I was watching Allan Adams' lecture on energy eigenfunctions, and there's one part (around 43 minutes into the lecture) that confuses me.

Suppose we have the initial wave function $\Psi (x,0)$ such that $\hat{E}\,\Psi (x,0)=E \,\Psi (x,0)$ for some constant $E$. Then, plugging this into the Schrödinger equation, we'd get:

\begin{align}
i \hbar \frac{\partial}{\partial t} \Psi (x, 0) &= E \, \Psi (x,0) \\
 \frac{\partial}{\partial t} \Psi (x, 0) &= \frac{E}{i \hbar} \, \Psi (x,0) \tag{1}\\
\therefore \Psi (x, t) &= \exp\left({-i \frac{E\,t}{\hbar}}\right) \Psi(x,0) \tag{2}
\end{align}

I'm a bit confused about how to go from $(1)$ to $(2)$.

Now if we make the additional assumption that $\hat{E}\,\Psi (x,t)=E \,\Psi (x,t)$ for all $t$, then the Schrödinger equation becomes:
\begin{align}
 \frac{\partial}{\partial t} \Psi (x, t) &= \frac{E}{i \hbar} \, \Psi (x,t)
\end{align}

and I can solve this differential equation easily and get $(2)$. But from watching that part of the lecture, it seems we only need to assume a weaker statement - that the initial wave function is an energy eigenfunction. But then, it's not clear to me how I can get the solution $(2)$ from $(1)$. Am I missing something?

Update: Thanks for all the answers. After reading through the accompanying lecture note, we indeed need to assume that the energy operator is a constant over time. 

This is a tricky question because it asks about the meaning of words. People use the word "particle" to refer to various, not always well defined, notions in physics.

In the end, I think the simplest and more correct single way to categorize the terms is to interpret "particle" as "excitation of a field". For example, if someone says


  There are two electrons in this box"


I would mentally translate that to


  The electron field in this box has two units of excitation.


This is all much easier to think about if you're familiar with the so-called "second quantization".$^{[1]}$

Second quantization

Consider a one-dimensional infinite wall potential (i.e. "particle in a box"). The system has a set of discrete energy levels, which we can index as

$$\left\{ A, B, C, D, \ldots \right\}$$

If we have only one particle, we can denote its state as e.g. $|\Psi \rangle_1 = |B\rangle + |D\rangle$.$^{[2]}$ This is the so-called first quantization. If we have two particles, the situation is significantly more complex because, as you have probably learned, quantum particles are indistinguishable. You probably learned that you have to symmetrize (bosons) or antisymmetrize (fermions) the state vector to account for the fact that the particles are indistinguishable. For example, if you say that particle #1 is in state $|\Psi\rangle_1$ as written above, and particle #2 is in state $|\Psi\rangle_2=|C\rangle$, then the total system state is (assuming boson particles):

\begin{align}
\left \lvert \Phi \right \rangle
&= (|B\rangle_1 + |D\rangle_1)|C\rangle_2 + |C\rangle_1 (|B\rangle_2 + |D\rangle_2) \\
&= |B\rangle_1 |C\rangle_2 + |D\rangle_1 |C\rangle_2 + |C\rangle_1 |B\rangle_2 + |C\rangle_1 |D\rangle_2 \, .
\end{align}

This notation is horrible. In symmetrization/antisymmetrization you are basically saying:

"My notation contains information that it shouldn't, namely the independent states of particles which are actually indistinguishable, so let me add more terms to my notation to effectively remove the unwanted information."

This should seem really awkward and undesirable, and it is.

Let us consider an analogy for why the symmetrized state is such a bad representation. Consider a violin string with a set of vibrational modes. If we want to specify the state of the string, we enumerate the modes and specify the amplitude of each one, i.e. we write a Fourier series

$$\text{string displacement}(x) = \sum_{\text{mode }n=0}^{\infty}c_n \,\,\text{[shape of mode }n](x).$$

The vibrational modes are like the quantum eigenstates, and the amplitudes $c_n$ are like the number of particles in each state. With this analogy, the first quantization notation, in which we index over the particles and specify each one's state, is like indexing over units of amplitude and specifying each one's mode. That's obviously backwards. In particular, you now see why particles are indistinguishable. If a particle is just a unit of excitation of a quantum state, then just like units of amplitude of a vibrating string, it doesn't make any sense to say that the particle has identity. Units of excitation have no identity because they're just mathematical constructs to keep track of how excited a particular mode is.

A better way to specify a quantum state is to list each possible state and say how excited it is. In quantum mechanics, excitations come in discrete units $^{[3]}$, so we could specify a state like this:

$$|n_A\rangle_A |n_B\rangle_B |n_C\rangle_C |n_D\rangle_D$$

where $n_i$ is an integer. In this notation, the state $|\Psi\rangle_1$ from before is written

$$|\Psi\rangle_1 = |0\rangle_A |1\rangle_B |0\rangle_C |0\rangle_D +
|0\rangle_A |0\rangle_B |0\rangle_C |1\rangle_D.$$

For compactness this would often be written $|\Psi\rangle_1=|0100\rangle + |0001\rangle$. The more complex two particle state would be

$$\left \lvert \Phi \right \rangle = |0\rangle_A |1\rangle_B |1\rangle_C |0\rangle_D + |0\rangle_A |0\rangle_B |1\rangle_C |1\rangle_D$$

or, more compactly,

$$\left \lvert \Phi \right \rangle = |0110\rangle + |0011\rangle \, .$$

This is the so-called second quantization notation. Note that it has less terms than the first quantized version. This is because it doesn't need to undo information that it's not supposed to have.

Back to fields vs. particles

The second quantized notation is far better because it naturally accounts for the "indistinguishable" particles. But, what we really learned, is that particles are actually units of excitation of quantum states. In the field theory language, we'd say that the particle is a unit of excitation of the various modes of the field. I won't say that either fields or particles are more fundamental because one has little meaning without the other, but now that we understand what "particle" really means, the whole situation is hopefully much clearer to you.

P.S. I do hope you'll ask for clarification as needed.

[1] The term "second quantization" is stupid, so don't try to interpret it.

[2] We ignore normalization.

[3] Hence the term "quantum".
