## Constructing a vector from a spinor

The purpose of this post is to examine why the quantity

transforms like a vector under rotations. Here, is a 2 component spinor, and is one of the 3 Pauli matrices. The explanation follows essentially the arguments put forth by Sakurai in chapter 3 of his book on quantum mechanics.

**Spadework**

An arbitrary state ket in spin space can be written as

where and are the base “kets” (actually they are spinors, for the spin-1/2 case). This identification suggests that we can specify a general two component spinor as

Further, from Sakurai’s equations 3.2.30,

Therefore, we get the most important result:

Physically, this equation implies that the right hand side equals the expectation value of the spin operator in spinor-space (or in more precise terms, in the basis generated by the set of all possible ‘s). So, the quantity that we are examining is just the expectation value of the operator (modulo a constant factor of ).

**The argument..**

Now, Sakurai has shown that under a rotation of the state kets, the expectation value of transforms like the component of a classical vector. That is,

(This is equation 3.2.11 of Sakurai)

where are the elements of the 3 x 3 orthogonal matrix R that specifies this rotation.

The corresponding rotation of the state kets is carried out by the operator .

Therefore, transforms like a vector, and hence so does .

Thanks to David Angelaszek for a long, patient and extremely useful discussion, which led to the resolution of this argument. And of course, thanks to JJ Sakurai for forcing us to read between the lines 🙂

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