1. Physics 3313 - Lecture 15 Wednesday March 24, 2010 Dr. Andrew Brandt
Hydrogen Atom
HW 6 on Ch. 7 is assigned due Weds 3/31
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2. CHAPTER 7 The Hydrogen Atom
7.3 Quantum Numbers
7.4 Magnetic Effects on Atomic Spectra – The Normal Zeeman Effect
7.5 Intrinsic Spin
7.6 Energy Levels and Electron Probabilities
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3. 7.3: Quantum Numbers The three quantum numbers:
n Principal quantum number
ℓ Orbital angular momentum quantum number
mℓ Magnetic quantum number
The boundary conditions:
n = 1, 2, 3, 4, Integer
ℓ = 0, 1, 2, 3, , n − 1 Integer
mℓ = −ℓ, −ℓ + 1, , 0, 1, , ℓ − 1, ℓ Integer
The restrictions for quantum numbers:
n > 0
ℓ < n
|mℓ| ≤ ℓ
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4. Hydrogen Atom Wave Function
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5. Principal Quantum Number n
It results from the solution of R(r). The result for this quantized energy is
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6. Orbital Angular Momentum Quantum Number ℓ
It is associated with the R(r) and f(θ) parts of the wave function.
Classically, the orbital angular momentum with L = mvorbitalr.
ℓ is related to L by
In an ℓ = 0 state,
This disagrees with Bohr’s semi-classical “planetary” model of electrons orbiting a nucleus L = nħ.
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7. Orbital Angular Momentum Quantum Number ℓ
A certain energy level is degenerate with respect to ℓ when the energy is independent of ℓ.
Use letter names for the various ℓ values.
ℓ =
Letter = s p d f g h . . .
Atomic states are referred to by their n and ℓ.
A state with n = 2 and ℓ = 1 is called a 2p state.
The boundary conditions require n > ℓ.
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8. Magnetic Quantum Number mℓ
The angle  is a measure of the rotation about the z axis.
The solution for specifies that mℓ is an integer and related to the z component of L by
Only certain orientations of L are possible and this is called space quantization
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9. 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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10. Magnetic Quantum Number mℓ
Quantum mechanics allows L to be quantized along only one direction in space. Because of the relation L2 = Lx2 + Ly2 + Lz2 the knowledge of a second component would imply a knowledge of the third component because we know L .
We expect the average of the angular momentum components squared to be
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11. Angular Momentum
L cannot be aligned parallel with an external magnetic field (B) because Lz is always smaller than L (except when l=0)
In the absence of an external field the choice of the z axis is arbitrary (measure projection as in any direction)
Why only Lz quantized? What about Lx and Ly?
Suppose L were in z direction, then electron would be confined to x-y plane; this implies z position is known and pz is infinitely uncertain, which is not true if part of a hydrogen atom
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12. Precession of Angular Momentum
The direction of L is thus continually changing as it precesses around the z axis
(note average values of Lx =Ly =0 )
Therefore and it is only necessary to specify L and Lz
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13. Angular Momentum Operator
Consider angular momentum definition: so
We can define the angular momentum operator in cartesian and spherical coordinates:
with
gives similarly
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14. (Normal) 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 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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15. Magnetic Moment What about the magnitude of the magnetic moment?
The Bohr magneton is defined as
And the z component?
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16. 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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17. 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:
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18. Zeeman Effect
Note for l=2 get splitting into 5 lines, but due to selection rules, still only 3 different
energy photons
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19. Anomalous Zeeman Effect
The Normal Zeeman effect is the splitting of spectral lines into 2l+1 separate sub-lines giving 3-fold splitting of spectral lines
Some elements did not behave this way but split into different numbers of sub-lines
“Pauli why do you look so sad?”
“How can one look happy when he is thinking about the Anomalous Zeeman effect!”
Recall and
So electrons with different ml have different energy
What if you sent a beam of atoms in different ml states through a magnetic field
They would experience a force proportional to ml (if magnetic field not uniform)
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20. The Normal Zeeman Effect
An atomic beam of particles in the ℓ = 1 state pass through a magnetic field along the z direction.
The mℓ = +1 state will be deflected down, the mℓ = −1 state up, and the mℓ = 0 state will be undeflected.
If the space quantization were due to the magnetic quantum number mℓ, mℓ states is always odd (2ℓ + 1) and should produce an odd number of lines.
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21. Stern-Gerlach Experiment
Used silver atoms
in ground state:
l=0 so ml=0
Expect 1 line
Observe 2 lines
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22. 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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23. 4th Quantum Number So spin is the 4th quanutum number
Two 1s ground states and
In general
probability of measuring ms=+1/2 in Stern-Gerlach is |C2| for unpolarized beam
|C2|= |C’2|=1/2
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24. Spin Magnetic Moment Recall g=gyromagnetic ratio = -e/2m For spin and
gs/gl~2 deviations from 2 could indicated new physics
g-2 experiments
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