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diff --git a/source/know/concept/hilbert-space/index.md b/source/know/concept/hilbert-space/index.md index 42b9cb1..2a60896 100644 --- a/source/know/concept/hilbert-space/index.md +++ b/source/know/concept/hilbert-space/index.md @@ -18,22 +18,32 @@ is an abstract **vector space** with a notion of length and angle. An abstract **vector space** $$\mathbb{V}$$ is a generalization of the traditional concept of vectors as "arrows". It consists of a set of objects called **vectors** -which support the following (familiar) operations: +that support the following (familiar) operations: -+ **Vector addition**: the sum of two vectors $$V$$ and $$W$$, denoted by $$V + W$$. -+ **Scalar multiplication**: product of a vector $$V$$ with a scalar $$a$$, denoted by $$a V$$. ++ **Vector addition**: + the sum of two vectors $$V$$ and $$W$$, denoted by $$V + W$$. ++ **Scalar multiplication**: + product of a vector $$V$$ with a scalar $$a$$, denoted by $$a V$$. -In addition, for a given $$\mathbb{V}$$ to qualify as a proper vector -space, these operations must obey the following axioms: +In addition, for a given $$\mathbb{V}$$ to qualify as a proper vector space, +these operations must have the following (again familiar) properties: -+ **Addition is associative**: $$U + (V + W) = (U + V) + W$$ -+ **Addition is commutative**: $$U + V = V + U$$ -+ **Addition has an identity**: there exists a $$\mathbf{0}$$ such that $$V + 0 = V$$ -+ **Addition has an inverse**: for every $$V$$ there exists $$-V$$ so that $$V + (-V) = 0$$ -+ **Multiplication is associative**: $$a (b V) = (a b) V$$ -+ **Multiplication has an identity**: There exists a $$1$$ such that $$1 V = V$$ -+ **Multiplication is distributive over scalars**: $$(a + b)V = aV + bV$$ -+ **Multiplication is distributive over vectors**: $$a (U + V) = a U + a V$$ ++ **Addition is associative**: + $$U + (V + W) = (U + V) + W$$ ++ **Addition is commutative**: + $$U + V = V + U$$ ++ **Addition has an identity**: + there exists a $$\mathbf{0}$$ such that $$V + 0 = V$$ ++ **Addition has an inverse**: + for every $$V$$ there exists $$-V$$ so that $$V + (-V) = 0$$ ++ **Multiplication is associative**: + $$a (b V) = (a b) V$$ ++ **Multiplication has an identity**: + There exists a $$1$$ such that $$1 V = V$$ ++ **Multiplication is distributive over scalars**: + $$(a + b)V = aV + bV$$ ++ **Multiplication is distributive over vectors**: + $$a (U + V) = a U + a V$$ A set of $$N$$ vectors $$V_1, V_2, ..., V_N$$ is **linearly independent** if the only way to satisfy the following relation @@ -46,25 +56,28 @@ $$\begin{aligned} In other words, these vectors cannot be expressed in terms of each other. Otherwise, they would be **linearly dependent**. -A vector space $$\mathbb{V}$$ has **dimension** $$N$$ -if only up to $$N$$ of its vectors can be linearly indepedent. +$$\mathbb{V}$$ has **dimension** $$N$$ +if only up to $$N$$ of its vectors can be linearly independent. All other vectors in $$\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$$, -known as the **components** of $$V$$ in that basis: +Let $$\vu{e}_1, ..., \vu{e}_N$$ be a (generally not unique) +valid set of basis vectors of $$\mathbb{V}$$, +then any vector $$V$$ in that space can be **expanded** +in that basis according to unique weights $$v_n$$, +called the **components** of $$V$$ in that basis: $$\begin{aligned} V = \sum_{n = 1}^N v_n \vu{e}_n \end{aligned}$$ -Using these, the vector space operations can then be implemented as follows: +Using these components, +the operations of vector addition and scalar multiplication +can then be implemented as follows: $$\begin{gathered} V = \sum_{n = 1} v_n \vu{e}_n - \quad + \qquad W = \sum_{n = 1} w_n \vu{e}_n \\ \quad \implies \quad @@ -73,18 +86,24 @@ $$\begin{gathered} a V = \sum_{n = 1}^N a v_n \vu{e}_n \end{gathered}$$ +It is straightforward to see that this implementation satisfies the properties above. + ## Inner product -A given vector space $$\mathbb{V}$$ can be promoted to a **Hilbert space** or **inner product space** +A given vector space $$\mathbb{V}$$ can be promoted +to a **Hilbert space** or **inner product space** if it supports an operation $$\Inprod{U}{V}$$ called the **inner product**, which takes two vectors and returns a scalar, and has the following properties: -+ **Skew symmetry**: $$\Inprod{U}{V} = (\Inprod{V}{U})^*$$, where $${}^*$$ is the complex conjugate. -+ **Positive semidefiniteness**: $$\Inprod{V}{V} \ge 0$$, and $$\Inprod{V}{V} = 0$$ if $$V = \mathbf{0}$$. -+ **Linearity in second operand**: $$\Inprod{U}{(a V + b W)} = a \Inprod{U}{V} + b \Inprod{U}{W}$$. ++ **Skew symmetry**: + $$\Inprod{U}{V} = (\Inprod{V}{U})^*$$, where $${}^*$$ is the complex conjugate. ++ **Positive semidefiniteness**: + $$\Inprod{V}{V} \ge 0$$, and $$\Inprod{V}{V} = 0$$ if $$V = \mathbf{0}$$. ++ **Linearity in second operand**: + $$\Inprod{U}{(a V + b W)} = a \Inprod{U}{V} + b \Inprod{U}{W}$$. The inner product describes the lengths and angles of vectors, and in Euclidean space it is implemented by the dot product. @@ -93,34 +112,39 @@ The **magnitude** or **norm** $$|V|$$ of a vector $$V$$ is given by $$|V| = \sqrt{\Inprod{V}{V}}$$ and represents the real positive length of $$V$$. A **unit vector** has a norm of 1. -Two vectors $$U$$ and $$V$$ are **orthogonal** if their inner product -$$\Inprod{U}{V} = 0$$. If in addition to being orthogonal, $$|U| = 1$$ and -$$|V| = 1$$, then $$U$$ and $$V$$ are known as **orthonormal** vectors. +Two vectors $$U$$ and $$V$$ are **orthogonal** +if their inner product $$\Inprod{U}{V} = 0$$. +If, in addition to being orthogonal, $$|U| = 1$$ and $$|V| = 1$$, +then $$U$$ and $$V$$ are known as **orthonormal** vectors. -Orthonormality is desirable for basis vectors, so if they are -not already like that, it is common to manually turn them into a new -orthonormal basis using e.g. the [Gram-Schmidt method](/know/concept/gram-schmidt-method). +Orthonormality is desirable for basis vectors, +so if they are not already like that, +it is common to manually turn them into a new orthonormal basis, +using e.g. the [Gram-Schmidt method](/know/concept/gram-schmidt-method). -As for the implementation of the inner product, it is given by: +The implementation of the inner product in terms of components and basis vectors +is as follows, which can easily be shown to satisfy the properties above: $$\begin{gathered} V = \sum_{n = 1}^N v_n \vu{e}_n - \quad + \qquad W = \sum_{n = 1}^N w_n \vu{e}_n \\ \quad \implies \quad \Inprod{V}{W} = \sum_{n = 1}^N \sum_{m = 1}^N v_n^* w_m \Inprod{\vu{e}_n}{\vu{e}_j} \end{gathered}$$ -If the basis vectors $$\vu{e}_1, ..., \vu{e}_N$$ are already -orthonormal, this reduces to: +If the basis vectors $$\vu{e}_1, ..., \vu{e}_N$$ are already orthonormal, +this reduces to: $$\begin{aligned} \Inprod{V}{W} = \sum_{n = 1}^N v_n^* w_n \end{aligned}$$ -As it turns out, the components $$v_n$$ are given by the inner product -with $$\vu{e}_n$$, where $$\delta_{nm}$$ is the Kronecker delta: +This suggests a way to calculate the components $$v_n$$: +taking the inner product of $$V$$ with a basis vector $$\vu{e}_n$$ +"picks out" the corresponding component $$v_n$$. +Let $$\delta_{nm}$$ be the Kronecker delta: $$\begin{aligned} \Inprod{\vu{e}_n}{V} = \sum_{m = 1}^N \delta_{nm} v_m = v_n @@ -134,40 +158,46 @@ As the dimensionality $$N$$ tends to infinity, things may or may not change significantly, depending on whether $$N$$ is **countably** or **uncountably** infinite. -In the former case, not much changes: the infinitely many **discrete** -basis vectors $$\vu{e}_n$$ can all still be made orthonormal as usual, -and as before: +In the former case, not much changes: +the infinitely many **discrete** basis vectors $$\vu{e}_n$$ +can all still be made orthonormal as usual, and as before: $$\begin{aligned} V = \sum_{n = 1}^\infty v_n \vu{e}_n \end{aligned}$$ -A good example of such a countably-infinitely-dimensional basis are the -solution eigenfunctions of a [Sturm-Liouville problem](/know/concept/sturm-liouville-theory/). +A good example of such a countably-infinitely-dimensional basis +are the solution eigenfunctions of +a [Sturm-Liouville problem](/know/concept/sturm-liouville-theory/). -However, if the dimensionality is uncountably infinite, the basis -vectors are **continuous** and cannot be labeled by $$n$$. For example, all -complex functions $$f(x)$$ defined for $$x \in [a, b]$$ which -satisfy $$f(a) = f(b) = 0$$ form such a vector space. -In this case $$f(x)$$ is expanded as follows, where $$x$$ is a basis vector: +However, if the dimensionality is uncountably infinite, +the basis vectors are **continuous** and cannot be labeled by $$n$$. +For example, all complex functions $$f(x)$$ defined on the interval $$x \in [a, b]$$ +satisfying the boundary condition $$f(a) = f(b) = 0$$, form such a vector space. +In this case, every value of $$f(x)$$ is the component of +an abstract vector $$\Ket{f}$$ with respect to a basis vector $$\Ket{x}$$: $$\begin{aligned} - f(x) = \int_a^b \Inprod{x}{f} \dd{x} + f(x) = \Inprod{x}{f} \end{aligned}$$ -Similarly, the inner product $$\Inprod{f}{g}$$ must also be redefined as -follows: +The inner product $$\Inprod{f}{g}$$ must be redefined as follows, +effectively turning the sum over a discrete basis +into an integral over a continuous basis: $$\begin{aligned} \Inprod{f}{g} = \int_a^b f^*(x) \: g(x) \dd{x} \end{aligned}$$ -The concept of orthonormality must be also weakened. A finite function -$$f(x)$$ can be normalized as usual, but the basis vectors $$x$$ themselves -cannot, since each represents an infinitesimal section of the real line. +The concept of orthonormality must be also weakened. +A finite function $$f(x)$$ can be normalized as usual, +but the basis vectors $$x$$ themselves cannot, +since each represents an infinitesimal section of the real line. +So how to proceed? -The rationale in this case is that action of the identity operator $$\hat{I}$$ must -be preserved, which is given here in [Dirac notation](/know/concept/dirac-notation/): +The rationale in this case is that the action +of the identity operator $$\hat{I}$$ must be preserved, +which is given here in [Dirac notation](/know/concept/dirac-notation/): $$\begin{aligned} \hat{I} = \int_a^b \Ket{\xi} \Bra{\xi} \dd{\xi} @@ -181,8 +211,9 @@ $$\begin{aligned} = \int_a^b \Inprod{x}{\xi} f(\xi) \dd{\xi} \end{aligned}$$ -Since we want the latter integral to reduce to $$f(x)$$, it is plain to see that -$$\Inprod{x}{\xi}$$ can only be a [Dirac delta function](/know/concept/dirac-delta-function/), +Since we want the latter integral to reduce to $$f(x)$$, +it is plain to see that $$\Inprod{x}{\xi}$$ can only be +a [Dirac delta function](/know/concept/dirac-delta-function/), i.e $$\Inprod{x}{\xi} = \delta(x - \xi)$$: $$\begin{aligned} @@ -191,12 +222,13 @@ $$\begin{aligned} = f(x) \end{aligned}$$ -Consequently, $$\Inprod{x}{\xi} = 0$$ if $$x \neq \xi$$ as expected for an -orthogonal set of vectors, but if $$x = \xi$$ the inner product -$$\Inprod{x}{\xi}$$ is infinite, unlike earlier. +Consequently, $$\Inprod{x}{\xi} = 0$$ if $$x \neq \xi$$ +as expected for an orthogonal set of vectors, +but if $$x = \xi$$ then the inner product $$\Inprod{x}{\xi}$$ is infinite, +unlike earlier. -Technically, because the basis vectors $$x$$ cannot be normalized, they -are not members of a Hilbert space, but rather of a superset called a -**rigged Hilbert space**. Such vectors have no finite inner product with -themselves, but do have one with all vectors from the actual Hilbert -space. +Technically, because the basis vectors $$x$$ cannot be normalized, +they are not members of a Hilbert space, +but rather of a superset called a **rigged Hilbert space**. +Such vectors have no finite inner product with themselves, +but do have one with all vectors from the actual Hilbert space. |