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Review quantum basics Hermitian operators have realvalued observables

step inference rule input feed output step validity (as per SymPy)
1
  • 0000111981: declare initial expression
  • number of inputs: 0; feeds: 0; outputs: 1
  • Eq.~\ref{eq:#1} is an initial equation.
  1. 9999999975
    \(\langle \psi| \hat{A} |\psi \rangle = \langle a \rangle\)
 no validation is available for declarations
2
  • 0000111696: conjugate transpose both sides
  • number of inputs: 1; feeds: 0; outputs: 1
  • Conjugate transpose of both sides of Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.
  1. 9999999975
    \(\langle \psi| \hat{A} |\psi \rangle = \langle a \rangle\)
  1. 2394935835
    \(\left(\langle\psi| \hat{A} |\psi \rangle \right)^+ = \left(\langle a \rangle\right)^+\)
 recognized infrule but not yet supported
3
  • 0000111890: distribute conjugate transpose to factors
  • number of inputs: 1; feeds: 0; outputs: 1
  • Distribute conjugate transpose to factors in Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.
  1. 2394935835
    \(\left(\langle\psi| \hat{A} |\psi \rangle \right)^+ = \left(\langle a \rangle\right)^+\)
  1. 1010393913
    \(\langle \psi| \hat{A}^+ |\psi \rangle = \langle a \rangle^*\)
 recognized infrule but not yet supported
4
  • 0000111104: declare assumption
  • number of inputs: 0; feeds: 0; outputs: 1
  • Eq.~\ref{eq:#1} is an assumption.
  1. 9294858532
    \(\hat{A}^+ = \hat{A}\)
 no validation is available for declarations
5
  • 0000111634: substitute RHS of expr 1 into expr 2
  • number of inputs: 2; feeds: 0; outputs: 1
  • Substitute RHS of Eq.~\ref{eq:#1} into Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.
  1. 1010393913
    \(\langle \psi| \hat{A}^+ |\psi \rangle = \langle a \rangle^*\)
  1. 9294858532
    \(\hat{A}^+ = \hat{A}\)
  1. 4948934890
    \(\langle \psi| \hat{A} |\psi \rangle = \langle a \rangle^*\)
 Not evaluated due to missing term in SymPy
6
  • 0000111634: substitute RHS of expr 1 into expr 2
  • number of inputs: 2; feeds: 0; outputs: 1
  • Substitute RHS of Eq.~\ref{eq:#1} into Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.
  1. 9999999975
    \(\langle \psi| \hat{A} |\psi \rangle = \langle a \rangle\)
  1. 4948934890
    \(\langle \psi| \hat{A} |\psi \rangle = \langle a \rangle^*\)
  1. 2848934890
    \(\langle a \rangle^* = \langle a \rangle\)
 Not evaluated due to missing term in SymPy
7
  • 0000111341: declare final expression
  • number of inputs: 1; feeds: 0; outputs: 0
  • Eq.~\ref{eq:#1} is one of the final equations.
  1. 2848934890
    \(\langle a \rangle^* = \langle a \rangle\)
 no validation is available for declarations

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