Integral Representation of Hermitian Conjugation

Question

My Quantum Physics textbook asserts that, $$⟨u│u⟩=\int_{-∞}^{∞}\left|u\right|^{2}dx$$ Where the term, $$⟨u│u⟩$$ denotes the product of u and its Hermitian conjugate.

What I am confused about is where did this "representation" come from? And how can it be derived mathematically? I don't see how this integral "representation" makes any mathematical sense.

Answer 1

In quantum mechanics, the expression $\langle u | v \rangle$ is used to denote the inner-product of two vectors $u$ and $v$ (or, as a physicist might insist, $|u\rangle$ and $|v\rangle$). What exactly this inner product is depends on the context.

When $u,v$ are vectors $u = (u_1,\dots,u_n)$ and $v = (v_1,\dots v_n)$ over $\Bbb C$, their inner-product is defined by $$ \langle u|v\rangle = \sum_{k=1}^n u_k^* v_k. $$ When $u,v$ are functions $u,v: \Bbb R \to \Bbb C$, their inner-product is defined by $$ \langle u|v \rangle = \int_{-\infty}^\infty u^*(x) v(x)\,dx. $$ So, what exactly $\langle u|v \rangle$ means depends on how it is defined for the given context. In all cases though, this function has the properties that define an inner-product.

Answer 2

$⟨u│u⟩=\int_{-∞}^{∞}u(x) \overline{u(x)}dx$, hence $⟨u│u⟩=\int_{-∞}^{∞}\left|u(x)\right|^{2}dx$, since for a complex number $z$ we have $z \overline{z}=|z|^2.$