Angular Momentum
We begin this chapter with the consideration of the orbital angular momentum. This gives rise to a general definition of angular momenta. We derive the eigenvalue spectrum of the orbital angular momentum with an algebraic method. After a brief presentatio
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Angular Momentum
Apart from the Hamiltonian, the angular momentum operator is of one of the most important Hermitian operators in quantum mechanics. In this chapter we consider its eigenvalues and eigenfunctions in more detail.
We begin this chapter with the consideration of the orbital angular momentum. This gives rise to a general definition of angular momenta. We derive the eigenvalue spectrum of the orbital angular momentum with an algebraic method. After a brief presentation of the eigenfunctions of the orbital angular momentum in the position representation, we outline some concepts for the addition of angular momenta.
16.1 Orbital Angular Momentum Operator The orbital angular momentum is given by l = r × p.
(16.1)
As we have seen in Chap. 3, Vol. 1, it is not necessary to symmetrize for the translation into quantum mechanics (spatial representation). It follows directly that l= or, in components, lx =
i
r × ∇, i
y
∂ ∂ −z ∂z ∂y
(16.2) (16.3)
plus cyclic permutations (x → y → z → x → · · · ). All the components of l are observables.
J. Pade, Quantum Mechanics for Pedestrians 2: Applications and Extensions, Undergraduate Lecture Notes in Physics, DOI: 10.1007/978-3-319-00813-4_16, © Springer International Publishing Switzerland 2014
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16 Angular Momentum
We know that one can measure two variables simultaneously if the corresponding operators commute. What about the components of the orbital angular momentum? A short calculation (see the exercises) yields l x , l y = il z ; l y , l z = il x ; l z , l x = il y ,
(16.4)
or, compactly, l x , l y = il z and cyclic permutations. This term can be written still more compactly with the Levi-Civita symbol (permutation symbol, epsilon tensor) εi jk (see also Appendix F, Vol. 1): εi jk
⎧ i jk is an even permutation of 123 ⎨ 1 = −1 if i jk is an odd permutation of 123 ⎩ 0 otherwise
namely as
lk εi jk . li , l j = i
(16.5)
(16.6)
k
Each component of the orbital angular momentum commutes with l2 = l x2 + l y2 + l z2 (see the exercises):
l x , l2 = l y , l2 = l z , l2 = 0.
(16.7)
16.2 Generalized Angular Momentum, Spectrum We now generalize these facts by the following definition: A vector operator1 J is a (generalized) angular momentum operator , if its components are observables and satisfy the commutation relation
Jx , Jy = iJz .
(16.8)
and its cyclic permutations.2 It follows that J2 = Jx2 + Jy2 + Jz2 commutes with all the components:
1
Jx , J2 = 0; Jy , J2 = 0; Jz , J2 = 0.
(16.9)
Instead of J, one often finds j (whereby this is of course not be confused with the probability current density). In this section, we denote the operator by J and the eigenvalue by j. 2 The factor is due to the choice of units, and would be replaced by a different constant if one were to choose different units. The essential factor is i.
16.2 Generalized Angular Momentum, Spectrum
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The task that we address now is the calculation of the angular momentum spectrum. We will deduce that the a
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