2.5: Explicit form of the spin-1/2 rotation operator
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- 20882
For spin-1/2, the rotation operator
can be written as an explicit 2\(\times\) 2 matrix. This is accomplished by expanding the exponential into a Taylor series:
Note that
Thus, the Taylor series becomes:
\(\displaystyle \exp\left(-i{\alpha \over 2}\stackrel{\rightarrow}{\sigma}\cdot{\hat{\bf n}}\right)\) | \(\textstyle =\) | ||
\(\textstyle =\) | |||
\(\textstyle =\) |
Thus,
As a 2\(\times\)2 matrix,
so that the rotation operator becomes
Now consider the example of \(\alpha = 2\pi\). In this case, it is easy to see that the rotation operator reduces to
Interestingly, a rotation through an angle \(2\pi\) of a spin state returns the state to its original value but causes it to pick up an overall phase factor
While this phase factor cannot affect any physical property, it is, nevertheless observable in the experiment depicted below:
\(\vert\chi\rangle\), is split by a partially reflecting material into two beams. One of these is sent through a magnetic field region tuned to generate a rotation by \(\alpha = 2\pi\) of the spin state, so that the new state is \(\vert\chi'\rangle\). The beams are then brought back together and allowed to interfere. The overlap, \(\langle \chi'\vert\chi\rangle=-1\) is measured, which will yield the over phase factor \(-1\).