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Review quantum basics orthogonality

step inference rule input feed output step validity (as per SymPy)
1
  • 111981: declare initial expression
  • number of inputs: 0; feeds: 0; outputs: 1
  • Eq.~\\ref{eq:#1} is an initial equation.
  1. 9596004948
    \(x=\langle\psi_{\alpha}| \hat{A} |\psi_{\beta}\rangle\)
 no validation is available for declarations
2
  • 111946: apply operator to ket
  • number of inputs: 1; feeds: 0; outputs: 1
  • Apply operator in Eq.~\\ref{eq:#1} to ket; yields Eq.~\\ref{eq:#2}.
  1. 9596004948
    \(x=\langle\psi_{\alpha}| \hat{A} |\psi_{\beta}\rangle\)
  1. 1010393944
    \(x=\langle\psi_{\alpha}| a_{\beta} |\psi_{\beta} \rangle\)
 recognized infrule but not yet supported
3
  • 111390: apply operator to bra
  • number of inputs: 1; feeds: 0; outputs: 1
  • Apply operator in Eq.~\\ref{eq:#1} to bra; yields Eq.~\\ref{eq:#2}.
  1. 9596004948
    \(x=\langle\psi_{\alpha}| \hat{A} |\psi_{\beta}\rangle\)
  1. 1395858355
    \(x=\langle \psi_{\alpha}| a_{\alpha} |\psi_{\beta}\rangle\)
 recognized infrule but not yet supported
4
  • 111457: simplify
  • number of inputs: 1; feeds: 0; outputs: 1
  • Simplify Eq.~\\ref{eq:#1}; yields Eq.~\\ref{eq:#2}.
  1. 1010393944
    \(x=\langle\psi_{\alpha}| a_{\beta} |\psi_{\beta} \rangle\)
  1. 2394240499
    \(x=a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle\)
 valid
5
  • 111457: simplify
  • number of inputs: 1; feeds: 0; outputs: 1
  • Simplify Eq.~\\ref{eq:#1}; yields Eq.~\\ref{eq:#2}.
  1. 1395858355
    \(x=\langle \psi_{\alpha}| a_{\alpha} |\psi_{\beta}\rangle\)
  1. 3943939590
    \(x=a_{\alpha} \langle \psi_{\alpha}| \psi_{\beta}\rangle\)
 invalid syntax (<string>, line 0)
6
  • 111355: LHS of expr 1 equals LHS of expr 2
  • number of inputs: 2; feeds: 0; outputs: 1
  • LHS of Eq.~\\ref{eq:#1} is equal to LHS of Eq.~\\ref{eq:#2}; yields Eq.~\\ref{eq:#3}.
  1. 2394240499
    \(x=a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle\)
  1. 3943939590
    \(x=a_{\alpha} \langle \psi_{\alpha}| \psi_{\beta}\rangle\)
  1. 1203938249
    \(a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle=a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangle\)
 invalid syntax (<string>, line 0)
7
  • 111282: subtract X from both sides
  • number of inputs: 1; feeds: 1; outputs: 1
  • Subtract $#1$ from both sides of Eq.~\\ref{eq:#2}; yields Eq.~\\ref{eq:#3}.
  1. 1203938249
    \(a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle=a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangle\)
  1. 0005395034
    \(a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangle\)
  1. 3924948349
    \(a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle - a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangle=0\)
 name 'Bra' is not defined
8
  • 111728: combine like terms
  • number of inputs: 1; feeds: 0; outputs: 1
  • Combine like terms in Eq.~\\ref{eq:#1}; yields Eq.~\\ref{eq:#2}.
  1. 3924948349
    \(a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle - a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangle=0\)
  1. 2394935831
    \(( a_{\beta} - a_{\alpha} ) \langle \psi_{\alpha} | \psi_{\beta} \rangle=0\)
 recognized infrule but not yet supported
9
  • 111341: declare final expression
  • number of inputs: 1; feeds: 0; outputs: 0
  • Eq.~\\ref{eq:#1} is one of the final equations.
  1. 2394935831
    \(( a_{\beta} - a_{\alpha} ) \langle \psi_{\alpha} | \psi_{\beta} \rangle=0\)
 no validation is available for declarations


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