{
"schema_version": 1,
"kind": "ConceptView",
"producer": "concept-view-1",
"concept_id": "urn:uuid:3a23f505-a2e2-4be4-92e0-ccc73f9a600a",
"labels": {
"zh-CN": "海森堡不确定性定理",
"en": "Heisenberg uncertainty theorem"
},
"alternative_labels": [],
"scope_note": "",
"identity_status": "active",
"identity_revision": "b99565c5ca149ddc93b69e72d27f8ce35390fe64615b2d7c7dd085291118ba49",
"occurrences": [
{
"concept_id": "urn:uuid:3a23f505-a2e2-4be4-92e0-ccc73f9a600a",
"book_id": "szekeres",
"edition_id": "source-snapshot-bc3404156f6e",
"lang": "zh-CN",
"route": "zh/chapter-14/01-basic-concepts",
"anchor": "first-6bb255744d1d4ebd815a-zh",
"chapter": "14",
"section": "14.1",
"offset": 14121,
"range": [
20443,
21877
],
"source_revision": "aa7ae5d33878e4eddbd8c7943c808eda392dd1072d2000abb456f9ac1793e9dd",
"source_category": "translation",
"review_status": "unreviewed",
"role": "theorem",
"role_basis": {
"kind": "registered-source-environment",
"registered_position": {
"route": "zh/chapter-14/01-basic-concepts",
"lang": "zh-CN",
"section": "14.1",
"signal": "named-theorem",
"label": "海森堡不确定性定理",
"english": "Heisenberg uncertainty theorem",
"offset": 12714
}
},
"label": "> [!theorem] 定理 14.1 · 海森堡不确定性定理 (Heisenberg uncertainty theorem)",
"context": {
"range": [
20443,
21877
],
"quote": "\n\n> [!theorem] 定理 14.1 · 海森堡不确定性定理 (Heisenberg uncertainty theorem)\n>\n> （海森堡）设 $A$ 和 $B$ 为两个厄米算符，则对任意态 $| \\psi \\rangle$\n>\n> \n>\n> $$\n> \\Delta A \\Delta B \\geq \\frac {1}{2} | \\langle [ A, B ] \\rangle |\\tag{14.6}\n> $$\n>\n> 其中 $[ A , B ] = A B - B A$ 是两个算符的对易子\n\n> [!proof]- 证明\n>\n> 设\n>\n> $$\n> | \\psi_ {1} \\rangle = (A - \\langle A \\rangle I) | \\psi \\rangle , \\quad | \\psi_ {2} \\rangle = (B - \\langle B \\rangle I) | \\psi \\rangle ,\n> $$\n>\n> 使得 $\\Delta A = \\| \\psi _ { 1 } \\|$ 和 $\\Delta B = \\| \\psi _ { 2 } \\|$。利用柯西–施瓦茨不等式，\n>\n> $$\n> \\begin{aligned}\n> \\Delta A \\Delta B &= \\| \\psi_ {1} \\| \\| \\psi_ {2} \\| \\geq \\big | \\langle \\psi_ {1} | \\psi_ {2} \\rangle \\big | \\\\[0.4em]\n> &\\geq \\big | \\mathrm{Im} \\langle \\psi_ {1} | \\psi_ {2} \\rangle \\big | \\\\[0.4em]\n> &= \\Big | \\frac {1}{2 i} \\big (\\langle \\psi_ {1} | \\psi_ {2} \\rangle - \\langle \\psi_ {2} | \\psi_ {1} \\rangle \\big) \\Big |.\n> \\end{aligned}\n> $$\n>\n> 现在\n>\n> $$\n> \\begin{aligned}\n> \\langle \\psi_ {1} | \\psi_ {2} \\rangle &= \\langle (A - \\langle A \\rangle I) \\psi | (B - \\langle B \\rangle I) \\psi \\rangle \\\\[0.4em]\n> &= \\langle \\psi | A B | \\psi \\rangle - \\langle A \\rangle \\langle B \\rangle .\n> \\end{aligned}\n> $$\n>\n> 因此\n>\n> $$\n> \\Delta A \\Delta B \\geq \\frac {1}{2} \\big | \\langle \\psi | A B - B A | \\psi \\rangle \\big | = \\frac {1}{2} \\big | \\langle [ A, B ] \\rangle \\big |.\n> $$\n",
"quote_sha256": "3ec656458fce61ff082c2dea3770393cc46161f09306e3200a6a9c507bca172d",
"environment": "theorem",
"excerpt_omitted": false
},
"parallel_readings": [
{
"route": "en/chapter-14/01-basic-concepts",
"lang": "en",
"source_category": "original_transcription",
"basis": "registered-book-edition-section-language-context; not-concept-equivalence"
}
],
"id": "occ-279908991a2e8134a49d28dc"
},
{
"concept_id": "urn:uuid:3a23f505-a2e2-4be4-92e0-ccc73f9a600a",
"book_id": "szekeres",
"edition_id": "source-snapshot-bc3404156f6e",
"lang": "en",
"route": "en/chapter-14/01-basic-concepts",
"anchor": "first-6bb255744d1d4ebd815a-en",
"chapter": "14",
"section": "14.1",
"offset": 20780,
"range": [
26631,
28176
],
"source_revision": "5ffc1a138ecca9c924f0557a73a687627be309ac35b887823d07e23731a53f3b",
"source_category": "original_transcription",
"review_status": "unreviewed",
"role": "theorem",
"role_basis": {
"kind": "registered-source-environment",
"registered_position": {
"route": "en/chapter-14/01-basic-concepts",
"lang": "en",
"section": "14.1",
"signal": "named-theorem",
"label": "Heisenberg uncertainty theorem",
"english": "Heisenberg uncertainty theorem",
"offset": 19293
}
},
"label": "> [!theorem] Theorem 14.1 · Heisenberg uncertainty theorem",
"context": {
"range": [
26631,
28176
],
"quote": "\n\n> [!theorem] Theorem 14.1 · Heisenberg uncertainty theorem\n>\n> (Heisenberg) Let $A$ and $B$ be two hermitian operators, thenfor any state $| \\psi \\rangle$\n>\n> \n>\n> $$\n> \\Delta A \\Delta B \\geq \\frac {1}{2} | \\langle [ A, B ] \\rangle |\\tag{14.6}\n> $$\n>\n> where $[ A , B ] = A B - B A$ is the commutator of the two operators\n\n> [!proof]- Proof\n>\n> Let\n>\n> $$\n> | \\psi_ {1} \\rangle = (A - \\langle A \\rangle I) | \\psi \\rangle , \\quad | \\psi_ {2} \\rangle = (B - \\langle B \\rangle I) | \\psi \\rangle ,\n> $$\n>\n> so that $\\Delta A = \\| \\psi _ { 1 } \\|$ and $\\Delta B = \\| \\psi _ { 2 } \\|$ . Using the Cauchy–Schwarz inequality,\n>\n> $$\n> \\begin{aligned}\n> \\Delta A \\Delta B &= \\| \\psi_ {1} \\| \\| \\psi_ {2} \\| \\geq \\big | \\langle \\psi_ {1} | \\psi_ {2} \\rangle \\big | \\\\[0.4em]\n> &\\geq \\big | \\mathrm{Im} \\langle \\psi_ {1} | \\psi_ {2} \\rangle \\big | \\\\[0.4em]\n> &= \\Big | \\frac {1}{2 i} \\big (\\langle \\psi_ {1} | \\psi_ {2} \\rangle - \\langle \\psi_ {2} | \\psi_ {1} \\rangle \\big) \\Big |.\n> \\end{aligned}\n> $$\n>\n> Now\n>\n> $$\n> \\begin{aligned}\n> \\langle \\psi_ {1} | \\psi_ {2} \\rangle &= \\langle (A - \\langle A \\rangle I) \\psi | (B - \\langle B \\rangle I) \\psi \\rangle \\\\[0.4em]\n> &= \\langle \\psi | A B | \\psi \\rangle - \\langle A \\rangle \\langle B \\rangle .\n> \\end{aligned}\n> $$\n>\n> Hence\n>\n> $$\n> \\Delta A \\Delta B \\geq \\frac {1}{2} \\big | \\langle \\psi | A B - B A | \\psi \\rangle \\big | = \\frac {1}{2} \\big | \\langle [ A, B ] \\rangle \\big |.\n> $$\n",
"quote_sha256": "a7900cd058c0ba981fe3ca6465e3766c3584691f00818b9886df333b357da601",
"environment": "theorem",
"excerpt_omitted": false
},
"parallel_readings": [
{
"route": "zh/chapter-14/01-basic-concepts",
"lang": "zh-CN",
"source_category": "translation",
"basis": "registered-book-edition-section-language-context; not-concept-equivalence"
}
],
"id": "occ-506219a4c2214f15a33fb3f8"
}
],
"registered_positions": [
{
"route": "en/chapter-14/01-basic-concepts",
"lang": "en",
"section": "14.1",
"signal": "named-theorem",
"label": "Heisenberg uncertainty theorem",
"english": "Heisenberg uncertainty theorem",
"offset": 19293
},
{
"route": "zh/chapter-14/01-basic-concepts",
"lang": "zh-CN",
"section": "14.1",
"signal": "named-theorem",
"label": "海森堡不确定性定理",
"english": "Heisenberg uncertainty theorem",
"offset": 12714
}
],
"statements": [],
"resolved_references": [
{
"raw": "href=\"#theorem-14-1\"",
"href": "#theorem-14-1",
"range": [
20486,
20506
],
"relation": "references",
"target_route": "zh/chapter-14/01-basic-concepts",
"target_anchor": "theorem-14-1",
"resolved": true,
"reason": null,
"source_revision": "aa7ae5d33878e4eddbd8c7943c808eda392dd1072d2000abb456f9ac1793e9dd",
"source_route": "zh/chapter-14/01-basic-concepts",
"occurrence_id": "occ-279908991a2e8134a49d28dc"
},
{
"raw": "href=\"#theorem-14-1\"",
"href": "#theorem-14-1",
"range": [
26674,
26694
],
"relation": "references",
"target_route": "en/chapter-14/01-basic-concepts",
"target_anchor": "theorem-14-1",
"resolved": true,
"reason": null,
"source_revision": "5ffc1a138ecca9c924f0557a73a687627be309ac35b887823d07e23731a53f3b",
"source_route": "en/chapter-14/01-basic-concepts",
"occurrence_id": "occ-506219a4c2214f15a33fb3f8"
}
],
"reading_order": [
"occ-279908991a2e8134a49d28dc",
"occ-506219a4c2214f15a33fb3f8"
],
"recorded_dependencies": [],
"reading_paths": [],
"coverage": {
"registered_positions": 2,
"located_occurrences": 2,
"whole_book_search_complete": false,
"registered_positions_truncated": false,
"independent_human_review_implied": false
},
"unresolved": [],
"base_revision": "d27fbd2fabff4d6a1e5993597d7e24959afb1a059c08a610761d062a4b91d1ef",
"optional_enrichments": []
}