I am currently studying the textbook Mathematical methods of quantum optics by Ravinder R. Puri. When discussing the postulates of quantum mechanics, the author introduces the quantum-mechanical version of the Schwarz inequality as follows:
An important consequence of the axioms defining the scalar product is the Schwarz inequality
$$\langle \phi \mid \phi \rangle \langle \psi \mid \psi \rangle \ge \langle \phi \mid \psi \rangle \langle \psi \mid \phi \rangle, \tag{1.5}$$
where the equality holds if and only if the two vectors in question are linearly dependent i.e. if
$$\mid \psi \rangle = \mu \mid \phi \rangle, \tag{1.6}$$
$\mu$ being a complex number. In order to establish this, show that the minimum value of $\langle \Psi(\mu) \mid \Psi(\mu) \rangle$, where $\mid \Psi \rangle \ = \ \mid \psi \rangle – \mu \mid \phi \rangle$, as a function of $\mu$ is $\langle \psi \mid \psi \rangle – |\langle \psi \mid \phi \rangle |^2 / \langle \phi \mid \phi \rangle$. The requirement that this value, due to axiom 3 of the scalar product, be positive leads to the Schwarz inequality in (1.5). Also, according to the axiom 4 above, $\langle \Psi(\mu) \mid \Psi(\mu) \rangle = 0$ iff $\mid \Psi(\mu) \rangle – 0$ i.e. iff (1.6) holds. it may be verified easily that (1.5) then holds with equality. In a similar way we can derive the generalized Schwarz inequality
$$\det(\langle \psi_\mu \mid \psi_\nu \rangle ) \ge 0, \tag{1.7}$$
where $\det(\langle \psi_\mu \mid \psi_\nu \rangle )$ is the determinant of the matrix constituted by the elements $\det\langle \psi_\mu \mid \psi_\nu \rangle$, $\mu, \nu = 1, \dots, n$. Invoking the fact that the determinant of a matrix is zero if its rows (or columns) are linearly dependent, it follows that the equality in (1.7) holds iff $\mid \psi_\mu \rangle$ are linearly dependent.
Axiom 3 is as follows:
$$\langle \psi \mid \psi \rangle > 0$$
Axiom 4 is as follows:
$$\langle \psi \mid \psi \rangle = 0 \ \text{if and only if $\mid \psi \rangle = 0$}$$
My goal is to prove this case of the Schwarz inequality for myself.
In researching this problem, I found this webpage. The author claims that
$$| (\psi, \phi) |^2 \le (\psi, \psi)(\phi, \phi).$$
Trying to connect this with (1.5), I get
$$\langle \phi \mid \phi \rangle \langle \psi \mid \psi \rangle \ge \langle \phi \mid \psi \rangle \langle \psi \mid \phi \rangle = \langle \psi \mid \phi \rangle^* \langle \psi \mid \phi \rangle \ge 0,$$
where $*$ is the complex conjugate.
So then I wonder: Is it true that $\langle \psi \mid \phi \rangle^* \langle \psi \mid \phi \rangle \ge 0$? It isn't clear to me that this is true. So my objective now is to prove that this is true. To simplify things, I will first try to prove that it is true for the 1-dimensional case:
$\psi$ and $\phi$ are complex numbers, right? So let $\psi = x_1 + i y_1$ and $\phi = x_2 + i y_2$.
$$\begin{align} \langle \psi \mid \phi \rangle &= (x_1 – i y_1) \cdot (x_2 + i y_2) \\ &= x_1 x_2 – i x_1 y_2 – i x_2 y_1 + y_1 y_2 \\ &= (x_1 x_2 + y_1 y_2) – i(x_1 y_2 + x_2 y_1) \end{align}$$
$$\langle \psi \mid \phi \rangle^* = (x_1 x_2 + y_1 y_2) + i (x_1 y_2 + x_2 y_1)$$
$$\begin{align} \langle \psi \mid \phi \rangle^* \langle \psi \mid \phi \rangle &= [(x_1 x_2 + y_1 y_2) + i (x_1 y_2 + x_2 y_1)] \cdot [(x_1 x_2 + y_1 y_2) – i(x_1 y_2 + x_2 y_1)] \\ &= (x_1 x_2 + y_1 y_2) (x_1 x_2 + y_1 y_2) + i (x_1 x_2 + y_1 y_2)(x_1 y_2 + x_2 y_1) – i (x_1 y_2 + x_2 y_1)(x_1 x_2 + y_1 y_2) – (x_2 y_1 + y_1 y_2)(x_1 y_2 + x_2 y_1) \\ &= (x_1^2 x_2^2 + 2 x_1 x_2 y_1 y_2 + y_1^2 y_2^2) + i (x_1^2 x_2 y_2 + x_1 x_2^2 y_1 + x_1 y_1 y_2^2 + x_2 y_2 y_1^2) – i (x_1^2 x_2 y_2 + x_1 y_1 y_2^2 + x_2^2 x_1 y_1 + x_2 y_1^2 y_2) – (x_1 x_2 y_1 y_2 + x_2^2 y_1^2 + y_2^2 y_1 x_1 + y_1^2 y_2 x_2) \\ &= (x_1^2 x_2^2 + 2 x_1 x_2 y_1 y_2 + y_1^2 y_2^2) – (x_1 x_2 y_1 y_2 + x_2^2 y_1^2 + y_2^2 y_1 x_1 + y_1^2 y_2 x_2) \end{align}$$
I'm unsure of how to proceed from here. Have I made an error somewhere? Am I going about this incorrectly?
I would greatly appreciate it if people would please take the time to review my this.
Best Answer
No, $|\phi\rangle$ and $|\psi\rangle$ are not complex numbers. They're element in a Hilbert space and $\langle \phi |\psi\rangle$ is their inner product. So it's $\langle \phi |\psi\rangle$ which is a complex number. And then $$\langle \phi |\psi\rangle^\star\langle \phi |\psi\rangle=|\langle \phi |\psi\rangle|^2$$ is the norm of a complex number so it's a real non negative value.