# Unitary time evolution operator for hamiltonian

I have homework, which I've seen solution with which I have some problem.

The homework is to find time evolution of state $$|\psi(t)\rangle$$, if the hamiltonian of the state is $$\hat{\textbf{H}}=\epsilon \hat{\textbf{P}}$$, where $$\hat{\textbf{P}}$$ is projection operator and $$\epsilon$$ some energy dim constant.

So, we can begin with equation $$|\psi(t)\rangle = \hat{\textbf{U}}|\psi(0)\rangle,$$ where $$\hat{\textbf{U}}=\mathrm{e}^{-it\hat{\textbf{H}}}$$.

Then we can write $$\hat{\textbf{U}}|\psi(0)\rangle = \mathrm{e}^{-it\hat{\textbf{H}}} |\psi(0)\rangle = \mathrm{e}^{-it\epsilon\hat{\textbf{P}}} |\psi(0)\rangle = \left(\sum_{n=0}^\infty \frac{1}{n!}(it\epsilon\hat{\textbf{P}})^n\right)|\psi(0)\rangle = \left[ \hat{1} + \sum_{n=1}^\infty \frac{1}{n!}(it\epsilon)^n \hat{\textbf{P}} \right] |\psi(0)\rangle = \left[\hat{1}+ \left(\sum_{n=1}^\infty \frac{1}{n!}(it\epsilon)^n + \hat{1} - \hat{1} \right) \hat{\textbf{P}}\right]|\psi(0)\rangle = \left[\hat{1}+ \left(\sum_{n=0}^\infty \frac{1}{n!}(it\epsilon)^n - \hat{1} \right) \hat{\textbf{P}}\right]|\psi(0)\rangle = \left[\hat{1}- \hat{\textbf{P}} + \sum_{n=0}^\infty \frac{1}{n!}(it\epsilon)^n \hat{\textbf{P}}\right]|\psi(0)\rangle = \left[\hat{\textbf{Q}} + \sum_{n=0}^\infty \frac{1}{n!}(it\epsilon)^n \hat{\textbf{P}}\right]|\psi(0)\rangle = \left[\hat{\textbf{Q}} + \hat{\textbf{P}} \mathrm{e}^{i\epsilon t} \right]|\psi(0)\rangle.$$

And ok, I understand i mathematically, but I don't get why we can't end this lot earlier like here $$\hat{\textbf{U}}|\psi(0)\rangle = \mathrm{e}^{-it\hat{\textbf{H}}} |\psi(0)\rangle = \mathrm{e}^{-it\epsilon\hat{\textbf{P}}} |\psi(0)\rangle = \left(\sum_{n=0}^\infty \frac{1}{n!}(it\epsilon\hat{\textbf{P}})^n\right)|\psi(0)\rangle = \left(\sum_{n=0}^\infty \frac{1}{n!}(it\epsilon)^n\hat{\textbf{P}}\right)|\psi(0)\rangle = \hat{\textbf{P}}\mathrm{e}^{i\epsilon t} |\psi(0)\rangle.$$

Is this something with index? I have to first rewrite sum to get rid of $$\hat{\textbf{P}} = \hat{1}$$ and then pull out $$\hat{\textbf{P}}$$ from sum and then minus unitary operator?

Note that $$\hat{\mathbf{P}}^n = \hat{\mathbf{P}}$$ only holds if $$n\neq 0$$, so your manipulation is incorrect. You need to extract the $$n=0$$ case before you replace $$\hat{\mathbf P}^n$$ with $$\hat{\mathbf P}$$, which is why the additional steps in the correct solution are required.