Angular Momentum from Its Commutation Relations
Hermitian angular-momentum operators obey commutation relations that determine finite ladders of states. Raising and lowering operators change the projection quantum number by one. The allowed j values are nonnegative integers or half-integers; orbital angular momentum uses integers.
What a learner can do afterwards
- Builds the ladder of states from the commutators without solving a differential equation
- Explains why the eigenvalues step by one and may be half-integer
- Matches the algebraic result to the spherical harmonics found the analytic way
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Start with Hermitian angular-momentum operators satisfying [Jx,Jy]=i hbar Jz and cyclic versions. Define J²=Jx²+Jy²+Jz² and J±=Jx±iJy. J² commutes with each component. Label joint eigenstates of J² and Jz by |j,m>, with eigenvalues hbar²j(j+1) and hbar m. The component commutators start the algebraic derivation without an orbital differential equation. The j=0 state has all components sharp.
The relation [Jz,J±]=±hbar J± fixes the size of a nonzero ladder step. Acting on |j,m>, JzJ± gives hbar(m±1) times the new vector. Thus J+ raises m by one and J- lowers it by one. The change in the Jz eigenvalue is ±hbar, not ±hbar/2. At an endpoint, the relevant ladder action gives zero instead of a new state.
† means adjoint. Hermiticity gives J+†=J-. Thus ||J±|j,m>||²=<j,m|J∓J±|j,m>. Expand J∓J±=J²-Jz²∓hbar Jz. In normalized states, insert the eigenvalues to get hbar²[j(j+1)-m(m±1)]. Norms cannot be negative. J+ annihilates the top m=j; J- annihilates the bottom m=-j. These endpoints and unit steps determine a finite ladder's projections, without an orbital differential equation or a line spectrum.
To list a ladder, start at m=j, subtract one repeatedly, and stop at m=-j. Include both endpoints. There are 2j steps and 2j+1 states. For j=1, the list is 1,0,-1 and the count is three. Apply the same listing rule to half-integer j; successive entries still differ by one.
The number of steps from j to -j is 2j, so 2j must be a nonnegative integer. Thus j may be integer or half-integer. Orbital angular momentum has integer j, usually written l. Intrinsic spin may be integer or half-integer. An electron has spin 1/2; an integer-only rule would exclude its observed two spin projections.
For orbital motion, spherical harmonics Y_lm have L² and Lz eigenvalues hbar²l(l+1) and hbar m, with integer l and m=-l,...,l. They realize the integer orbital ladders. They do not describe intrinsic spin states or disprove half-integer representations. Intrinsic spin is a different representation of the same angular-momentum algebra.
Commutators build ladders in unit steps, giving integer and half-integer angular momentum.
Original lesson, practice questions and flashcards authored for this dataset.
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