Avoided crossing

In quantum physics and quantum chemistry, an avoided crossing (sometimes called intended crossing, non-crossing or anticrossing) is the phenomenon where two eigenvalues of an Hermitian matrix representing a quantum observable and depending on N continuous real parameters cannot become equal in value ('cross') except on a manifold of N-2 dimensions. In the case of a diatomic molecule (with one parameter, namely the bond length), this means that the eigenvalues cannot cross at all. In the case of a triatomic molecule, this means that the eigenvalues can coincide only at a single point (see conical intersection). In quantum physics and quantum chemistry, an avoided crossing (sometimes called intended crossing, non-crossing or anticrossing) is the phenomenon where two eigenvalues of an Hermitian matrix representing a quantum observable and depending on N continuous real parameters cannot become equal in value ('cross') except on a manifold of N-2 dimensions. In the case of a diatomic molecule (with one parameter, namely the bond length), this means that the eigenvalues cannot cross at all. In the case of a triatomic molecule, this means that the eigenvalues can coincide only at a single point (see conical intersection). This is particularly important in quantum chemistry. In the Born–Oppenheimer approximation, the electronic molecular Hamiltonian is diagonalized on a set of distinct molecular geometries (the obtained eigenvalues are the values of the adiabatic potential energy surfaces). The geometries for which the potential energy surfaces are avoiding to cross are the locus where the Born–Oppenheimer approximation fails. Study of a two-level system is of vital importance in quantum mechanics because it embodies simplification of many of physically realizable systems. The effect of perturbation on a two-state system Hamiltonian is manifested through avoided crossings in the plot of individual energy vs energy difference curve of the eigenstates. The two-state Hamiltonian can be written as The eigenvalues of which are E 1 {displaystyle extstyle E_{1}} and E 2 {displaystyle extstyle E_{2}} and the eigenvectors, ( 1 0 ) {displaystyle extstyle {egin{pmatrix}1\0end{pmatrix}}} and ( 0 1 ) {displaystyle extstyle {egin{pmatrix}0\1end{pmatrix}}} . These two eigenvectors designate the two states of the system. If the system is prepared in either of the states it would remain in that state. If E 1 {displaystyle extstyle E_{1}} happens to be equal to E 2 {displaystyle E_{2}} there will be a twofold degeneracy in the Hamiltonian. In that case any superposition of the degenerate eigenstates is evidently another eigenstate of the Hamiltonian. Hence the system prepared in any state will remain in that forever. However, when subjected to an external perturbation, the matrix elements of the Hamiltonian change. For the sake of simplicity we consider a perturbation with only off diagonal elements. Since the overall Hamiltonian must be Hermitian we may simply write the new Hamiltonian