DOCSLIB.ORG

The Hydrogen Atom As an Entangled Electron–Proton System Paolo Tommasini, Eddy Timmermans, and A

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Hydrogen Atom As an Entangled Electron–Proton System Paolo Tommasini, Eddy Timmermans, and A The hydrogen atom as an entangled electron–proton system Paolo Tommasini, Eddy Timmermans, and A. F. R. de Toledo Piza Citation: American Journal of Physics 66, 881 (1998); doi: 10.1119/1.18977 View online: http://dx.doi.org/10.1119/1.18977 View Table of Contents: http://scitation.aip.org/content/aapt/journal/ajp/66/10?ver=pdfcov Published by the American Association of Physics Teachers Advertisement: [This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 143.107.129.11 On: Mon, 14 Oct 2013 15:27:46 The hydrogen atom as an entangled electron–proton system Paolo Tommasini and Eddy Timmermans Institute for Theoretical Atomic and Molecular Physics, Harvard–Smithsonian Center for Astrophysics, Cambridge, Massachusetts 02138 A. F. R. de Toledo Piza Instituto de Fisica, Universidade de Sa˜o Paulo, C. P. 66318, Sa˜o Paulo, SP 05315-970, Brazil ~Received 22 October 1997; accepted 6 February 1998! We illustrate the description of correlated subsystems by studying the simple two-body hydrogen atom. We study the entanglement of the electron and proton coordinates in the exact analytical solution. This entanglement, which we quantify in the framework of the density matrix formalism, describes correlations in the electron–proton motion. © 1998 American Association of Physics Teachers. I. INTRODUCTION II. SUBSYSTEMS AND QUANTUM CORRELATIONS Independent particle models often provide a good starting In very broad terms, giving the state of a physical system point to describe the physics of many-particle systems. In means providing the necessary information to evaluate all its observables quantitatively. The states of a quantum mechani- these models, the individual particles behave as independent cal system are frequently represented by vectors of unit norm particles that move in a potential field that accounts for the in a Hilbert space H, in general of infinite dimension ~hence interaction with the other particles in an average sense. How- their usual designation as ‘‘state vectors’’!. If the system is ever, depending on the systems and the properties that are in a state uC&, with ^CuC&51, the outcome of measurements studied, the results of the independent particle calculations of an observable quantity A has an expectation value ^A& are not always sufficiently accurate. In that case it is neces- equal to sary to correct for the fact that the motion of a single particle depends on the positions of the other particles, rather than on ^A&5^CuAˆ uC&, ~1! some average density. Consequently, in a system of interact- ˆ ing particles, the probability of finding two particles with where we use a caret, ‘‘A,’’ to distinguish the operator rep- given positions or momenta is not simply the product of the resenting the observable from the observable itself. If the vectors w , i51,2,... are a basis of the Hilbert space ,we single-particle probabilities: We say that the particles are u i& H ‘‘correlated.’’ can use the completeness relation, 15(muwm&^wmu, to write the expectation value as In a more general context, the problem can be formulated as the description of interacting subsystems ~the single par- ticle in the many-particle system being the example of the ^A&5 ^Cuwm&^wmuAuwn&^wnuC&. ~2! m(,n subsystem!. A convenient theoretical framework to deal with this problem in quantum mechanics is provided by the den- In matrix notation, the mn-matrix element of the Aˆ operator sity matrix theory, which is almost as old as quantum me- ˆ ˆ in the uw& basis is equal to Amn5^wmuAuwn&. The remaining chanics itself.1,2 The ever-present interest in density matrix ingredients of Eq. ~2! can be collected as rnm5^wnuC& theory, in spite of its long history, is justified by the power of 1,4,5 3^Cuwm&, known as the density matrix, which is recog- its description and the importance of its applications, which nized as the matrix representing the so-called density opera- extends to the study of the very foundations of quantum tor, r5uC Cu, in the chosen basis. The expectation value 1–3 &^ theory. In this article, we briefly review some important of Eq. ~2! can then be written as aspects of density matrix theory and we illustrate its use for describing correlations of interacting subsystems by studying ˆ ^A&5( Amnrnm , ~3! the simple, exactly solvable system of the hydrogen atom. m,n 881 Am. J. Phys. 66 ~10!, October 1998 © 1998 American Association of Physics Teachers 881 [This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 143.107.129.11 On: Mon, 14 Oct 2013 15:27:46 which is the trace of the product of the A and r matrices, An important limiting situation is that in which the overall ^A&5Tr(Aˆ r). In this expression, the state of the system is state vector uC& is itself a product of a u vector and a v represented by the density matrix or, equivalently, by the vector, uC&5uu&uv&. For these product states, the density density operator r. The form of this operator, together with matrix r5uC&^Cu becomes the normalization property of the underlying state vector uC&, lead to some important properties, which are of course r5uu&uv&^vu^uu5rurv, ~6! shared by the corresponding density matrices. Namely, the density operator is a non-negative ~meaning that ^furuf& and the reduced density ru is simply given by uu&^uu. This is >0 for any vector uf&! self-adjoint operator (r5r†), of unit just of the form of a density operator associated with the trace @Tr(r)51#, and idempotent ~meaning that r5r2, state vector uu&, which can then also be used to describe the which follows from uC&^CuC&^Cu5uC&^Cu!. state of the subsystem. In general, the state vector uC& cannot The use of state vectors uC& to describe the state of a be factored in this fashion, however. In this case one says quantum system is, however, not general enough to cover that the two subsystems are in an ‘‘entangled’’ state. The many frequently occurring situations. As we shall discuss density matrix of an entangled state does not factor and ap- below, a suitable generalization is provided by the density pears as a series of u and v products: The subsystems are matrix language, if only we relax the idempotency require- correlated. The number of terms gives some indication of the ment. departure from the uncorrelated density matrix, but as the Often, when probing a system, only part of the total sys- number of terms depends on the choice of basis in the u and tem is subjected to a measurement. For example, in a typical v spaces, this is not a good measure of correlation or en- scattering experiment only the scattered particle is detected. tanglement. A way out of this difficulty is, however, pro- The target system is left behind in a final state that could be vided by a simple and remarkable result due to Schmidt7 ~see different from its initial state, but this state is not observed also Refs. 2, 8, and 9! which essentially identifies a ‘‘natu- directly. It is then natural to divide the total system into the ral’’ basis ~in the sense of being determined by the structure target system and the single-particle system of the scatterer. of the entangled state itself! in which the description of the Also, in many experiments involving mesoscopic, or semi- entanglement is achieved with maximum simplicity. macroscopic, quantum devices, it is very hard, if not impos- The Schmidt basis is a product basis, in the usual sense sible, to ensure efficient isolation from the environment, that it is written as the tensor product of two particular bases, which then plays the role of a second subsystem coupled to one for the H space and another for the H space. In order 6 u v the system of interest. Mathematically, such partitioning of to find these two bases one looks for the eigenvectors of the the system into subsystems means factoring the Hilbert space reduced density matrices. If the state is not entangled, uC& H into the tensor product of the corresponding subspaces, 5uu&uv&, ru and rv each have one single eigenvector of H5Hu ^ Hv . If the Hu and Hv spaces are spanned, re- nonzero eigenvalue, uu& for ru and uv& for rv, and the nu- spectively, by the bases uui&, i51,2,... and uv j&, j51,2,..., merical value of the corresponding eigenvalues is equal to 1. then H is spanned by the product states uui&uv j&. Accord- One then completes the basis in each subspace by including ingly, a state vector uC& describing the state of the system enough additional orthonormal vectors which are moreover can be expanded as orthogonal to the single ‘‘relevant’’ vector with nonzero ei- genvalue. Each one of these additional basis vectors is then an eigenvector of the corresponding reduced density with uC&5( di, juui&uv j&. ~4! i, j eigenvalue zero, and the large degree of arbitrariness in choosing them reflects the large degeneracy of the corre- Suppose now that one wishes to describe the state of one sponding eigenvalue zero. If, on the other hand, the sub- of the subsystems alone, say subsystem u. The observable systems are correlated, then ru and rv have a spectrum of quantities of this subsystem are represented by operators nonvanishing eigenvalues. Interestingly, as we prove below, which act nontrivially on the vectors of Hu while acting on the eigenvalues of the ru and rv operators are the same.
Load more

Recommended publications
  • 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, .
    [Show full text]
  • 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…..∞.
    [Show full text]
  • Further Quantum Physics
    Further Quantum Physics Concepts in quantum physics and the structure of hydrogen and helium atoms Prof Andrew Steane January 18, 2005 2 Contents 1 Introduction 7 1.1 Quantum physics and atoms . 7 1.1.1 The role of classical and quantum mechanics . 9 1.2 Atomic physics—some preliminaries . .... 9 1.2.1 Textbooks...................................... 10 2 The 1-dimensional projectile: an example for revision 11 2.1 Classicaltreatment................................. ..... 11 2.2 Quantum treatment . 13 2.2.1 Mainfeatures..................................... 13 2.2.2 Precise quantum analysis . 13 3 Hydrogen 17 3.1 Some semi-classical estimates . 17 3.2 2-body system: reduced mass . 18 3.2.1 Reduced mass in quantum 2-body problem . 19 3.3 Solution of Schr¨odinger equation for hydrogen . ..... 20 3.3.1 General features of the radial solution . 21 3.3.2 Precisesolution.................................. 21 3.3.3 Meanradius...................................... 25 3.3.4 How to remember hydrogen . 25 3.3.5 Mainpoints.................................... 25 3.3.6 Appendix on series solution of hydrogen equation, off syllabus . 26 3 4 CONTENTS 4 Hydrogen-like systems and spectra 27 4.1 Hydrogen-like systems . 27 4.2 Spectroscopy ........................................ 29 4.2.1 Main points for use of grating spectrograph . ...... 29 4.2.2 Resolution...................................... 30 4.2.3 Usefulness of both emission and absorption methods . 30 4.3 The spectrum for hydrogen . 31 5 Introduction to fine structure and spin 33 5.1 Experimental observation of fine structure . ..... 33 5.2 TheDiracresult ..................................... 34 5.3 Schr¨odinger method to account for fine structure . 35 5.4 Physical nature of orbital and spin angular momenta .
    [Show full text]
  • 1.2 Formulation of the Schrödinger Equation for the Hydrogen Atom
    2 1 Hydrogen orbitals 1.2 Formulation of the Schrodinger¨ equation for the hydrogen atom In this initial treatment, we will make some practical approximations and simplifi- cations. Since we are for the moment only trying to establish the general form of the hydrogenic wave functions, this will suffice. To start with, we will assume that the nucleus has zero extension. We place the origin at its position, and we ignore the centre-of-mass motion. This reduces the two-body problem to a single particle, the electron, moving in a central-field potential. To take the finite mass of the nu- cleus into account, we replace the electron mass with the reduced mass, m, of the two-body problem. Moreover, we will in this chapter ignore the effect on the wave function of relativistic effects, which automatically implies that we ignore the spins of the electron and of the nucleus. This makes us ready to formulate the Hamilto- nian. The potential is the classical Coulomb interaction between two particles of op- posite charges. With spherical coordinates, and with r as the radial distance of the electron from the origin, this is: Ze2 V(r) = − ; (1.1) 4pe0 r with Z being the charge state of the nucleus. The Schrodinger¨ equation is: h¯ 2 − ∇2y(r) +V(r)y = Ey(r) ; (1.2) 2m where the Laplacian in spherical coordinates is: 1 ¶ ¶ 1 ¶ ¶ 1 ¶ 2 ∇2 = r2 + sinq + : (1.3) r2 ¶r ¶r r2 sinq ¶q ¶q r2 sin2 q ¶j2 Since the potential is purely central, the solution to (1.2) can be factorised into a radial and an angular part, y(r;q;j) = R(r)Y(q;j).
    [Show full text]
  • 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.
    [Show full text]
  • Importance of Hydrogen Atom
    Hydrogen Atom Dragica Vasileska Arizona State University Importance of Hydrogen Atom • Hydrogen is the simplest atom • The quantum numbers used to characterize the allowed states of hydrogen can also be used to describe (approximately) the allowed states of more complex atoms – This enables us to understand the periodic table • The hydrogen atom is an ideal system for performing precise comparisons of theory and experiment – Also for improving our understanding of atomic structure • Much of what we know about the hydrogen atom can be extended to other single-electron ions – For example, He+ and Li2+ Early Models of the Atom • ’ J.J. Thomson s model of the atom – A volume of positive charge – Electrons embedded throughout the volume • ’ A change from Newton s model of the atom as a tiny, hard, indestructible sphere “ ” watermelon model Experimental tests Expect: 1. Mostly small angle scattering 2. No backward scattering events Results: 1. Mostly small scattering events 2. Several backward scatterings!!! Early Models of the Atom • ’ Rutherford s model – Planetary model – Based on results of thin foil experiments – Positive charge is concentrated in the center of the atom, called the nucleus – Electrons orbit the nucleus like planets orbit the sun ’ Problem: Rutherford s model “ ” ’ × – The size of the atom in Rutherford s model is about 1.0 10 10 m. (a) Determine the attractive electrical force between an electron and a proton separated by this distance. (b) Determine (in eV) the electrical potential energy of the atom. “ ” ’ × – The size of the atom in Rutherford s model is about 1.0 10 10 m. (a) Determine the attractive electrical force between an electron and a proton separated by this distance.
    [Show full text]
  • The Hydrogen Atom: Consideration of the Electron Self-Field Arxiv
    The hydrogen atom: consideration of the electron self-field Biguaa L.V ∗ Kassandrov V.V † ∗ Quantum technology center Moscow State University y Institute of Gravitation and Cosmology, Peoples’ Friendship University of Russia January 8, 2021 Keywords: Spinor electrodynamics. Dirac-Maxwell system of equations. Solitonlike solu- tions. Spectrum of characteristics. “Bohrian” law for binding energies PACS: 03.50.-z; 03.65.-w; 03.65.-Pm 1 THE HYDROGEN ATOM AND CLASSICAL FIELD MODELS OF ELEMENTARY PARTICLES It is well known that the successful description of the hydrogen atom, first in Bohr’s theory, and then via solutions to the Schr¨odinger equation was one of the main motivations for the development of quantum theory. Later, Dirac described almost perfectly the observed hydrogen spectrum using his relativistic generalization of the Schr¨odingerequation. A slight deviation from experiment (the Lamb shift between 2s1=2 and 2p1=2 levels in the hydrogen atom) arXiv:2101.02202v1 [quant-ph] 5 Jan 2021 discovered later was interpreted as the effect of electron interaction with vacuum fluctuations of electromagnetic field and explained in the framework of quantum electrodynamics (QED) using second quantization. Nonetheless, the problem of the description of the hydrogen atom still attracts attention of researchers, and, in some sense, is a “touchstone” for many new theoretical developments. In particular, it is possible to obtain the energy spectrum via purely algebraic methods [1]. Theories of the ”supersymmetric” hydrogen atom are also widespread [2]. In the framework of the classical ideas, the spectrum can be explained by the absence of radiation for the electron ∗E-mail: [email protected] †E-mail: [email protected] 1 motion along certain “stable” orbits.
    [Show full text]
  • The Hydrogen Atom
    Quantum Mechanics: The Hydrogen Atom 13th April 2011 I. The Hydrogen Atom In this next section, we will tie together the elements of the last several sections to arrive at a complete description of the hydrogen atom. This will cul- minate in the definition of the hydrogen-atom orbitals and associated energies. From these functions, taken as a complete basis, we will be able to construct approximations to more complex wave functions for more complex molecules. Thus, the work of the last few lectures has fundamentally been aimed at estab- lishing a foundation for more complex problems in terms of exact solutions for smaller, model problems. II. The Radial Function We will start by reiterating the Schrodinger equation in 3D spherical coordi- nates as (refer to any standard text to get the transformation from Cartesian to spherical coordinate reference systems). Here, we have not placed the constraint of a constant distance separting the masses of the rigid rotor (refer to last lec- ture); furthermore, we will keep in the formulation the potential V (r; θ; φ) for generality. Thus, in spherical polar coordinates, H^ (r; θ; φ) (r; θ; φ) = E (r; θ; φ) be- comes: E (r; θ; φ) = −h¯2 1 @ @ 1 @ @ 1 @2 r2 + sinθ + + V (r; θ; φ) (r; θ; φ) 2µ r2 @r @r r2sinθ @θ @θ r2sin2θ @φ2 Now, for the hydrogen atom, with one electron found in "orbits" (note the quotes!) around the nucleus of charge +1, we can include an electrostatic poten- tial which is essentially the Coulomb potential between a positive and negative charge: −Ze2 V (jrj) = V (r) = 4πor where Z is the nuclear charge (i.e, +1 for the nucleus of a hydrogen atom).
    [Show full text]
  • The Hydrogen Atom Atomic Emission Spectroscopy
    The Hydrogen atom Atomic Emission Spectroscopy the atomic emission spectrum is the result of many atoms emitting light simultaneously. a spectrometer Absorption and Emission Spectroscopy Light will be absorbed or emitted only if ∆E = hν The spectrum is QUANTIZED! ∆E51=24G ∆E41=15G ∆E31=8G ∆E21=3G G= h2/8mL2 Atomic Emission Spectroscopy The emission spectra of H, He and Hg are all quantized! Energy eigenvectors/eigenstates is the eigenstate vector solution to the Schrödinger equation. is the energy eigenvalue associated with this eigenstate vector. Hydrogen eigenstates/wavefunctions State Vectors Three Dimensional Wavefunction (Hydrogen Atom) The spatial component of the eigenstate vector can be represented by a three dimensional wavefunction ψ(r) = ψ(x,y,z). The Hydrogen Atom Hamiltonian V(r) is called the Coulomb potential. WHEN we solve the Schrödinger equation, we always obtain two things: 1. a set of eigenstate energies, En. Example: PIAB n = 1, 2, 3... WHEN we solve the Schrödinger equation, we always obtain two things: 1. a set of eigenstate energies, En. 2. a set of eigenstate vectors |ψn>. Example: PIAB n = 1, 2, 3... Eigenstate vectors are labeled with the quantum number n. WHEN we solve the Schrödinger equation, we always obtain two things: 1. a set of eigenstate energies, En. Hydrogen atom solution: n = 1, 2, 3... WHEN we solve the Schrödinger equation, we always obtain two things: 1. a set of eigenstate energies, En. 2. a set of eigenstate vectors |ψn>. Hydrogen atom solution: n = 1, 2, 3... State vectors are labeled with the quantum number n... AND three more: l, ml, m s WHEN we solve the Schrödinger equation, we always obtain two things: 1.
    [Show full text]
  • Name___Making Models of Atoms & Isotopes
    Hydrogen Models 1 Name______________________ Making Models of Atoms & Isotopes Problem: To build 3 dimensional models of the hydrogen atom. Background Information: An atom is defined as Re-write the main ideas of the background a small particle that makes up most types of matter. Atoms are so small it would take about 1 information here: million of them lined up in a row to equal the thickness of a human hair. Atoms are made up ____________________________________ of even smaller particles. The largest of these ____________________________________ particles are protons, neutrons and electrons. The identity of a type of matter depends on the ____________________________________ number of protons in the nucleus of an atom of that type of matter. All atoms of the same type of ____________________________________ matter have the same number of protons in the ____________________________________ nucleus. For example, all carbon atoms have six protons. ____________________________________ Not all atoms of the same type of matter have the ____________________________________ same number of neutrons. Most carbon atoms ____________________________________ have 6 neutrons, although some have more and some have less. Atoms of the same type of ____________________________________ matter that have different numbers of neutrons are called isotopes. Most types of matter have ____________________________________ isotopes. The atomic mass, or mass number, can be used to identify isotopes. The mass number of a type of matter is the number of protons and neutrons in one atom of that matter. In our example of carbon, 6 protons + 6 neutrons = 12 particles in the nucleus; the mass number = 12. The name for this isotope is Carbon 12. The symbol for Carbon 12 is: Mass Number 12 C A tomic Number 6 Diagrams of carbon isotopes: M.
    [Show full text]
  • Hydrogen Atomic and Molecular Hydrogen
    Hydrogen Hydrogen, the simplest atom, combines with itself to make the simplest molecule, H2. Most other elements in the periodic table combine with hydrogen to make compounds. The chemical bonds between hydrogen and other elements result from sharing electrons. The electrons are shared equally in non-polar covalent bonds and unequally in polar covalent bonds. The chemical energy stored in covalent bonds can be transformed into heat or electrical energy. Outline • Atomic and Molecular Hydrogen • Hydrogen Compounds • Energy from Hydrogen • Homework Atomic and Molecular Hydrogen Electronic Structure of Atomic Hydrogen We can represent a hydrogen atom with the symbol H and a dot that stands for the single electron in the neutral atom. In its lowest energy form, the electron occupies the 1s orbital, spherically symmetrical around the hydrogen nucleus. As you saw in the section on electronic configuration, all atoms have many possible energy levels for their electrons. The lowest energy state (ground state) of an atom has its electrons filling orbitals beginning from lowest energy, with 2 electrons per orbital. When energy is added to the ground state atom, an electron can be promoted to one of the higher energy orbitals. The energy added must be exactly the same as the energy between the orbital occupied by the electron and one of the higher energy, unoccupied orbitals. Below you can see an orbital energy diagram showing the ground state hydrogen atom on the left. When hydrogen absorbs a quantity of energy exactly equal to ΔE1, the electron goes from the orbital in the first shell (n = 1) to an orbital in the second shell (n = 2).
    [Show full text]
  • 28-1 Line Spectra and the Hydrogen Atom Figure 28.1 Gives Some Examples of the Line Spectra Emitted by Atoms of Gas
    28-1 Line Spectra and the Hydrogen Atom Figure 28.1 gives some examples of the line spectra emitted by atoms of gas. The atoms are typically excited by applying a high voltage across a glass tube that contains a particular gas. By observing the light through a diffraction grating, the light is separated into a set of wavelengths that characterizes the element. The spectrum of light emitted by excited hydrogen atoms is shown Figure 28.1: Line spectra from hydrogen (top) and helium in Figure 28.1(a). The decoding of the (bottom). A line spectrum is like the fingerprint of an element. hydrogen spectrum represents one of Astronomers, for instance, can determine what a star is made of by the great scientific mystery stories. carefully examining the spectrum of light emitted by the star. First on the scene was the Swiss mathematician and schoolteacher, Johann Jakob Balmer (1825 – 1898), who published an equation in 1885 giving the wavelengths in the visible spectrum emitted by hydrogen. The Swedish physicist Johannes Rydberg (1854 – 1919) followed up on Balmer’s work in 1888 with a more general equation that predicted all the wavelengths of light emitted by hydrogen: , (Eq. 28.1: The Rydberg equation for the hydrogen spectrum) 7 –1 where R = 1.097 ! 10 m is the Rydberg constant, and the two n’s are integers, with n2 greater than n1. Neither Balmer nor Rydberg had a physical explanation to justify their equations, however, so the search was on for such a physical explanation. The breakthrough was made by the Danish physicist Niels Bohr (1885 – 1962), who showed that if the angular momentum of the electron in a hydrogen atom was quantized in a particular way (related to Planck’s constant, in fact), that the energy levels for an electron within the hydrogen atom were also quantized, with the energies of the electrons being given by: , (Eq.
    [Show full text]