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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?

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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.

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