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review derivation: quantum basics orthogonality

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Index Inference Rule Input latex Feeds latex Output latex step validity dimension check unit check notes
3 apply operator to bra
  1. 9596004948; locally 3849595:
    \(x = \langle\psi_{\alpha}| \hat{A} |\psi_{\beta}\rangle\)
    \(pdg_{1464} = pdg_{5598} {\left\langle pdg_{4679}\right|} {\left|pdg_{2090}\right\rangle }\)
  1. 1395858355; locally 4349300:
    \(x = \langle \psi_{\alpha}| a_{\alpha} |\psi_{\beta}\rangle\)
    \(pdg_{1464} = pdg_{2427} {\left\langle pdg_{4679}\right|} {\left|pdg_{2090}\right\rangle }\)
no check performed 9596004948:
1395858355:
9596004948:
1395858355:
4 simplify
  1. 1010393944; locally 4940359:
    \(x = \langle\psi_{\alpha}| a_{\beta} |\psi_{\beta} \rangle\)
    \(pdg_{1464} = pdg_{7752} {\left\langle pdg_{4679}\right|} {\left|pdg_{2090}\right\rangle }\)
  1. 2394240499; locally 2409402:
    \(x = a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle\)
    \(pdg_{1464} = pdg_{7752} {\left\langle pdg_{4679}\right|} {\left|pdg_{2090}\right\rangle }\)
valid 1010393944:
2394240499:
1010393944:
2394240499:
9 declare final expr
  1. 2394935831; locally 3494855:
    \(( a_{\beta} - a_{\alpha} ) \langle \psi_{\alpha} | \psi_{\beta} \rangle = 0\)
    \(\left(- pdg_{2427} + pdg_{7752}\right) {\left\langle pdg_{4679}\right|} {\left|pdg_{2090}\right\rangle } = 0\)
no validation is available for declarations 2394935831: error for dim with 2394935831
 2394935831: N/A
1 declare initial expr
  1. 9596004948; locally 3849595:
    \(x = \langle\psi_{\alpha}| \hat{A} |\psi_{\beta}\rangle\)
    \(pdg_{1464} = pdg_{5598} {\left\langle pdg_{4679}\right|} {\left|pdg_{2090}\right\rangle }\)
no validation is available for declarations 9596004948:
9596004948:
2 apply operator to ket
  1. 9596004948; locally 3849595:
    \(x = \langle\psi_{\alpha}| \hat{A} |\psi_{\beta}\rangle\)
    \(pdg_{1464} = pdg_{5598} {\left\langle pdg_{4679}\right|} {\left|pdg_{2090}\right\rangle }\)
  1. 1010393944; locally 4940359:
    \(x = \langle\psi_{\alpha}| a_{\beta} |\psi_{\beta} \rangle\)
    \(pdg_{1464} = pdg_{7752} {\left\langle pdg_{4679}\right|} {\left|pdg_{2090}\right\rangle }\)
no check performed 9596004948:
1010393944:
9596004948:
1010393944:
7 subtract X from both sides
  1. 1203938249; locally 3495045:
    \(a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle = a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangle\)
    \(\text{True}\)
  1. 0005395034:
    \(a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangle\)
    \(pdg_{2427} {\left\langle pdg_{4679}\right|} {\left|pdg_{2090}\right\rangle }\)
  1. 3924948349; locally 4939583:
    \(a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle - a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangle = 0\)
    \(pdg_{7752}\)
failed 1203938249:
3924948349: no LHS/RHS split
 1203938249:
3924948349: N/A
5 simplify
  1. 1395858355; locally 4349300:
    \(x = \langle \psi_{\alpha}| a_{\alpha} |\psi_{\beta}\rangle\)
    \(pdg_{1464} = pdg_{2427} {\left\langle pdg_{4679}\right|} {\left|pdg_{2090}\right\rangle }\)
  1. 3943939590; locally 4934893:
    \(x = a_{\alpha} \langle \psi_{\alpha}| \psi_{\beta}\rangle\)
    \(pdg_{2427}\)
Nothing to split 1395858355:
3943939590:
1395858355:
3943939590:
6 LHS of expr 1 equals LHS of expr 2
  1. 2394240499; locally 2409402:
    \(x = a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle\)
    \(pdg_{1464} = pdg_{7752} {\left\langle pdg_{4679}\right|} {\left|pdg_{2090}\right\rangle }\)
  2. 3943939590; locally 4934893:
    \(x = a_{\alpha} \langle \psi_{\alpha}| \psi_{\beta}\rangle\)
    \(pdg_{2427}\)
  1. 1203938249; locally 3495045:
    \(a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle = a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangle\)
    \(\text{True}\)
Nothing to split 2394240499:
3943939590:
1203938249:
2394240499:
3943939590:
1203938249:
8 combine like terms
  1. 3924948349; locally 4939583:
    \(a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle - a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangle = 0\)
    \(pdg_{7752}\)
  1. 2394935831; locally 3494855:
    \(( a_{\beta} - a_{\alpha} ) \langle \psi_{\alpha} | \psi_{\beta} \rangle = 0\)
    \(\left(- pdg_{2427} + pdg_{7752}\right) {\left\langle pdg_{4679}\right|} {\left|pdg_{2090}\right\rangle } = 0\)
Nothing to split 3924948349: no LHS/RHS split
2394935831: error for dim with 2394935831
 3924948349: N/A
2394935831: N/A
Physics Derivation Graph: Steps for quantum basics orthogonality

Symbols for this derivation

See also all 227 symbols
symbol ID category latex scope dimension name value Used in derivations references
7752 variable a_{\beta}
\(a_{\beta}\)		 complex dimensionless none 5
2090 variable | \psi_{\beta} \rangle
\(| \psi_{\beta} \rangle\)		 complex dimensionless ket 7
2427 variable a_{\alpha}
\(a_{\alpha}\)		 complex dimensionless none 4
1464 variable x
\(x\)		 ['real'] dimensionless 140
5598 variable \hat{A}
\(\hat{A}\)		 real dimensionless observerable operator 3
4679 variable \langle \psi_{\alpha} |
\(\langle \psi_{\alpha} |\)		 complex dimensionless bra 7
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