5.2: Properties of Spin Angular Momentum
- Page ID
- 1205
Let us denote the three components of the spin angular momentum of a particle by the Hermitian operators \( {\bf S} \times {\bf S} = {\rm i}\,\hbar \, {\bf S}.\)
We can also define the operator
Thus, it is possible to find simultaneous eigenstates of \( S_z\). These are denoted \( S_z \,\vert s, s_z\rangle\)
Spin angular momentum clearly has many properties in common with orbital angular momentum. However, there is one vitally important difference. Spin angular momentum operators cannot be expressed in terms of position and momentum operators, like in Equations \ref{290}-\ref{292}, because this identification depends on an analogy with classical mechanics, and the concept of spin is purely quantum mechanical: i.e., it has no analogy in classical physics. Consequently, the restriction that the quantum number of the overall angular momentum must take integer values is lifted for spin angular momentum, since this restriction (found in Sections 4.3 and 4.4) depends on Equations \ref{290}-\ref{292}. In other words, the spin quantum number \( S_z \,\vert\pm \rangle\)
They are also properly normalized and complete, so that
It is easily verified that the Hermitian operators defined by
satisfy the commutation relations \ref{297}-\ref{299} (with the \( S_j\) ). The operator \( S^2 = \frac{3\,\hbar^2}{4}.\)
It is also easily demonstrated that \( S_z\), defined in this manner, satisfy the eigenvalue relations \ref{422}-\ref{423}. Equations \ref{427}-\ref{430} constitute a realization of the spin operators \( S^2\) (for a spin one-half particle) in spin space (i.e., the Hilbert sub-space consisting of kets which correspond to the different spin states of the particle).
Contributors
Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)
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