Abstract
In this chapter, we focus on a topic usually called the application of algebraic methods or Lie algebras within quantum mechanics. We have already demonstrated that one can determine very efficiently the spectrum of the harmonic oscillator owing to the closure of the set of three operators, namely the Hamiltonian \(\hat{\mathsf{H}}\) and the ladder operators \(\hat{\mathsf{a}}\) and \(\hat{\mathsf{a}}^{+}\), under the operation of commutation. We now show that this method can be extended to the problem of angular momentum, the addition of angular momenta, the hydrogen atom, and a free particle.
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Notes
- 1.
- 2.
To be more precise, we should express the compound angular momentum \(\hat{\mathsf{\boldsymbol{J}}}\) as a sum of tensor products \(\hat{\mathsf{\boldsymbol{J}}} =\hat{ \mathsf{\boldsymbol{J}}}_{1} \otimes \mathbf{1}_{2} + \mathbf{1}_{1} \otimes \hat{\mathsf{\boldsymbol{J}}}_{2}\) to avoid ambiguity when applying \(\hat{\mathsf{\boldsymbol{J}}}\) on the state \(\left \vert j_{1},m_{1}\right>\left \vert j_{2},m_{2}\right>\).
- 3.
The reader can easily prove by himself that they all commute with each other.
- 4.
We inserted the word particles into quotation marks since one can have in mind the spin and orbital states of the same particle as well.
- 5.
There are (2j 1 + 1)(2j 2 + 1) such products due to the restriction on the possible values of the projections m 1 and m 2.
- 6.
Recall, see Eq. (3.72), that we are considering particle of unit mass.
- 7.
The derivation in this section is inspired by that in [1].
- 8.
This conclusion is true only for the so-called dipole radiation, see Sect. 6.2.4.
- 9.
The normal Zeeman effect occurs under the very same conditions with the only exception that the spin-orbit interaction is negligible. The terminology, stemming from the historical development, was coined somewhat unfortunately, though.
- 10.
- 11.
It is a common habit within literature to call these functions the Sturmian functions, or the Sturmians, see for example [2].
References
B.G. Adams, J. ČÞek, J. Paldus, Adv. Quantum Chem. 19, 1 (1988)
J. Avery, Hyperspherical Harmonics and Generalized Sturmians (Kluwer Academic, Boston, 2000)
J. ČÞek, J. Paldus, Int. J. Quantum Chem. 12, 875 (1977)
W. Greiner, B. Müller, Quantum Mechanics: Symmetries (Springer, Berlin, 1997)
S. Hatamian et al., Phys. Rev. Lett. 58, 1833 (1987)
I. Lindgren, J. Morrison, Atomic Many-Body Theory (Springer, Berlin, 1982)
H.J. Lipkin, Lie Groups for Pedestrians (North Holland, Amsterdam, 1965)
A. Messiah, Quantum Mechanics (North Holland, Amsterdam, 1961)
D.H. Perkins, Introduction to High Energy Physics (Addison-Wesley, Reading, 1987)
L. Schiff, Quantum Mechanics (McGraw Hill College, New York, 1968)
J. Schwinger, Quantum Mechanics: Symbolism of Atomic Measurement (Springer, Berlin, 2003)
J. Schwinger et al., Classical Electrodynamics (Perseus Books, Cambridge, 1998)
P.E.S. Wormer, J. Paldus, Adv. Quantum Chem. 51, 59 (2006)
A. Zee, Group Theory in a Nutshell for Physicists (Princeton University Press, Princeton, 2016)
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Zamastil, J., Benda, J. (2017). Treasures Hidden in Commutators. In: Quantum Mechanics and Electrodynamics. Springer, Cham. https://doi.org/10.1007/978-3-319-65780-6_4
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DOI: https://doi.org/10.1007/978-3-319-65780-6_4
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