garden/inner-product.md

Inner Product

• operates on two vectors and produces a scalar
• notation `\braket{x|y}`
  • `\bra{x}` is a bra
  • `\ket{y}` is a ket
• matrix-multiplication, associates with outer-product
• follows laws
  1. `\braket{0|y} = 0` and `\braket{x|0} = 0`
  2. `\braket{x + y|z} = \braket{x|z} + \braket{y|z}` and `\braket{x|y + z} = \braket{x|y} + \braket{x|z}`
  3. `\braket{cx|y} = \overline{c}\braket{x|y}` and `\braket{x|cy} = \braket{x|y}c`
  4. `\braket{x|y} = \overline{\braket{y|x}}`
• the inner product `\braket{x|y}` is antilinear in `x` and linear in `y`
  • if starting with two kets, take the conjugate transpose of the former: `\bra{x} = \ket{x}^\dagger = \overline{\ket{x}}^\intercal = \overline{\ket{x}^\intercal}`