Physics Derivation Graph navigation Sign in

review derivation: quantum basics orthogonality

This page contains three views of the steps in the derivation: d3js, graphviz PNG, and a table.


Hold the mouse over a node to highlight that node and its neighbors. You can zoom in/out. You can pan the image. You can move nodes by clicking and dragging.

Notes for this derivation:

Options
Alternate views of this derivation:
Edit this content:    

To edit a step, click on the number in the "Index" column in the table below

Clicking on the step index will take you to the page where you can edit that step.

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
2090 variable | \psi_{\beta} \rangle
\(| \psi_{\beta} \rangle\)
complex dimensionless ket 7
2427 variable a_{\alpha}
\(a_{\alpha}\)
complex dimensionless none 4
5598 variable \hat{A}
\(\hat{A}\)
real dimensionless observerable operator 3
7752 variable a_{\beta}
\(a_{\beta}\)
complex dimensionless none 5
1464 variable x
\(x\)
['real'] dimensionless 140
4679 variable \langle \psi_{\alpha} |
\(\langle \psi_{\alpha} |\)
complex dimensionless bra 7
MESSAGES: