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Schrödinger Representation

November 18, 2008
From A. Messiah’s Quantum Mechanics p.722, Dover (1999)

Knowing that our quantum system is in a certain dynamical state at time t_0 we wish to determine its state at a later time t. The problem, therefore, is to determine as exactly as possible the operator U(t,t_0) describing the evolution in time of the dynamical states of the system in the Schrödinger “representation” (picture). This operator U(t,t_0) is completely determined once the Hamiltonian H(t) of the system is given.

In the Schrödinger representation, the operator U(t,t_0) is defined by the integral equation (or Fredholm equation):

\displaystyle U(t,t_0)=1-i \hbar^{-1} \int_{t_0}^t H(t') U(t',t_0) dt'

an equivalent expression of this law is the Schrödinger equation:

\displaystyle i\hbar\frac{\mathrm{d}}{\mathrm{d} t}U(t,t_0)=H(t)U(t,t_0)

with the initial condition:

\displaystyle U(t_0,t_0)=1

The solution of these equations is the central problem of the theory, because they express the fundamental law of evolution of the quantum system.

Since H(t) is hermitean, Uis unitary:

\displaystyle U^{\dagger}(t,t')U(t,t')=U(t,t')U^{\dagger}(t,t')=1

whence

\displaystyle U^{\dagger}(t,t')=U(t',t)

so, it is possible to write:

\displaystyle U(t,t_0)=1-i \hbar^{-1} \int_{t_0}^t U(t,t') H(t') dt'

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