuum! … A “ justification” of the Schroedinger eqn this represents matter waves hmm Note: I cheated a bit, since K is really K(x) potential energy dx2ψ () 2 m =−[()]()E −Uxψ x dx22= This is the 1-D eqn. We should really use a 3-D version (in this class we will stick to 1-D) what we need to do is to find appropriate U(x) which describe real situations and then solve for ψ(x) Boundary Conditions We have to require some things about ψ(x) since P(x)= ψ2 ψ(x) must be continuous ψ(x) →0 as x→∞ and x→-∞ ψ(x)=0 in areas it should not be ψ(x) must be normalized (i.e. its total probability = 1 or 100%) PROBLEM-SOLVING STRATEGY Quantum-mechanics problems MODEL Determine a potential-energy function that describes the particle’s interactions. Make simplifying assumptions. VISUALIZE The potential-energy curve is the pictorial representation. Draw the potential-energy curve. Identify known information.
    [Show full text]
  • Massless and Massive Particle-In-A-Box States in Single-And Bi
    Massless and massive particle-in-a-box states in single-and bi- layer graphene Sungjae Cho and Michael S. Fuhrer Department of Physics and Center for Nanophysics and Advanced Materials, University of Maryland, College Park, Maryland 20742, USA Electron transport through short, phase-coherent metal-graphene-metal devices occurs via resonant transmission through particle-in-a-box-like states defined by the atomically-sharp metal leads. we study the spectrum of particle-in- a-box states for single- and bi-layer graphene, corresponding to massless and massive two-dimensional (2d) Fermions.The density of states D as a function of particle number n shows the expected relations D(n) ~ n1/2 for massless 2d Fermions (electrons in single-layer graphene) and D(n) ~ constant for massive 2d Fermions (electrons in bi-layer graphene). The single parameters of the massless and massive dispersion relations are found, Fermi velocity vF = 1.1 x 6 * 10 m/s and effective mass m = 0.032 me, where me is the electron mass, in excellent agreement with theoretical expectations. The problem of a wavelike particle confined to a hard-walled box is one of the most basic problems in quantum mechanics. The spectra of the particle-in-a- box are strikingly different for massive and massless particles: massless particles (e.g. photons) have energies which depend linearly on quantum number, while the energies of massive particles (e.g. free electrons) depend quadratically on quantum number. For Fermions in two dimensions (2d) this leads to a density of single-particle states D ≡ dn/dE, where E is the particle energy and n the particle density, which varies as the square root of particle density, i.e.
    [Show full text]
  • Semiconductor Physics Syllabus Code: BSC-PHY-103G Introduction to Subject
    Semiconductor Physics Syllabus code: BSC-PHY-103G Introduction to subject A semiconductor is a material that has a resistivity value in between that of a conductor and an insulator. A semiconductor material is a device that allows the passage of electric current through it. Consider the device CPU, it consists of many numbers of transistors within it and these transistors contain the semiconductor material which allows the passage of current and all these are controlled by a switch. Here in a transistor, the passage of electric current is controlled by the actions based on the state of the switch whether it is on or off. Hence such a device, which allows the current to pass through them partially is called a semiconductor device. Examples of Semiconductor Devices • These devices are said to be neither good insulators nor good conductors, hence the name ‘Semi Conductors’. The semiconductor examples include the following: • op-amps • resistors • capacitors • diodes • Transistors Syllabus UNIT – I: Electronic Materials Free electron theory, Density of states and energy band diagrams, Kronig-Penny model (to introduce origin of band gap), Energy bands in solids, E-k diagram, Direct and indirect band gaps, Types of electronic materials: metals, semiconductors, and insulators, Density of states, Occupation probability, Fermi level, Effective mass, Phonons. UNIT – II: Semiconductors Intrinsic and extrinsic semiconductors, Dependence of Fermi level on carrier-concentration and temperature (equilibrium carrier statistics), Carrier generation and recombination, Carrier transport: diffusion and drift, p-n junction, Metal- semiconductor junction (Ohmic and Schottky), Semiconductor materials of interest for optoelectronic devices. UNIT – III :Light-Semiconductor Interaction Optical transitions in bulk semiconductors: absorption, spontaneous emission, and stimulated emission; Joint density of states, Density of states for photons, Transition rates (Fermi's golden rule), Optical loss and gain; Photovoltaic effect, Exciton, Drude model.
    [Show full text]
  • Confinement, Average Forces, and the Ehrenfest Theorem for a One
    PRAMANA c Indian Academy of Sciences Vol. 80, No. 5 — journal of May 2013 physics pp. 797–810 Confinement, average forces, and the Ehrenfest theorem for a one-dimensional particle SALVATOREDEVINCENZO Escuela de Física, Facultad de Ciencias, Universidad Central de Venezuela, A.P. 47145, Caracas 1041-A, Venezuela E-mail: [email protected] MS received 11 July 2012; revised 14 October 2012; accepted 16 November 2012 Abstract. The topics of confinement, average forces, and the Ehrenfest theorem are examined for a particle in one spatial dimension. Two specific cases are considered: (i) A free particle mov- ing on the entire real line, which is then permanently confined to a line segment or ‘a box’ (this situation is achieved by taking the limit V0 →∞in a finite well potential). This case is called ‘a particle-in-an-infinite-square-well-potential’. (ii) A free particle that has always been moving inside a box (in this case, an external potential is not necessary to confine the particle, only bound- ary conditions). This case is called ‘a particle-in-a-box’. After developing some basic results for the problem of a particle in a finite square well potential, the limiting procedure that allows us to obtain the average force of the infinite square well potential from the finite well potential prob- lem is re-examined in detail. A general expression is derived for the mean value of the external classical force operator for a particle-in-an-infinite-square-well-potential, Fˆ. After calculating sim- ilar general expressions for the mean value of the position (Xˆ ) and momentum (Pˆ) operators, the Ehrenfest theorem for a particle-in-an-infinite-square-well-potential (i.e., dXˆ /dt =Pˆ/M and dPˆ/dt =Fˆ) is proven.
    [Show full text]