Unitary transformation (quantum mechanics)

In quantum mechanics, the Schrödinger equation describes how a system changes with time. It does this by relating changes in the state of system to the energy in the system (given by an operator called the Hamiltonian). Therefore, once the Hamiltonian is known, the time dynamics are in principle known. All that remains is to plug the Hamiltonian into the Schrödinger equation and solve for the system state as a function of time. Often, however, the Schrödinger equation is difficult to solve (even with a computer). Therefore, physicists have developed mathematical techniques to simplify these problems and clarify what is happening physically. One such technique is to apply a unitary transformation to the Hamiltonian. Doing so can result in a simplified version of the Schrödinger equation which nonetheless has the same solution as the original. A unitary transformation (or frame change) can be expressed in terms of a time-dependent Hamiltonian and unitary operator . Under this change, the Hamiltonian transforms as: The Schrödinger equation applies to the new Hamiltonian. Solutions to the untransformed and transformed equations are also related by . Specifically, if the wave function satisfies the original equation, then will satisfy the new equation. Recall that by the definition of a unitary matrix, . Beginning with the Schrödinger equation, we can therefore insert at will. In particular, inserting it after and also premultiplying both sides by , we get Next, note that by the product rule, Inserting another and rearranging, we get Finally, combining (1) and (2) above results in the desired transformation: If we adopt the notation to describe the transformed wave function, the equations can be written in a clearer form. For instance, can be rewritten as which can be rewritten in the form of the original Schrödinger equation, The original wave function can be recovered as . Unitary transformations can be seen as a generalization of the interaction (Dirac) picture.