Observable on a qubit

Any observable on a Qubit can be written as

A^=a0σ0+a⃗σ1,2,3⃗\begin{gather*} \hat{A}=a_0\sigma_0+\vec{a}\vec{\sigma_{1,2,3}} \end{gather*}

You can think of a0σ0a_0\sigma_0 as the scalar value and a⃗σ1,2,3⃗\vec{a}\vec{\sigma_{1,2,3}} as it helps with the Bloch Sphere.

This can be written in spherical units too

n^=(nx,ny,nz)=(sin⁡θcos⁡ψ,sin⁡θsin⁡ψ,cos⁡θ)\begin{gather*} \hat{n}=(n_x,n_y,n_z)=(\sin\theta\cos\psi,\sin\theta\sin\psi,\cos\theta) \end{gather*}

spin along n^\hat{n} can be described as S^=ℏ2σ⃗\hat{S}=\frac{\hbar}{2}\vec{\sigma}. This is known as the "spin along the unit vector n^\hat{n}". To do this we project S^\hat{S} onto n^\hat{n}.

S^n=S^⋅n^=ℏ2σ⃗⋅n^=ℏ2(nxσx+nyσy+nzσz)=nx ⁣(0110)+ny ⁣(0−ii0)+nz ⁣(100−1)\begin{gather*} \hat{S}_n=\hat{S}\cdot\hat{n}=\frac{\hbar}{2}\vec{\sigma}\cdot\hat{n}\\ =\frac{\hbar}{2}\left(n_x\sigma_x+n_y\sigma_y+n_z\sigma_z\right)\\ =n_x\!\begin{pmatrix}0&1\\1&0\end{pmatrix} + n_y\!\begin{pmatrix}0&-i\\i&0\end{pmatrix} + n_z\!\begin{pmatrix}1&0\\0&-1\end{pmatrix} \end{gather*} =(nznx−inynx+iny−nz)= \begin{pmatrix} n_z & n_x - i n_y \\ n_x + i n_y & -n_z \end{pmatrix} ⇒S^n^=ℏ2(nznx−inynx+iny−nz).\Rightarrow\hat{S}_{\hat{n}} = \frac{\hbar}{2}\begin{pmatrix} n_z & n_x - i n_y \\ n_x + i n_y & -n_z \end{pmatrix}.

We work out the eigenvalues by doing

det(S^n^−λI)=0det(\hat{S}_{\hat{n}} - \lambda I) = 0 ⇒(ℏ2nz−λ)(−ℏ2nz−λ)−(ℏ2)2(nx−iny)(nx+iny)=0\Rightarrow\left(\tfrac{\hbar}{2}n_z-\lambda\right)\left(-\tfrac{\hbar}{2}n_z-\lambda\right) - \left(\tfrac{\hbar}{2}\right)^2(n_x - i n_y)(n_x + i n_y) = 0 ⇒λ2−(ℏ2)2(nx2+ny2+nz2)=0\Rightarrow\lambda^2 - \left(\tfrac{\hbar}{2}\right)^2(n_x^2+n_y^2+n_z^2) = 0 λ2=(ℏ2)2∣n^∣2⇒λ=±ℏ2∣n^∣\lambda^2=\left(\tfrac{\hbar}{2}\right)^2|\hat{n}|^2\Rightarrow\lambda=\pm\tfrac{\hbar}{2}|\hat{n}|

Because n^\hat{n} is a unit vector,

λ=±ℏ2\lambda=\pm\tfrac{\hbar}{2}