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review derivation: derivation of Schrodinger Equation

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Index Inference Rule Input latex Feeds latex Output latex step validity dimension check unit check notes
14 replace scalar with vector
  1. 9999999870; locally 4948325:
    \(\frac{p}{\hbar} = k\)
    \(\)
  1. 9999998870; locally 2948487:
    \(\frac{ \vec{p}}{\hbar} = \vec{k}\)
    \(\)
no check performed 9999999870:
9999998870:
9999999870:
9999998870:
17 substitute RHS of expr 1 into expr 2
  1. 3948574226; locally 2100421:
    \(\psi( \vec{r},t) = \psi_0 \exp\left(i\left(\frac{ \vec{p}\cdot\vec{r}}{\hbar} - \omega t \right) \right)\)
    \(\)
  2. 9999999961; locally 4499582:
    \(\frac{E}{\hbar} = \omega\)
    \(\)
  1. 3948574228; locally 1291313:
    \(\psi( \vec{r},t) = \psi_0 \exp\left(i\left(\frac{ \vec{p}\cdot\vec{r}}{\hbar} - \frac{E t}{\hbar} \right) \right)\)
    \(\)
LHS diff is -pdg9489(pdg9472, pdg1467) + pdg4931/pdg1054 RHS diff is pdg2321 - pdg8330*pdg2718(pdg4621((pdg1134*pdg9472 - pdg1467*pdg6238)/pdg1054)) 3948574226:
9999999961:
3948574228:
3948574226:
9999999961:
3948574228:
1 declare initial expr
  1. 3121513111; locally 2934848:
    \(k = \frac{2 \pi}{\lambda}\)
    \(\)
no validation is available for declarations 3121513111:
3121513111:
19 declare initial expr
  1. 1029039903; locally 1039948:
    \(p = m v\)
    \(\)
no validation is available for declarations 1029039903:
1029039903:
11 substitute RHS of expr 1 into expr 2
  1. 1020394900; locally 1203491:
    \(p = h/\lambda\)
    \(\)
  2. 3121234211; locally 1039485:
    \(\frac{k}{2\pi} = \lambda\)
    \(\)
  1. 3121234212; locally 2901049:
    \(p = \frac{h k}{2\pi}\)
    \(\)
LHS diff is -pdg1134 + pdg5321/(2*pdg3141) RHS diff is pdg1115 - pdg4413*pdg5321/(2*pdg3141) 1020394900: dimensions are consistent
3121234211:
3121234212:
1020394900: N/A
3121234211:
3121234212:
3 declare initial expr
  1. 9999999960; locally 2949002:
    \(\hbar = h/(2 \pi)\)
    \(\)
no validation is available for declarations 9999999960:
9999999960:
7 substitute RHS of expr 1 into expr 2
  1. 3147472131; locally 2939402:
    \(\frac{\omega}{2 \pi} = f\)
    \(\)
  2. 1020394902; locally 3499522:
    \(E = h f\)
    \(\)
  1. 4147472132; locally 2949821:
    \(E = \frac{h \omega}{2 \pi}\)
    \(\)
valid 3147472131:
1020394902:
4147472132:
3147472131:
1020394902:
4147472132:
18 simplify
  1. 3948574228; locally 1291313:
    \(\psi( \vec{r},t) = \psi_0 \exp\left(i\left(\frac{ \vec{p}\cdot\vec{r}}{\hbar} - \frac{E t}{\hbar} \right) \right)\)
    \(\)
  1. 3948574230; locally 1305534:
    \(\psi( \vec{r},t) = \psi_0 \exp\left(\frac{i}{\hbar}\left( \vec{p}\cdot\vec{r} - E t \right) \right)\)
    \(\)
LHS diff is 0 RHS diff is pdg8330*(-pdg2718(pdg4621*(pdg1134*pdg9472 - pdg1467*pdg6238)/pdg1054) + pdg2718(pdg4621((pdg1134*pdg9472 - pdg1467*pdg6238)/pdg1054))) 3948574228:
3948574230:
3948574228:
3948574230:
24 partially differentiate with respect to
  1. 3948574230; locally 1305534:
    \(\psi( \vec{r},t) = \psi_0 \exp\left(\frac{i}{\hbar}\left( \vec{p}\cdot\vec{r} - E t \right) \right)\)
    \(\)
  1. 0006544644:
    \(t\)
    \(\)
  1. 3948574233; locally 2364546:
    \(\frac{\partial}{\partial t} \psi( \vec{r},t) = \psi_0 \frac{\partial}{\partial t}\exp\left(i\left(\frac{ \vec{p}\cdot\vec{r}}{\hbar} - \frac{E t}{\hbar} \right) \right)\)
    \(\)
no check performed 3948574230:
3948574233:
3948574230:
3948574233:
20 declare initial expr
  1. 4298359835; locally 1353583:
    \(E = \frac{1}{2}m v^2\)
    \(\)
no validation is available for declarations 4298359835:
4298359835:
15 declare initial expr
  1. 3948574224; locally 3940505:
    \(\psi( \vec{r},t) = \psi_0 \exp\left(i\left( \vec{k}\cdot\vec{r} - \omega t \right) \right)\)
    \(\)
no validation is available for declarations 3948574224:
3948574224:
13 divide both sides by
  1. 9999999962; locally 1039013:
    \(p = \hbar k\)
    \(\)
  1. 0001304952:
    \(\hbar\)
    \(\)
  1. 9999999870; locally 4948325:
    \(\frac{p}{\hbar} = k\)
    \(\)
valid 9999999962:
9999999870:
9999999962:
9999999870:
22 multiply RHS by unity
  1. 4298359835; locally 1353583:
    \(E = \frac{1}{2}m v^2\)
    \(\)
  1. 0002342425:
    \(m/m\)
    \(\)
  1. 4298359845; locally 2326309:
    \(E = \frac{1}{2m}m^2 v^2\)
    \(\)
valid 4298359835:
4298359845:
4298359835:
4298359845:
32 simplify
  1. 4394958389; locally 4938589:
    \(\vec{ \nabla}\cdot \left( \vec{ \nabla} \psi( \vec{r},t) \right) = \frac{i}{\hbar} \vec{ \nabla}\cdot\left( \vec{p} \psi( \vec{r},t) \right)\)
    \(\)
  1. 1648958381; locally 1495034:
    \(\nabla^2 \psi \left( \vec{r},t \right) = \frac{i}{\hbar} \vec{p} \cdot \left( \vec{ \nabla} \psi( \vec{r},t) \right)\)
    \(\)
failed 4394958389:
1648958381:
4394958389:
1648958381:
21 raise both sides to power
  1. 1029039903; locally 1039948:
    \(p = m v\)
    \(\)
  1. 0002239424:
    \(2\)
    \(\)
  1. 1029039904; locally 1432042:
    \(p^2 = m^2 v^2\)
    \(\)
no check is performed 1029039903:
1029039904:
1029039903:
1029039904:
23 substitute RHS of expr 1 into expr 2
  1. 1029039904; locally 1432042:
    \(p^2 = m^2 v^2\)
    \(\)
  2. 4298359845; locally 2326309:
    \(E = \frac{1}{2m}m^2 v^2\)
    \(\)
  1. 4298359851; locally 3576787:
    \(E = \frac{p^2}{2m}\)
    \(\)
LHS diff is 0 RHS diff is (-pdg1134**2 + pdg1357**2*pdg5156**2)/(2*pdg5156) 1029039904:
4298359845:
4298359851:
1029039904:
4298359845:
4298359851:
28 apply gradient to scalar function
  1. 3948574230; locally 1305534:
    \(\psi( \vec{r},t) = \psi_0 \exp\left(\frac{i}{\hbar}\left( \vec{p}\cdot\vec{r} - E t \right) \right)\)
    \(\)
  1. 3948574230; locally 5577584:
    \(\psi( \vec{r},t) = \psi_0 \exp\left(\frac{i}{\hbar}\left( \vec{p}\cdot\vec{r} - E t \right) \right)\)
    \(\)
no check performed 3948574230:
3948574230:
3948574230:
3948574230:
37 declare initial expr
  1. 1158485859; locally 2344324:
    \(\frac{-\hbar^2}{2m} \nabla^2 = {\cal H}\)
    \(\)
no validation is available for declarations 1158485859:
1158485859:
25 substitute RHS of expr 1 into expr 2
  1. 3948574233; locally 2364546:
    \(\frac{\partial}{\partial t} \psi( \vec{r},t) = \psi_0 \frac{\partial}{\partial t}\exp\left(i\left(\frac{ \vec{p}\cdot\vec{r}}{\hbar} - \frac{E t}{\hbar} \right) \right)\)
    \(\)
  2. 3948574230; locally 1305534:
    \(\psi( \vec{r},t) = \psi_0 \exp\left(\frac{i}{\hbar}\left( \vec{p}\cdot\vec{r} - E t \right) \right)\)
    \(\)
  1. 3948571256; locally 5345567:
    \(\frac{\partial}{\partial t} \psi( \vec{r},t) = \frac{-i}{\hbar}E \psi( \vec{r},t)\)
    \(\)
LHS diff is pdg9489(pdg9472, pdg1467) - Derivative(pdg9489(pdg9472, pdg1467), pdg1467) RHS diff is (pdg1054*pdg8330*pdg2718(pdg4621*(pdg1134*pdg9472 - pdg1467*pdg6238)/pdg1054) + pdg4621*pdg6238*pdg9489(pdg9472, pdg1467))/pdg1054 3948574233:
3948574230:
3948571256:
3948574233:
3948574230:
3948571256:
10 divide both sides by
  1. 3121513111; locally 2934848:
    \(k = \frac{2 \pi}{\lambda}\)
    \(\)
  1. 0001209482:
    \(2 \pi\)
    \(\)
  1. 3121234211; locally 1039485:
    \(\frac{k}{2\pi} = \lambda\)
    \(\)
LHS diff is 0 RHS diff is -pdg1115 + 1/pdg1115 3121513111:
3121234211:
3121513111:
3121234211:
2 declare initial expr
  1. 3131211131; locally 9214650:
    \(\omega = 2 \pi f\)
    \(\)
no validation is available for declarations 3131211131:
3131211131:
8 substitute RHS of expr 1 into expr 2
  1. 9999999960; locally 2949002:
    \(\hbar = h/(2 \pi)\)
    \(\)
  2. 4147472132; locally 2949821:
    \(E = \frac{h \omega}{2 \pi}\)
    \(\)
  1. 9999999965; locally 3741728:
    \(E = \omega \hbar\)
    \(\)
valid 9999999960:
4147472132:
9999999965:
9999999960:
4147472132:
9999999965:
16 substitute RHS of expr 1 into expr 2
  1. 3948574224; locally 3940505:
    \(\psi( \vec{r},t) = \psi_0 \exp\left(i\left( \vec{k}\cdot\vec{r} - \omega t \right) \right)\)
    \(\)
  2. 9999998870; locally 2948487:
    \(\frac{ \vec{p}}{\hbar} = \vec{k}\)
    \(\)
  1. 3948574226; locally 2100421:
    \(\psi( \vec{r},t) = \psi_0 \exp\left(i\left(\frac{ \vec{p}\cdot\vec{r}}{\hbar} - \omega t \right) \right)\)
    \(\)
LHS diff is -pdg9489(pdg9472, pdg1467) + pdg2046/pdg1054 RHS diff is pdg7394 - pdg8330*pdg2718(pdg4621(-pdg1467*pdg2321 + pdg1134*pdg9472/pdg1054)) 3948574224:
9999998870:
3948574226:
3948574224:
9999998870:
3948574226:
12 substitute RHS of expr 1 into expr 2
  1. 3121234212; locally 2901049:
    \(p = \frac{h k}{2\pi}\)
    \(\)
  2. 9999999960; locally 2949002:
    \(\hbar = h/(2 \pi)\)
    \(\)
  1. 9999999962; locally 1039013:
    \(p = \hbar k\)
    \(\)
LHS diff is pdg1054 - pdg1134 RHS diff is -pdg1054*pdg5321 + pdg4413/(2*pdg3141) 3121234212:
9999999960:
9999999962:
3121234212:
9999999960:
9999999962:
38 substitute LHS of expr 1 into expr 2
  1. 1158485859; locally 2344324:
    \(\frac{-\hbar^2}{2m} \nabla^2 = {\cal H}\)
    \(\)
  2. 9958485859; locally 1304924:
    \(\frac{-\hbar^2}{2m} \nabla^2 \psi \left( \vec{r},t \right) = i \hbar \frac{\partial}{\partial t} \psi( \vec{r},t)\)
    \(\)
  1. 2258485859; locally 2456546:
    \({\cal H} \psi \left( \vec{r},t \right) = i \hbar \frac{\partial}{\partial t} \psi( \vec{r},t)\)
    \(\)
Nothing to split 1158485859:
9958485859:
2258485859:
1158485859:
9958485859:
2258485859:
31 apply divergence
  1. 5985371230; locally 5535257:
    \(\vec{ \nabla} \psi( \vec{r},t) = \frac{i}{\hbar} \vec{p} \psi( \vec{r},t)\)
    \(\)
  1. 4394958389; locally 4938589:
    \(\vec{ \nabla}\cdot \left( \vec{ \nabla} \psi( \vec{r},t) \right) = \frac{i}{\hbar} \vec{ \nabla}\cdot\left( \vec{p} \psi( \vec{r},t) \right)\)
    \(\)
failed 5985371230:
4394958389:
5985371230:
4394958389:
26 substitute RHS of expr 1 into expr 2
  1. 4298359851; locally 3576787:
    \(E = \frac{p^2}{2m}\)
    \(\)
  2. 3948571256; locally 5345567:
    \(\frac{\partial}{\partial t} \psi( \vec{r},t) = \frac{-i}{\hbar}E \psi( \vec{r},t)\)
    \(\)
  1. 4348571256; locally 2495835:
    \(\frac{\partial}{\partial t} \psi( \vec{r},t) = \frac{-i}{\hbar}\frac{p^2}{2 m} \psi( \vec{r},t)\)
    \(\)
LHS diff is 0 RHS diff is pdg4621*(pdg1134**2 - 2*pdg5156*pdg6238)*pdg9489(pdg9472, pdg1467)/(2*pdg1054*pdg5156) 4298359851:
3948571256:
4348571256:
4298359851:
3948571256:
4348571256:
9 divide both sides by
  1. 9999999965; locally 3741728:
    \(E = \omega \hbar\)
    \(\)
  1. 0003949921:
    \(\hbar\)
    \(\)
  1. 9999999961; locally 4499582:
    \(\frac{E}{\hbar} = \omega\)
    \(\)
valid 9999999965:
9999999961:
9999999965:
9999999961:
5 declare initial expr
  1. 1020394902; locally 3499522:
    \(E = h f\)
    \(\)
no validation is available for declarations 1020394902:
1020394902:
6 divide both sides by
  1. 3131211131; locally 9214650:
    \(\omega = 2 \pi f\)
    \(\)
  1. 0002940021:
    \(2 \pi\)
    \(\)
  1. 3147472131; locally 2939402:
    \(\frac{\omega}{2 \pi} = f\)
    \(\)
valid 3131211131:
3147472131:
3131211131:
3147472131:
33 substitute RHS of expr 1 into expr 2
  1. 5985371230; locally 5535257:
    \(\vec{ \nabla} \psi( \vec{r},t) = \frac{i}{\hbar} \vec{p} \psi( \vec{r},t)\)
    \(\)
  2. 1648958381; locally 1495034:
    \(\nabla^2 \psi \left( \vec{r},t \right) = \frac{i}{\hbar} \vec{p} \cdot \left( \vec{ \nabla} \psi( \vec{r},t) \right)\)
    \(\)
  1. 2648958382; locally 1049553:
    \(\nabla^2 \psi \left( \vec{r},t \right) = \frac{i}{\hbar} \vec{p} \cdot \left( \frac{i}{\hbar} \vec{p} \psi( \vec{r},t) \right)\)
    \(\)
Nothing to split 5985371230:
1648958381:
2648958382:
5985371230:
1648958381:
2648958382:
34 simplify
  1. 2648958382; locally 1049553:
    \(\nabla^2 \psi \left( \vec{r},t \right) = \frac{i}{\hbar} \vec{p} \cdot \left( \frac{i}{\hbar} \vec{p} \psi( \vec{r},t) \right)\)
    \(\)
  1. 2395958385; locally 4959593:
    \(\nabla^2 \psi \left( \vec{r},t \right) = \frac{-p^2}{\hbar} \psi( \vec{r},t)\)
    \(\)
Nothing to split 2648958382:
2395958385:
2648958382:
2395958385:
27 multiply both sides by
  1. 4348571256; locally 2495835:
    \(\frac{\partial}{\partial t} \psi( \vec{r},t) = \frac{-i}{\hbar}\frac{p^2}{2 m} \psi( \vec{r},t)\)
    \(\)
  1. 0002436656:
    \(i \hbar\)
    \(\)
  1. 4341171256; locally 3429538:
    \(i \hbar \frac{\partial}{\partial t} \psi( \vec{r},t) = \frac{p^2}{2 m} \psi( \vec{r},t)\)
    \(\)
LHS diff is 0 RHS diff is -pdg1134**2*(pdg4621**2 + 1)*pdg9489(pdg9472, pdg1467)/(2*pdg5156) 4348571256:
4341171256:
4348571256:
4341171256:
35 multiply both sides by
  1. 2395958385; locally 4959593:
    \(\nabla^2 \psi \left( \vec{r},t \right) = \frac{-p^2}{\hbar} \psi( \vec{r},t)\)
    \(\)
  1. 0005938585:
    \(\frac{-\hbar^2}{2m}\)
    \(\)
  1. 5868688585; locally 4349493:
    \(\frac{-\hbar^2}{2m} \nabla^2 \psi \left( \vec{r},t \right) = \frac{p^2}{2m} \psi( \vec{r},t)\)
    \(\)
LHS diff is 0 RHS diff is pdg1134**2*(pdg1054 - 1)*pdg9489(pdg9472, pdg1467)/(2*pdg5156) 2395958385:
5868688585:
2395958385:
5868688585:
30 substitute RHS of expr 1 into expr 2
  1. 4943571230; locally 3454565:
    \(\vec{ \nabla} \psi( \vec{r},t) = \frac{i}{\hbar} \vec{p} \psi_0 \exp\left(\frac{i}{\hbar}\left( \vec{p}\cdot\vec{r} - E t \right) \right)\)
    \(\)
  2. 3948574230; locally 1305534:
    \(\psi( \vec{r},t) = \psi_0 \exp\left(\frac{i}{\hbar}\left( \vec{p}\cdot\vec{r} - E t \right) \right)\)
    \(\)
  1. 5985371230; locally 5535257:
    \(\vec{ \nabla} \psi( \vec{r},t) = \frac{i}{\hbar} \vec{p} \psi( \vec{r},t)\)
    \(\)
failed 4943571230:
3948574230:
5985371230:
4943571230:
3948574230:
5985371230:
29 simplify
  1. 3948574230; locally 5577584:
    \(\psi( \vec{r},t) = \psi_0 \exp\left(\frac{i}{\hbar}\left( \vec{p}\cdot\vec{r} - E t \right) \right)\)
    \(\)
  1. 4943571230; locally 3454565:
    \(\vec{ \nabla} \psi( \vec{r},t) = \frac{i}{\hbar} \vec{p} \psi_0 \exp\left(\frac{i}{\hbar}\left( \vec{p}\cdot\vec{r} - E t \right) \right)\)
    \(\)
failed 3948574230:
4943571230:
3948574230:
4943571230:
39 declare final expr
  1. 2258485859; locally 2456546:
    \({\cal H} \psi \left( \vec{r},t \right) = i \hbar \frac{\partial}{\partial t} \psi( \vec{r},t)\)
    \(\)
no validation is available for declarations 2258485859:
2258485859:
36 LHS of expr 1 equals LHS of expr 2
  1. 4341171256; locally 3429538:
    \(i \hbar \frac{\partial}{\partial t} \psi( \vec{r},t) = \frac{p^2}{2 m} \psi( \vec{r},t)\)
    \(\)
  2. 5868688585; locally 4349493:
    \(\frac{-\hbar^2}{2m} \nabla^2 \psi \left( \vec{r},t \right) = \frac{p^2}{2m} \psi( \vec{r},t)\)
    \(\)
  1. 9958485859; locally 1304924:
    \(\frac{-\hbar^2}{2m} \nabla^2 \psi \left( \vec{r},t \right) = i \hbar \frac{\partial}{\partial t} \psi( \vec{r},t)\)
    \(\)
Nothing to split 4341171256:
5868688585:
9958485859:
4341171256:
5868688585:
9958485859:
4 declare initial expr
  1. 1020394900; locally 1203491:
    \(p = h/\lambda\)
    \(\)
no validation is available for declarations 1020394900: dimensions are consistent
 1020394900: N/A
Physics Derivation Graph: Steps for derivation of Schrodinger Equation

Symbols for this derivation

See also all 227 symbols
symbol ID category latex scope dimension name value Used in derivations references
3141 constant \pi
\(\pi\)		 ['real'] dimensionless pi 3.1415   dimensionless
 72
1115 variable \lambda
\(\lambda\)		 ['real']
  • length: 1
 wavelength 5
9472 variable \vec{r}
\(\vec{r}\)		 real
  • length: 1
 radius vector
  • str_note
24
6799 variable {\cal H}
\({\cal H}\)		 complex dimensionless operator 1
1134 variable p
\(p\)		 ['real']
  • length: 1
  • mass: 1
  • time: -1
 momentum 15
4413 variable h
\(h\)		 real
  • length: 2
  • mass: 1
  • time: -1
 Planck's constant 5
2718 constant \exp
\(\exp\)		 ['real'] dimensionless e 2.71828   unitless
 8
6238 variable E
\(E\)		 real dimensionless electric field 20
8330 variable \psi_0
\(\psi_0\)		 complex dimensionless amplitude of wavefunction
  • str_note
6
4201 variable f
\(f\)		 ['real']
  • time: -1
 frequency 8
9489 variable \psi
\(\psi\)		 complex dimensionless none
  • str_note
27
4931 variable E
\(E\)		 ['real']
  • length: 2
  • mass: 1
  • time: -2
 energy 10
2321 variable \omega
\(\omega\)		 ['real']
  • time: -1
 angular frequency 26
5321 variable k
\(k\)		 ['real']
  • length: -1
 angular wavenumber 13
4621 variable i
\(i\)		 ['imaginary'] dimensionless imaginary unit 74
1357 variable v
\(v\)		 ['real']
  • length: 1
  • time: -1
 velocity 83
1054 constant \hbar
\(\hbar\)		 ['real']
  • length: 2
  • mass: 1
  • time: -1
 Reduced Planck's constant 1.0545718*10^{-34}   meter^2 kilogram second^-1
 33
1467 variable t
\(t\)		 ['real']
  • time: 1
 time 121
7394 variable \vec{k}
\(\vec{k}\)		 real
  • length: -1
 wavenumber 1
2046 variable \vec{p}
\(\vec{p}\)		 real
  • length: 1
  • mass: 1
  • time: -1
 momentum 5
5156 variable m
\(m\)		 ['real']
  • mass: 1
 mass 69
MESSAGES: