(Algebra) Quantum Theory - Cauchy-Schwartz inequality proof

[FatPhysicsBoy]

Homework Statement

Given two arbitrary vectors $|\phi_{1}\rangle$ and $|\phi_{2}\rangle$ belonging to the inner product space $\mathcal{H}$, the Cauchy-Schwartz inequality states that:

$|\langle\phi_{1}|\phi_{2}\rangle|^{2} \leq \langle\phi_{1}|\phi_{1}\rangle \langle\phi_{2}|\phi_{2}\rangle$.

Consider $|\Psi\rangle = |\phi_{1}\rangle + \lambda|\phi_{2}\rangle$

where $\lambda$ is a complex number that can be written as $\lambda = a + ib$.

a) Write an expression for $\langle\Psi|\Psi\rangle \geq 0$ as a function of $\lambda$ then rewrite as a function of a and b ($f(a,b)$).

b) Show that the value of $\lambda$ that minimises $\langle\Psi|\Psi\rangle$ is:

$\lambda_{min} = -\frac{\langle\phi_{2}|\phi_{1}\rangle}{\langle\phi_{2}|\phi_{2}\rangle}$.

Hint: Compute the derivatives of $f(a,b)$ wrt a and b, solve these to get $a_{min}$ and $b_{min}$ and then compute $\lambda_{min}$.

Homework Equations

N/A

The Attempt at a Solution

I get $\langle\Psi|\Psi\rangle = \langle\phi_{1}|\phi_{1}\rangle + a(\langle\phi_{1}|\phi_{2}\rangle + \langle\phi_{2}|\phi_{1}\rangle) + ib(\langle\phi_{1}|\phi_{2}\rangle - \langle\phi_{2}\phi_{1}\rangle) + (a^{2} + b^{2})\langle\phi_{2}|\phi_{2}\rangle = f(a,b)$. However I can only show:

$\lambda_{min} = -\frac{\langle\phi_{1}|\phi_{2}\rangle}{\langle\phi_{2}|\phi_{2}\rangle}$,

by combining Re and I am parts in $f(a,b)$ as follows (and finding the relevant derivatives etc.):

$\langle\Psi|\Psi\rangle = \langle\phi_{1}|\phi_{1}\rangle + 2a\textrm{Re}(\langle\phi_{1}|\phi_{2}\rangle) + 2b\textrm{Im}(\langle\phi_{1}|\phi_{2}\rangle) + (a^{2} + b^{2})\langle\phi_{2}|\phi_{2}\rangle$.

This is the only way I understand how to do it, however, in the solutions for this problem the collection of Re and I am parts is done as follows which I don't understand (in particular the imaginary part):

$\langle\Psi|\Psi\rangle = \langle\phi_{1}|\phi_{1}\rangle + 2a\textrm{Re}(\langle\phi_{2}|\phi_{1}\rangle) + 2b\textrm{Im}(\langle\phi_{2}|\phi_{1}\rangle) + (a^{2} + b^{2})\langle\phi_{2}|\phi_{2}\rangle$

Thank you

 

[FatPhysicsBoy]

Anyone? Parts 1) and 2) can pretty much be ignored they just provide context for the problem.. I think ultimately it's just a complex number/conjugation question which I haven't understood properly.