## 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

symbol ID category latex scope dimension name value Used in derivations references
2090 variable | \psi_{\beta} \rangle
$$| \psi_{\beta} \rangle$$
complex dimensionless ket
7
4679 variable \langle \psi_{\alpha} |
$$\langle \psi_{\alpha} |$$
complex dimensionless bra
7
7752 variable a_{\beta}
$$a_{\beta}$$
complex dimensionless none
5
5598 variable \hat{A}
$$\hat{A}$$
real dimensionless observerable operator
3
1464 variable x
$$x$$
['real'] dimensionless 140
2427 variable a_{\alpha}
$$a_{\alpha}$$
complex dimensionless none
4
MESSAGES:
• local variable 'all_df' referenced before assignment
• in step 4838499: cannot split into LHS/RHS