In Quantum Mechanics is it possible to apply time evolution operator to wavefunction? If I consider a wavefunction that is the superposition of Hamiltonian eigenfunctions, for example like: $$\psi(x)=\frac{1}{\sqrt{2}}\psi_1(x)+\frac{1}{\sqrt{2}}\psi_2(x)$$ with $\hat{H}\psi_1(x)=E_1\psi_1(x)$ and $\hat{H}\psi_2(x)=E_2\psi_2(x)$ and I want to know the time evolution of the state $\psi(x)$, can I apply the time evolution op. $e^{-i\frac{\hat{H}}{\hbar}t}$ directly to $\psi(x)$ so that $\psi(x,t)=\frac{1}{\sqrt{2}}e^{-i\frac{E_1}{\hbar}t}\psi_1(x)+\frac{1}{\sqrt{2}}e^{-i\frac{E_2}{\hbar}t}\psi_2(x)$?
I have this doubt 'cause usually I apply the time evolution op. only on the kets. In the case of eigenfunctions instead of eigenkets what I need to do is "apply" the Hamiltonian in Schroedinger representation to $\psi_1(x)$ and $\psi_2(x)$.
Is this right?
 A: Using kets, because the evolution operator is linear, we can write :
$$|\psi(t) \rangle = \frac{1}{\sqrt 2}\left(e^{-iHt/\hbar} |\psi_1\rangle + e^{-iHt/\hbar}|\psi_2\rangle\right) = \frac{1}{\sqrt{2}} \left(e^{-iE_1t/\hbar} |\psi_1\rangle + e^{-iE_2t/\hbar}|\psi_2\rangle\right) $$
Taking the product of this equation with $\langle x|$, we get (usingsince $\psi(x,t) = \langle x|\psi(t)\rangle$ and linearity) :
$$\psi(x,t) = \frac{1}{\sqrt{2}}\left(e^{-iE_1t/\hbar} \psi_1(x)+ e^{-iE_2t/\hbar}\psi_2(x)\right)$$
A: This is correct. You can easily derive it from bra-ket notation by projection unto a position state, let
$$
|\psi(t_0) \rangle = c_1|\psi_1\rangle + c_2 |\psi_2\rangle 
$$
We have $\hat U(t)|\psi_n\rangle = \exp \left (-\frac{i}{\hbar }E_nt \right )|\psi_n\rangle$,  so
$$\begin{aligned}
\hat U(t) |\psi(t_0) \rangle &= \hat U(t)(c_1|\psi_1\rangle + c_2 |\psi_2\rangle )\\
&= c_1\hat U(t)|\psi_1\rangle + c_2 \hat U(t)|\psi_2\rangle \\
&= c_1 \exp \left (-\frac{i}{\hbar }E_1t \right )|\psi_1\rangle + c_2 \exp \left (-\frac{i}{\hbar }E_2t \right )|\psi_2\rangle \\
\end{aligned}$$
Now project unto the position basis,
$$\begin{aligned}
\langle x|\hat U(t) |\psi(t_0) \rangle & = c_1 \exp \left (-\frac{i}{\hbar }E_1t \right )\langle x|\psi_1\rangle + c_2 \exp \left (-\frac{i}{\hbar }E_2t \right )\langle x|\psi_2\rangle \\
\langle x|\psi(t)\rangle = \psi(x,t) &= c_1 \exp \left (-\frac{i}{\hbar }E_1t \right )\psi_2(x) + c_2 \exp \left (-\frac{i}{\hbar }E_2t \right ) \psi_2(x) \\
\end{aligned}$$
Answer to question in comments:
Lets look at the case where the ket is not an eigenstate of the Hamilton operator, we have
$$
\hat U(t)|\psi\rangle = \exp(-\frac{i}{\hbar}\hat H t)|\psi\rangle 
$$
You can derive the expression in position basis by insertion of a resolution of the identity $\hat 1 = \int dx'|x'\rangle \langle x'| $ and projection,
$$
\langle x|\hat U \hat 1|\psi\rangle =\int dx' \langle x|\hat U|x'\rangle \langle x'|\psi\rangle
$$
we can simplify the expression once by inserting the matrix elements of the time evolution operator $\langle x|\hat U|x'\rangle $ with respect to the position basis. The matrix elements are
$$
\langle x|\hat U|x'\rangle = \exp\left(-\frac{i}{\hbar}H(x)t\right)\delta(x-x')
$$
with $H(x)= \frac{-\hbar^2}{2m}\nabla^2_x +V(x)$. Inserting this into our former expression yields
$$\begin{aligned}
\int dx' \langle x|\hat U|x'\rangle \langle x'|\psi\rangle &= \int dx'  \exp\left(-\frac{i}{\hbar}H(x)t\right)\delta(x-x')\psi(x') \\
\langle x |\hat U(t)|\psi\rangle&=\exp\left(-\frac{i}{\hbar}H(x)t\right)\psi(x)
\end{aligned}$$
Everything is still linear, but you cannot replace the Hamiltonian with the energy if your states aren't eigenfunctions. So a sum of states would look like this,
$$
\psi(x, t) = c_1\exp\left(-\frac{i}{\hbar}H(x)t\right)\psi_1(x) + c_2\exp\left(-\frac{i}{\hbar}H(x)t\right)\psi_2(x) 
$$

Typically you can replace your propagator by $\exp\left(-\frac{i}{\hbar}H(x)t\right)$ in expressions like $\langle x |\hat U|\psi\rangle$ as long as your Hamilton operator is diagonal with respect to the position basis and time independent.
