Free Electron Model
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Fermi- Dirac Statistics
Prof. Surajit Dhara Guest Teacher, Dept. Of Physics, Narajole Raj College GE2T (Thermal physics and Statistical Mechanics) , Topic :- Fermi- Dirac Statistics Fermi- Dirac Statistics Basic features: The basic features of the Fermi-Dirac statistics: 1. The particles are all identical and hence indistinguishable. 2. The fermions obey (a) the Heisenberg’s uncertainty relation and also (b) the exclusion principle of Pauli. 3. As a consequence of 2(a), there exists a number of quantum states for a given energy level and because of 2(b) , there is a definite a priori restriction on the number of fermions in a quantum state; there can be simultaneously no more than one particle in a quantum state which would either remain empty or can at best contain one fermion. If , again, the particles are isolated and non-interacting, the following two additional condition equations apply to the system: ∑ ; Thermodynamic probability: Consider an isolated system of N indistinguishable, non-interacting particles obeying Pauli’s exclusion principle. Let particles in the system have energies respectively and let denote the degeneracy . So the number of distinguishable arrangements of particles among eigenstates in the ith energy level is …..(1) Therefore, the thermodynamic probability W, that is, the total number of s eigenstates of the whole system is the product of ……(2) GE2T (Thermal physics and Statistical Mechanics), Topic :- Fermi- Dirac Statistics: Circulated by-Prof. Surajit Dhara, Dept. Of Physics, Narajole Raj College FD- distribution function : Most probable distribution : From eqn(2) above , taking logarithm and applying Stirling’s theorem, For this distribution to represent the most probable distribution , the entropy S or klnW must be maximum, i.e. -
Unit 1 Old Quantum Theory
UNIT 1 OLD QUANTUM THEORY Structure Introduction Objectives li;,:overy of Sub-atomic Particles Earlier Atom Models Light as clectromagnetic Wave Failures of Classical Physics Black Body Radiation '1 Heat Capacity Variation Photoelectric Effect Atomic Spectra Planck's Quantum Theory, Black Body ~diation. and Heat Capacity Variation Einstein's Theory of Photoelectric Effect Bohr Atom Model Calculation of Radius of Orbits Energy of an Electron in an Orbit Atomic Spectra and Bohr's Theory Critical Analysis of Bohr's Theory Refinements in the Atomic Spectra The61-y Summary Terminal Questions Answers 1.1 INTRODUCTION The ideas of classical mechanics developed by Galileo, Kepler and Newton, when applied to atomic and molecular systems were found to be inadequate. Need was felt for a theory to describe, correlate and predict the behaviour of the sub-atomic particles. The quantum theory, proposed by Max Planck and applied by Einstein and Bohr to explain different aspects of behaviour of matter, is an important milestone in the formulation of the modern concept of atom. In this unit, we will study how black body radiation, heat capacity variation, photoelectric effect and atomic spectra of hydrogen can be explained on the basis of theories proposed by Max Planck, Einstein and Bohr. They based their theories on the postulate that all interactions between matter and radiation occur in terms of definite packets of energy, known as quanta. Their ideas, when extended further, led to the evolution of wave mechanics, which shows the dual nature of matter -
Quantum Theory of the Hydrogen Atom
Quantum Theory of the Hydrogen Atom Chemistry 35 Fall 2000 Balmer and the Hydrogen Spectrum n 1885: Johann Balmer, a Swiss schoolteacher, empirically deduced a formula which predicted the wavelengths of emission for Hydrogen: l (in Å) = 3645.6 x n2 for n = 3, 4, 5, 6 n2 -4 •Predicts the wavelengths of the 4 visible emission lines from Hydrogen (which are called the Balmer Series) •Implies that there is some underlying order in the atom that results in this deceptively simple equation. 2 1 The Bohr Atom n 1913: Niels Bohr uses quantum theory to explain the origin of the line spectrum of hydrogen 1. The electron in a hydrogen atom can exist only in discrete orbits 2. The orbits are circular paths about the nucleus at varying radii 3. Each orbit corresponds to a particular energy 4. Orbit energies increase with increasing radii 5. The lowest energy orbit is called the ground state 6. After absorbing energy, the e- jumps to a higher energy orbit (an excited state) 7. When the e- drops down to a lower energy orbit, the energy lost can be given off as a quantum of light 8. The energy of the photon emitted is equal to the difference in energies of the two orbits involved 3 Mohr Bohr n Mathematically, Bohr equated the two forces acting on the orbiting electron: coulombic attraction = centrifugal accelleration 2 2 2 -(Z/4peo)(e /r ) = m(v /r) n Rearranging and making the wild assumption: mvr = n(h/2p) n e- angular momentum can only have certain quantified values in whole multiples of h/2p 4 2 Hydrogen Energy Levels n Based on this model, Bohr arrived at a simple equation to calculate the electron energy levels in hydrogen: 2 En = -RH(1/n ) for n = 1, 2, 3, 4, . -
Solid State Physics II Level 4 Semester 1 Course Content
Solid State Physics II Level 4 Semester 1 Course Content L1. Introduction to solid state physics - The free electron theory : Free levels in one dimension. L2. Free electron gas in three dimensions. L3. Electrical conductivity – Motion in magnetic field- Wiedemann-Franz law. L4. Nearly free electron model - origin of the energy band. L5. Bloch functions - Kronig Penney model. L6. Dielectrics I : Polarization in dielectrics L7 .Dielectrics II: Types of polarization - dielectric constant L8. Assessment L9. Experimental determination of dielectric constant L10. Ferroelectrics (1) : Ferroelectric crystals L11. Ferroelectrics (2): Piezoelectricity L12. Piezoelectricity Applications L1 : Solid State Physics Solid state physics is the study of rigid matter, or solids, ,through methods such as quantum mechanics, crystallography, electromagnetism and metallurgy. It is the largest branch of condensed matter physics. Solid-state physics studies how the large-scale properties of solid materials result from their atomic- scale properties. Thus, solid-state physics forms the theoretical basis of materials science. It also has direct applications, for example in the technology of transistors and semiconductors. Crystalline solids & Amorphous solids Solid materials are formed from densely-packed atoms, which interact intensely. These interactions produce : the mechanical (e.g. hardness and elasticity), thermal, electrical, magnetic and optical properties of solids. Depending on the material involved and the conditions in which it was formed , the atoms may be arranged in a regular, geometric pattern (crystalline solids, which include metals and ordinary water ice) , or irregularly (an amorphous solid such as common window glass). Crystalline solids & Amorphous solids The bulk of solid-state physics theory and research is focused on crystals. -
Vibrational Quantum Number
Fundamentals in Biophotonics Quantum nature of atoms, molecules – matter Aleksandra Radenovic [email protected] EPFL – Ecole Polytechnique Federale de Lausanne Bioengineering Institute IBI 26. 03. 2018. Quantum numbers •The four quantum numbers-are discrete sets of integers or half- integers. –n: Principal quantum number-The first describes the electron shell, or energy level, of an atom –ℓ : Orbital angular momentum quantum number-as the angular quantum number or orbital quantum number) describes the subshell, and gives the magnitude of the orbital angular momentum through the relation Ll2 ( 1) –mℓ:Magnetic (azimuthal) quantum number (refers, to the direction of the angular momentum vector. The magnetic quantum number m does not affect the electron's energy, but it does affect the probability cloud)- magnetic quantum number determines the energy shift of an atomic orbital due to an external magnetic field-Zeeman effect -s spin- intrinsic angular momentum Spin "up" and "down" allows two electrons for each set of spatial quantum numbers. The restrictions for the quantum numbers: – n = 1, 2, 3, 4, . – ℓ = 0, 1, 2, 3, . , n − 1 – mℓ = − ℓ, − ℓ + 1, . , 0, 1, . , ℓ − 1, ℓ – –Equivalently: n > 0 The energy levels are: ℓ < n |m | ≤ ℓ ℓ E E 0 n n2 Stern-Gerlach experiment If the particles were classical spinning objects, one would expect the distribution of their spin angular momentum vectors to be random and continuous. Each particle would be deflected by a different amount, producing some density distribution on the detector screen. Instead, the particles passing through the Stern–Gerlach apparatus are deflected either up or down by a specific amount. -
Quantum Mechanics Electromotive Force
Quantum Mechanics_Electromotive force . Electromotive force, also called emf[1] (denoted and measured in volts), is the voltage developed by any source of electrical energy such as a batteryor dynamo.[2] The word "force" in this case is not used to mean mechanical force, measured in newtons, but a potential, or energy per unit of charge, measured involts. In electromagnetic induction, emf can be defined around a closed loop as the electromagnetic workthat would be transferred to a unit of charge if it travels once around that loop.[3] (While the charge travels around the loop, it can simultaneously lose the energy via resistance into thermal energy.) For a time-varying magnetic flux impinging a loop, theElectric potential scalar field is not defined due to circulating electric vector field, but nevertheless an emf does work that can be measured as a virtual electric potential around that loop.[4] In a two-terminal device (such as an electrochemical cell or electromagnetic generator), the emf can be measured as the open-circuit potential difference across the two terminals. The potential difference thus created drives current flow if an external circuit is attached to the source of emf. When current flows, however, the potential difference across the terminals is no longer equal to the emf, but will be smaller because of the voltage drop within the device due to its internal resistance. Devices that can provide emf includeelectrochemical cells, thermoelectric devices, solar cells and photodiodes, electrical generators,transformers, and even Van de Graaff generators.[4][5] In nature, emf is generated whenever magnetic field fluctuations occur through a surface. -
The Quantum Mechanical Model of the Atom
The Quantum Mechanical Model of the Atom Quantum Numbers In order to describe the probable location of electrons, they are assigned four numbers called quantum numbers. The quantum numbers of an electron are kind of like the electron’s “address”. No two electrons can be described by the exact same four quantum numbers. This is called The Pauli Exclusion Principle. • Principle quantum number: The principle quantum number describes which orbit the electron is in and therefore how much energy the electron has. - it is symbolized by the letter n. - positive whole numbers are assigned (not including 0): n=1, n=2, n=3 , etc - the higher the number, the further the orbit from the nucleus - the higher the number, the more energy the electron has (this is sort of like Bohr’s energy levels) - the orbits (energy levels) are also called shells • Angular momentum (azimuthal) quantum number: The azimuthal quantum number describes the sublevels (subshells) that occur in each of the levels (shells) described above. - it is symbolized by the letter l - positive whole number values including 0 are assigned: l = 0, l = 1, l = 2, etc. - each number represents the shape of a subshell: l = 0, represents an s subshell l = 1, represents a p subshell l = 2, represents a d subshell l = 3, represents an f subshell - the higher the number, the more complex the shape of the subshell. The picture below shows the shape of the s and p subshells: (notice the electron clouds) • Magnetic quantum number: All of the subshells described above (except s) have more than one orientation. -
4 Nuclear Magnetic Resonance
Chapter 4, page 1 4 Nuclear Magnetic Resonance Pieter Zeeman observed in 1896 the splitting of optical spectral lines in the field of an electromagnet. Since then, the splitting of energy levels proportional to an external magnetic field has been called the "Zeeman effect". The "Zeeman resonance effect" causes magnetic resonances which are classified under radio frequency spectroscopy (rf spectroscopy). In these resonances, the transitions between two branches of a single energy level split in an external magnetic field are measured in the megahertz and gigahertz range. In 1944, Jevgeni Konstantinovitch Savoiski discovered electron paramagnetic resonance. Shortly thereafter in 1945, nuclear magnetic resonance was demonstrated almost simultaneously in Boston by Edward Mills Purcell and in Stanford by Felix Bloch. Nuclear magnetic resonance was sometimes called nuclear induction or paramagnetic nuclear resonance. It is generally abbreviated to NMR. So as not to scare prospective patients in medicine, reference to the "nuclear" character of NMR is dropped and the magnetic resonance based imaging systems (scanner) found in hospitals are simply referred to as "magnetic resonance imaging" (MRI). 4.1 The Nuclear Resonance Effect Many atomic nuclei have spin, characterized by the nuclear spin quantum number I. The absolute value of the spin angular momentum is L =+h II(1). (4.01) The component in the direction of an applied field is Lz = Iz h ≡ m h. (4.02) The external field is usually defined along the z-direction. The magnetic quantum number is symbolized by Iz or m and can have 2I +1 values: Iz ≡ m = −I, −I+1, ..., I−1, I. -
Chapter 4: Bonding in Solids and Electronic Properties
Chapter 4: Bonding in Solids and Electronic Properties Free electron theory Consider free electrons in a metal – an electron gas. •regards a metal as a box in which electrons are free to move. •assumes nuclei stay fixed on their lattice sites surrounded by core electrons, while the valence electrons move freely through the solid. •Ignoring the core electrons, one can treat the outer electrons with a quantum mechanical description. •Taking just one electron, the problem is reduced to a particle in a box. Electron is confined to a line of length a. Schrodinger equation 2 2 2 d d 2me E 2 (E V ) 2 2 2me dx dx Electron is not allowed outside the box, so the potential is ∞ outside the box. Energy is quantized, with quantum numbers n. 2 2 2 n2h2 h2 n n n In 1d: E In 3d: E a b c 2 8m a2 b2 c2 8mea e 1 2 2 2 2 Each set of quantum numbers n , n , h n n n a b a b c E 2 2 2 and nc will give rise to an energy level. 8me a b c •In three dimensions, there are multiple combination of energy levels that will give the same energy, whereas in one-dimension n and –n are equal in energy. 2 2 2 2 2 2 na/a nb/b nc/c na /a + nb /b + nc /c 6 6 6 108 The number of states with the 2 2 10 108 same energy is known as the degeneracy. 2 10 2 108 10 2 2 108 When dealing with a crystal with ~1020 atoms, it becomes difficult to work out all the possible combinations. -
Lecture 24. Degenerate Fermi Gas (Ch
Lecture 24. Degenerate Fermi Gas (Ch. 7) We will consider the gas of fermions in the degenerate regime, where the density n exceeds by far the quantum density nQ, or, in terms of energies, where the Fermi energy exceeds by far the temperature. We have seen that for such a gas μ is positive, and we’ll confine our attention to the limit in which μ is close to its T=0 value, the Fermi energy EF. ~ kBT μ/EF 1 1 kBT/EF occupancy T=0 (with respect to E ) F The most important degenerate Fermi gas is 1 the electron gas in metals and in white dwarf nε()(),, T= f ε T = stars. Another case is the neutron star, whose ε⎛ − μ⎞ exp⎜ ⎟ +1 density is so high that the neutron gas is ⎝kB T⎠ degenerate. Degenerate Fermi Gas in Metals empty states ε We consider the mobile electrons in the conduction EF conduction band which can participate in the charge transport. The band energy is measured from the bottom of the conduction 0 band. When the metal atoms are brought together, valence their outer electrons break away and can move freely band through the solid. In good metals with the concentration ~ 1 electron/ion, the density of electrons in the electron states electron states conduction band n ~ 1 electron per (0.2 nm)3 ~ 1029 in an isolated in metal electrons/m3 . atom The electrons are prevented from escaping from the metal by the net Coulomb attraction to the positive ions; the energy required for an electron to escape (the work function) is typically a few eV. -
A Relativistic Electron in a Coulomb Potential
A Relativistic Electron in a Coulomb Potential Alfred Whitehead Physics 518, Fall 2009 The Problem Solve the Dirac Equation for an electron in a Coulomb potential. Identify the conserved quantum numbers. Specify the degeneracies. Compare with solutions of the Schrödinger equation including relativistic and spin corrections. Approach My approach follows that taken by Dirac in [1] closely. A few modifications taken from [2] and [3] are included, particularly in regards to the final quantum numbers chosen. The general strategy is to first find a set of transformations which turn the Hamiltonian for the system into a form that depends only on the radial variables r and pr. Once this form is found, I solve it to find the energy eigenvalues and then discuss the energy spectrum. The Radial Dirac Equation We begin with the electromagnetic Hamiltonian q H = p − cρ ~σ · ~p − A~ + ρ mc2 (1) 0 1 c 3 with 2 0 0 1 0 3 6 0 0 0 1 7 ρ1 = 6 7 (2) 4 1 0 0 0 5 0 1 0 0 2 1 0 0 0 3 6 0 1 0 0 7 ρ3 = 6 7 (3) 4 0 0 −1 0 5 0 0 0 −1 1 2 0 1 0 0 3 2 0 −i 0 0 3 2 1 0 0 0 3 6 1 0 0 0 7 6 i 0 0 0 7 6 0 −1 0 0 7 ~σ = 6 7 ; 6 7 ; 6 7 (4) 4 0 0 0 1 5 4 0 0 0 −i 5 4 0 0 1 0 5 0 0 1 0 0 0 i 0 0 0 0 −1 We note that, for the Coulomb potential, we can set (using cgs units): Ze2 p = −eΦ = − o r A~ = 0 This leads us to this form for the Hamiltonian: −Ze2 H = − − cρ ~σ · ~p + ρ mc2 (5) r 1 3 We need to get equation 5 into a form which depends not on ~p, but only on the radial variables r and pr. -
Chapter 13 Ideal Fermi
Chapter 13 Ideal Fermi gas The properties of an ideal Fermi gas are strongly determined by the Pauli principle. We shall consider the limit: k T µ,βµ 1, B � � which defines the degenerate Fermi gas. In this limit, the quantum mechanical nature of the system becomes especially important, and the system has little to do with the classical ideal gas. Since this chapter is devoted to fermions, we shall omit in the following the subscript ( ) that we used for the fermionic statistical quantities in the previous chapter. − 13.1 Equation of state Consider a gas ofN non-interacting fermions, e.g., electrons, whose one-particle wave- functionsϕ r(�r) are plane-waves. In this case, a complete set of quantum numbersr is given, for instance, by the three cartesian components of the wave vector �k and thez spin projectionm s of an electron: r (k , k , k , m ). ≡ x y z s Spin-independent Hamiltonians. We will consider only spin independent Hamiltonian operator of the type ˆ 3 H= �k ck† ck + d r V(r)c r†cr , �k � where thefirst and the second terms are respectively the kinetic and th potential energy. The summation over the statesr (whenever it has to be performed) can then be reduced to the summation over states with different wavevectork(p=¯hk): ... (2s + 1) ..., ⇒ r � �k where the summation over the spin quantum numberm s = s, s+1, . , s has been taken into account by the prefactor (2s + 1). − − 159 160 CHAPTER 13. IDEAL FERMI GAS Wavefunctions in a box. We as- sume that the electrons are in a vol- ume defined by a cube with sidesL x, Ly,L z and volumeV=L xLyLz.