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How to solve Schrödinger equation back in time to find past wavefunction from which present wavefunction has been evolved?

i.e.

Suppose, at present or at this moment I know $\psi_{present}(r)$. Since Schrödinger equation is fully deterministic, there must be a unique wavefunction $\psi_{past}(r)$ from which $\psi_{present}(r)$ has been evolved.

How to find $\psi_{past}(r)$ ?

Edit (after John Rennie's answer):

Are following steps correct then ?

Let us call $ (ih/2\pi) = k$, then Schrodinger eq. is

$\partial\psi(r,t=present)/\partial t = H_{op}\psi(r,t=present)/k$

$ H_{op}\psi(r,t=present)$ will be a function of $r$ only; $t$ is fixed i.e. present instantaneous time.

Let $ H_{op}\psi(r,t=present) = \phi(r)$

So we get, $d\psi/dt = \phi/k$; now it is not partial time derivative.

Integrating both sides, w.r.t. time, from i.e. present to past:

$ \psi = (1/k)\phi(r)\int_{present}^{past} dt$

i.e. $ \psi = (1/k)\phi(r) (t_{past}-t_{present})$

atom
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1 Answers1

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The Schrodinger equation tells you:

$$ i\hbar\frac{d\psi}{dt} = \mathcal H\psi $$

So simply calculate $\mathcal H\psi/i\hbar$ and integrate.

John Rennie
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