Fine Structure of Hydrogen like atoms

Gives an idea about what fine structure is and the origin of fine structure in hydrogen-like atoms. It includes the derivation which will be essential in understanding the splitting of spectral lines.

1. Fine Structure of Hydrogen
like atoms
CB.SC.I5PHY16041
SREERAG K
14 September 2020

2. Energy levels of Hydrogen like atoms
n =2
n =3
n =4
n =5
n =6
Balmer series (Visible light)
-3.4 eV
-1.511 eV
-0.85 eV
-0.544 eV
-0.377 eV
656 nm486 nm434 nm410 nm
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3. Fine structure
0.016 nm
The difference between the spectral lines is about 0.016
nm which corresponds to an energy of 0.000045 eV
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4. Explanation
•This was one of the first experimental evidence for electron spin.
•The splitting of spectral lines was attributed to an interaction
between electron spin (S) and the orbitalangular momentum
(L).
•This accompanied with the relativistic correction will
completely explain the fine structure that was observed in the
case of hydrogen
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5. Spin orbit coupling
•Interaction between the electron’s spin magnetic moment, and
the proton’s orbital magnetic field.
+e
-e
B,L
Imagine electron in orbit around the nucleus; from
the electron’s point of view the proton is circling
around it.
Orbiting positive charge sets up a magnetic field in
the electron frame, which exerts a torque on the
spinning electron tending to align magnetic moment
along the direction of the field.
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6. The Hamiltonian for such an interaction is given by
Magnetic moment of electron
Magnetic field produced by
proton(experienced by
electron)
We need to find both the magnetic moment of electron
and the magnetic field produced by the proton
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μ
B

7. Magnetic field of proton
•Picture the proton as a continuous current loop, its magnetic
field can then be calculated from the Biot-Savart Law.
r
I = e/T is the effective current
T is the time period of the orbit
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8. Substituting the expression of T in equation 1 we get
We can manipulate this equation
Orbital angular momentum of electron
This is the magnetic field experienced by the electron
(1)
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9. Magnetic dipole moment of electron
•Consider a charge q smeared out around a ring of radius r,
which rotates about the axis with period T.
•The magnetic dipole moment(I*A) of such a ring is given by
•Angular momentum is the moment of inertia times the angular
velocity
(2)
(3)
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10. We can now express magnetic dipole moment in terms of the spin angular momentum.
Dividing equations 2 and 3 we get
Since S and μare in the same direction we can write
This was purely a classical calculation. The actual value of electron’s magnetic
moment is twice the classical value
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11. Now that we have equations for both magnetic field of proton and magnetic moment
of electron we can find the Hamiltonian.
The electrons rest frame is not a inertial frame it accelerates as the electron orbit
around the nucleus. So in order to account for this we need to do a kinematic
correction, known as the Thomas Precession. In this context it throws in a factor of 1/2
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12. Expectation of H
sogives the correction energy.
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13. Relativistic correction
•The first term in the Hamiltonian represents the Kinetic energy
•This equation is the classical expression of KE. The relativistic
formula is
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14. The KE can be expressed in terms of momentum instead of velocity
Since (p/mc) is very small we can expand the equation in powers of (p/mc)
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15. So the lowest order correction to Hamiltonian is
According to the first order perturbation theory the correction E
nis given by the
expectation value of H
rin the unperturbed state.
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16. Total correction in Energy
•Adding the contributions from both Spin-orbit couplingand
relativistic correctionwe get the complete correction in energy.
+
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17. When l =1:j = 1/2 , j= 3/2
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18. Fine structure of hydrogen atom
n = 1
n = 2
n = 3
n = 4
l = 0
(s)
l = 1
(p)
l = 2
(d)
l = 3
(f)
j = 1/2
j = 3/2
j = 5/2
j = 7/2
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Spin orbit couplingand relativistic correction
shifts the degeneracy from l to j

20. References
•Introduction to Quantum Mechanics by David J. Griffiths
•Hyper physics -Hydrogen Fine Structure
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Thank You