Quantum Computation and Quantum Information

[G.F.Again]

Hey all, I'm a beginner of the Quantum Computation and Quantum Information. For a long time, I feel very confuse about the question bellow. Could you do me a favor and show me the proof? Many thanks!
Question:
Show that any qubit can be expressed as
psi=exp(iγ)[cos(θ/2)0+exp(iΦ)sin(θ/2)1]
for real numbers γ,θ and Φ. The phase factor exp(iγ) has no observational effect and can be dropped.
And then show that there is a one to one correspondence between qubits
psi=cos(θ/2)0+exp(iΦ)sin(θ/2)1
and the points on the unit sphere in R(3) called the Bloch sphere, with and as the spherical coordinates of a point of the sphere.

 

[DavidK]

Start by expressing any normalized state vector as $$|\psi \rangle = a |0\rangle + b|1\rangle$$, where normalization entails $$a^2 + b^2 =1$$. $$a,b$$ are complex numbers, which can always be expressed as $$a = r_a e^{i\gamma_a}$$, $$b = r_b e^{i\gamma_b}$$. This means that $$a^2 + b^2 =r_a^2+r_b^2=1$$, and since $$r_a, r_b$$ are positive we can express them as $$r_a = \cos(\theta /2), r_b= \sin(\theta/2)$$ where $$0 \leq \theta \leq \pi$$. Consequently, we can write $$|\psi \rangle = \cos(\theta /2) e^{i\gamma_a} |0\rangle + \sin(\theta/2) e^{i\gamma_b} |1\rangle = e^{i\gamma}(\cos(\theta /2)|0\rangle + \sin(\theta/2) e^{i\Phi} |1\rangle )$$, where $$\gamma = \gamma_a, \Phi = \gamma_b-\gamma_a$$.

 

[G.F.Again]

Thank you, Davidk!

 

[qtp]

how is it that you can say r_a = cos (theta/2) when r_a is a radius with a real length and cos(theta) is a dimensionless quantity?