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Review derivation of Schrodinger Equation

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. 3121513111
    \(k=\frac{2 \pi}{\lambda}\)
 no validation is available for declarations
2
  • 111981: declare initial expression
  • number of inputs: 0; feeds: 0; outputs: 1
  • Eq.~\\ref{eq:#1} is an initial equation.
  1. 3131211131
    \(\omega=2 \pi f\)
 no validation is available for declarations
3
  • 111981: declare initial expression
  • number of inputs: 0; feeds: 0; outputs: 1
  • Eq.~\\ref{eq:#1} is an initial equation.
  1. 9999999960
    \(\hbar=h/(2 \pi)\)
 no validation is available for declarations
4
  • 111981: declare initial expression
  • number of inputs: 0; feeds: 0; outputs: 1
  • Eq.~\\ref{eq:#1} is an initial equation.
  1. 1020394900
    \(p=h/\lambda\)
 no validation is available for declarations
5
  • 111981: declare initial expression
  • number of inputs: 0; feeds: 0; outputs: 1
  • Eq.~\\ref{eq:#1} is an initial equation.
  1. 1020394902
    \(E=h f\)
 no validation is available for declarations
6
  • 111975: divide both sides by
  • number of inputs: 1; feeds: 1; outputs: 1
  • Divide both sides of Eq.~\\ref{eq:#2} by $#1$; yields Eq.~\\ref{eq:#3}.
  1. 3131211131
    \(\omega=2 \pi f\)
  1. 0002940021
    \(2 \pi\)
  1. 3147472131
    \(\frac{\omega}{2 \pi}=f\)
 valid
7
  • 111634: 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. 1020394902
    \(E=h f\)
  1. 3147472131
    \(\frac{\omega}{2 \pi}=f\)
  1. 4147472132
    \(E=\frac{h \omega}{2 \pi}\)
 LHS diff is pdg0002321/(2*pdg0003141) - pdg0004931 RHS diff is -pdg0002321*pdg0004413/(2*pdg0003141) + pdg0004201
8
  • 111634: 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. 9999999960
    \(\hbar=h/(2 \pi)\)
  1. 4147472132
    \(E=\frac{h \omega}{2 \pi}\)
  1. 9999999965
    \(E=\omega \hbar\)
 valid
9
  • 111975: divide both sides by
  • number of inputs: 1; feeds: 1; outputs: 1
  • Divide both sides of Eq.~\\ref{eq:#2} by $#1$; yields Eq.~\\ref{eq:#3}.
  1. 9999999965
    \(E=\omega \hbar\)
  1. 0003949921
    \(\hbar\)
  1. 9999999961
    \(\frac{E}{\hbar}=\omega\)
 valid
10
  • 111975: divide both sides by
  • number of inputs: 1; feeds: 1; outputs: 1
  • Divide both sides of Eq.~\\ref{eq:#2} by $#1$; yields Eq.~\\ref{eq:#3}.
  1. 3121513111
    \(k=\frac{2 \pi}{\lambda}\)
  1. 0001209482
    \(2 \pi\)
  1. 3121234211
    \(\frac{k}{2\pi}=\lambda\)
 Algebraic error: LHS diff is 0, RHS diff is -pdg0001115 + 1/pdg0001115
11
  • 111634: 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. 3121234211
    \(\frac{k}{2\pi}=\lambda\)
  1. 1020394900
    \(p=h/\lambda\)
  1. 3121234212
    \(p=\frac{h k}{2\pi}\)
 LHS diff is 0 RHS diff is 2*pdg0003141*pdg0004413/pdg0005321 - pdg0004413*pdg0005321/(2*pdg0003141)
12
  • 111634: 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. 3121234212
    \(p=\frac{h k}{2\pi}\)
  1. 9999999960
    \(\hbar=h/(2 \pi)\)
  1. 9999999962
    \(p=\hbar k\)
 LHS diff is pdg0001054 - pdg0001134 RHS diff is -pdg0001054*pdg0005321 + pdg0004413/(2*pdg0003141)
13
  • 111975: divide both sides by
  • number of inputs: 1; feeds: 1; outputs: 1
  • Divide both sides of Eq.~\\ref{eq:#2} by $#1$; yields Eq.~\\ref{eq:#3}.
  1. 9999999962
    \(p=\hbar k\)
  1. 0001304952
    \(\hbar\)
  1. 9999999870
    \(\frac{p}{\hbar}=k\)
 valid
14
  • 111215: replace scalar with vector
  • number of inputs: 1; feeds: 0; outputs: 1
  • Replace scalar variables in Eq.~\\ref{eq:#1} with equivalent vector variables; yields Eq.~\\ref{eq:#2}.
  1. 9999999870
    \(\frac{p}{\hbar}=k\)
  1. 9999998870
    \(\frac{ \vec{p}}{\hbar}=\vec{k}\)
 recognized infrule but not yet supported
15
  • 111981: declare initial expression
  • number of inputs: 0; feeds: 0; outputs: 1
  • Eq.~\\ref{eq:#1} is an initial equation.
  1. 3948574224
    \(\psi( \vec{r},t)=\psi_0 \exp\left(i\left( \vec{k}\cdot\vec{r} - \omega t \right) \right)\)
 no validation is available for declarations
16
  • 111634: 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. 9999998870
    \(\frac{ \vec{p}}{\hbar}=\vec{k}\)
  1. 3948574224
    \(\psi( \vec{r},t)=\psi_0 \exp\left(i\left( \vec{k}\cdot\vec{r} - \omega t \right) \right)\)
  1. 3948574226
    \(\psi( \vec{r},t)=\psi_0 \exp\left(i\left(\frac{ \vec{p}\cdot\vec{r}}{\hbar} - \omega t \right) \right)\)
 LHS diff is 0 RHS diff is pdg0008330*(-pdg0002718(pdg0004621((-pdg0001054*pdg0001467*pdg0002321 + pdg0001134*pdg0009472)/pdg0001054)) + pdg0002718(pdg0004621(-pdg0001467*pdg0002321 + dot(pdg0005321, pdg0009472))))
17
  • 111634: 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. 9999999961
    \(\frac{E}{\hbar}=\omega\)
  1. 3948574226
    \(\psi( \vec{r},t)=\psi_0 \exp\left(i\left(\frac{ \vec{p}\cdot\vec{r}}{\hbar} - \omega t \right) \right)\)
  1. 3948574228
    \(\psi( \vec{r},t)=\psi_0 \exp\left(i\left(\frac{ \vec{p}\cdot\vec{r}}{\hbar} - \frac{E t}{\hbar} \right) \right)\)
 LHS diff is 0 RHS diff is pdg0008330*(pdg0002718(pdg0004621((pdg0001134*pdg0009472 - pdg0001467*pdg0004931)/pdg0001054)) - pdg0002718(pdg0004621((pdg0001134*pdg0009472 - pdg0001467*pdg0006238)/pdg0001054)))
18
  • 111457: simplify
  • number of inputs: 1; feeds: 0; outputs: 1
  • Simplify Eq.~\\ref{eq:#1}; yields Eq.~\\ref{eq:#2}.
  1. 3948574228
    \(\psi( \vec{r},t)=\psi_0 \exp\left(i\left(\frac{ \vec{p}\cdot\vec{r}}{\hbar} - \frac{E t}{\hbar} \right) \right)\)
  1. 3948574230
    \(\psi( \vec{r},t)=\psi_0 \exp\left(\frac{i}{\hbar}\left( \vec{p}\cdot\vec{r} - E t \right) \right)\)
 LHS diff is 0 RHS diff is pdg0008330*(-pdg0002718(pdg0004621*(pdg0001134*pdg0009472 - pdg0001467*pdg0006238)/pdg0001054) + pdg0002718(pdg0004621((pdg0001134*pdg0009472 - pdg0001467*pdg0006238)/pdg0001054)))
19
  • 111981: declare initial expression
  • number of inputs: 0; feeds: 0; outputs: 1
  • Eq.~\\ref{eq:#1} is an initial equation.
  1. 1029039903
    \(p=m v\)
 no validation is available for declarations
20
  • 111981: declare initial expression
  • number of inputs: 0; feeds: 0; outputs: 1
  • Eq.~\\ref{eq:#1} is an initial equation.
  1. 4298359835
    \(E=\frac{1}{2}m v^2\)
 no validation is available for declarations
21
  • 111483: raise both sides to power
  • number of inputs: 1; feeds: 1; outputs: 1
  • Raise both sides of Eq.~\\ref{eq:#2} to $#1$; yields Eq.~\\ref{eq:#3}.
  1. 1029039903
    \(p=m v\)
  1. 0002239424
    \(2\)
  1. 1029039904
    \(p^2=m^2 v^2\)
 recognized infrule but not yet supported
22
  • 111646: multiply RHS by unity
  • number of inputs: 1; feeds: 1; outputs: 1
  • Multiply RHS of Eq.~\\ref{eq:#2} by 1, which in this case is $#1$; yields Eq.~\\ref{eq:#3}
  1. 4298359835
    \(E=\frac{1}{2}m v^2\)
  1. 0002342425
    \(m/m\)
  1. 4298359845
    \(E=\frac{1}{2m}m^2 v^2\)
 valid
23
  • 111634: 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. 4298359845
    \(E=\frac{1}{2m}m^2 v^2\)
  1. 1029039904
    \(p^2=m^2 v^2\)
  1. 4298359851
    \(E=\frac{p^2}{2m}\)
 LHS diff is pdg0001134**2 - pdg0004931 RHS diff is (-pdg0001134**2/2 + pdg0001357**2*pdg0005156**3)/pdg0005156
24
  • 111680: partially differentiate with respect to
  • number of inputs: 1; feeds: 1; outputs: 1
  • Partially differentiate Eq.~\\ref{eq:#2} with respect to $#1$; yields Eq.~\\ref{eq:#3}.
  1. 3948574230
    \(\psi( \vec{r},t)=\psi_0 \exp\left(\frac{i}{\hbar}\left( \vec{p}\cdot\vec{r} - E t \right) \right)\)
  1. 0006544644
    \(t\)
  1. 3948574233
    \(\frac{\partial}{\partial t} \psi( \vec{r},t)=\psi_0 \frac{\partial}{\partial t}\exp\left(i\left(\frac{ \vec{p}\cdot\vec{r}}{\hbar} - \frac{E t}{\hbar} \right) \right)\)
 recognized infrule but not yet supported
25
  • 111634: 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. 3948574233
    \(\frac{\partial}{\partial t} \psi( \vec{r},t)=\psi_0 \frac{\partial}{\partial t}\exp\left(i\left(\frac{ \vec{p}\cdot\vec{r}}{\hbar} - \frac{E t}{\hbar} \right) \right)\)
  1. 3948574230
    \(\psi( \vec{r},t)=\psi_0 \exp\left(\frac{i}{\hbar}\left( \vec{p}\cdot\vec{r} - E t \right) \right)\)
  1. 3948571256
    \(\frac{\partial}{\partial t} \psi( \vec{r},t)=\frac{-i}{\hbar}E \psi( \vec{r},t)\)
 Type Tuple cannot be instantiated; use tuple() instead
26
  • 111634: 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. 3948571256
    \(\frac{\partial}{\partial t} \psi( \vec{r},t)=\frac{-i}{\hbar}E \psi( \vec{r},t)\)
  1. 4298359851
    \(E=\frac{p^2}{2m}\)
  1. 4348571256
    \(\frac{\partial}{\partial t} \psi( \vec{r},t)=\frac{-i}{\hbar}\frac{p^2}{2 m} \psi( \vec{r},t)\)
 Type Tuple cannot be instantiated; use tuple() instead
27
  • 111182: multiply both sides by
  • number of inputs: 1; feeds: 1; outputs: 1
  • Multiply both sides of Eq.~\\ref{eq:#2} by $#1$; yields Eq.~\\ref{eq:#3}.
  1. 4348571256
    \(\frac{\partial}{\partial t} \psi( \vec{r},t)=\frac{-i}{\hbar}\frac{p^2}{2 m} \psi( \vec{r},t)\)
  1. 0002436656
    \(i \hbar\)
  1. 4341171256
    \(i \hbar \frac{\partial}{\partial t} \psi( \vec{r},t)=\frac{p^2}{2 m} \psi( \vec{r},t)\)
 Type Tuple cannot be instantiated; use tuple() instead
28
  • 111531: apply gradient to scalar function
  • number of inputs: 1; feeds: 0; outputs: 1
  • Apply gradient to both sides of Eq.~\\ref{eq:#1}; yields Eq.~\\ref{eq:#2}.
  1. 3948574230
    \(\psi( \vec{r},t)=\psi_0 \exp\left(\frac{i}{\hbar}\left( \vec{p}\cdot\vec{r} - E t \right) \right)\)
  1. 3948574230
    \(\psi( \vec{r},t)=\psi_0 \exp\left(\frac{i}{\hbar}\left( \vec{p}\cdot\vec{r} - E t \right) \right)\)
 recognized infrule but not yet supported
29
  • 111457: simplify
  • number of inputs: 1; feeds: 0; outputs: 1
  • Simplify Eq.~\\ref{eq:#1}; yields Eq.~\\ref{eq:#2}.
  1. 3948574230
    \(\psi( \vec{r},t)=\psi_0 \exp\left(\frac{i}{\hbar}\left( \vec{p}\cdot\vec{r} - E t \right) \right)\)
  1. 4943571230
    \(\vec{ \nabla} \psi( \vec{r},t)=\frac{i}{\hbar} \vec{p} \psi_0 \exp\left(\frac{i}{\hbar}\left( \vec{p}\cdot\vec{r} - E t \right) \right)\)
 'Symbol' object is not callable
30
  • 111634: 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. 3948574230
    \(\psi( \vec{r},t)=\psi_0 \exp\left(\frac{i}{\hbar}\left( \vec{p}\cdot\vec{r} - E t \right) \right)\)
  1. 4943571230
    \(\vec{ \nabla} \psi( \vec{r},t)=\frac{i}{\hbar} \vec{p} \psi_0 \exp\left(\frac{i}{\hbar}\left( \vec{p}\cdot\vec{r} - E t \right) \right)\)
  1. 5985371230
    \(\vec{ \nabla} \psi( \vec{r},t)=\frac{i}{\hbar} \vec{p} \psi( \vec{r},t)\)
 'Symbol' object is not callable
31
  • 111463: apply divergence
  • number of inputs: 1; feeds: 0; outputs: 1
  • Apply divergence to both sides of Eq.~\\ref{eq:#1}; yields Eq.~\\ref{eq:#2}.
  1. 5985371230
    \(\vec{ \nabla} \psi( \vec{r},t)=\frac{i}{\hbar} \vec{p} \psi( \vec{r},t)\)
  1. 4394958389
    \(\vec{ \nabla}\cdot \left( \vec{ \nabla} \psi( \vec{r},t) \right)=\frac{i}{\hbar} \vec{ \nabla}\cdot\left( \vec{p} \psi( \vec{r},t) \right)\)
 recognized infrule but not yet supported
32
  • 111457: simplify
  • number of inputs: 1; feeds: 0; outputs: 1
  • Simplify Eq.~\\ref{eq:#1}; yields Eq.~\\ref{eq:#2}.
  1. 4394958389
    \(\vec{ \nabla}\cdot \left( \vec{ \nabla} \psi( \vec{r},t) \right)=\frac{i}{\hbar} \vec{ \nabla}\cdot\left( \vec{p} \psi( \vec{r},t) \right)\)
  1. 1648958381
    \(\nabla^2 \psi \left( \vec{r},t \right)=\frac{i}{\hbar} \vec{p} \cdot \left( \vec{ \nabla} \psi( \vec{r},t) \right)\)
 'Symbol' object has no attribute 'dot'
33
  • 111634: 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. 1648958381
    \(\nabla^2 \psi \left( \vec{r},t \right)=\frac{i}{\hbar} \vec{p} \cdot \left( \vec{ \nabla} \psi( \vec{r},t) \right)\)
  1. 5985371230
    \(\vec{ \nabla} \psi( \vec{r},t)=\frac{i}{\hbar} \vec{p} \psi( \vec{r},t)\)
  1. 2648958382
    \(\nabla^2 \psi \left( \vec{r},t \right)=\frac{i}{\hbar} \vec{p} \cdot \left( \frac{i}{\hbar} \vec{p} \psi( \vec{r},t) \right)\)
 invalid syntax (<string>, line 0)
34
  • 111457: simplify
  • number of inputs: 1; feeds: 0; outputs: 1
  • Simplify Eq.~\\ref{eq:#1}; yields Eq.~\\ref{eq:#2}.
  1. 2648958382
    \(\nabla^2 \psi \left( \vec{r},t \right)=\frac{i}{\hbar} \vec{p} \cdot \left( \frac{i}{\hbar} \vec{p} \psi( \vec{r},t) \right)\)
  1. 2395958385
    \(\nabla^2 \psi \left( \vec{r},t \right)=\frac{-p^2}{\hbar} \psi( \vec{r},t)\)
 invalid syntax (<string>, line 0)
35
  • 111182: multiply both sides by
  • number of inputs: 1; feeds: 1; outputs: 1
  • Multiply both sides of Eq.~\\ref{eq:#2} by $#1$; yields Eq.~\\ref{eq:#3}.
  1. 2395958385
    \(\nabla^2 \psi \left( \vec{r},t \right)=\frac{-p^2}{\hbar} \psi( \vec{r},t)\)
  1. 0005938585
    \(\frac{-\hbar^2}{2m}\)
  1. 5868688585
    \(\frac{-\hbar^2}{2m} \nabla^2 \psi \left( \vec{r},t \right)=\frac{p^2}{2m} \psi( \vec{r},t)\)
 RHS arithmetic error. Diff: pdg0001134**2*(pdg0001054 - 1)*pdg0009489(pdg0009472, pdg0001467)/(2*pdg0005156)
36
  • 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. 5868688585
    \(\frac{-\hbar^2}{2m} \nabla^2 \psi \left( \vec{r},t \right)=\frac{p^2}{2m} \psi( \vec{r},t)\)
  1. 4341171256
    \(i \hbar \frac{\partial}{\partial t} \psi( \vec{r},t)=\frac{p^2}{2 m} \psi( \vec{r},t)\)
  1. 9958485859
    \(\frac{-\hbar^2}{2m} \nabla^2 \psi \left( \vec{r},t \right)=i \hbar \frac{\partial}{\partial t} \psi( \vec{r},t)\)
 Type Tuple cannot be instantiated; use tuple() instead
37
  • 111981: declare initial expression
  • number of inputs: 0; feeds: 0; outputs: 1
  • Eq.~\\ref{eq:#1} is an initial equation.
  1. 1158485859
    \(\frac{-\hbar^2}{2m} \nabla^2={\cal H}\)
 no validation is available for declarations
38
  • 111556: substitute LHS of expr 1 into expr 2
  • number of inputs: 2; feeds: 0; outputs: 1
  • Substitute LHS of Eq.~\\ref{eq:#1} into Eq.~\\ref{eq:#2}; yields Eq.~\\ref{eq:#3}.
  1. 1158485859
    \(\frac{-\hbar^2}{2m} \nabla^2={\cal H}\)
  1. 9958485859
    \(\frac{-\hbar^2}{2m} \nabla^2 \psi \left( \vec{r},t \right)=i \hbar \frac{\partial}{\partial t} \psi( \vec{r},t)\)
  1. 2258485859
    \({\cal H} \psi \left( \vec{r},t \right)=i \hbar \frac{\partial}{\partial t} \psi( \vec{r},t)\)
 invalid syntax (<string>, line 0)
39
  • 111341: declare final expression
  • number of inputs: 1; feeds: 0; outputs: 0
  • Eq.~\\ref{eq:#1} is one of the final equations.
  1. 2258485859
    \({\cal H} \psi \left( \vec{r},t \right)=i \hbar \frac{\partial}{\partial t} \psi( \vec{r},t)\)
 no validation is available for declarations


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