How does the addition of two wavefunctions develop in time?

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?:

$\mid \psi_1exp(\frac{-i E_1}{\hbar}*t)\mid^2 = \mid\psi_1^{(*)}\psi_1\mid = \mid\psi_1\mid^2* exp(\frac{-i E_1}{\hbar}t) * exp(\frac{i E_1}{\hbar}t) = \mid\psi_1\mid^2$

$\mid \psi_1exp(\frac{-i E_1}{\hbar}t) + \psi_2exp(\frac{-i E_2}{\hbar}t)\mid ^2 = \mid\psi_1\mid^2 + \mid\psi_2\mid^2 + 2 \mid\psi_1 \psi_2exp(\frac{-i (E_2-E_1)}{\hbar}t) \mid$

meaning that: $\mid\Psi _1(t)+\Psi _2(t)\mid$ ocilliates with $\frac{\hbar}{(E_2-E_2)}$ ?

• How can those wave functions be solutions of the time independent Schrodinger equation, if they do not depend on the spatial coordinates? Commented May 20, 2017 at 10:14
• Sorry I was just to lazy to write $\Psi(r,t)$, but you are right, it should have been said that the wavefunctions depended on the spartial coordinates. Any how it does not change the problem. Commented May 20, 2017 at 10:17
• I think what you wrote is right. What's the question? Commented May 20, 2017 at 10:43
• If what I wrote was correct, which seems to be the case, which answers my question. thx! Commented May 20, 2017 at 10:46
• Commented May 20, 2017 at 13:11

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 $E_2 - E_1$.
• The question was: was my calculation correct. Did I properly undersand the calculation behind the superposition squared. Your correction of the $\mid\Psi_1+\Psi_2\mid^2$ answered any doubt I had Commented May 20, 2017 at 11:55