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Review quantum basics orthogonality

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. 9596004948
    \(x = \langle\psi_{\alpha}| \hat{A} |\psi_{\beta}\rangle\)
 no validation is available for declarations
2
  • 0000111946: 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
  • 0000111390: 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
  • 0000111457: 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
  • 0000111457: 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
    \(x = a_{\alpha} \langle \psi_{\alpha}| \psi_{\beta}\rangle\)
 Not evaluated due to missing term in SymPy
6
  • 0000111355: 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. 3943939590
    \(x = a_{\alpha} \langle \psi_{\alpha}| \psi_{\beta}\rangle\)
  2. 2394240499
    \(x = a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle\)
  1. 1203938249
    \(a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle = a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangle\)
 Not evaluated due to missing term in SymPy
7
  • 0000111282: 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
    \(a_{\beta} \langle \psi_{\alpha} | \psi_{\beta} \rangle - a_{\alpha} \langle \psi_{\alpha} | \psi_{\beta} \rangle = 0\)
 Not evaluated due to missing term in SymPy
8
  • 0000111728: 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
    \(( a_{\beta} - a_{\alpha} ) \langle \psi_{\alpha} | \psi_{\beta} \rangle = 0\)
 Not evaluated due to missing term in SymPy
9
  • 0000111341: 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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