Angular momentum algebra

Canonical commutation relation

the canonical commutation relation is the relation between canonical conjugate quantities (quantities which are related by definition such that one is the Fourier transform of another), for example:

[x,p_x] = i\hbar

between the position x and momentum px in the x direction of a point particle in one dimension, where [x,px] = xpx − pxx is the commutator of x and pxi is the imaginary unit, and ħ is the reducedPlanck’s constant h /2π . This relation is attributed to Max Born, and it was noted by E. Kennard (1927) to imply the Heisenberg uncertainty principle.

By contrast, in classical physics, all observables commute and the commutator would be zero. However, an analogous relation exists, which is obtained by replacing the commutator with the Poisson bracket multiplied by i ħ:

\{x,p\} = 1 \,      .

This observation led Dirac to propose that the quantum counterparts \hat f,\hat g of classical observables fg satisfy

[\hat f,\hat g]= i\hbar\widehat{\{f,g\}} \, .

According to the standard mathematical formulation of quantum mechanics, quantum observables such as x and p should be represented as self-adjoint operators on some Hilbert space. It is relatively easy to see that two operators satisfying the canonical commutation relations cannot both be bounded. The canonical commutation relations can be made tamer by writing them in terms of the (bounded) unitary operators e − ikx and e − iap. The result is the so-called Weyl relations. The uniqueness of the canonical commutation relations between position and momentum is guaranteed by theStone-von Neumann theorem. The group associated with the commutation relations is called the Heisenberg group.

Angular momentum operators

 [{L_x}, {L_y}] = i \hbar \epsilon_{xyz} {L_z},

where εxyz is the Levi-Civita symbol and simply reverses the sign of the answer under pairwise interchange of the indices. An analogous relation holds for the spin operators.

All such nontrivial commutation relations for pairs of operators lead to corresponding uncertainty relations (H. P. Robertson[2]), involving positive semi-definite expectation contributions by their respective commutators and anticommutators. In general, for two Hermitian operators A and B, consider expectation values in a system in the state ψ, the variances around the corresponding expectation values being (ΔA)2 ≡ 〈 (A −<A>)2 〉, etc.


 \Delta  A \, \Delta  B \geq  \frac{1}{2} \sqrt{ \left|\left\langle\left[{A},{B}\right]\right\rangle \right|^2 + \left|\left\langle\left\{ A-\langle A\rangle ,B-\langle B\rangle  \right\} \right\rangle \right|^2} ,

where [A,B] ≡ ABBA is the commutator of A and B, and {A,B} ≡ AB+BA is the anticommutator. This follows through use of the Cauchy–Schwarz inequality, since |〈A2〉| |〈B2〉| ≥ |〈AB〉|2, and AB = ([A,B] + {A,B}) /2 ; and similarly for the shifted operators A−〈A〉 and B−〈B〉 . Judicious choices for A and B yield Heisenberg’s familiar uncertainty relation, for x and p, as usual; or, here, Lx and Ly , in angular momentum multiplets, ψ = |lm 〉 , useful constraints such as l (l+1) ≥ m (m+1), and hence l ≥ m, among others.


Lie Groups for Pedestrians

Modern Quantum Mechanics (2nd Edition)


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