$$ \begin{aligned} {\rm \LARGE I:} \;&\left \{ \begin{aligned} &\text{The pure state of a quantum system are described by the unit vectors} \\ &\text{up to phase (rays) of a Hilbert space} \end{aligned} \right . \\ &\small \begin{aligned} \textbf{📝\color{CornflowerBlue} Note:}\;&\text{thus the phase is doesnt change the physics, ie $\ket{\psi}$ and $e^{i\alpha}\ket{\psi}$} \\ &\text{describe the same object} \end{aligned} \\ \\ {\rm \LARGE II:} \;&\left \{ \begin{aligned} &\text{The state space of a composite system (of distinguishable components)} \\ &\text{is the tensor product of the component state spaces.} \end{aligned} \right . \\ &\small \begin{aligned} \textbf{💃\color{OrangeRed} Example:}\;&\text{state space of an electron $\ket{e}$ and a proton $\ket{p}$ is the tensor} \\ &\text{product }\ket{e}\otimes\ket{p} \end{aligned} \\ \\ {\rm \LARGE III:} \;&\left \{ \begin{aligned} &\text{The time evolution from $ t_0 $ to $t_1$ of the state of a closed quantum} \\ &\text{system is given by a unitary operator }U(t_0,t_1) \end{aligned} \right . \\ &\small \begin{aligned} \textbf{💃\color{OrangeRed} Example:}\;&\text{state at time $t_0$ is $\ket{\psi(t_0)}$ then the state at $t_1>t_0$ is } \\ &\ket{\psi(t_1)}=U(t_0,t_1)\ket{\psi(t_0)} \end{aligned} \\ \\ {\rm \LARGE IV:} \;&\left \{ \begin{aligned} &\text{Every physical observable $A$ corresponds to a hermitian operator $\hat A$} \\ &\text{on the state space. The eigenvalues of $A$ are the only values that can } \\ &\text{be obtained for $A$ by experiment.} \end{aligned} \right . \\ \\ {\rm \LARGE V_a:} \;&\left \{ \begin{aligned} &\text{A measurement $M_B$ is a set of projections} \\ &\text{$M=\{ P_{\Lambda_1}=\sum_{l_1} \ket{l_1}\bra{l_1},\ldots,P_{\Lambda_m}=\sum_{l_m} \ket{l_m}\bra{l_m} \} $ corresponding to a} \\ &\text{partition of a basis $ B=\{\{ \ket{l_1} \}{l_1},\cdots , \{\ket{l_m}\}{l_m}\}= $} \\ &\text{$\{ \{ \ket{l_{1,0}}\ldots \ket{l_{1,n_1-1}}\},\ldots , \{ \ket{l_{m,0} } \ldots \ket{l_{m,n_m-1}}\}\}$ of the state space} \end{aligned} \right . \\ &\small \begin{aligned} \textbf{📝\color{CornflowerBlue} Note:}\;&\text{the identity $P_{\Lambda_1} + \ldots + P_{\Lambda_m}=I=\sum_k \ket{k}\bra{k}$} \end{aligned} \\ \\ {\rm \LARGE V_b:} \;&\left \{ \begin{aligned} &\text{Performing a measurement } M_B \text{ on a state } \ket{\psi} \text{nondeterministically }\\ &\text{sends } \ket{\psi} \text{ to one of the outcomes:} \\ &N_1^{-1}P_{\Lambda_1}\ket{\psi} = N_1^{-1}\sum_{l_1}\ket{l_1}\bra{l_1}\ket{\psi} \quad \text{with probability } \|P_{\Lambda_1}\ket{\psi}\|^2,\\ &\quad\vdots\\ &N_m^{-1}P_{\Lambda_m}\ket{\psi} = N_m^{-1}\sum_{l_m}\ket{l_m}\bra{l_m}\ket{\psi} \quad \text{with probability } \|P_{\Lambda_m}\ket{\psi}\|^2. \end{aligned} \right . \\ &\small \begin{aligned} \textbf{📝\color{CornflowerBlue} Note:}\;&\text{Each outcome is a projection and renormalization of } \ket{\psi}\text{ by } P_{\Lambda_j} \\ &\text{ (the projection postulate).} \end{aligned} \\ \\ {\rm \LARGE V_c:} \;&\left \{ \begin{aligned} &\text{Performing an observation of an observable } A \text{ (i.e. an experiment}\\ &\text{ to measure } A\text{)} \text{ means performing } M_B, \text{where } B \text{ is a basis of}\\ & \text{eigenvectors of } A \text{ partitioned according to the eigenspaces of } A.\\ &\text{For each outcome } N_j^{-1}P_{\Lambda_j}\ket{\psi}\text{, the measured value of } A \text{ is the} \\ &\text{eigenvalue } v_j \text{ corresponding to the eigenspace } \Lambda_j. \end{aligned} \right . \\ &\small \begin{aligned} \textbf{📝\color{CornflowerBlue} Note:}\;&\text{The total probability is } \sum_j \|P_{\Lambda_j}\ket{\psi}\|^2 = 1\text{, since } \ket{\psi}\text{ is a unit vector.} \\ &\text{and the expectation value is $\bra{\psi}A\ket{\psi}=\sum_j v_j \| P_{\Lambda_j} \ket{\psi} \|^2$ } \end{aligned}

\end{aligned} $$

💼 Case: consider an observable $A$ corresponding to $\hat A=\ket{4}4 \bra{4}+\ket{5a}5 \bra{5a}+\ket{5b}5\bra{5b}$

Lets now calculate the probabilities of finding these eigenvalues for $\ket{\psi} = \frac 13 \ket{4}+\frac 23 \ket{5a} + \frac 23 \ket{5b}$

📖 Definition:

Electrons

🧽 Assumptions:

  1. For this course we will consider only electrons (spin-$\frac 12$ particles) and more specifically consider only the spin which we describe as vector $\ket{\psi}$ in $Q$

  2. We only care about spin direction so we set the vector to be normalised and thus has 3 DoF in 3D space (due to phase term) so is completely defined by $\ket\psi=e^{i\gamma}[a,b]^T$ with $|a|^2+|b|^2=1$

  3. We are assuming physical results so the spin doesn't go back after $4\pi$ but $2\pi$ instead. In other words we are not in Hilbert space but in a Bloch sphere so $\ket{\psi}=[e^{i\gamma}\cos(\frac \theta 2 ), e^{i(\gamma+\phi)} \sin (\frac \theta 2)]^T$

  4. We don’t care about non-physical terms so a general phase of $e^{i\gamma}$ is irrelevant and we get 2 DoF

    $$ \boxed{\ket{\psi}=\ket{\psi}=[\cos(\tfrac \theta 2 ), e^{i\phi} \sin (\tfrac \theta 2)]^T \quad \forall\theta\in[0,\pi]\: \& \; \forall \phi\in[0,2\pi)} $$

Lets now analyse what we measure when studying spin