diff options
Diffstat (limited to 'latex/know/concept/dirac-notation/source.md')
| -rw-r--r-- | latex/know/concept/dirac-notation/source.md | 12 |
1 files changed, 6 insertions, 6 deletions
diff --git a/latex/know/concept/dirac-notation/source.md b/latex/know/concept/dirac-notation/source.md index 7b384ab..47aa370 100644 --- a/latex/know/concept/dirac-notation/source.md +++ b/latex/know/concept/dirac-notation/source.md @@ -14,13 +14,13 @@ and therefore cannot be added, but every bra has a corresponding ket and vice versa. Bras and kets can only be combined in two ways: the *inner product* -$\braket{V | W}$, which returns a scalar, and the *outer product* +$\braket{V}{W}$, 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: $$\begin{aligned} - \braket{V | W} = \braket{W | V}^* + \braket{V}{W} = \braket{W}{V}^* \end{aligned}$$ So far, nothing has been said about the actual representation of bras or @@ -40,10 +40,10 @@ $$\begin{aligned} \end{bmatrix} \end{aligned}$$ -The inner product $\braket{V | W}$ is then just the familiar dot product $V \cdot W$: +The inner product $\braket{V}{W}$ is then just the familiar dot product $V \cdot W$: $$\begin{gathered} - \braket{V | W} + \braket{V}{W} = \begin{bmatrix} v_1^* & \cdots & v_N^* @@ -91,7 +91,7 @@ $$\begin{aligned} Consequently, the inner product is simply the following familiar integral: $$\begin{gathered} - \braket{f | g} + \braket{f}{g} = F[g(x)] = \int_a^b f^*(x) \: g(x) \dd{x} \end{gathered}$$ @@ -115,5 +115,5 @@ $$\begin{aligned} \\ &= \Big( \int_a^b u^*(x) \: f(x) \dd{x} \Big) \Big( \int_a^b g^*(\xi) \: w(\xi) \dd{\xi} \Big) \\ - &= \braket{u | f} \braket{g | w} + &= \braket{u}{f} \braket{g}{w} \end{aligned}$$ |