Lecture #2: August 25, 2020 Goal Is to Define Electrons in Atoms
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The Balmer Series
The Balmer Series Introduction Historically, the spectral lines of hydrogen have been categorized as six distinct series. The visible portion of the hydrogen spectrum is contained in the Balmer series, named after Johann Balmer who discovered an empirical relationship for calculating its wave- lengths [2]. In 1885 Balmer discovered that by labeling the four visible lines of the hydorgen spectrum with integers n, (n = 3, 4, 5, 6) (See Table 1 and Figure 1) each wavelength λ could be calculated from the relationship n2 λ = B , (1) n2 − 4 where B = 364.56 nm. Johannes Rydberg generalized Balmer’s result to include all of the wavelengths of the hydrogen spectrum. The Balmer formula is more commonly re–expressed in the form of the Rydberg formula 1 1 1 = RH − , (2) λ 22 n2 where n =3, 4, 5 . and the Rydberg constant RH =4/B. The purpose of this experiment is to determine the wavelengths of the visible spec- tral lines of hydrogen using a spectrometer and to calculate the Balmer constant B. Procedure The spectrometer used in this experiment is shown in Fig. 1. Adjust the diffraction grating so that the normal to its plane makes a small angle α to the incident beam of light. This is shown schematically in Fig. 2. Since α u 0, the angles between the first and zeroth order intensity maxima on either side, θ and θ0 respectively, are related to the wavelength λ of the incident light according to [1] λ = d sin(φ) , (3) accurate to first order in α. Here, d is the separation between the slits of the grating, and θ0 + θ φ = (4) 2 1 Color Wavelength [nm] Integer [n] Violet2 410.2 6 Violet1 434.0 5 blue 486.1 4 red 656.3 3 Table 1: The wavelengths and integer associations of the visible spectral lines of hy- drogen shown in Fig. -
October 16, 2014
October 16, 2014 Chapter 5: Electrons in Atoms Honors Chemistry Bohr Model Niels Bohr, a young Danish physicist and a student of Rutherford improved Rutherford's model. Bohr proposed that an electron is found only in specific circular paths, or orbits, around the nucleus. Each electron orbit has a fixed energy. Energy levels: the fixed energies of an electron. Quantum: The amount of energy required to move an electron from one energy level to another level. Light Electrons can jump from one energy level to another. The term quantum leap can be used to describe an The modern quantum model grew out of the study of light. 1. Light as a wave abrupt change in energy. 2. Light as a particle -Newton was the first to try to explain the behavior of light by assuming that light consists of particles. October 16, 2014 Light as a Wave 1. Wavelength-shortest distance between equivalent points on a continuous wave. symbol: λ (Greek letter lambda) 2. Frequency- the number of waves that pass a point per second. symbol: ν (Greek letter nu) 3. Amplitude-wave's height from the origin to the crest or trough. 4. Speed-all EM waves travel at the speed of light. symbol: c=3.00 x 108 m/s in a vacuum The speed of light Calculate the following: 8 1. The wavelength of radiation with a frequency of 1.50 x 1013 c=3.00 x 10 m/s Hz. Does this radiation have a longer or shorter wavelength than red light? (2.00 x 10-5 m; longer than red) c=λν 2. -
Principal, Azimuthal and Magnetic Quantum Numbers and the Magnitude of Their Values
268 A Textbook of Physical Chemistry – Volume I Principal, Azimuthal and Magnetic Quantum Numbers and the Magnitude of Their Values The Schrodinger wave equation for hydrogen and hydrogen-like species in the polar coordinates can be written as: 1 휕 휕휓 1 휕 휕휓 1 휕2휓 8휋2휇 푍푒2 (406) [ (푟2 ) + (푆푖푛휃 ) + ] + (퐸 + ) 휓 = 0 푟2 휕푟 휕푟 푆푖푛휃 휕휃 휕휃 푆푖푛2휃 휕휙2 ℎ2 푟 After separating the variables present in the equation given above, the solution of the differential equation was found to be 휓푛,푙,푚(푟, 휃, 휙) = 푅푛,푙. 훩푙,푚. 훷푚 (407) 2푍푟 푘 (408) 3 푙 푘=푛−푙−1 (−1)푘+1[(푛 + 푙)!]2 ( ) 2푍 (푛 − 푙 − 1)! 푍푟 2푍푟 푛푎 √ 0 = ( ) [ 3] . exp (− ) . ( ) . ∑ 푛푎0 2푛{(푛 + 푙)!} 푛푎0 푛푎0 (푛 − 푙 − 1 − 푘)! (2푙 + 1 + 푘)! 푘! 푘=0 (2푙 + 1)(푙 − 푚)! 1 × √ . 푃푚(퐶표푠 휃) × √ 푒푖푚휙 2(푙 + 푚)! 푙 2휋 It is obvious that the solution of equation (406) contains three discrete (n, l, m) and three continuous (r, θ, ϕ) variables. In order to be a well-behaved function, there are some conditions over the values of discrete variables that must be followed i.e. boundary conditions. Therefore, we can conclude that principal (n), azimuthal (l) and magnetic (m) quantum numbers are obtained as a solution of the Schrodinger wave equation for hydrogen atom; and these quantum numbers are used to define various quantum mechanical states. In this section, we will discuss the properties and significance of all these three quantum numbers one by one. Principal Quantum Number The principal quantum number is denoted by the symbol n; and can have value 1, 2, 3, 4, 5…..∞. -
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. -
Performance of Numerical Approximation on the Calculation of Two-Center Two-Electron Integrals Over Non-Integer Slater-Type Orbitals Using Elliptical Coordinates
Performance of numerical approximation on the calculation of two-center two-electron integrals over non-integer Slater-type orbitals using elliptical coordinates A. Bağcı* and P. E. Hoggan Institute Pascal, UMR 6602 CNRS, University Blaise Pascal, 24 avenue des Landais BP 80026, 63177 Aubiere Cedex, France [email protected] Abstract The two-center two-electron Coulomb and hybrid integrals arising in relativistic and non- relativistic ab-initio calculations of molecules are evaluated over the non-integer Slater-type orbitals via ellipsoidal coordinates. These integrals are expressed through new molecular auxiliary functions and calculated with numerical Global-adaptive method according to parameters of non-integer Slater- type orbitals. The convergence properties of new molecular auxiliary functions are investigated and the results obtained are compared with results found in the literature. The comparison for two-center two- electron integrals is made with results obtained from one-center expansions by translation of wave- function to same center with integer principal quantum number and results obtained from the Cuba numerical integration algorithm, respectively. The procedures discussed in this work are capable of yielding highly accurate two-center two-electron integrals for all ranges of orbital parameters. Keywords: Non-integer principal quantum numbers; Two-center two-electron integrals; Auxiliary functions; Global-adaptive method *Correspondence should be addressed to A. Bağcı; e-mail: [email protected] 1 1. Introduction The idea of considering the principal quantum numbers over Slater-type orbitals (STOs) in the set of positive real numbers firstly introduced by Parr [1] and performed for the He atom and single- center calculations on the H2 molecule to demonstrate that improved accuracy can be achieved in Hartree-Fock-Roothaan (HFR) calculations. -
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. -
Rydberg Constant and Emission Spectra of Gases
Page 1 of 10 Rydberg constant and emission spectra of gases ONE WEIGHT RECOMMENDED READINGS 1. R. Harris. Modern Physics, 2nd Ed. (2008). Sections 4.6, 7.3, 8.9. 2. Atomic Spectra line database https://physics.nist.gov/PhysRefData/ASD/lines_form.html OBJECTIVE - Calibrating a prism spectrometer to convert the scale readings in wavelengths of the emission spectral lines. - Identifying an "unknown" gas by measuring its spectral lines wavelengths. - Calculating the Rydberg constant RH. - Finding a separation of spectral lines in the yellow doublet of the sodium lamp spectrum. INSTRUCTOR’S EXPECTATIONS In the lab report it is expected to find the following parts: - Brief overview of the Bohr’s theory of hydrogen atom and main restrictions on its application. - Description of the setup including its main parts and their functions. - Description of the experiment procedure. - Table with readings of the vernier scale of the spectrometer and corresponding wavelengths of spectral lines of hydrogen and helium. - Calibration line for the function “wavelength vs reading” with explanation of the fitting procedure and values of the parameters of the fit with their uncertainties. - Calculated Rydberg constant with its uncertainty. - Description of the procedure of identification of the unknown gas and statement about the gas. - Calculating resolution of the spectrometer with the yellow doublet of sodium spectrum. INTRODUCTION In this experiment, linear emission spectra of discharge tubes are studied. The discharge tube is an evacuated glass tube filled with a gas or a vapor. There are two conductors – anode and cathode - soldered in the ends of the tube and connected to a high-voltage power source outside the tube. -
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. -
Rydberg Excitation of Single Atoms for Applications in Quantum Information and Metrology Aaron Hankin
University of New Mexico UNM Digital Repository Physics & Astronomy ETDs Electronic Theses and Dissertations 1-28-2015 Rydberg Excitation of Single Atoms for Applications in Quantum Information and Metrology Aaron Hankin Follow this and additional works at: https://digitalrepository.unm.edu/phyc_etds Recommended Citation Hankin, Aaron. "Rydberg Excitation of Single Atoms for Applications in Quantum Information and Metrology." (2015). https://digitalrepository.unm.edu/phyc_etds/23 This Dissertation is brought to you for free and open access by the Electronic Theses and Dissertations at UNM Digital Repository. It has been accepted for inclusion in Physics & Astronomy ETDs by an authorized administrator of UNM Digital Repository. For more information, please contact [email protected]. Aaron Hankin Candidate Physics and Astronomy Department This dissertation is approved, and it is acceptable in quality and form for publication: Approved by the Dissertation Committee: Ivan Deutsch , Chairperson Carlton Caves Keith Lidke Grant Biedermann Rydberg Excitation of Single Atoms for Applications in Quantum Information and Metrology by Aaron Michael Hankin B.A., Physics, North Central College, 2007 M.S., Physics, Central Michigan Univeristy, 2009 DISSERATION Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy Physics The University of New Mexico Albuquerque, New Mexico December 2014 iii c 2014, Aaron Michael Hankin iv Dedication To Maiko and our unborn daughter. \There are wonders enough out there without our inventing any." { Carl Sagan v Acknowledgments The experiment detailed in this manuscript evolved rapidly from an empty lab nearly four years ago to its current state. Needless to say, this is not something a graduate student could have accomplished so quickly by him or herself. -
23-Chapt-6-Quantum2
The Nature of Energy • For atoms and molecules, one does not observe a continuous spectrum, as one gets from a ? white light source. • Only a line spectrum of discrete wavelengths is Electronic Structure observed. of Atoms © 2012 Pearson Education, Inc. The Players Erwin Schrodinger Werner Heisenberg Louis Victor De Broglie Neils Bohr Albert Einstein Max Planck James Clerk Maxwell Neils Bohr Explained the emission spectrum of the hydrogen atom on basis of quantization of electron energy. Neils Bohr Explained the emission spectrum of the hydrogen atom on basis of quantization of electron energy. Emission spectrum Emitted light is separated into component frequencies when passed through a prism. 397 410 434 486 656 wavelenth in nm Hydrogen, the simplest atom, produces the simplest emission spectrum. In the late 19th century a mathematical relationship was found between the visible spectral lines of hydrogen the group of hydrogen lines in the visible range is called the Balmer series 1 1 1 = R – λ ( 22 n2 ) Rydberg 1.0968 x 107m–1 Constant Johannes Rydberg Bohr Solution the electron circles nucleus in a circular orbit imposed quantum condition on electron energy only certain “orbits” allowed energy emitted is when electron moves from higher energy state (excited state) to lower energy state the lowest electron energy state is ground state ground state of hydrogen atom n = 5 n = 4 n = 3 n = 2 n = 1 e- excited state of hydrogen atom n = 5 n = 4 n = 3 e- n = 2 n = 1 hν excited state of hydrogen atom n = 5 n = 4 n = 3 n = 2 e- n = 1 hν excited state of hydrogen atom n = 5 n = 4 n = 3 e- n = 2 n = 1 hν Emission spectrum Emitted light is separated into component frequencies when passed through a prism. -
Quantum Spin Systems and Their Local and Long-Time Properties Carolee Wheeler Faculty Advisor: Robert Sims
URA Project Proposal Quantum Spin Systems and Their Local and Long-Time Properties Carolee Wheeler Faculty Advisor: Robert Sims In quantum mechanics, spin is an important concept having to do with atomic nuclei, hadrons, and elementary particles. Spin may be thought of as a measure of a particle’s rotation about its axis. However, spin differs from orbital angular momentum in the sense that the particles may carry integer or half-integer quantum numbers, i.e. 0, 1/2, 1, 3/2, 2, etc., whereas orbital angular momentum may only take integer quantum numbers. Furthermore, the spin of a charged elementary particle is related to a magnetic dipole moment. All quantum mechanic particles have an inherent spin. This is due to the fact that elementary particles (such as photons, electrons, or quarks) cannot be divided into smaller entities. In other words, they cannot be viewed as particles that are made up of individual, smaller particles that rotate around a common center. Thus, the spin that elementary particles carry is an intrinsic property [1]. An important characteristic of spin in quantum mechanics is that it is quantized. The magnitude of spin takes values S = h s(s + )1 , with h being the reduced Planck’s constant and s being the spin quantum number (a non- negative integer or half-integer). Spin may also be viewed in composite particles, and is calculated by summing the spins of the constituent particles. In the case of atoms and molecules, spin is the sum of the spins of unpaired electrons [1]. Particles with spin can possess a magnetic moment. -
The Principal Quantum Number the Azimuthal Quantum Number The
To completely describe an electron in an atom, four quantum numbers are needed: energy (n), angular momentum (ℓ), magnetic moment (mℓ), and spin (ms). The Principal Quantum Number This quantum number describes the electron shell or energy level of an atom. The value of n ranges from 1 to the shell containing the outermost electron of that atom. For example, in caesium (Cs), the outermost valence electron is in the shell with energy level 6, so an electron incaesium can have an n value from 1 to 6. For particles in a time-independent potential, as per the Schrödinger equation, it also labels the nth eigen value of Hamiltonian (H). This number has a dependence only on the distance between the electron and the nucleus (i.e. the radial coordinate r). The average distance increases with n, thus quantum states with different principal quantum numbers are said to belong to different shells. The Azimuthal Quantum Number The angular or orbital quantum number, describes the sub-shell and gives the magnitude of the orbital angular momentum through the relation. ℓ = 0 is called an s orbital, ℓ = 1 a p orbital, ℓ = 2 a d orbital, and ℓ = 3 an f orbital. The value of ℓ ranges from 0 to n − 1 because the first p orbital (ℓ = 1) appears in the second electron shell (n = 2), the first d orbital (ℓ = 2) appears in the third shell (n = 3), and so on. This quantum number specifies the shape of an atomic orbital and strongly influences chemical bonds and bond angles.