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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 212 symbols
symbol ID category latex scope dimension name value Used in derivations references
1357 variable v
\(v\)
['real']
  • length: 1
  • time: -1
velocity 80
6799 variable {\cal H}
\({\cal H}\)
complex dimensionless operator 1
4201 variable f
\(f\)
['real']
  • time: -1
frequency 8
5156 variable m
\(m\)
['real']
  • mass: 1
mass 49
8330 variable \psi_0
\(\psi_0\)
complex dimensionless amplitude of wavefunction
  • str_note
6
5321 variable k
\(k\)
['real']
  • length: -1
angular wavenumber 13
2046 variable \vec{p}
\(\vec{p}\)
real
  • length: 1
  • time: -1
  • mass: 1
momentum 5
3141 constant \pi
\(\pi\)
['real']
pi 3.1415   dimensionless
55
1115 variable \lambda
\(\lambda\)
['real']
  • length: 1
wavelength 5
9489 variable \psi
\(\psi\)
complex dimensionless none
  • str_note
27
7394 variable \vec{k}
\(\vec{k}\)
real
  • length: -1
wavenumber 1
6238 variable E
\(E\)
real dimensionless electric field 20
1467 variable t
\(t\)
['real']
  • time: 1
time 115
2718 constant \exp
\(\exp\)
['real']
e 2.71828   unitless
8
4413 variable h
\(h\)
real
  • length: 2
  • time: -1
  • mass: 1
Planck's constant 5
1134 variable p
\(p\)
['real']
  • length: 1
  • time: -1
  • mass: 1
momentum 15
4621 variable i
\(i\)
['imaginary']
imaginary unit 74
2321 variable \omega
\(\omega\)
['real']
  • time: -1
angular frequency 15
1054 constant \hbar
\(\hbar\)
['real']
  • length: 2
  • time: -1
  • mass: 1
Reduced Planck's constant 1.0545718*10^{-34}   meter^2 kilogram second^-1
33
4931 variable E
\(E\)
['real']
  • length: 2
  • time: -2
  • mass: 1
energy 10
9472 variable \vec{r}
\(\vec{r}\)
real
  • length: 1
radius vector
  • str_note
24
MESSAGES: