[NOTES/QM-09001] Unitary Operator for Time Evolution

The description of state of a quantum mechanical system at one time is by a  vector, called state vector, in a Hilbert space. As the system evolves this state vector will change.We will see that  general requirements  lead to time evolution governed by unitary operator and for short times by a hermitian operator \(H\) which will be identified with Hamiltonian of the system.

Unitary nature of time evolution

Let $\ket{\psi t_0}$ represent the state of system at time $t_0$ and $\ket{\psi t}$ represent the state at time $t$. We assume that $\ket{\psi t_0}$ determines the state at time $t$ completely. The principle of superposition should apply at these two times $t_0$ and $t$. If we have a relation at time $t_0$ \begin{equation}\label{EQ02} \ket{\psi t_0} = \alpha\ket{\chi t_0} + \beta\ket{\phi t_0} \end{equation}
between three possible states,$\ket{\psi}, \ket{\chi}, \ket{\phi}$, the same relation must hold at all times $t > t_0$ when the system is left undisturbed \begin{equation}\label{EQ03} \ket{\psi (t)} = \alpha \ket{\chi t} + \beta\ket{\phi t} \end{equation}
Thus if we write
\begin{equation}\label{EQ03A} \ket{\psi t} = U(t,t_0)\ket{\psi t_0}\qquad\mbox{etc.} \end{equation}
Then $U(t,t_0)$ must be a linear operator independent of $\psi$. Obviously $U$ must reduce to the identity operator at time $t=t_0$ \begin{equation} U(t_0,t_0) = I\,\,\,. \end{equation}
Next we demand that the norm of vector $\ket{\psi t}$ should not change with time and hence \begin{equation}\label{EQ05} \innerproduct{\psi t}{\psi t} = \innerproduct{\psi t_0}{\psi t_0} \qquad\mbox{for all $t$} \end{equation}
The above requirements \eqref{EQ02} and \eqref{EQ05}, respectively, imply
\begin{equation} \matrixelement{\psi t_0}{U^\dagger U}{\psi t_0} = \innerproduct{\psi t_0}{\psi t_0} \qquad\mbox{ for all } \ket{\psi t_0} \end{equation}
It, therefore, follows that the operator $U$ must be must be a unitary unitary operator.
\begin{equation} UU^\dagger = U^\dagger U = I \end{equation}

Differential Equation for time evolution

We shall now derive a differential equation satisfied by the state vector at time $t$. We, therefore, compute \begin{eqnarray} {d\over dt} \ket{\psi t} &=& \lim_{\Delta t\to 0}{ \ket{\psi t+\Delta t)} - \ket{\psi t}\over\Delta t}\nonumber\\ &=& \lim_{\Delta t\to 0} {(U(t+\Delta t,t)-I) \over\Delta t} \ket{\psi t}\\ \mbox{or}~~{d\over dt} \ket{\psi t} &=& \hat{X}\ket{\psi t} \label{EQ08}\end{eqnarray}
where 
\begin{equation} \hat{X}(t) = \lim_{\Delta t\to 0} {U(t+\Delta t,t)-I\over\Delta t}= {d\over dt} U(t,t^\prime)|_{t^\prime=t} \end{equation}
The operator $\hat{X}$ can be shown to be anti-hermitian. Hence with notation \( (i\hbar) X \equiv H(t)\), the operator \(H(t)\) will be hermitian. We therefore write \eqref{EQ08} as \begin{equation} i\hbar{d\over dt} \ket{\psi t} = \hat{H}(t)\ket{\psi t} \end{equation} where \begin{equation} \hat{H}(t) ={1\over i\hbar} \dd{t} U(t,t^\prime)|_{t^\prime=t} \end{equation}
Thus the time evolution of a quantum system is governed by the equation
\begin{equation}\label{EQ17} i\hbar {\partial\over\partial t}\ket{\psi t} = \hat{H}(t)\ket{\psi t} \end{equation} Using correspondence with classical mechanics, Dirac shows that the operator $\hat{H}$ represents the energy,  i.e. the Hamiltonian, of the system. Using \eqref{EQ03A} in \eqref{EQ17} we get
\begin{equation} i\hbar\pp{t} U(t,t_0)\ket{\psi t_0} = \hat{H}(t)U(t,t_0)\ket{\psi t_0} \end{equation}
This equation must hold for all vectors $|\psi t_0\rangle$. Hence the time evolution operator $U$ must satisfy the differential equation
\begin{equation} i\hbar \pp{t} U(t,t_0) = \hat{H}(t)U(t,t_0) \,\,\,. \end{equation} It may be remarked that that \eqref{EQ17} has a wide applicability to all quantum systems, and not just quantum mechanics of point particles, a fact  not emphasized in most text books.