CH107%20Special%20Topics

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CH107 Special Topics
Part A The Bohr Model of the Hydrogen Atom
(First steps in Quantization of the Atom) Part
B Waves and Wave Equations (the Electron as a
Wave Form) Part C Particle (such as an Electron)
in a Box (Square Well) and Similar
Situations
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A. The Bohr Model of the Hydrogen Atom
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Setting a Goal for Part A

• You will learn how Bohr imposed Plancks
  hypothesis on a classical description of the
  Rutherford model of the hydrogen atom and used
  this model to explain the emission and absorption
  spectra of hydrogen (Z 1) and other single
  electron atomic species (Z 2, 3, etc).

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Objective for Part A

• Describe how Bohr applied the quantum hypothesis
  of Planck and classical physics to build a model
  for the hydrogen atom (and other single electron
  species) and how this can be used to explain and
  predict atomic line spectra.

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Setting the Scene

• At the turn of the 19th/20th centuries, classical
  
• physics (Newtonian mechanics
• and Maxwellian wave theory) were unable
• explain a number of observations relating to
• atomic phenomena
• Line spectra of atoms (absorption or emission)
• Black body radiation
• The photoelectric effect
• The stability of the Rutherford (nuclear) atom

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The Beginnings of Bohrs Model

• In 1913, Niels Bohr, a student of Ernest
• Rutherford, put forward the idea of
• superimposing the quantum principal of Max
• Planck (1901) on the nuclear model of the H
• atom.
• He believed that exchange of energy in
• quanta (E hn) could explain the lines in the
• absorption and emission spectra of H and
• other elements.

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Basic Assumptions (Postulates) of Bohrs Original
Model of the H Atom

• The electron moves in a circular path around the
  nucleus.
• The energy of the electron can assume only
  certain quantized values.
• Only orbits of angular momentum equal to integral
  multiples of h/2p are allowed (meru nh,
• which is equation (1) n 1, 2, 3, )
• 4. The atom can absorb or emit electromagnetic
  radiation (hn) only when the electron transfers
  between stable orbits.

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First Steps Balance of Forces
The centripetal force of circular motion balances
the electrostatic force of attraction
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Determination of Potential Energy (PE or V) and
Kinetic Energy (KE)
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Determination of Total Energy and Calculation of u
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Determination of Total Energy
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Determination of Total Energy - Continued
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Energy Level Diagram for the Bohr H Atom
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Determination of Orbit Radius
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Determination of Orbit Radius - Continued
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Electronic Radius Diagram of Bohr H Atom
Each electronic radius corresponds to an energy
level with the same quantum number
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The Line Spectra of Hydrogen

• The major triumph of the Bohr model of the
• H atom was its ability to explain and predict
• the wavelength of the lines in the absorption
• and emission spectra of H, for the first time.
• Bohr postulated that the H spectra are
• obtained from transitions of the electron
• between stable energy levels, by absorbing or
• emitting a quantum of radiation.

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Electronic Transitions and Spectra in the Bohr H
atom
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Qualitative Explanation of Emission Spectra of
H in Different Regions of the Electromagnetic
Spectrum
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Quantitative Explanation of Line Spectra of
Hydrogen and Hydrogen-Like Species
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.Continued
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Conclusion

• For the H atom (Z 1), the predicted
• emission spectrum associated with nf 1
• corresponds to the Lyman series of lines in
• the ultraviolet region.
• That associated with nf 2 corresponds to
• the Balmer series in the visible region, and
• so on.
• Likewise, the lines in all the absorption
• spectra can be predicted by Bohrs equations.

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Aftermath the Successes and Failures of Bohrs
Model

• For the first time, Bohr was able to give a
• theoretical explanation of the stability of the
• Rutherford H atom, and of the line spectra of
• hydrogen and other single electron species
• (e.g. He, Li2, etc).
• However, Bohrs theory failed totally with
  two-and
• many-electron atoms, even after several drastic
• modifications. Also, the imposition of
  quantization
• on an otherwise classical description was uneasy.
• Clearly a new theory was needed!

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B. Waves and Wave Equations
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Setting a Goal for Part B

• You will learn how to express equations for wave
  motions in both sine/cosine terms and second
  order derivative terms.
• You will learn how de Broglies matter wave
  hypothesis can be incorporated into a wave
  equation to give Schrödinger-type equations.
• You will learn qualitatively how the Schrödinger
  equation can be solved for the H atom and what
  the solutions mean.

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Objective for Part B

• Describe wave forms in general and matter (or
  particle) waves in particular, and how the
  Schrödinger equation for a 1-dimensional particle
  can be constructed.
• Describe how the Schrödinger equation can be
  applied to the H atom, and the meaning of the
  sensible solutions to this equation.

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Basic Wave Equations
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Characteristics of a Travelling Wave
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Characteristics of a Standing Wave
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Wave Form Related to Vibration and Circular Motion
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The Dual Nature of Matter de Broglies Matter
Waves

• We have seen that Bohrs model of the H atom
  could not be used on multi-electron atoms. Also,
  the theory was an uncomfortable mixture of
  classical and modern ideas.
• These (and other) problems forced scientists to
  look for alternative theories.
• The most important of the new theories was that
  of Louis de Broglie, who suggested all matter had
  wave-like character.

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de Broglies Matter Waves

• De Broglie suggested that the wave-like character
  of matter could be expressed by the equation (5),
  for any object of mass m, moving with velocity v.
  
• Since kinetic energy (Ek 1/2mv2) can be written
  as
• De Broglies matter wave expression can thus be
• written

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de Broglies Matter Waves, Continued

• Since h is very small, the de Broglie wavelength
  will be too small to measure for high mass, fast
  objects, but not for very light objects. Thus the
  wave character is significant only for atomic
  particles such as electrons, neutrons and
  protons.
• De Broglies equation (5) can be derived from
• equations representing the energy of photons
  (from Einstein E mc2 and Planck E hc/l)
• and also
• (2) equations representing the electron in the
  Bohr H atom as a standing wave (mevr nh/2p nl
  2pr)

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The Electron in an Atom as a Standing Wave
An important suggestion of de Broglie was that
the electron in the Bohr H atom could be
considered as a circular standing wave
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Differential Form of Wave Equations

• Consider a one-dimensional standing wave. If we
  suppose that the value y(x) of the wave form at
  any point x to be the wave function y(x), then we
  have, according to equation (4)
• Of particular interest is the curvature of the
  wave function the way that the gradient of the
  gradient of the plot of y versus x varies. This
  is the second derivative of y with respect to x.

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Differential Form of Wave Equations, Continued

• Thus
• Equation (9) is a second order differential
  equation
• whose solutions are of the form given by equation
  
• (7).

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Differential Wave Equation for a One-Dimensional
de Broglie Particle Wave

• We now consider the differential wave equation
  for
• a one-dimensional particle with both kinetic
  energy
• (Ek) and potential energy (V(x)).

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The Schrödinger Equation for a Particle Moving in
One Dimension

• Equation (10) shows the relationship between the
• second derivative of a wave function and the
  kinetic
• energy of the particle it represents.
• If external forces are present (e.g. due to the
• presence of fixed charges, as in an atom), then a
  
• potential energy term V(x) must be added.
• Since E(total) Ek V(x), substituting for Ek
  in
• Equation (10) gives
• This is the Schrödinger equation for a particle
• moving in one dimension.

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The Schrödinger Equation for the Hydrogen Atom

• Erwin Schrödinger (1926) was the first to act
  upon de
• Broglies idea of the electron in a hydrogen atom
  
• behaving as a standing wave. The resulting
  equation
• (12) is analogous to equation (11)
• It represents the wave form in three dimensions
  and is
• thus a second-order partial differential
  equation.

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The Schrödinger Equation, Continued

• The general solution of equations like equation
• (12) had been determined in the 19th century (by
• Laguerre and Legendre).
• The equations are more easily solved if
• expressed in terms of spherical polar coordinates
  
• (r,q,f), rather than in cartesian coordinates
  (x,y,z), in
• which case,

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Erwin Schrödinger
Students I hope you are staying awake while the
professor talks about my work! Love, Erwin
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Spherical Polar Coordinates
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Solutions of the Schrödinger Equation for the
Hydrogen Atom

• The number of solutions to the Schrödinger
  equation is infinite.
• By assuming certain properties of y (the wave
  function) - boundary conditions relevant to the
  physical nature of the H atom - only solutions
  meaningful to the H atom are selected.
• These sensible solutions for y (originally called
  specific quantum states, now orbitals) can be
  expressed as the product of a radial function
  R(r) and an angular function Y(q,f), both of
  which include integers, known as quantum numbers
  n, l and m (or ml ).

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Solutions of the Schrödinger Equation for the
Hydrogen Atom, Continued

• Y(r,q,f) Rnl(r)Ylm(q,f) (13)
• The radial function R is a polynomial in r of
  degree n 1 (highest power r(n-1), called a
  Laguerre polynomial) multiplied by an exponential
  function of the type e(-r/na0) or e(-?/n), where
  a0 is the Bohr radius.
• The angular function Y consists of products of
  polynomials in sin? and cos? (called Legendre
  polynomials) multiplied by a complex exponential
  function of the type e(im?).

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Solutions of the Schrödinger Equation for the
Hydrogen Atom, Continued

• The principal quantum number is n (like the Bohr
  quantum number 1, 2, 3,), whereas the other
  two quantum numbers both depend on n.
• l 0 to n - 1 (in integral values).
• m -l through 0 to l (again in integral
  values).
• The energies of the specific quantum states (or
  orbitals) depend only on n for the H atom (but
  not for many-electron atoms) and are numerically
  the same as those for the Bohr H atom.

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Orbitals

• Orbitals where l 0 are called s orbitals
  those with l 1 are known as p orbitals and
  those with l 2 are known as d orbitals.
• When n 1, l m 0 only there is only one 1s
  orbital.
• When n 2, l can be 0 again (one 2s orbital),
  but l can also be 1, in which case m -1, 0 or
  1 (corresponding to three p orbitals).
• When n 3, l can be 0 (one 3s orbital) and 1
  (three 3p orbitals) again, but can also be 2,
  whence m can be 2, -1, 0, 1 or 2
  (corresponding to five d orbitals).

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Energy Levels of the H atom
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The 1s Wave Function of H and Corresponding
Pictorial Representation
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The 2pz Wave Function of H and Corresponding
Pictorial Representation
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C. Particle in a One-Dimensional Box
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Setting a Goal for Part C

• You will learn how the Schrödinger equation can
  be applied to one of the simplest problems a
  particle in a one-dimensional box or energy well.
• You will learn how to calculate the energies of
  various quantum states associated with this
  system.
• You will learn how extend these ideas to three
  dimensions.

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Objective for Part C

• Describe how the Schrödinger equation can be
  applied to a particle in a one-dimensional box
  (and similar situations) and how the energies of
  specific quantum states can be calculated.

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Particle in a One-Dimensional Box

• The simplest model to which the Schrödinger
  equation can be applied is the particle (such as
  a 1-D electron) in a one-dimensional box or
  potential energy well.
• The potential energy of the particle is 0 when it
  is in the box and ? beyond the boundaries of the
  box clearly the particle is totally confined to
  the box.
• All its energy will thus be kinetic energy.

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Defining the Problem
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Setting up the Schrödinger Equation
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Solution of the Schrödinger Equation and use of
the First Boundary Condition
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Evaluation of the Constant k and Use of the 2nd
Boundary Condition
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Determination of the Energy Levels
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Determination of the Constant A
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A Particle in a Three-Dimensional Box

• The arguments in the previous slides can be
  extended to a particle confined in a 3D box of
  lengths Lx, Ly and Lz.
• Within the box, V(x,y,z) 0 outside the cube it
  is ?
• A quantum number is needed for each dimension and
  the Schrödinger equation includes derivatives
  with respect to each coordinate.
• The allowed energies for the particle are given by

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Calculation of Energies of a Particle in a 3-D
Box
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.Continued
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Calculation of Energy Spacing in Different
Situations
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Continued