wiki:adiabatic-theorem
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| wiki:adiabatic-theorem [August 16, 2026 at 17:05] – created - external edit 127.0.0.1 | wiki:adiabatic-theorem [August 19, 2026 at 16:12] (current) – external edit 127.0.0.1 | ||
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| Let $H(t)$ have instantaneous eigenstates and eigenvalues | Let $H(t)$ have instantaneous eigenstates and eigenvalues | ||
| - | $$H(t)\, | + | $$H(t)\, |
| These are defined pointwise in time: at each instant, freeze the Hamiltonian and diagonalise it. Nothing about $\lvert n(t)\rangle$ requires the system to actually be in that state. | These are defined pointwise in time: at each instant, freeze the Hamiltonian and diagonalise it. Nothing about $\lvert n(t)\rangle$ requires the system to actually be in that state. | ||
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| The gap is smallest at $t = 0$, where it equals $2\Delta$, and the sweep rate $v$ is what the system is being asked to keep up with. Below, the left panel is the spectrum with the state' | The gap is smallest at $t = 0$, where it equals $2\Delta$, and the sweep rate $v$ is what the system is being asked to keep up with. Below, the left panel is the spectrum with the state' | ||
| - | < | + | < |
| // Two level sweep through an avoided crossing, integrated directly: | // Two level sweep through an avoided crossing, integrated directly: | ||
| // i dc/dt = H c, H = [[v t, D], [D, -v t]], hbar = 1 | // i dc/dt = H c, H = [[v t, D], [D, -v t]], hbar = 1 | ||
| Line 181: | Line 181: | ||
| </ | </ | ||
| - | Each pass raises the sweep rate and then starts over. At small $v$ the green curve dips at the crossing and comes back to one. At large $v$ it dips and stays down: the system went straight through, keeping its old character rather than its old energy level, and the run has failed. | + | Each pass raises the sweep rate and then starts over, and the reset button takes it back to the slowest pass. At small $v$ the green curve dips at the crossing and comes back to one. At large $v$ it dips and stays down: the system went straight through, keeping its old character rather than its old energy level, and the run has failed. |
| ## Landau and Zener put a number on it | ## Landau and Zener put a number on it | ||
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| The oldest version of the theorem is a spin in a magnetic field that slowly changes direction. The spin precesses about $\vec B$ at the Larmor frequency, and if $\vec B$ turns much more slowly than that, the spin's cone of precession is dragged along with it. | The oldest version of the theorem is a spin in a magnetic field that slowly changes direction. The spin precesses about $\vec B$ at the Larmor frequency, and if $\vec B$ turns much more slowly than that, the spin's cone of precession is dragged along with it. | ||
| - | Both panels below start with the spin aligned to the field. Only the rotation rate differs. | + | Both panels below start with the spin aligned to the field. Only the rotation rate differs. The right panel wanders further the longer it runs, so this one carries a reset button as well: it reloads the sketch and puts both spins back on the field. |
| - | < | + | < |
| // Bloch vector precessing about a field that rotates about z: | // Bloch vector precessing about a field that rotates about z: | ||
| // dr/dt = B x r | // dr/dt = B x r | ||
| Line 331: | Line 331: | ||
| \end{document} | \end{document} | ||
| - | < | + | < |
| // The field traces a closed loop; the phase accumulated is half the | // The field traces a closed loop; the phase accumulated is half the | ||
| // solid angle enclosed, and does not care how fast the loop is walked. | // solid angle enclosed, and does not care how fast the loop is walked. | ||
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