Momentum Eigenfunction Addition

[Safinaz]

Homework Statement

$\psi_1$ and $\psi_2$ are momentum eigenfunctions corresponding to
different momentum eigenvalues $p_1 \not= p_2$. Is $\psi_1$ + $\psi_2$ also momentum eigenfunction ?

Homework Equations

Is the right answer[/B]

Yes
No
It Depends ?

The Attempt at a Solution

I think yes, because

$$ \frac{h}{i} \frac{d}{dx} \psi_1 = p_1 \psi_1, $$
$$ \frac{h}{i} \frac{d}{dx} \psi_2 = p_2 \psi_2, $$
Then
$$ \frac{h}{i} \frac{d}{dx} (\psi_1+\psi_2) = p_1+p_2 (\psi_1+\psi_2), $$

is valid also

 

[blue_leaf77]

Think again how you should calculate
$$ \frac{h}{i} \frac{d}{dx} (\psi_1+\psi_2)$$
from
$$ \frac{h}{i} \frac{d}{dx} \psi_1 = p_1 \psi_1, $$
$$ \frac{h}{i} \frac{d}{dx} \psi_2 = p_2 \psi_2, $$

 

[Safinaz]

Ya ..

$$ \frac{h}{i} \frac{d}{dx} (\psi_1+\psi_2) = (p_1 \psi_1+ p_2 \psi_2), $$
so $(\psi_1+\psi_2)$ is not an Eigenfunction .

 

[blue_leaf77]

No, it's not.