## 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

symbol ID category latex scope dimension name value Used in derivations references
9329 variable |\psi \rangle
$$|\psi \rangle$$
complex dimensionless ket
1
9139 variable a
$$a$$
['real'] dimensionless 45
5598 variable \hat{A}
$$\hat{A}$$
real dimensionless observerable operator
3
4065 variable \langle \psi|
$$\langle \psi|$$
complex dimensionless bra
4
MESSAGES:
• local variable 'all_df' referenced before assignment
• in step 2394942: unable to eval AST for "Equality(Bra('pdg4065')*Dagger(Operator('pdg5598'))*Ket('pdg9329'),conjugate(E(Symbol('pdg9139'))))" aka "sympy.Equality(Bra('pdg4065')*Dagger(Operator('pdg5598'))*Ket('pdg9329'),conjugate(E(sympy.Symbol('pdg9139'))))"
• in step 2485909: unable to eval AST for "Equality(Bra('pdg4065')*Dagger(Operator('pdg5598'))*Ket('pdg9329'),conjugate(E(Symbol('pdg9139'))))" aka "sympy.Equality(Bra('pdg4065')*Dagger(Operator('pdg5598'))*Ket('pdg9329'),conjugate(E(sympy.Symbol('pdg9139'))))"
• in step 2485909: unable to eval AST for "Equality(Bra('pdg4065')*Dagger(Operator('pdg5598'))*Ket('pdg9329'),conjugate(E(Symbol('pdg9139'))))" aka "sympy.Equality(Bra('pdg4065')*Dagger(Operator('pdg5598'))*Ket('pdg9329'),conjugate(E(sympy.Symbol('pdg9139'))))"
• unable to eval AST for "Equality(Bra('pdg4065')*Dagger(Operator('pdg5598'))*Ket('pdg9329'),conjugate(E(Symbol('pdg9139'))))" aka "sympy.Equality(Bra('pdg4065')*Dagger(Operator('pdg5598'))*Ket('pdg9329'),conjugate(E(sympy.Symbol('pdg9139'))))"