Superselection Rule Results from Schrodinger Equation

[StevieTNZ]

From what I've gathered, certain physical states cannot exist in superposition. When applying the Schrödinger 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?

 

[strangerep]

From what I've gathered, certain physical states cannot exist in superposition. When applying the Schrödinger 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?

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

 

[Demystifier]

We already have a similar thread:
https://www.physicsforums.com/showthread.php?t=548541

 

[StevieTNZ]

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

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

We already have a similar thread:
https://www.physicsforums.com/showthread.php?t=548541

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

 

[strangerep]

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

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

The (time-dependent) Schrödinger 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?)

 

[StevieTNZ]

The (time-dependent) Schrödinger 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?)

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

 

[strangerep]

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

By definition, if two states $\psi_1,\psi_2$ belong to different superselection sectors, then
$$\def\<{\langle}<br /> \def\>{\rangle}<br /> \<\psi_1|A|\psi_2\> = 0$$
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.