1. Physics 3313 – Review 2 Wednesday May 5, 2010 Dr. Andrew Brandt
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2. Hydrogen Atom Wave Function
n= 0, 1, 2, 3,…
l=0, 1, 2, …n-1
-l  ml  l
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3. Orbital (l) and Magnetic (ml) Quantum Numbers
l is related to orbital angular momentum; angular momentum is quantized and conserved, but since h is so small, often don’t notice quantization
Electron orbiting nucleus is a small current loop and has a magnetic field, so an electron with angular momentum interacts with an external magnetic field
The magnetic quantum number ml specifies the direction of L (which is a vector—right hand rule) and gives the component of L in the direction of the magnetic field Lz
Five ml values for l=2 correspond to five different orientations of angular momentum vector.
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4. Probabilities
Probability of finding electron in hydrogen atom in a spherical shell between r and r+dr is given by
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5. Example Verify that the average value of 1/r for a 1s electron is 1/a0
combining these gives:
The angular integral gives 2(2)=4 
Need integration by parts:

6. Selection Rules For a transition to occur the overlap integral
must be non-zero, since we have seen that transitions are proportional to this integral. If the integral is zero, the transition between the two states is forbidden.
By evaluating transitions that occur we obtain the following selection rules:
This is consistent with the photon carrying an angular
momentum of
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7. Zeeman Effect
The Zeeman effect is the splitting of spectral lines into separate sub-lines when atoms radiate in a magnetic field , with a spacing of the lines dependent on the strength of the magnetic field (first observed by Peter Zeeman in 1896). Can this be explained by Quantum Mechanics?
Start the journey by considering a particle with mass m and charge e moving with a speed v in a circular orbit of radius r , we define its magnetic moment  as
We would like to relate the magnetic moment to the angular momentum L=mvr
d=vt so if the particle traverses a circle once in a time t=T=1/f
(where  or f, the frequency, is the inverse of the period T); it travels a distance 2r so and so
why negative? for a positron
but for an electron they are anti-parallel
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8. Magnetic Moment What about the magnitude of the magnetic moment?
The Bohr magneton is defined as
And the z component?
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9. Torque and Energy 
What happens if you put a magnetic moment in a magnetic field?
Yup: maximum torque when 
effect is to rotate dipole, therefore a dipole has energy due
to its position in a magnetic field.
By convention B is in the +z direction:
Combine with gives
Energy of atomic state depends on magnetic quantum number (not just n) if atom is in a magnetic field !
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10. Zeeman Effect
So since ml has 2l+1 different values, a given state with angular momentum l will split into 2l+1 different sub-states, each separated in energy by
The explanation of the Zeeman effect is one of the triumphs of quantum theory: f0 f0 f0 f0 f0
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11. Stern-Gerlach Experiment
Used silver atoms
in ground state:
l=0 so ml=0
Expect 1 line
Observe 2 lines
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12. Spin
Intrinsic angular momentum of electron independent of orbital angular momentum
Like revolution and rotation
2s+1=2 -> s=1/2 (always ½ for electron)
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13. Pauli Exclusion Principal
Symmetric solution:
Anti-symmetric solution:
Suppose particles are in the same state a=b
Conclude that electrons must have anti-symmetric wave functions since observed to not be in same quantum state
Particles with odd half-integer spin (1/2,3/2,…) like electrons, neutrons, and protons are fermions and obey the PEP and are described by Fermi-Dirac statistics
Spin 0 or integer spin (photons, alpha particles) are symmetric under exchange and do not obey the exclusion principle; these are termed bosons since they follow
Bose-Einstein Statistics
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14. Periodic Table
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15. Deriving Chemistry
The structure of atoms with multiple electrons is determined by two principles
A system is stable when the total energy is minimum
Only one electron at a time is allowed in a particular quantum state
Much can be learned about atomic structure by considering each electron as though it exists in a constant electric field +Ze decreased by the partial screening of the nucleus by electrons that are between the electron under consideration and the nucleus
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16. Consider Sodium
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17. Shells and Sub-shells
Electrons with same n roughly at same distance, occupy same shell
n=1 K n=2 L n=3 M n=4 N n=5 O
within each shell there are sub-shells with same angular momentum,
small l states are less shielded than big l states, so energy has small increase with l
sodium z=11: 1s2 2s2 2p6 3s1 principal quantum number identifies shell, letter identifies sub-shell, super script is number of electrons in a given subshell
n= electrons n=2 8 electrons n=3 3s2 3p6 3d10 Ne=2x(2l+1)
18 electrons total electrons in a shell is 2n2
notion of shells and sub-shells fits perfectly into pattern of periodic table:
closed shell electrons are tightly bound (noble gasses), alkali metals have valence electron far from nucleus, easily ionized
1s2 2s2 2p6 3s2 3p6 4s2 3d10 4p6 5s2 4d10 5p6 6s2 4f14 (transition elements chemical properties determined by 4s electrons)
Hund’s rule parallel spins more separated, energetically favorable leads to ferromagnetism
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18. Adding Angular Momenta
what is the
total angular
momentum
when I add the
spin angular momentum
of an electron to its
orbital angular momentum?
l=1 +s=1/2?
3/2?
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19. Total Angular Momentum (J)
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20. Energy Approach for H2+
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21. Rotational Energy Classically QM adds quantization of angular momentum
J= allowed?
yes, since other forms of energy exist
J=1
J=2
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22. Rotational Frequency For diatomic molecules selection rule
from angular momentum of photon
Frequency of absorbed photon
Spectrum is thus equally spaced lines separated by
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23. Vibrational Energy Model the motion as simple harmonic oscillator
κis the effective spring constant measured in N/m
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24. Stat Mech, Nuclear, etc. see HW lecture notes 5/5/2010
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