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review derivation: quantum basics Hermitian operators have realvalued observables

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Index Inference Rule Input latex Feeds latex Output latex step validity dimension check unit check notes
4 declare assumption
  1. 9294858532; locally 2484892:
    \(\hat{A}^+ = \hat{A}\)
    \(\)
no validation is available for declarations 9294858532:
9294858532:
3 distribute conjugate transpose to factors
  1. 2394935835; locally 2495954:
    \(\left(\langle\psi| \hat{A} |\psi \rangle \right)^+ = \left(\langle a \rangle\right)^+\)
    \(\)
  1. 1010393913; locally 2390094:
    \(\langle \psi| \hat{A}^+ |\psi \rangle = \langle a \rangle^*\)
    \(\)
Nothing to split 2394935835: no LHS/RHS split
1010393913:
2394935835: N/A
1010393913:
5 substitute RHS of expr 1 into expr 2
  1. 9294858532; locally 2484892:
    \(\hat{A}^+ = \hat{A}\)
    \(\)
  2. 1010393913; locally 2390094:
    \(\langle \psi| \hat{A}^+ |\psi \rangle = \langle a \rangle^*\)
    \(\)
  1. 4948934890; locally 2494040:
    \(\langle \psi| \hat{A} |\psi \rangle = \langle a \rangle^*\)
    \(\)
failed 9294858532:
1010393913:
4948934890:
9294858532:
1010393913:
4948934890:
6 substitute RHS of expr 1 into expr 2
  1. 4948934890; locally 2494040:
    \(\langle \psi| \hat{A} |\psi \rangle = \langle a \rangle^*\)
    \(\)
  2. 9999999975; locally 3402919:
    \(\langle \psi| \hat{A} |\psi \rangle = \langle a \rangle\)
    \(\)
  1. 2848934890; locally 4930585:
    \(\langle a \rangle^* = \langle a \rangle\)
    \(\)
Nothing to split 4948934890:
9999999975: no LHS/RHS split
2848934890:
4948934890:
9999999975: N/A
2848934890:
7 declare final expr
  1. 2848934890; locally 4930585:
    \(\langle a \rangle^* = \langle a \rangle\)
    \(\)
no validation is available for declarations 2848934890:
2848934890:
1 declare initial expr
  1. 9999999975; locally 3402919:
    \(\langle \psi| \hat{A} |\psi \rangle = \langle a \rangle\)
    \(\)
no validation is available for declarations 9999999975: no LHS/RHS split
9999999975: N/A
2 conjugate transpose both sides
  1. 9999999975; locally 3402919:
    \(\langle \psi| \hat{A} |\psi \rangle = \langle a \rangle\)
    \(\)
  1. 2394935835; locally 2495954:
    \(\left(\langle\psi| \hat{A} |\psi \rangle \right)^+ = \left(\langle a \rangle\right)^+\)
    \(\)
Nothing to split 9999999975: no LHS/RHS split
2394935835: no LHS/RHS split
9999999975: N/A
2394935835: N/A
Physics Derivation Graph: Steps for quantum basics Hermitian operators have realvalued observables

Symbols for this derivation

See also all 212 symbols
symbol ID category latex scope dimension name value Used in derivations references
4065 variable \langle \psi|
\(\langle \psi|\)
complex dimensionless bra 4
9139 variable a
\(a\)
['real']
45
9329 variable |\psi \rangle
\(|\psi \rangle\)
complex dimensionless ket 1
5598 variable \hat{A}
\(\hat{A}\)
real
observerable operator 3
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