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Diffstat (limited to 'latex/know/concept/hilbert-space/source.md')
| -rw-r--r-- | latex/know/concept/hilbert-space/source.md | 13 |
1 files changed, 6 insertions, 7 deletions
diff --git a/latex/know/concept/hilbert-space/source.md b/latex/know/concept/hilbert-space/source.md index 7d2ea05..3f6ceb5 100644 --- a/latex/know/concept/hilbert-space/source.md +++ b/latex/know/concept/hilbert-space/source.md @@ -41,12 +41,11 @@ other. Otherwise, they would be **linearly dependent**. A vector space $\mathbb{V}$ has **dimension** $N$ if only up to $N$ of its vectors can be linearly indepedent. All other vectors in -$\mathbb{V}$ can then be written as a **linear combination** of these $N$ -so-called **basis vectors**. +$\mathbb{V}$ can then be written as a **linear combination** of these $N$ **basis vectors**. Let $\vu{e}_1, ..., \vu{e}_N$ be the basis vectors, then any vector $V$ in the same space can be **expanded** in the basis according to -the unique "weights" $v_n$, known as the **components** of the vector $V$ +the unique weights $v_n$, known as the **components** of $V$ in that basis: $$\begin{aligned} @@ -89,9 +88,9 @@ Two vectors $U$ and $V$ are **orthogonal** if their inner product $\braket{U}{V} = 0$. If in addition to being orthogonal, $|U| = 1$ and $|V| = 1$, then $U$ and $V$ are known as **orthonormal** vectors. -Orthonormality is a desirable property for basis vectors, so if they are +Orthonormality is desirable for basis vectors, so if they are not already orthonormal, it is common to manually derive a new -orthonormal basis from them using e.g. the Gram-Schmidt method. +orthonormal basis from them using e.g. the [Gram-Schmidt method](/know/concept/gram-schmidt-method). As for the implementation of the inner product, it is given by: @@ -157,7 +156,7 @@ The concept of orthonormality must be also weakened. A finite function $f(x)$ can be normalized as usual, but the basis vectors $x$ themselves cannot, since each represents an infinitesimal section of the real line. -The rationale in this case is that the identity operator $\hat{I}$ must +The rationale in this case is that action of the identity operator $\hat{I}$ must be preserved, which is given here in [Dirac notation](/know/concept/dirac-notation/): $$\begin{aligned} @@ -172,7 +171,7 @@ $$\begin{aligned} = \int_a^b \braket{x}{\xi} f(\xi) \dd{\xi} \end{aligned}$$ -For the latter integral to turn into $f(x)$, it is clear that +For the latter integral to turn into $f(x)$, it is plain to see that $\braket{x}{\xi}$ must be a [Dirac delta function](/know/concept/dirac-delta-function/), i.e $\braket{x}{\xi} = \delta(x - \xi)$: |