1. General Physics (PHY 2140) Lecture 16 Modern Physics Atomic Physics
Early models of the atom
Atomic spectra
Bohr’s theory of hydrogen
De Broglie wavelength in the atom
Quantum mechanics and Spin
Chapter 28
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2. Lightning Review Last lecture: Quantum physics Wave function
Uncertainty relations
Review Problem:
If matter has a wave structure, why is this not observable in our daily experiences?
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(baseball)  m

3. 27.9 The Uncertainty Principle
When measurements are made, the experimenter is always faced with experimental uncertainties in the measurements
Classical mechanics offers no fundamental barrier to ultimate refinements in measurements
Classical mechanics would allow for measurements with arbitrarily small uncertainties
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4. The Uncertainty Principle
Quantum mechanics predicts that a barrier to measurements with ultimately small uncertainties does exist
In 1927 Heisenberg introduced the uncertainty principle
If a measurement of position of a particle is made with precision Δx and a simultaneous measurement of linear momentum is made with precision Δp, then the product of the two uncertainties can never be smaller than h/4
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5. The Uncertainty Principle
Mathematically,
It is physically impossible to measure simultaneously the exact position and the exact linear momentum of a particle
Another form of the principle deals with energy and time:
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6. Thought Experiment – the Uncertainty Principle
A thought experiment for viewing an electron with a powerful microscope
In order to see the electron, at least one photon must bounce off it
During this interaction, momentum is transferred from the photon to the electron
Therefore, the light that allows you to accurately locate the electron changes the momentum of the electron
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7. Problem: macroscopic uncertainty
A 50.0-g ball moves at 30.0 m/s. If its speed is measured to an accuracy of 0.10%, what is the minimum uncertainty in its position?
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8. Thus, uncertainty relation implies
A 50.0-g ball moves at 30.0 m/s. If its speed is measured to an accuracy of 0.10%, what is the minimum uncertainty in its position?
Notice that the ball is non-relativistic. Thus, p = mv, and uncertainty in measuring momentum is
Given:
v = 30 m/s
Dv/v = 0.10%
m = 50.0 g
Find:
dx = ?
Thus, uncertainty relation implies
Too small to measure!!
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9. Problem: Macroscopic measurement
A 0.50-kg block rests on the icy surface of a frozen pond, which we can assume to be frictionless. If the location of the block is measured to a precision of 0.50 cm, what speed must the block acquire because of the measurement process?
Recall:
and
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10. Atomic physics: Chapter 28
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11. 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+
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12. 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
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13. Experimental tests Expect: Mostly small angle scattering
No backward scattering events
Results:
Mostly small scattering events
Several backward scatterings!!!
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14. 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
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15. 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.
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16. Electron and proton interact via the Coulomb force Given:
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.
Electron and proton interact via the Coulomb force
Given:
r = 1.0 × 10–10 m
Find:
F = ?
PE = ?
Potential energy is
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17. Difficulties with the Rutherford Model
Atoms emit certain discrete characteristic frequencies of electromagnetic radiation
The Rutherford model is unable to explain this phenomena
Rutherford’s electrons are undergoing a centripetal acceleration and so should radiate electromagnetic waves of the same frequency, thus leading to electron “falling on a nucleus” in about seconds!!!
The radius should steadily decrease as this radiation is given off
The electron should eventually spiral into the nucleus
It doesn’t
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18. 28.2 Emission Spectra
A gas at low pressure has a voltage applied to it
A gas emits light characteristic of the gas
When the emitted light is analyzed with a spectrometer, a series of discrete bright lines is observed
Each line has a different wavelength and color
This series of lines is called an emission spectrum
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19. Emission Spectrum of Hydrogen
The wavelengths of hydrogen’s spectral lines can be found from
RH is the Rydberg constant
RH = x 107 m-1
n is an integer, n = 1, 2, 3, …
The spectral lines correspond to
different values of n
A.k.a. Balmer series
Examples of spectral lines
n = 3, λ = nm
n = 4, λ = nm
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20. Absorption Spectra
An element can also absorb light at specific wavelengths
An absorption spectrum can be obtained by passing a continuous radiation spectrum through a vapor of the gas
The absorption spectrum consists of a series of dark lines superimposed on the otherwise continuous spectrum
The dark lines of the absorption spectrum coincide with the bright lines of the emission spectrum
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21. Applications of Absorption Spectrum
The continuous spectrum emitted by the Sun passes through the cooler gases of the Sun’s atmosphere
The various absorption lines can be used to identify elements in the solar atmosphere
Led to the discovery of helium (from helios)
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22. 28.3 The Bohr Theory of Hydrogen
In 1913 Bohr provided an explanation of atomic spectra that includes some features of the currently accepted theory
His model includes both classical and non-classical ideas
His model included an attempt to explain why the atom was stable
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23. Bohr’s Assumptions for Hydrogen
The electron moves in circular orbits around the proton under the influence of the Coulomb force of attraction
The Coulomb force produces the centripetal acceleration
Only certain electron orbits are stable
These are the orbits in which the atom does not emit energy in the form of electromagnetic radiation
Therefore, the energy of the atom remains constant and classical mechanics can be used to describe the electron’s motion
Radiation is emitted by the atom when the electron “jumps” from a more energetic initial state to a lower state
The “jump” cannot be treated classically
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24. Bohr’s Assumptions More on the electron’s “jump”:
The frequency emitted in the “jump” is related to the change in the atom’s energy
It is generally not the same as the frequency of the electron’s orbital motion
The size of the allowed electron orbits is determined by a condition imposed on the electron’s orbital angular momentum
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25. Results The total energy of the atom Newton’s law
This can be used to rewrite kinetic energy as
Thus, the energy can also be expressed as
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26. Bohr Radius The radii of the Bohr orbits are quantized ( )
(from quantized angular momentum: mevr = nħ and Newtons law)
This shows that the electron can only exist in certain allowed orbits determined by the integer n
When n = 1, the orbit has the smallest radius, called the Bohr radius, ao
ao = nm
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27. Radii and Energy of Orbits
A general expression for the radius of any orbit in a hydrogen atom is
rn = n2 ao
The energy of any orbit is
En = eV/ n2
The lowest energy state is called the ground state
This corresponds to n = 1
Energy is –13.6 eV
The next energy level has an energy of –3.40 eV
The energies can be compiled in an energy level diagram
The ionization energy is the energy needed to completely remove the electron from the atom (n = ∞, E = 0)
The ionization energy for hydrogen is 13.6 eV
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28. Energy Level Diagram
The value of RH from Bohr’s analysis is in excellent agreement with the experimental value
A more generalized equation can be used to find the wavelengths of any spectral lines
For the Balmer series, nf = 2
For the Lyman series, nf = 1
Whenever a transition occurs between a state, ni and another state, nf (where ni > nf), a photon is emitted
The photon has a frequency f = (Ei – Ef)/h and wavelength λ
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29. Problem: Transitions in the Bohr’s model
A photon is emitted as a hydrogen atom undergoes a transition from the n = 6 state to the n = 2 state. Calculate the energy and the wavelength of the emitted photon.
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30. With ni = 6 and nf = 2 we have: Given: ni = 6 nf = 2
A photon is emitted as a hydrogen atom undergoes a transition from the n = 6 state to the n = 2 state. Calculate the energy and the wavelength of the emitted photon.
From or
With ni = 6 and nf = 2 we have:
Given:
ni = 6
nf = 2
Find:
l = ?
Eg = ?
The Photon energy is:
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31. Bohr’s Correspondence Principle
Bohr’s Correspondence Principle states that quantum mechanics is in agreement with classical physics when the energy differences between quantized levels are very small
Similar to having Newtonian Mechanics be a special case of relativistic mechanics when v << c
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32. Successes of the Bohr Theory
Explained several features of the hydrogen spectrum
Accounts for Balmer and other series
Predicts a value for RH that agrees with the experimental value
Gives an expression for the radius of the atom
Predicts energy levels of hydrogen
Gives a model of what the atom looks like and how it behaves
Can be extended to “hydrogen-like” atoms
Those with one electron
Ze2 needs to be substituted for e2 in equations
Z is the atomic number of the element
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33. Recall Bohr’s Assumptions
Only certain electron orbits are stable. Radiation is emitted by the atom when the electron “jumps” from a more energetic initial state to a lower state
The size of the allowed electron orbits is determined by a condition imposed on the electron’s orbital angular momentum
Why is that?
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34. Modifications of the Bohr Theory – Elliptical Orbits
Sommerfeld extended the results to include elliptical orbits
Retained the principle quantum number, n
Added the orbital quantum number, ℓ
ℓ ranges from 0 to n-1 in integer steps
All states with the same principle quantum number are said to form a shell
The states with given values of n and ℓ are said to form a subshell
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35. Modifications of the Bohr Theory – Zeeman Effect and fine structure
Another modification was needed to account for the Zeeman effect
The Zeeman effect is the splitting of spectral lines in a strong magnetic field
This indicates that the energy of an electron is slightly modified when the atom is immersed in a magnetic field
A new quantum number, m ℓ, called the orbital magnetic quantum number, had to be introduced
m ℓ can vary from - ℓ to + ℓ in integer steps
High resolution spectrometers show that spectral lines are, in fact, two very closely spaced lines, even in the absence of a magnetic field
This splitting is called fine structure
Another quantum number, ms, called the spin magnetic quantum number, was introduced to explain the fine structure
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36. 28.5 de Broglie Waves
One of Bohr’s postulates was the angular momentum of the electron is quantized, but there was no explanation why the restriction occurred
de Broglie assumed that the electron orbit would be stable only if it contained an integral number of electron wavelengths
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37. de Broglie Waves in the Hydrogen Atom
In this example, three complete wavelengths are contained in the circumference of the orbit
In general, the circumference must equal some integer number of wavelengths
but
, so
This was the first convincing argument that the wave nature of matter was at the heart of the behavior of atomic systems
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38. QUICK QUIZ 1
In an analysis relating Bohr's theory to the de Broglie wavelength of electrons, when an electron moves from the n = 1 level to the n = 3 level, the circumference of its orbit becomes 9 times greater. This occurs because (a) there are 3 times as many wavelengths in the new orbit, (b) there are 3 times as many wavelengths and each wavelength is 3 times as long, (c) the wavelength of the electron becomes 9 times as long, or (d) the electron is moving 9 times as fast.
(b). The circumference of the orbit is n times the de Broglie wavelength (2πr = nλ), so there are three times as many wavelengths in the n = 3 level as in the n = 1 level.
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39. 28.6 Quantum Mechanics and the Hydrogen Atom
One of the first great achievements of quantum mechanics was the solution of the wave equation for the hydrogen atom
The significance of quantum mechanics is that the quantum numbers and the restrictions placed on their values arise directly from the mathematics and not from any assumptions made to make the theory agree with experiments
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40. Problem: wavelength of the electron
Determine the wavelength of an electron in the third excited orbit of the hydrogen atom, with n = 4.
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41. Determine the wavelength of an electron in the third excited orbit of the hydrogen atom, with n = 4.
Recall that de Broglie’s wavelength of electron depends on its momentum, l = h/(mev). Let us find it,
Given:
n = 4
Find:
le = ?
Recall that
Thus,
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42. Quantum Number Summary
The values of n can increase from 1 in integer steps
The values of ℓ can range from 0 to n-1 in integer steps
The values of m ℓ can range from -ℓ to ℓ in integer steps
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43. QUICK QUIZ 2
How many possible orbital states are there for (a) the n = 3 level of hydrogen? (b) the n = 4 level?
The quantum numbers associated with orbital states are n, , and m . For a specified value of n, the allowed values of  range from 0 to n – 1. For each value of , there are (2  + 1) possible values of m. (a) If n = 3, then  = 0, 1, or 2. The number of possible orbital states is then [2(0) + 1] + [2(1) + 1] + [2(2) + 1] = = 9. (b) If n = 4, one additional value of  is allowed ( = 3) so the number of possible orbital states is now 9 + [2(3) + 1] = = 16 (each is n2)
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44. Spin Magnetic Quantum Number
It is convenient to think of the electron as spinning on its axis
The electron is not physically spinning
There are two directions for the spin
Spin up, ms = ½
Spin down, ms = -½
There is a slight energy difference between the two spins and this accounts for the Zeeman effect
Sodium D lines are an example of this splitting.
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45. Electron Clouds
The graph shows the solution to the wave equation for hydrogen in the ground state
The curve peaks at the Bohr radius
The electron is not confined to a particular orbital distance from the nucleus
The probability of finding the electron at the Bohr radius is a maximum
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46. Electron Clouds
The wave function for hydrogen in the ground state is symmetric
The electron can be found in a spherical region surrounding the nucleus
The result is interpreted by viewing the electron as a cloud surrounding the nucleus
The densest regions of the cloud represent the highest probability for finding the electron
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