Why Does the Bloch Sphere Use Theta Over Two for Qubit Representation?

[maverick280857]

I am reading Quantum Computation and Quantum Information by Nelson and Chuang myself and came across the Bloch Sphere representation of a quibit on page 15 (equation 1.4) as:

$$|\psi> = \cos\frac{\theta}{2} |0> + e^{i\psi}\sin\frac{\theta}{2} |1>$$

I have two questions:

1. What is the motivation behind such a representation (other than the fact that the sum of the squares of the coefficients of $|0>$ and $|1>$ equals 1)?

2. Why use $\theta/2$ rather than $\theta$?

 

[Hurkyl]

1. What is the motivation behind such a representation (other than the fact that the sum of the squares of the coefficients of $|0>$ and $|1>$ equals 1)?

It really is just spherical coordinates.

I suppose the fact that the parameter space is decomposed into a part that affects measurements in done this basis ($\theta$) and a part that does not ($\psi$) is an extra source of convenience.

2. Why use $\theta/2$ rather than $\theta$?

Aesthetic reasons. For example, look at expectation calculations. Or maybe the author already gave $\theta$ a meaning, so he has to use this to be consistent.

 

[maverick280857]

It really is just spherical coordinates.

Yes, I thought so too, that's why I asked about $\theta/2$. Thanks.