Review Derivation: Physics Derivation Graph

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Review derivation of Schrodinger Equation

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reference:

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.			3121513111 $k = \frac{2 \pi}{\lambda}$	no validation is available for declarations
2	0000111981: declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\ref{eq:#1} is an initial equation.			3131211131 $\omega = 2 \pi f$	no validation is available for declarations
3	0000111981: declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\ref{eq:#1} is an initial equation.			9999999960 $\hbar = h/(2 \pi)$	no validation is available for declarations
4	0000111981: declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\ref{eq:#1} is an initial equation.			1020394900 $p = h/\lambda$	no validation is available for declarations
5	0000111981: declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\ref{eq:#1} is an initial equation.			1020394902 $E = h f$	no validation is available for declarations
6	0000111975: 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}.	3131211131 $\omega = 2 \pi f$	0002940021 $2 \pi$	3147472131 $\frac{\omega}{2 \pi} = f$	valid
7	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}.	1020394902 $E = h f$ 3147472131 $\frac{\omega}{2 \pi} = f$		4147472132 $E = \frac{h \omega}{2 \pi}$	LHS diff is pdg0002321/(2pdg0003141) - pdg0004931 RHS diff is -pdg0002321pdg0004413/(2*pdg0003141) + pdg0004201
8	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}.	4147472132 $E = \frac{h \omega}{2 \pi}$ 9999999960 $\hbar = h/(2 \pi)$		9999999965 $E = \omega \hbar$	LHS diff is pdg0001054 - pdg0004931 RHS diff is -pdg0001054pdg0002321 + pdg0004413/(2pdg0003141)
9	0000111975: 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}.	9999999965 $E = \omega \hbar$	0003949921 $\hbar$	9999999961 $\frac{E}{\hbar} = \omega$	valid
10	0000111975: 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}.	3121513111 $k = \frac{2 \pi}{\lambda}$	0001209482 $2 \pi$	3121234211 $\frac{k}{2\pi} = \lambda$	Algebraic error: LHS diff is 0, RHS diff is -pdg0001115 + 1/pdg0001115
11	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}.	3121234211 $\frac{k}{2\pi} = \lambda$ 1020394900 $p = h/\lambda$		3121234212 $p = \frac{h k}{2\pi}$	LHS diff is 0 RHS diff is 2pdg0003141pdg0004413/pdg0005321 - pdg0004413pdg0005321/(2pdg0003141)
12	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}.	9999999960 $\hbar = h/(2 \pi)$ 3121234212 $p = \frac{h k}{2\pi}$		9999999962 $p = \hbar k$	valid
13	0000111975: 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}.	9999999962 $p = \hbar k$	0001304952 $\hbar$	9999999870 $\frac{p}{\hbar} = k$	valid
14	0000111215: 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}.	9999999870 $\frac{p}{\hbar} = k$		9999998870 $\frac{ \vec{p}}{\hbar} = \vec{k}$	recognized infrule but not yet supported
15	0000111981: declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\ref{eq:#1} is an initial equation.			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	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}.	9999998870 $\frac{ \vec{p}}{\hbar} = \vec{k}$ 3948574224 $\psi( \vec{r},t) = \psi_0 \exp\left(i\left( \vec{k}\cdot\vec{r} - \omega t \right) \right)$		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((-pdg0001054pdg0001467pdg0002321 + pdg0001134pdg0009472)/pdg0001054)) + pdg0002718(pdg0004621(-pdg0001467*pdg0002321 + dot(pdg0005321, pdg0009472))))
17	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}.	9999999961 $\frac{E}{\hbar} = \omega$ 3948574226 $\psi( \vec{r},t) = \psi_0 \exp\left(i\left(\frac{ \vec{p}\cdot\vec{r}}{\hbar} - \omega t \right) \right)$		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((pdg0001134pdg0009472 - pdg0001467pdg0004931)/pdg0001054)) - pdg0002718(pdg0004621((pdg0001134pdg0009472 - pdg0001467*pdg0006238)/pdg0001054)))
18	0000111457: simplify number of inputs: 1; feeds: 0; outputs: 1 Simplify Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.	3948574228 $\psi( \vec{r},t) = \psi_0 \exp\left(i\left(\frac{ \vec{p}\cdot\vec{r}}{\hbar} - \frac{E t}{\hbar} \right) \right)$		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(pdg0001134pdg0009472 - pdg0001467pdg0006238)/pdg0001054) + pdg0002718(pdg0004621((pdg0001134pdg0009472 - pdg0001467pdg0006238)/pdg0001054)))
19	0000111981: declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\ref{eq:#1} is an initial equation.			1029039903 $p = m v$	no validation is available for declarations
20	0000111981: declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\ref{eq:#1} is an initial equation.			4298359835 $E = \frac{1}{2}m v^2$	no validation is available for declarations
21	0000111483: 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}.	1029039903 $p = m v$	0002239424 $2$	1029039904 $p^2 = m^2 v^2$	recognized infrule but not yet supported
22	0000111646: 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}	4298359835 $E = \frac{1}{2}m v^2$	0002342425 $m/m$	4298359845 $E = \frac{1}{2m}m^2 v^2$	valid
23	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}.	4298359845 $E = \frac{1}{2m}m^2 v^2$ 1029039904 $p^2 = m^2 v^2$		4298359851 $E = \frac{p^2}{2m}$	LHS diff is pdg00011342 - pdg0004931 RHS diff is (-pdg00011342/2 + pdg0001357*2pdg0005156**3)/pdg0005156
24	0000111680: 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}.	3948574230 $\psi( \vec{r},t) = \psi_0 \exp\left(\frac{i}{\hbar}\left( \vec{p}\cdot\vec{r} - E t \right) \right)$	0006544644 $t$	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	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}.	3948574230 $\psi( \vec{r},t) = \psi_0 \exp\left(\frac{i}{\hbar}\left( \vec{p}\cdot\vec{r} - E t \right) \right)$ 3948574233 $\frac{\partial}{\partial t} \psi( \vec{r},t) = \psi_0 \frac{\partial}{\partial t}\exp\left(i\left(\frac{ \vec{p}\cdot\vec{r}}{\hbar} - \frac{E t}{\hbar} \right) \right)$		3948571256 $\frac{\partial}{\partial t} \psi( \vec{r},t) = \frac{-i}{\hbar}E \psi( \vec{r},t)$	LHS diff is 0 RHS diff is (pdg0004621pdg0006238pdg0009489(pdg0009472, pdg0001467) - pdg0006238pdg0008330Derivative(pdg0002718(pdg0004621((pdg0001134pdg0009472 - pdg0001467pdg0006238)/pdg0001054)), pdg0004621((pdg0001134pdg0009472 - pdg0001467pdg0006238)/pdg0001054))Subs(Derivative(pdg0004621(_xi_1), _xi_1), _xi_1, (pdg0001134pdg0009472 - pdg0001467*pdg0006238)/pdg0001054))/pdg0001054
26	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}.	3948571256 $\frac{\partial}{\partial t} \psi( \vec{r},t) = \frac{-i}{\hbar}E \psi( \vec{r},t)$ 4298359851 $E = \frac{p^2}{2m}$		4348571256 $\frac{\partial}{\partial t} \psi( \vec{r},t) = \frac{-i}{\hbar}\frac{p^2}{2 m} \psi( \vec{r},t)$	LHS diff is pdg0004931 - Derivative(pdg0009489(pdg0009472, pdg0001467), pdg0001467) RHS diff is pdg0001134*2(pdg0001054 + pdg0004621pdg0009489(pdg0009472, pdg0001467))/(2pdg0001054*pdg0005156)
27	0000111182: 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}.	4348571256 $\frac{\partial}{\partial t} \psi( \vec{r},t) = \frac{-i}{\hbar}\frac{p^2}{2 m} \psi( \vec{r},t)$	0002436656 $i \hbar$	4341171256 $i \hbar \frac{\partial}{\partial t} \psi( \vec{r},t) = \frac{p^2}{2 m} \psi( \vec{r},t)$	RHS arithmetic error. Diff: pdg0001134*2(-pdg0004621*2 - 1)pdg0009489(pdg0009472, pdg0001467)/(2*pdg0005156)
28	0000111531: 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}.	3948574230 $\psi( \vec{r},t) = \psi_0 \exp\left(\frac{i}{\hbar}\left( \vec{p}\cdot\vec{r} - E t \right) \right)$		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	0000111457: simplify number of inputs: 1; feeds: 0; outputs: 1 Simplify Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.	3948574230 $\psi( \vec{r},t) = \psi_0 \exp\left(\frac{i}{\hbar}\left( \vec{p}\cdot\vec{r} - E t \right) \right)$		4943571230 $\vec{ \nabla} \psi( \vec{r},t) = \frac{i}{\hbar} \vec{p} \psi_0 \exp\left(\frac{i}{\hbar}\left( \vec{p}\cdot\vec{r} - E t \right) \right)$	LHS diff is (1 - nabla)pdg0009489(pdg0009472, pdg0001467) RHS diff is pdg0008330pdg0002718(pdg0001134pdg0004621pdg0009472/pdg0001054 - pdg0001467pdg0004621pdg0006238/pdg0001054) - pdg0002046pdg0004621pdg0008330exp(-pdg0001467pdg0004621pdg0006238/pdg0001054 + pdg0002046pdg0004621*pdg0009472/pdg0001054)/pdg0001054
30	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}.	3948574230 $\psi( \vec{r},t) = \psi_0 \exp\left(\frac{i}{\hbar}\left( \vec{p}\cdot\vec{r} - E t \right) \right)$ 4943571230 $\vec{ \nabla} \psi( \vec{r},t) = \frac{i}{\hbar} \vec{p} \psi_0 \exp\left(\frac{i}{\hbar}\left( \vec{p}\cdot\vec{r} - E t \right) \right)$		5985371230 $\vec{ \nabla} \psi( \vec{r},t) = \frac{i}{\hbar} \vec{p} \psi( \vec{r},t)$	LHS diff is 0 RHS diff is pdg0002046pdg0004621pdg0008330exp(-pdg0001467pdg0004621pdg0006238/pdg0001054 + pdg0002046pdg0004621pdg0009472/pdg0001054)/pdg0001054 - pdg0002046pdg0004621*pdg0009489(pdg0009472, pdg0001467)/pdg0001054
31	0000111463: 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}.	5985371230 $\vec{ \nabla} \psi( \vec{r},t) = \frac{i}{\hbar} \vec{p} \psi( \vec{r},t)$		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	0000111457: simplify number of inputs: 1; feeds: 0; outputs: 1 Simplify Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.	4394958389 $\vec{ \nabla}\cdot \left( \vec{ \nabla} \psi( \vec{r},t) \right) = \frac{i}{\hbar} \vec{ \nabla}\cdot\left( \vec{p} \psi( \vec{r},t) \right)$		1648958381 $\nabla^2 \psi \left( \vec{r},t \right) = \frac{i}{\hbar} \vec{p} \cdot \left( \vec{ \nabla} \psi( \vec{r},t) \right)$
33	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}.	1648958381 $\nabla^2 \psi \left( \vec{r},t \right) = \frac{i}{\hbar} \vec{p} \cdot \left( \vec{ \nabla} \psi( \vec{r},t) \right)$ 5985371230 $\vec{ \nabla} \psi( \vec{r},t) = \frac{i}{\hbar} \vec{p} \psi( \vec{r},t)$		2648958382 $\nabla^2 \psi \left( \vec{r},t \right) = \frac{i}{\hbar} \vec{p} \cdot \left( \frac{i}{\hbar} \vec{p} \psi( \vec{r},t) \right)$	Not evaluated due to missing term in SymPy
34	0000111457: simplify number of inputs: 1; feeds: 0; outputs: 1 Simplify Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.	2648958382 $\nabla^2 \psi \left( \vec{r},t \right) = \frac{i}{\hbar} \vec{p} \cdot \left( \frac{i}{\hbar} \vec{p} \psi( \vec{r},t) \right)$		2395958385 $\nabla^2 \psi \left( \vec{r},t \right) = \frac{-p^2}{\hbar} \psi( \vec{r},t)$	Not evaluated due to missing term in SymPy
35	0000111182: 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}.	2395958385 $\nabla^2 \psi \left( \vec{r},t \right) = \frac{-p^2}{\hbar} \psi( \vec{r},t)$	0005938585 $\frac{-\hbar^2}{2m}$	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)/(2pdg0005156)
36	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}.	5868688585 $\frac{-\hbar^2}{2m} \nabla^2 \psi \left( \vec{r},t \right) = \frac{p^2}{2m} \psi( \vec{r},t)$ 4341171256 $i \hbar \frac{\partial}{\partial t} \psi( \vec{r},t) = \frac{p^2}{2 m} \psi( \vec{r},t)$		9958485859 $\frac{-\hbar^2}{2m} \nabla^2 \psi \left( \vec{r},t \right) = i \hbar \frac{\partial}{\partial t} \psi( \vec{r},t)$	Not evaluated due to missing term in SymPy
37	0000111981: declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\ref{eq:#1} is an initial equation.			1158485859 $\frac{-\hbar^2}{2m} \nabla^2 = {\cal H}$	no validation is available for declarations
38	0000111556: 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}.	9958485859 $\frac{-\hbar^2}{2m} \nabla^2 \psi \left( \vec{r},t \right) = i \hbar \frac{\partial}{\partial t} \psi( \vec{r},t)$ 1158485859 $\frac{-\hbar^2}{2m} \nabla^2 = {\cal H}$		2258485859 ${\cal H} \psi \left( \vec{r},t \right) = i \hbar \frac{\partial}{\partial t} \psi( \vec{r},t)$	Not evaluated due to missing term in SymPy
39	0000111341: declare final expression number of inputs: 1; feeds: 0; outputs: 0 Eq.~\ref{eq:#1} is one of the final equations.	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

Symbols used in derivation of Schrodinger Equation

0000002718: $ \exp $
0000001054: $ \hbar $
0000001115: $ \lambda $
0000002321: $ \omega $
0000003141: $ \pi $
0000009489: $ \psi $
0000008330: $ \psi_0 $
0000007394: $ \vec{k} $
0000002046: $ \vec{p} $
0000009472: $ \vec{r} $
0000006238: $ E $
0000004931: $ E $
0000004201: $ f $
0000004413: $ h $
0000004621: $ i $
0000005321: $ k $
0000005156: $ m $
0000001134: $ p $
0000001467: $ t $
0000001357: $ v $
0000006799: $ {\cal H} $

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0.006 seconds for pdg_app/to_review_derivation fa7819f6-8e06-46ba-b861-bf9953d1b762
0.017 seconds for compute/get_dict_of_steps_in_derivation: steps_in_this_derivation06266f12-9573-4571-9ad4-efdbc1dcfda0
0.005 seconds for compute/input_feed_output_infrule_for_step: get_inference_rule_connected_to_step_ID06266f12-9573-4571-9ad4-efdbc1dcfda0
0.006 seconds for compute/input_feed_output_infrule_for_step: get_expressions_from_step_id_and_expr_type HAS_INPUT06266f12-9573-4571-9ad4-efdbc1dcfda0
0.005 seconds for compute/input_feed_output_infrule_for_step: get_expressions_from_step_id_and_expr_type, HAS_FEED06266f12-9573-4571-9ad4-efdbc1dcfda0
0.005 seconds for compute/input_feed_output_infrule_for_step: get_expressions_from_step_id_and_expr_type, HAS_OUTPUT06266f12-9573-4571-9ad4-efdbc1dcfda0
0.005 seconds for compute/get_dict_of_steps_in_derivation: get_sequence_index_for_step06266f12-9573-4571-9ad4-efdbc1dcfda0
0.007 seconds for pdg_app/ fa7819f6-8e06-46ba-b861-bf9953d1b762

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.			3121513111 \(k = \frac{2 \pi}{\lambda}\)	no validation is available for declarations
2	0000111981: declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\ref{eq:#1} is an initial equation.			3131211131 \(\omega = 2 \pi f\)	no validation is available for declarations
3	0000111981: declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\ref{eq:#1} is an initial equation.			9999999960 \(\hbar = h/(2 \pi)\)	no validation is available for declarations
4	0000111981: declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\ref{eq:#1} is an initial equation.			1020394900 \(p = h/\lambda\)	no validation is available for declarations
5	0000111981: declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\ref{eq:#1} is an initial equation.			1020394902 \(E = h f\)	no validation is available for declarations
6	0000111975: 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}.	3131211131 \(\omega = 2 \pi f\)	0002940021 \(2 \pi\)	3147472131 \(\frac{\omega}{2 \pi} = f\)	valid
7	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}.	1020394902 \(E = h f\) 3147472131 \(\frac{\omega}{2 \pi} = f\)		4147472132 \(E = \frac{h \omega}{2 \pi}\)	LHS diff is pdg0002321/(2pdg0003141) - pdg0004931 RHS diff is -pdg0002321pdg0004413/(2*pdg0003141) + pdg0004201
8	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}.	4147472132 \(E = \frac{h \omega}{2 \pi}\) 9999999960 \(\hbar = h/(2 \pi)\)		9999999965 \(E = \omega \hbar\)	LHS diff is pdg0001054 - pdg0004931 RHS diff is -pdg0001054pdg0002321 + pdg0004413/(2pdg0003141)
9	0000111975: 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}.	9999999965 \(E = \omega \hbar\)	0003949921 \(\hbar\)	9999999961 \(\frac{E}{\hbar} = \omega\)	valid
10	0000111975: 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}.	3121513111 \(k = \frac{2 \pi}{\lambda}\)	0001209482 \(2 \pi\)	3121234211 \(\frac{k}{2\pi} = \lambda\)	Algebraic error: LHS diff is 0, RHS diff is -pdg0001115 + 1/pdg0001115
11	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}.	3121234211 \(\frac{k}{2\pi} = \lambda\) 1020394900 \(p = h/\lambda\)		3121234212 \(p = \frac{h k}{2\pi}\)	LHS diff is 0 RHS diff is 2pdg0003141pdg0004413/pdg0005321 - pdg0004413pdg0005321/(2pdg0003141)
12	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}.	9999999960 \(\hbar = h/(2 \pi)\) 3121234212 \(p = \frac{h k}{2\pi}\)		9999999962 \(p = \hbar k\)	valid
13	0000111975: 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}.	9999999962 \(p = \hbar k\)	0001304952 \(\hbar\)	9999999870 \(\frac{p}{\hbar} = k\)	valid
14	0000111215: 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}.	9999999870 \(\frac{p}{\hbar} = k\)		9999998870 \(\frac{ \vec{p}}{\hbar} = \vec{k}\)	recognized infrule but not yet supported
15	0000111981: declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\ref{eq:#1} is an initial equation.			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	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}.	9999998870 \(\frac{ \vec{p}}{\hbar} = \vec{k}\) 3948574224 \(\psi( \vec{r},t) = \psi_0 \exp\left(i\left( \vec{k}\cdot\vec{r} - \omega t \right) \right)\)		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((-pdg0001054pdg0001467pdg0002321 + pdg0001134pdg0009472)/pdg0001054)) + pdg0002718(pdg0004621(-pdg0001467*pdg0002321 + dot(pdg0005321, pdg0009472))))
17	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}.	9999999961 \(\frac{E}{\hbar} = \omega\) 3948574226 \(\psi( \vec{r},t) = \psi_0 \exp\left(i\left(\frac{ \vec{p}\cdot\vec{r}}{\hbar} - \omega t \right) \right)\)		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((pdg0001134pdg0009472 - pdg0001467pdg0004931)/pdg0001054)) - pdg0002718(pdg0004621((pdg0001134pdg0009472 - pdg0001467*pdg0006238)/pdg0001054)))
18	0000111457: simplify number of inputs: 1; feeds: 0; outputs: 1 Simplify Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.	3948574228 \(\psi( \vec{r},t) = \psi_0 \exp\left(i\left(\frac{ \vec{p}\cdot\vec{r}}{\hbar} - \frac{E t}{\hbar} \right) \right)\)		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(pdg0001134pdg0009472 - pdg0001467pdg0006238)/pdg0001054) + pdg0002718(pdg0004621((pdg0001134pdg0009472 - pdg0001467pdg0006238)/pdg0001054)))
19	0000111981: declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\ref{eq:#1} is an initial equation.			1029039903 \(p = m v\)	no validation is available for declarations
20	0000111981: declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\ref{eq:#1} is an initial equation.			4298359835 \(E = \frac{1}{2}m v^2\)	no validation is available for declarations
21	0000111483: 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}.	1029039903 \(p = m v\)	0002239424 \(2\)	1029039904 \(p^2 = m^2 v^2\)	recognized infrule but not yet supported
22	0000111646: 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}	4298359835 \(E = \frac{1}{2}m v^2\)	0002342425 \(m/m\)	4298359845 \(E = \frac{1}{2m}m^2 v^2\)	valid
23	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}.	4298359845 \(E = \frac{1}{2m}m^2 v^2\) 1029039904 \(p^2 = m^2 v^2\)		4298359851 \(E = \frac{p^2}{2m}\)	LHS diff is pdg00011342 - pdg0004931 RHS diff is (-pdg00011342/2 + pdg0001357*2pdg0005156**3)/pdg0005156
24	0000111680: 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}.	3948574230 \(\psi( \vec{r},t) = \psi_0 \exp\left(\frac{i}{\hbar}\left( \vec{p}\cdot\vec{r} - E t \right) \right)\)	0006544644 \(t\)	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	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}.	3948574230 \(\psi( \vec{r},t) = \psi_0 \exp\left(\frac{i}{\hbar}\left( \vec{p}\cdot\vec{r} - E t \right) \right)\) 3948574233 \(\frac{\partial}{\partial t} \psi( \vec{r},t) = \psi_0 \frac{\partial}{\partial t}\exp\left(i\left(\frac{ \vec{p}\cdot\vec{r}}{\hbar} - \frac{E t}{\hbar} \right) \right)\)		3948571256 \(\frac{\partial}{\partial t} \psi( \vec{r},t) = \frac{-i}{\hbar}E \psi( \vec{r},t)\)	LHS diff is 0 RHS diff is (pdg0004621pdg0006238pdg0009489(pdg0009472, pdg0001467) - pdg0006238pdg0008330Derivative(pdg0002718(pdg0004621((pdg0001134pdg0009472 - pdg0001467pdg0006238)/pdg0001054)), pdg0004621((pdg0001134pdg0009472 - pdg0001467pdg0006238)/pdg0001054))Subs(Derivative(pdg0004621(_xi_1), _xi_1), _xi_1, (pdg0001134pdg0009472 - pdg0001467*pdg0006238)/pdg0001054))/pdg0001054
26	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}.	3948571256 \(\frac{\partial}{\partial t} \psi( \vec{r},t) = \frac{-i}{\hbar}E \psi( \vec{r},t)\) 4298359851 \(E = \frac{p^2}{2m}\)		4348571256 \(\frac{\partial}{\partial t} \psi( \vec{r},t) = \frac{-i}{\hbar}\frac{p^2}{2 m} \psi( \vec{r},t)\)	LHS diff is pdg0004931 - Derivative(pdg0009489(pdg0009472, pdg0001467), pdg0001467) RHS diff is pdg0001134*2(pdg0001054 + pdg0004621pdg0009489(pdg0009472, pdg0001467))/(2pdg0001054*pdg0005156)
27	0000111182: 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}.	4348571256 \(\frac{\partial}{\partial t} \psi( \vec{r},t) = \frac{-i}{\hbar}\frac{p^2}{2 m} \psi( \vec{r},t)\)	0002436656 \(i \hbar\)	4341171256 \(i \hbar \frac{\partial}{\partial t} \psi( \vec{r},t) = \frac{p^2}{2 m} \psi( \vec{r},t)\)	RHS arithmetic error. Diff: pdg0001134*2(-pdg0004621*2 - 1)pdg0009489(pdg0009472, pdg0001467)/(2*pdg0005156)
28	0000111531: 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}.	3948574230 \(\psi( \vec{r},t) = \psi_0 \exp\left(\frac{i}{\hbar}\left( \vec{p}\cdot\vec{r} - E t \right) \right)\)		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	0000111457: simplify number of inputs: 1; feeds: 0; outputs: 1 Simplify Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.	3948574230 \(\psi( \vec{r},t) = \psi_0 \exp\left(\frac{i}{\hbar}\left( \vec{p}\cdot\vec{r} - E t \right) \right)\)		4943571230 \(\vec{ \nabla} \psi( \vec{r},t) = \frac{i}{\hbar} \vec{p} \psi_0 \exp\left(\frac{i}{\hbar}\left( \vec{p}\cdot\vec{r} - E t \right) \right)\)	LHS diff is (1 - nabla)pdg0009489(pdg0009472, pdg0001467) RHS diff is pdg0008330pdg0002718(pdg0001134pdg0004621pdg0009472/pdg0001054 - pdg0001467pdg0004621pdg0006238/pdg0001054) - pdg0002046pdg0004621pdg0008330exp(-pdg0001467pdg0004621pdg0006238/pdg0001054 + pdg0002046pdg0004621*pdg0009472/pdg0001054)/pdg0001054
30	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}.	3948574230 \(\psi( \vec{r},t) = \psi_0 \exp\left(\frac{i}{\hbar}\left( \vec{p}\cdot\vec{r} - E t \right) \right)\) 4943571230 \(\vec{ \nabla} \psi( \vec{r},t) = \frac{i}{\hbar} \vec{p} \psi_0 \exp\left(\frac{i}{\hbar}\left( \vec{p}\cdot\vec{r} - E t \right) \right)\)		5985371230 \(\vec{ \nabla} \psi( \vec{r},t) = \frac{i}{\hbar} \vec{p} \psi( \vec{r},t)\)	LHS diff is 0 RHS diff is pdg0002046pdg0004621pdg0008330exp(-pdg0001467pdg0004621pdg0006238/pdg0001054 + pdg0002046pdg0004621pdg0009472/pdg0001054)/pdg0001054 - pdg0002046pdg0004621*pdg0009489(pdg0009472, pdg0001467)/pdg0001054
31	0000111463: 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}.	5985371230 \(\vec{ \nabla} \psi( \vec{r},t) = \frac{i}{\hbar} \vec{p} \psi( \vec{r},t)\)		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	0000111457: simplify number of inputs: 1; feeds: 0; outputs: 1 Simplify Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.	4394958389 \(\vec{ \nabla}\cdot \left( \vec{ \nabla} \psi( \vec{r},t) \right) = \frac{i}{\hbar} \vec{ \nabla}\cdot\left( \vec{p} \psi( \vec{r},t) \right)\)		1648958381 \(\nabla^2 \psi \left( \vec{r},t \right) = \frac{i}{\hbar} \vec{p} \cdot \left( \vec{ \nabla} \psi( \vec{r},t) \right)\)
33	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}.	1648958381 \(\nabla^2 \psi \left( \vec{r},t \right) = \frac{i}{\hbar} \vec{p} \cdot \left( \vec{ \nabla} \psi( \vec{r},t) \right)\) 5985371230 \(\vec{ \nabla} \psi( \vec{r},t) = \frac{i}{\hbar} \vec{p} \psi( \vec{r},t)\)		2648958382 \(\nabla^2 \psi \left( \vec{r},t \right) = \frac{i}{\hbar} \vec{p} \cdot \left( \frac{i}{\hbar} \vec{p} \psi( \vec{r},t) \right)\)	Not evaluated due to missing term in SymPy
34	0000111457: simplify number of inputs: 1; feeds: 0; outputs: 1 Simplify Eq.~\ref{eq:#1}; yields Eq.~\ref{eq:#2}.	2648958382 \(\nabla^2 \psi \left( \vec{r},t \right) = \frac{i}{\hbar} \vec{p} \cdot \left( \frac{i}{\hbar} \vec{p} \psi( \vec{r},t) \right)\)		2395958385 \(\nabla^2 \psi \left( \vec{r},t \right) = \frac{-p^2}{\hbar} \psi( \vec{r},t)\)	Not evaluated due to missing term in SymPy
35	0000111182: 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}.	2395958385 \(\nabla^2 \psi \left( \vec{r},t \right) = \frac{-p^2}{\hbar} \psi( \vec{r},t)\)	0005938585 \(\frac{-\hbar^2}{2m}\)	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)/(2pdg0005156)
36	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}.	5868688585 \(\frac{-\hbar^2}{2m} \nabla^2 \psi \left( \vec{r},t \right) = \frac{p^2}{2m} \psi( \vec{r},t)\) 4341171256 \(i \hbar \frac{\partial}{\partial t} \psi( \vec{r},t) = \frac{p^2}{2 m} \psi( \vec{r},t)\)		9958485859 \(\frac{-\hbar^2}{2m} \nabla^2 \psi \left( \vec{r},t \right) = i \hbar \frac{\partial}{\partial t} \psi( \vec{r},t)\)	Not evaluated due to missing term in SymPy
37	0000111981: declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\ref{eq:#1} is an initial equation.			1158485859 \(\frac{-\hbar^2}{2m} \nabla^2 = {\cal H}\)	no validation is available for declarations
38	0000111556: 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}.	9958485859 \(\frac{-\hbar^2}{2m} \nabla^2 \psi \left( \vec{r},t \right) = i \hbar \frac{\partial}{\partial t} \psi( \vec{r},t)\) 1158485859 \(\frac{-\hbar^2}{2m} \nabla^2 = {\cal H}\)		2258485859 \({\cal H} \psi \left( \vec{r},t \right) = i \hbar \frac{\partial}{\partial t} \psi( \vec{r},t)\)	Not evaluated due to missing term in SymPy
39	0000111341: declare final expression number of inputs: 1; feeds: 0; outputs: 0 Eq.~\ref{eq:#1} is one of the final equations.	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