Schrödinger picture

In 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 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.

The Time Evolution Operator

Definition

The time evolution operator $$U(t,t_0) is defined as::$$ | psi(t) angle = U(t,t_0) | psi(t_0) angle

That is, this operator when acting on the state ket at $$t_0 gives the state ket at a later time $$t . For bras, we have::$$langle psi(t) | = langle psi(t_0) |U^{dagger}(t,t_0)

Properties

Property 1

The 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,:$$langle psi(t)| psi(t) angle = langle psi(t_0)|U^{dagger}(t,t_0)U(t,t_0)| psi(t_0) angle = langle psi(t_0) | psi(t_0) angle

:Therefore $$U^{dagger}(t,t_0)U(t,t_0)=I

Property 2

Clearly $$U(t_0,t_0) = I, the Identity operator. As::$$ | psi(t_0) angle = U(t_0,t_0) | psi(t_0) angle

Property 3

Also time evolution from $$t_0 to $$t may be viewed as time evolution from $$t_0 to an intermediate time $$t_1 and from $$t_1 to the final time $$t . therefore:: $$U(t,t_0) = U(t,t_1)U(t_1,t_0)

Differential Equation for Time Evolution Operator

We drop the $$t_0 index in the time evolution operator with the convention that $$t_0=0 and write it as $$U(t) . The Schrödinger equation can be written as::$$i hbar {d over dt} U(t) | psi_e (0) angle = H U(t)| psi_e (0) angle

Here " H " is the Hamiltonian for the system. As $$ | psi(0) angle is a constant ket( it is the state ket at $$t=0 ), we see that the time evolution operator obeys the Schrödinger equation: i.e.:$$i hbar {d over dt} U(t) = H U(t)

If the Hamiltonian is independent of time, the solution to the above equation is::$$U(t) = e^{-iHt / hbar}

Where we have also used the fact that at $$t=0, U(t) must reduce to the identity operator. Therefore we get: :$$| psi(t) angle = e^{-iHt / hbar} | psi(0) angle .

Note that $$ | psi(0) angle is an arbitrary ket. However, if the initial ket is an eigenstate of the Hamiltonian, with eigenvalue " a " , we get::$$| psi(t) angle = e^{-iat / hbar} | psi(0) angle .

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::$$U(t) = e^{-i/hbar intlimits _0^t H(t^'), dt^'}

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.

ee also

*Hamilton–Jacobi equation
*interaction picture

Further reading

* "Principles of Quantum Mechanics" by R. Shankar, Plenum Press.