oscillator, initially in its ground state, , remains in the ground state as the potential energy curve is compressed; the functional form of the state adapting to the slowly varying conditions. For a rapidly increased spring constant, the system undergoes a diabatic process in which the system has no time to adapt its functional form to the changing conditions. While the final state must look identical to the initial state for a process occurring over a vanishing time period, there is no eigenstate of the new Hamiltonian, , that resembles the initial state. The final state is composed of a linear superposition of many different eigenstates of which sum to reproduce the form of the initial state. Avoided curve crossing For a more widely applicable example, consider a 2-level atom subjected to an external magnetic field. The states, labelled and using bra–ket notation, can be thought of as atomic angular-momentum states, each with a particular geometry. For reasons that will become clear these states will henceforth be referred to as the diabatic states. The system wavefunction can be represented as a linear combination of the diabatic states: With the field absent, the energetic separation of the diabatic states is equal to ; the energy of state increases with increasing magnetic field (a low-field-seeking state), while the energy of state decreases with increasing magnetic field (a high-field-seeking state). Assuming the magnetic-field dependence is linear, the Hamiltonian matrix for the system with the field applied can be written where is the magnetic moment of the atom, assumed to be the same for the two diabatic states, and is some time-independent coupling between the two states. The diagonal elements are the energies of the diabatic states ( and ), however, as is not a diagonal matrix, it is clear that these states are not eigenstates of the new Hamiltonian that includes the magnetic field contribution. The eigenvectors of the matrix are the eigenstates of the system, which we will label and , with corresponding eigenvalues It is important to realise that the eigenvalues and are the only allowed outputs for any individual measurement of the system energy, whereas the diabatic energies and correspond to the expectation values for the energy of the system in the diabatic states and . Figure 2 shows the dependence of the diabatic and adiabatic energies on the value of the magnetic field; note that for non-zero coupling the