Spin Angular Momentum

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Alternatively, they can be derived as discussed. However, n can be arbitrarily large, so at least one operator cannot be bounded, and the dimension of the underlying Hilbert space cannot be finite. There are two properties of this quantity.

The is and obeys the canonical quantization relations defining the for , where is the. The inner products are and Therefore, and or This is the general uncertainty relation that we sought. All have a characteristic spin, which is usually nonzero. This is where the Levi symbol comes in to say that.

Spin Angular Momentum

Spin Angular Momentum The goal of this section is to introduce the spin angular momentumas a generalized angular momentum operator that satisfies the general commutation relations. The main difference between the angular momentaandis that can have half-integer quantum numbers. Note: Remember that the quantization rules established by the commutation relations did not rule out the possibility of half-integer values for see page 46. However, such possibility was ruled out by the periodicity requirement,associated with the eigenfunctions of and. Since the spin eigenfunctions i. Electron Spin: A particular case of half-integer spin is the spin angular momentum of an electron with seefor Goudsmit's historical recount of the discovery of the electron spin. In discussing the spin properties of a particle we adopt the notationand. The spin functions and are eigenfunctions of with eigenvalues andrespectively. These eigenfunctions are normalized according to, whereandare the probabilities that a measurement of yields the valueandrespectively, when the system is described by state. Exercise 27: Consider an electron localized at a crystal site. Assume that the spin is the only degree of freedom of the system and that due to the angular momentum commutation relations the electron has a magnetic moment, Consider that initially i. Compute the expectation values of and at time. Addition of Angular Momenta Since depends on spatial coordinates and does not, then the two operators commute i. It is, therefore, angular momentum commutation relations that the components of the total angular momentum.

Note, I may have mixed up the order of things, but I think its still right. The inner products are and Therefore, and or This is the general uncertainty relation that we sought. In other words, to get a precise energy value, we need to wait for a long time. This is different from a 360° rotation of the internal spin state of the particle, which might or might not be the same as no rotation at all. The commutation relation between these two observables is just so We have thus obtained a formal proof of the. Furthermore, if the potential energy and the constraints of the system are time-independent, then H is not only conserved, but H is the energy of the system. Let us now derive the commutation relations for the. A Modern Approach to Quantum Mechanics. However, such possibility was ruled out by the periodicity requirement, , associated with the eigenfunctions of and.