# Wave function collapse and a general wave function

Action of an operator on a state vector collapses the wave function to any of the eigenstate of that operator , So we get resulting state of the system as some base ket. But mathematically action of operator on a state vector change state vector to another state vector which is the superposition of all basis eigenkets, So how can we incorporate collapse of wave function and action of operator?

## 2 Answers

Not all operators transform a ket into a superposition of kets: for instance $$\hat \Pi=\vert\psi\rangle\langle\psi\vert$$ acting on a state $$\vert \Psi\rangle$$ will produce $$\vert\psi\rangle\langle\psi\vert\Psi\rangle$$, which is not a normalized ket and not a linear combination of kets either.

An operator like $$\hat \Pi$$ is a projection operator and is idempotent, i.e. $$\hat\Pi\hat \Pi=\hat \Pi$$; projection operators serve as basic prototypes for measurement operators.

You have a misunderstanding here. The action of an observable operator like $$x$$ or $$p= -i\frac{\partial}{\partial x}$$ does not change the wave function into a collapsed state. The operator which does that is a projector, like:

$$|x=1\rangle \langle x=1|$$

or
$$|x=-12.531\rangle \langle x=-12.531|$$

or

$$|p=0\rangle \langle p=0|$$

Which projector you apply depends on the measurement result and what you are measuring. The ones above apply to the results $$x=1$$, $$x=-12.531$$, and $$p=0$$, respectively (I left out units since they're not important to the point).

If you want to calculate these projectors: they are outer products of the eigenvectors of the corresponding operators. For example, $$\hat{x}|x=-12.531\rangle = -12.531 |x=-12.531\rangle$$

The projectors above will not leave you with a superposition (at least not in the x, x, and p basis, respectively). In general, whether a vector is in a superposition depends on the basis you express it in.