A Quantum Traversal

The rotation of nature

I would call this is an accumulation-free turnover of a system that remains integrated at every moment in time. The math here seems to be a crystal-clear depiction of the idea.

The time-dependent Schrödinger equation already tells us that quantum state evolves according to its Hamiltonian. We can use this to arrive at something that transforms an entire closed quantum system through time.

\begin{gathered} |\Psi(t)\rangle = U(t)|\Psi(0)\rangle \\[12pt] i\hbar\frac{d}{dt}|\Psi(t)\rangle = H|\Psi(t)\rangle \quad \longrightarrow \quad U(t) = e^{-\frac{i}{\hbar}Ht} \\[12pt] |\Psi(t)\rangle = e^{-\frac{i}{\hbar}Ht}|\Psi(0)\rangle \\[12pt] \end{gathered}

This something takes on the same fundamental form of the exponential to the imaginary, $e^{i\theta}$ , and this phase turning is in part dictated by its time-independent Hamiltonian.

We can then see that this, when applied to a closed quantum system, preserves the total probability while continuously evolving the relative phases of the components in accordance with time.

\begin{gathered} H|E_n\rangle = E_n|E_n\rangle \\[8pt] U(t)|E_n\rangle = e^{-\frac{i}{\hbar}Ht}|E_n\rangle = e^{-\frac{i}{\hbar}E_nt}|E_n\rangle \\[12pt] |\Psi(t)\rangle = U(t)|\Psi(0)\rangle = e^{-\frac{i}{\hbar}Ht} \sum_n c_n|E_n\rangle = \sum_n c_n e^{-\frac{i}{\hbar}E_nt}|E_n\rangle \\[8pt] \frac{ c_m e^{-\frac{i}{\hbar}E_mt} }{ c_n e^{-\frac{i}{\hbar}E_nt} } = \frac{c_m}{c_n} e^{-\frac{i}{\hbar}(E_m-E_n)t} \end{gathered}

\begin{gathered} H = \frac{\hbar\omega}{2}\sigma_z = \frac{\hbar\omega}{2} \begin{pmatrix} 1 & 0\\ 0 & -1 \end{pmatrix} \\[8pt] U(t) = e^{-\frac{i}{\hbar}Ht} = \begin{pmatrix} e^{-i\omega t/2} & 0\\ 0 & e^{+i\omega t/2} \end{pmatrix} \\[8pt] |\Psi(0)\rangle = \alpha|0\rangle+\beta|1\rangle = \begin{pmatrix} \alpha\\ \beta \end{pmatrix} \\[8pt] |\Psi(t)\rangle = U(t)|\Psi(0)\rangle \quad\longrightarrow\quad \alpha e^{-i\omega t/2}|0\rangle + \beta e^{+i\omega t/2}|1\rangle \\[8pt] \frac{\beta(t)}{\alpha(t)} = \frac{\beta}{\alpha}e^{i\omega t} \end{gathered}

The whole is encoded in the coherent relations among its parts, and the next move is incorporated into the present.