# Can we rewrite Schrodinger equation using observable variable?

1. Aug 18, 2013

### phdphysics

We know that in Schrodinger equation, Ψ is called wave function, which is not observable, while Ψ·Ψ* is the probability, which is observable.
can we rewirte the Schrodinger equation to a form without Ψ but only Ψ·Ψ*?

because I think, in this way can I figure out all conservations in the equation. Although I can make it with present Schrodinger equation, it's obvious that the Schrodinger equation will change if I make t→-t transformation.

Thanks~

2. Aug 18, 2013

### vanhees71

Write down the equation of motion for $\psi^*$ given that $\psi$ fulfills the usual Schrödinger equation. Then change $t \rightarrow -t$!

3. Aug 18, 2013

### phdphysics

you mean, this equation set(containing two equation, Ψ and Ψ*) does not change?
by the way, could you tell me, can Schrodinger equation be rewrited to Ψ·Ψ* mathemetically?
thanks

4. Aug 18, 2013

### tom.stoer

No, we can't.

Define

$\psi = R \, e^{iS}$

with two real variables R and S.

Then introduce

$\rho = \psi^\ast \psi = R^2$

Now we see that an equation in R or ρ is an equation in one single real variable, whereas the original equation was an equation in two independent real variables R and S.

Last edited: Aug 18, 2013
5. Aug 18, 2013

That's essentially the idea behind density functional theory (DFT).

6. Aug 18, 2013

### tom.stoer

But that's an approximation.

7. Aug 18, 2013

If you mean that the DF theory itself is an approximation - no, it's exact. Practical implementations are approximations, however.

8. Aug 19, 2013

### tom.stoer

But DFT works only for the ground state, whereas the SG works for all states including all bound and scattering states.

9. Aug 19, 2013