Extended bra–ket notation: Difference between revisions

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RTT mappings are typically thought of in terms of their ''rows''. This mapping <math>M</math> has two rows; following mathematical conventions, let's call them <math>𝒎_1</math> and <math>𝒎_2</math>. And so to notate this mapping in EBK, we can first imagine capturing the rows as bras like we would normally: <math>𝒎_1</math> = {{map|1 0 -4}} and <math>𝒎_2</math> = {{map|0 1 4}}. Then, to put them together, we can think of this matrix as a single column containing these two rows, or in other words, a ket containing the two bras: {{ket|<math>𝒎_1</math> <math>𝒎_2</math>}}, or fully written out, {{~~rket~~|{{map|1 0 -4}} {{map|0 1 4}}}}.

RTT mappings are typically thought of in terms of their ''rows''. This mapping <math>M</math> has two rows; following mathematical conventions, let's call them <math>𝒎_1</math> and <math>𝒎_2</math>. And so to notate this mapping in EBK, we can first imagine capturing the rows as bras like we would normally: <math>𝒎_1</math> = {{map|1 0 -4}} and <math>𝒎_2</math> = {{map|0 1 4}}. Then, to put them together, we can think of this matrix as a single column containing these two rows, or in other words, a ket containing the two bras: {{ket|<math>𝒎_1</math> <math>𝒎_2</math>}}, or fully written out, {{ket|{{map|1 0 -4}} {{map|0 1 4}}}}.

For another example, the canonical [[comma basis]] for 7-ET consists of the two commas 2187/2048 and 135/128, with PC-vectors {{vector|-11 7}} and {{vector|-7 3 1}}, respectively. As a matrix <math>\mathrm{C}</math>, we'd see this as:

Revision as of 23:24, 13 December 2022 view source Dave Keenan (talk \| contribs) autopatrolled, emailconfirmed 2,270 edits →Extensions: Changed "nesting" to "alternating", since "repeating" is a kind of nesting too. ← Older edit		Revision as of 23:34, 13 December 2022 view source Dave Keenan (talk \| contribs) autopatrolled, emailconfirmed 2,270 edits →Examples: Replaced curly-bracket rket with angle-bracket ket because the curly form hasn't yet been explained at this point in the article. Newer edit →
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	RTT mappings are typically thought of in terms of their ''rows''. This mapping <math>M</math> has two rows; following mathematical conventions, let's call them <math>𝒎_1</math> and <math>𝒎_2</math>. And so to notate this mapping in EBK, we can first imagine capturing the rows as bras like we would normally: <math>𝒎_1</math> = {{map\|1 0 -4}} and <math>𝒎_2</math> = {{map\|0 1 4}}. Then, to put them together, we can think of this matrix as a single column containing these two rows, or in other words, a ket containing the two bras: {{ket\|<math>𝒎_1</math> <math>𝒎_2</math>}}, or fully written out, {{~~rket~~\|{{map\|1 0 -4}} {{map\|0 1 4}}}}.		RTT mappings are typically thought of in terms of their ''rows''. This mapping <math>M</math> has two rows; following mathematical conventions, let's call them <math>𝒎_1</math> and <math>𝒎_2</math>. And so to notate this mapping in EBK, we can first imagine capturing the rows as bras like we would normally: <math>𝒎_1</math> = {{map\|1 0 -4}} and <math>𝒎_2</math> = {{map\|0 1 4}}. Then, to put them together, we can think of this matrix as a single column containing these two rows, or in other words, a ket containing the two bras: {{ket\|<math>𝒎_1</math> <math>𝒎_2</math>}}, or fully written out, {{ket\|{{map\|1 0 -4}} {{map\|0 1 4}}}}.

	For another example, the canonical [[comma basis]] for 7-ET consists of the two commas 2187/2048 and 135/128, with PC-vectors {{vector\|-11 7}} and {{vector\|-7 3 1}}, respectively. As a matrix <math>\mathrm{C}</math>, we'd see this as:		For another example, the canonical [[comma basis]] for 7-ET consists of the two commas 2187/2048 and 135/128, with PC-vectors {{vector\|-11 7}} and {{vector\|-7 3 1}}, respectively. As a matrix <math>\mathrm{C}</math>, we'd see this as: