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diff --git a/source/know/concept/dirac-notation/index.md b/source/know/concept/dirac-notation/index.md index 46cc325..2830a33 100644 --- a/source/know/concept/dirac-notation/index.md +++ b/source/know/concept/dirac-notation/index.md @@ -8,47 +8,49 @@ categories: layout: "concept" --- -**Dirac notation** is a notation to do calculations in a [Hilbert space](/know/concept/hilbert-space/) -without needing to worry about the space's representation. It is -basically the *lingua franca* of quantum mechanics. - -In Dirac notation there are **kets** $$\Ket{V}$$ from the Hilbert space -$$\mathbb{H}$$ and **bras** $$\Bra{V}$$ from a dual $$\mathbb{H}'$$ of the -former. Crucially, the bras and kets are from different Hilbert spaces -and therefore cannot be added, but every bra has a corresponding ket and -vice versa. +**Dirac notation** enables us to do calculations +in a general [Hilbert space](/know/concept/hilbert-space/) +without needing to worry about the space's representation. +It is the *lingua franca* of quantum mechanics. + +In Dirac notation there are +**kets** $$\ket{V}$$ from the Hilbert space $$\mathbb{H}$$ +and **bras** $$\bra{V}$$ from its dual space $$\mathbb{H}'$$. +Crucially, the bras and kets are from different Hilbert spaces +and therefore cannot be added, +but every bra has a corresponding ket and vice versa. Bras and kets can be combined in two ways: the **inner product** -$$\Inprod{V}{W}$$, which returns a scalar, and the **outer product** -$$\Ket{V} \Bra{W}$$, which returns a mapping $$\hat{L}$$ from kets $$\Ket{V}$$ -to other kets $$\Ket{V'}$$, i.e. a linear operator. Recall that the -Hilbert inner product must satisfy: +$$\inprod{V}{W}$$, which returns a scalar, and the **outer product** +$$\ket{V} \bra{W}$$, which returns a linear operator +that maps kets $$\ket{V}$$ to other kets $$\ket{V'}$$. +Recall that by definition the Hilbert inner product must satisfy: $$\begin{aligned} - \Inprod{V}{W} = \Inprod{W}{V}^* + \inprod{V}{W} = \inprod{W}{V}^* \end{aligned}$$ -So far, nothing has been said about the actual representation of bras or -kets. If we represent kets as $$N$$-dimensional columns vectors, the -corresponding bras are given by the kets' adjoints, i.e. their transpose -conjugates: +So far, nothing has been said about the actual representation of bras or kets. +If we represent kets as $$N$$-dimensional columns vectors, +the corresponding bras are given by the kets' adjoints, +i.e. their transpose conjugates: $$\begin{aligned} - \Ket{V} = + \ket{V} = \begin{bmatrix} v_1 \\ \vdots \\ v_N \end{bmatrix} \quad \implies \quad - \Bra{V} = + \bra{V} = \begin{bmatrix} v_1^* & \cdots & v_N^* \end{bmatrix} \end{aligned}$$ -The inner product $$\Inprod{V}{W}$$ is then just the familiar dot product $$V \cdot W$$: +The inner product $$\inprod{V}{W}$$ is then just the familiar dot product $$V \cdot W$$: $$\begin{gathered} - \Inprod{V}{W} + \inprod{V}{W} = \begin{bmatrix} v_1^* & \cdots & v_N^* @@ -60,10 +62,11 @@ $$\begin{gathered} = v_1^* w_1 + ... + v_N^* w_N \end{gathered}$$ -Meanwhile, the outer product $$\Ket{V} \Bra{W}$$ creates an $$N \cross N$$ matrix: +Meanwhile, the outer product $$\ket{V} \bra{W}$$ creates an $$N \cross N$$ matrix, +which can be thought of as applying an operation to any vector it multiplies: $$\begin{gathered} - \Ket{V} \Bra{W} + \ket{V} \bra{W} = \begin{bmatrix} v_1 \\ \vdots \\ v_N @@ -80,15 +83,14 @@ $$\begin{gathered} \end{bmatrix} \end{gathered}$$ -If the kets are instead represented by functions $$f(x)$$ of -$$x \in [a, b]$$, then the bras represent *functionals* $$F[u(x)]$$ which -take an unknown function $$u(x)$$ as an argument and turn it into a scalar -using integration: +If the kets are instead represented by continuous functions $$f(x)$$ of $$x \in [a, b]$$, +then the bras are *functionals* $$F[u(x)]$$ +that take an arbitrary function $$u(x)$$ as an argument and return a scalar: $$\begin{aligned} - \Ket{f} = f(x) + \ket{f} = f(x) \quad \implies \quad - \Bra{f} + \bra{f} = F[u(x)] = \int_a^b f^*(x) \: u(x) \dd{x} \end{aligned}$$ @@ -96,23 +98,25 @@ $$\begin{aligned} Consequently, the inner product is simply the following familiar integral: $$\begin{gathered} - \Inprod{f}{g} + \inprod{f}{g} = F[g(x)] = \int_a^b f^*(x) \: g(x) \dd{x} \end{gathered}$$ -However, the outer product becomes something rather abstract: +However, the outer product is then rather abstract: +a continuous analogue of a matrix: $$\begin{gathered} - \Ket{f} \Bra{g} + \ket{f} \bra{g} = f(x) \: G[u(x)] = f(x) \int_a^b g^*(\xi) \: u(\xi) \dd{\xi} \end{gathered}$$ -This result makes more sense if we surround it by a bra and a ket: +This maybe makes more sense if we surround it +by a bra $$\bra{u}$$ and a ket $$\ket{w}$$ and rearrange: $$\begin{aligned} - \Bra{u} \!\Big(\!\Ket{f} \Bra{g}\!\Big)\! \Ket{w} + \bra{u} \!\Big(\!\ket{f} \bra{g}\!\Big)\! \ket{w} &= U\big[f(x) \: G[w(x)]\big] = U\Big[ f(x) \int_a^b g^*(\xi) \: w(\xi) \dd{\xi} \Big] \\ @@ -120,7 +124,7 @@ $$\begin{aligned} \\ &= \Big( \int_a^b u^*(x) \: f(x) \dd{x} \Big) \Big( \int_a^b g^*(\xi) \: w(\xi) \dd{\xi} \Big) \\ - &= \Inprod{u}{f} \Inprod{g}{w} + &= \inprod{u}{f} \inprod{g}{w} \end{aligned}$$ |