eigenvalues of the Hamiltonian cannot be degenerate, and thus we have an avoided crossing. If an atom is initially in state in zero magnetic field (on the red curve, at the extreme left), an adiabatic increase in magnetic field will ensure the system remains in an eigenstate of the Hamiltonian throughout the process (follows the red curve). A diabatic increase in magnetic field will ensure the system follows the diabatic path (the dotted blue line), such that the system undergoes a transition to state . For finite magnetic field slew rates there will be a finite probability of finding the system in either of the two eigenstates. See below for approaches to calculating these probabilities. These results are extremely important in atomic and molecular physics for control of the energy-state distribution in a population of atoms or molecules. Mathematical statement Under a slowly changing Hamiltonian with instantaneous eigenstates and corresponding energies , a quantum system evolves from the initial state to the final state where the coefficients undergo the change of phase with the dynamical phase and geometric phase In particular, , so if the system begins in an eigenstate of , it remains in an eigenstate of during the evolution with a change of phase only. Proofs {| class="toccolours collapsible collapsed" width="80%" style="text-align:left" !Sakurai in Modern Quantum Mechanics |- | This proof is partly inspired by one given by Sakurai in Modern Quantum Mechanics. The instantaneous eigenstates and energies , by assumption, satisfy the time-independent Schrödinger equation at all times . Thus, they constitute a basis that can be used to expand the state at any time . The evolution of the system is governed by the time-dependent Schrödinger equation where (see ). Insert the expansion of , use , differentiate with the product rule, take the inner product with and use orthonormality of the eigenstates to obtain This coupled first-order differential equation is exact and expresses the time-evolution of the coefficients in terms of inner products between the eigenstates and the time-differentiated eigenstates. But it is possible to re-express the inner products for in terms of matrix elements of the time-differentiated Hamiltonian . To do so, differentiate both sides of the time-independent Schrödinger equation with respect to time using the product rule to get Again take the inner product with and use and orthonormality to find Insert this into the differential equation for the coefficients to obtain This