Quantum Mechanics (PHYS621)
Module III: Approximation Methods
Session 3: Zeeman Effect (Normal and Anomalous): Part 1
Zeeman Effect:
In the presence of an external uniform magnetic field, the energy levels of Hydrogen atom get
shifted. This energy shift is known as Zeeman effect. The Zeeman effect without considering the
spin of the electron is known as normal Zeeman effect. However, Zeeman effect with
consideration of spin of electron is defined as anomalous Zeeman effect.
Normal Zeeman Effect:
We know that the Hamiltonian of a particle of mass ‘’ and charge ‘q’ moving in a central
potential V(r) under the influence of a uniform magnetic field B.
̂ ( ) [ ( ) ]
Now we could write the Hamiltonian of the Hydrogen atom under the presence of external
uniform magnetic field B along the z-axis by replacing q with the electron’s charge q = -e.
̂ ( ) ................(1)
In this equation, Hamiltonian in the absence of magnetic field is denoted by ̂
The term ( ) can be ignored as it is too small for one electron atom (ex. Hydrogen
atom) even when B is strong.
̂ ̂ ̂ ...............(2)
wherein Bohr magneton (electron’s orbital magnetic dipole moment).
1
⸪ ̂ commutes with ̂ , the operators ̂ , ̂ and ̂ mutually commute, therefore they have a set
of common eigen functions ( ) ( ) ( )
The eigen values of ̂ are:
⟨ |̂| ⟩ ⟨ |̂ | ⟩ ⟨ |̂ | ⟩
.............(3)
Wherein, and denote ground state energy levels of H atom and Larmor
frequency, respectively.
Therefore, under the presence of uniform magnetic field, each level with angular momentum ‘l’
will split into (2l+1) equally spaced levels. The spacing between levels will be given by
. The equidistant splitting of the levels is known as ‘normal Zeeman effect’.
Figure. Normal Zeeman effect in Hydrogen atom. The energy levels are degenerate with respect
to ‘l’ and ‘m’ in case of B = 0. However, the degeneracy with respect to ‘m’ is removed but the
degeneracy with respect to ‘l’ persists in case of B 0. [Ref. N. Zettli, Quantum Mechanics:
Concepts and Applications].
2
As shown in splitting of levels under the presence of magnetic field, it is obvious that normal
Zeeman effect has partially removed the degeneracy of levels. The degeneracy with respect to ‘l’
still remains.
The degenerate levels are:
That means, the degeneracies of the levels corresponding to the same ‘n’ and ‘m’ but different
values of ‘l’ are not removed by the normal Zeeman effect i.e. with .
Note: Normal Zeeman effect show that each energy level splits into an odd number of (2l+1)
equally spaced levels. However, this does not agree with experimental observations. This is only
due to not taking spin of electron into account.
Text and References
L. I. Schiff, Quantum Mechanics, 3rd Edition (McGraw Hill Book Co., New York 1968).
E. Merzbacher, Quantum Mechanics, 3rd Edition (John Wiley and Sons, Inc1997).
A. K. Ghatak and S. Lokanathan, Quantum Mechanics: Theory and Applications, 5th
Edition, (Macmillan India, 2004).
Arno Bohm, Quantum Mechanics: Foundations and Applications, 3rd Edition (New York:
Springer-Verlag, 2003).
Quantum Mechanics by V. Devanathan. (Narosa Publishing House).
Nouredine Zettili, Quantum Mechanics: Concepts and Applications (John-Wiley,
Chichester, 2001).
J.L. Powell and B. Crasemann, Quantum Mechanics (Addison-Wesley Pubs. Co., 1965).
P. A. M. Dirac, The Principles of Quantum Mechanics (Oxford University Press,
Oxford, 1981).
J.J. Sakurai, Modern Quantum Mechanics (Pearson Education, Inc., 1994).
