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