# Is the collapsed wavefunction a solution of Time-dependent Schrodinger equation?

For measurement of any observable associated with the particle, should the wavefunction after collapse be a solution of the time-dependent Schrodinger equation?

A general solution of the time dependent schrodinger equation is given by superposition of eigenfunctions of the hamiltonian operator. Here by eigenfunctions, I mean the complete wavefunction as a function of spatial and time co - ordinates.

But since the wavefunction collapses to an eigenfunction of the operator of an obsevable when a measurement of the observable associated with the particle is made, What I want to ask is the following:

Should the collapsed wavefunction which is an eigenfunction of that operator be general solution of the Time - dependent schrodinger equation as well?

• The wavefunction is always a solution of the Schrödinger equation, but not always an eigenfunction of the Schrödinger equiation. Perhaps you could clarify exactly what you are asking and explain why specifically you're considering the time dependent Schrödinger equation. Nov 18, 2015 at 10:42
• For example, when the momentum of the particle in some potential energy distribution is measured, the wavefunction collapses to a wavefunction of the oscillatory kind A e ^ (ikx - wt). However this collpsed wavefunction is not a solution of the time - dependent schrodinger equation for this potential energy distribution. So is this collapse wavefunction allowed?? Nov 18, 2015 at 10:50
• ...is not what? Nov 18, 2015 at 10:53
• The collapsed state will be a superposition of the energy eigenfunctions and therefore a solution of the Schrodinger equation. Nov 18, 2015 at 11:19
• Can anyone precisely proof that any wavefunction (a function of both spatial and time co - ordinates) can be expressed as a superposition of energy eigenfunctions. If not possible, then are such collapsed wavefunctions allowed?? Nov 18, 2015 at 11:30

In the simple case of an instantaneous measurement (it never is), you would see a sharp discontinuity at the time of measurement $t^*$ for the partial wavefunction $\psi(t)$ of the measured system: for any $0\leq t<t^*$, the map is a (continuous in time) solution of the unperturbed Schrödinger equation with initial datum $\psi(0)$, and for any $t\geq t^*$ it is a (continuous in time) solution of the unperturbed Schrödinger equation with initial condition $\psi(t^*)$ (the wavefunction after measurement, that is in general different from $\lim_{t\uparrow t^* }\psi(t)$, since the measurement process perturbs instantaneously the system).