Why can $|\Psi (t=0)\rangle $ be written as a coherent superposition of some eigenkets?

One of the approaches to solve time dependent Schrodinger equation $i\hbar \frac{\partial |\Psi(t)\rangle}{\partial t} = \hat{H} |\Psi(t)\rangle $ is to solve the time-independent part $\hat{H} |\Psi\rangle = E |\Psi\rangle $ and using the eigen vectors & eigen values, find the $ |\Psi(t)\rangle $. In this we make use of the time propagator $\mathcal{U}(t,t_o)$ that acts on $|\Psi (t=0)\rangle $.

So, when we write $|\Psi (t)\rangle = \mathcal{U}(t,t_o) \ |\Psi(0)\rangle $ whys is it that we can write the wavefunction at t=0 as coherent superposition of eigen states of the time-independent schrodinger equation.

  • $\begingroup$ the propagator $U(t,t_0)$ acts on $| \psi(t_0) \rangle$ and not $| \psi(0) \rangle$ $\endgroup$ – lakehal May 17 '19 at 13:00

If we use Hermetian operators then we are guaranteed that the eigenvectors of the operator can form a complete orthogonal basis.

  • $\begingroup$ But why does the time independent equation yield a complete orthogonal basis for t=0 case only $\endgroup$ – EverydayFoolish May 17 '19 at 3:24
  • $\begingroup$ @EverydayFoolish I'm not sure I understand. Are you asking why $t=0$ is used instead of $t=1$ for example? This is true for all times $\endgroup$ – Aaron Stevens May 17 '19 at 3:27
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    $\begingroup$ Ok, so at any time it's a superposition of eigen states but the amplitudes of each state vary with time giving rise to dynamics of the system. $\endgroup$ – EverydayFoolish May 17 '19 at 4:05
  • $\begingroup$ @EverydayFoolish Yes that's right $\endgroup$ – Aaron Stevens May 17 '19 at 4:07
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    $\begingroup$ @EverydayFoolish It's not the variation of the amplitudes with time, but the variations of the phases with time that produce the dynamics. $\endgroup$ – Buzz May 17 '19 at 4:25

The Schrodinger equation is a first order equation in time, hence we need an initial condition for the wave function $\psi(\textbf{r},t=0)$ to solve for later time. The wave function at time $t=0$ is a superposition of whatever you want since it is arbitrary, like the initial position and velocity in classical mechanics. However, it is more practical to express it in the basis of the eigenstates of the Hamiltonian.


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