Proving Complex Wave Function: |ψ1 + ψ2|^2 in [0, 4α]

[jc09]

Show that for ψ1 , ψ2 ∈ C,
|ψ1 + ψ2 |^2
can take any value in [0, 4α], where
α = |ψ1 |^2 = |ψ2 |^ 2

I think the solution has something to do with triangular identies but I am not sure how to start this problem at all.

 

[quantumdude]

I think you can get away without using the triangle inequality. If $\psi_1$ and $\psi_2$ both have constant modulus $\sqrt{a}$ then the functions must be of the following form (assuming 1 spatial dimension and time independence).

$$\psi_1(x)=\sqrt{a}\exp\left(ik_1x+i\phi_1\right)$$

$$\psi_1(x)=\sqrt{a}\exp\left(ik_2x+i\phi_2\right)$$

Here $k_1,k_2,\phi_1,\phi_2\in\mathbb{R}$.

Can you do something with that?