# Superselection Rules

1. Jan 3, 2012

### StevieTNZ

From what I've gathered, certain physical states cannot exist in superposition. When applying the Schrodinger equation to a quantum system, do we get the relevant superselection rule results, or will it produce an answer giving two physical states in superposition, when according to the superselection rule cannot occur?

2. Jan 4, 2012

### strangerep

It wouldn't be much use to have a superselection rule which applies today, but doesn't apply tomorrow...

3. Jan 4, 2012

4. Jan 4, 2012

### StevieTNZ

I don't know what that response has to do with my question.....

I don't see any answer to my question in that thread. Unless I've overlooked it?

5. Jan 4, 2012

### strangerep

The (time-dependent) Schrodinger equation expresses how a wave function evolves continuously in time.

(If that's still not a relevant answer for you, perhaps you should elaborate your question in more detail?)

6. Jan 4, 2012

### StevieTNZ

But does the equation take into account superselection rules? Or will it produce superpositions of states that can't occur due to superselection rules?

7. Jan 5, 2012

### strangerep

By definition, if two states $\psi_1,\psi_2$ belong to different superselection sectors, then
$$\def\<{\langle} \def\>{\rangle} \<\psi_1|A|\psi_2\> = 0$$
for all observables $A$. This can be extended to unitaries $U = e^{iA}$ by expanding the exponential.

Now consider $U(t) = \exp(iHt)$ and consider $\psi_2$ at a specific time, say $t=0$. Then we have
$$0 ~=~ \<\psi_1| U(t) |\psi_2(0)\> ~=~ \<\psi_1|\psi_2(t)\>$$
which shows that $\psi_2(t)$ is still orthogonal to $\psi_1$, no matter what superposition of states $\psi_2(0)$ evolves into during the time interval t.