Adiabatic theorem of quantum mechanics

The adiabatic theorem of quantum mechanics , also Adiabatensatz of quantum mechanics called, says that the state of the system

${\displaystyle \psi (t)=e^{i\alpha (t)}\cdot \left|n(t)\right\rangle }$

remains in the -th eigenstate of to a good approximation in the course of the time evolution if the Hamilton operator of a system changes "slowly enough" (for example due to external influences). ${\displaystyle n}$ ${\displaystyle H(t)}$ ${\displaystyle H(t)}$

“Slow enough” means (for ) that ${\displaystyle m\neq k}$

${\displaystyle {\left|\left\langle m(t)\left|{\frac {d}{dt}}H(t)\right|k(t)\right\rangle \right|}\ll {\frac {\left|E_{k}(t)-E_{m}(t)\right|}{\Delta T_{km}}}}$

applies.

It is the characteristic time of the transition of the system from state to state and and are the on states and associated energy eigenvalues of the system. ${\displaystyle \Delta T_{km}}$ ${\displaystyle \left|k(t)\right\rangle }$ ${\displaystyle \left|m(t)\right\rangle }$ ${\displaystyle E_{k}(t)}$ ${\displaystyle E_{m}(t)}$ ${\displaystyle k}$ ${\displaystyle m}$

This means that the change in is slow compared to the natural time scale of the system, which is defined by transitions between the energetic eigenstates. ${\displaystyle H(t)}$

In the adiabatic limit , the changes are infinitely slow: ${\displaystyle H(t)}$

${\displaystyle \Delta T_{km}\longrightarrow \infty }$and thus follows: . ${\displaystyle \left|\left\langle m(t)\left|{\frac {d}{dt}}H(t)\right|k(t)\right\rangle \right|\longrightarrow 0}$

Examples in physics

The best-known example in physics is the Born-Oppenheimer approximation . Max Born and Robert Oppenheimer were able to show that the movement of the atomic nuclei (the change in ) can be neglected for the calculation of the changes in state of the electrons of a molecule . Put simply, the electrons move so quickly and the time they need to transition between two electron levels is so short that the movement of the (slow) atomic nuclei is irrelevant for a calculation. ${\displaystyle H(t)}$

history

The adiabatic theorem of quantum mechanics goes back to work by Max Born and Wladimir Alexandrowitsch Fock from 1928. However, a complete mathematical formulation was only achieved by Tosio Kato (1950) in connection with the perturbation theory of linear operators .

Michael Berry showed in 1984 that in the event of cyclical adiabatic changes in the parameters, the system does return to its initial state, but under certain circumstances receives a phase factor that depends on the geometry of the parameter space ( Berry phase ).

See also

• Adiabatic change of state

literature

• M. Born, V. Fock: Proof of the Adiabat Theorem . In: Journal of Physics . tape 51 , no. 3-4 , March 1928, pp. 165-180 , doi : 10.1007 / BF01343193 .
• Tosio Kato: On the Adiabatic Theorem of Quantum Mechanics . In: Journal of the Physical Society of Japan . tape 5 , no. 6 , 1950, pp. 435-439 , doi : 10.1143 / JPSJ.5.435 .
• VS Buslaev, EA Grinina: Remarks on the quantum adiabatic theorem . In: St. Petersburg Mathematical Journal . tape 16 , no. 04 , June 21, 2005, p. 639–648 (See also references given therein).