I'm a Physics student. While studying some quantum mechanics, I came across a problem: I want to minimize a certain real expression that contains a complex variable. The issue is, I do not know how to proceed, as the expression contains both the magnitude of the variable as well as its real part. WLOG, assume the expression I am looking at is the following: $$ H(\alpha) = k \cdot \frac{p + q \,(\alpha \cdot \bar{\alpha})}{(\alpha + \bar{\alpha})^\frac{5}{2}}$$
where $\alpha$ is the variable of interest and $\bar{\alpha}$ is its complex conjugate. The rest of the variables in the expression are reals. Moreover, $H$ itself is real, and I want to minimize it.
I tried looking at the cases where $\alpha$ is purely real and where it is purely imaginary, but they don't work out: the first leads to a contradiction where $\alpha$ is simultaneously real and imaginary, whereas the second results in division by zero. I also attempted expressing $\alpha$ as $\alpha = r + \mathbb{i}s$ where $r, s \in \mathbb{R}$ and treating the problem as a minimization problem in two variables, but it reduces to the aforementioned contradiction.
How do I do this?
With the expression that you have, $$H(r,s)=k\frac{p+q(r^2+s^2)}{(2r)^{5/2}}$$ If $kq>0$ then to minimize the expression you must have $s=0$. Then the problem reduces to a 1D minimization problem, which has a solution only if $p$ and $q$ have opposite signs. $$\frac{d}{dr}k\frac{p+qr^2}{2^{5/2}r^{5/2}}=0\\\frac{d}{dr}\frac{p+qr^2}{r^{5/2}}=0\\-\frac52pr^{-7/2}-\frac12qr^{-3/2}=0$$ Multiply with $-2r^{-7/2}$: $$5p+qr^2=0$$