Two time dependent wavefunctions:
$\Psi _1(t)= \psi_1*exp(\frac{-i * E_1}{\hbar}*t)$
$\Psi _2(t)= \psi_2*exp(\frac{-i * E_2}{\hbar}*t)$
Both a solution to the timeindependent (note "in") Schrödinger eq. with the same H. We know they are solutions. Furthermore $E_1$ and $E_2$ are different.
$\mid \psi_1exp(\frac{-i E_1}{\hbar}t) + \psi_2exp(\frac{-i E_2}{\hbar}t)\mid ^2 $
$= \mid \psi_1exp(\frac{-i E_1}{\hbar}*t)\mid^2 + \mid\psi_2exp(\frac{-i E_2}{\hbar}t)\mid^2 + 2 \mid\psi_1exp(\frac{-i E_1}{\hbar}t) \psi_2exp(\frac{-i E_2}{\hbar}t) \mid$
$= \mid \psi_1exp(\frac{-i E_1}{\hbar}*t)\mid^2 + \mid\psi_2exp(\frac{-i E_2}{\hbar}t)\mid^2 + 2 \mid\psi_1 \psi_2exp(\frac{-i (E_2-E_1)}{\hbar}t) \mid$
Is the following correct?:
$\mid \psi_1exp(\frac{-i E_1}{\hbar}*t)\mid^2 = \mid\psi_1^{(*)}\psi_1\mid = \mid\psi_1\mid^2* exp(\frac{-i E_1}{\hbar}t) * exp(\frac{i E_1}{\hbar}t) = \mid\psi_1\mid^2$
leading to:
$\mid \psi_1exp(\frac{-i E_1}{\hbar}t) + \psi_2exp(\frac{-i E_2}{\hbar}t)\mid ^2 = \mid\psi_1\mid^2 + \mid\psi_2\mid^2 + 2 \mid\psi_1 \psi_2exp(\frac{-i (E_2-E_1)}{\hbar}t) \mid$
meaning that:
$\mid\Psi _1(t)+\Psi _2(t)\mid $ ocilliates with $\frac{\hbar}{(E_2-E_2)}$ ?
Best Answer
You obviously mean that $$\begin{aligned} H\psi_1 &= E_1\psi_1,\\ H\psi_2 &= E_2\psi_2. \end{aligned}$$ Then, the solution of the Schrödinger equation (I will use units such that $\hbar=1$ throughout: old habit of a former theoretical physicist!!) $$H\Psi_k = i\dfrac{\partial \Psi_k}{\partial t}$$ with the initial condition $$\Psi_k(t=0) = \psi_k$$ is indeed your $$\Psi_k = \psi_k \exp(-iE_k t).$$ Then you consider a superposition of these two states, $\Psi = \Psi_1 + \Psi_2$, which is also solution of the Schrödinger equation for the initial condition $$\Psi(t=0) = \psi_1 + \psi_2,$$ i.e. a superposition of a state of energy $E_1$ and a state of energy $E_2$.
Your calculation of $|\Psi_1 + \Psi_2|^2$ is a wee bit incorrect: $$|\Psi_1 + \Psi_2|^2 = |\Psi_1|^2 + |\Psi_2|^2 + \underbrace{\Psi_1 {\Psi_2}^{\!*} + {\Psi_1}^{\!*}\Psi_2}_{2\Re\Psi_1{\Psi_2}^{\!*}}$$ where $\Re$ denotes the real part. That is to say $$|\Psi_1 + \Psi_2|^2 = |\psi_1|^2 + |\psi_2|^2+2\Re \psi_1{\psi_2}^{\!*}\exp[ i(E_2-E_1)t]$$ This does not change your conclusion though: oscillations with a pulsation $E_2 - E_1$.
Right, now that this is all clear and correct, could you elaborate on your question? I can't guess what you want to know!