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Schrödinger pictureIn 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 an unitary operator called the time-evolution operator. This differs from the Heisenberg picture where the states are constant while the observables evolve in time. The measurement statistics are the same in both pictures, as they should be. Additional recommended knowledge
The Time Evolution OperatorDefinitionThe time evolution operator U(t,t0) is defined as: That is, this operator when acting on the state ket at t0 gives the state ket at a later time t . For bras, we have: PropertiesProperty 1The time evolution operator must be unitary. This is because we demand that the norm of the state ket must not change with time. That is,
Property 2Clearly U(t0,t0) = I, the Identity operator. As: Property 3Also time evolution from t0 to t may be viewed as time evolution from t0 to an intermediate time t1 and from t1 to the final time t . therefore:
Differential Equation for Time Evolution OperatorWe drop the t0 index in the time evolution operator with the convention that t0 = 0 and write it as U(t) . The Schrodinger equation can be written as: Here H is the Hamiltonian for the system. As is a constant ket( it is the state ket at t = 0 ), we see that the time evolution operator obeys the Schrodinger 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 t = 0,U(t) 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 Heisenberg picture. See also interaction picture. Further reading
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This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Schrödinger_picture". A list of authors is available in Wikipedia. |