- Schrödinger picture
quantum mechanics, a state function is a linear combination (a superposition) of eigenstates. In the Schrödinger picture, the state of a system evolves with time, where the evolution for a closed quantum system is brought about by a unitary operator called the time-evolution operator. This differs from the Heisenberg picturewhere the states are constant while the observables evolve in time. The measurement statistics are the same in both pictures, as they should be.
The Time Evolution Operator
The time evolution operator is defined as::
Clearly = I, the
Identity operator. As::
Also time evolution from to may be viewed as time evolution from to an intermediate time and from to the final time . therefore::
Differential Equation for Time Evolution Operator
We drop the index in the time evolution operator with the convention that and write it as . The Schrödinger equation can be written as::
Here " H " is the Hamiltonian for the system. As is a constant ket( it is the state ket at ), we see that the time evolution operator obeys the Schrödinger equation: i.e.:
If the Hamiltonian is independent of time, the solution to the above equation is::
Where we have also used the fact that at must reduce to the identity operator. Therefore we get: :.
Note that is an arbitrary ket. However, if the initial ket is an eigenstate of the Hamiltonian, with eigenvalue " a " , we get::.
Thus we see that the eigenstates of the Hamiltonian are "stationary states", they only pick up an overall phase factor as they evolve with time.If the Hamiltonian is dependent on time, but the Hamiltonians at different time commute then, the time evolution operator can be written as::
The alternative to the Schrödinger picture is to switch to a rotating reference frame, which is itself being rotated by the propagator. Since the undulatory rotation is now being assumed by the reference frame itself, an undisturbed state function appears to be truly static. This is the
* "Principles of Quantum Mechanics" by R. Shankar, Plenum Press.
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