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Review particle in a 1D box

step inference rule input feed output validity (as per SymPy)
20
  • ID: 111886; change variable X to Y
  • number of inputs: 1; feeds: 2; outputs: 1
  • Change variable $#1$ to $#2$ in Eq.~\\ref{eq:#3}; yields Eq.~\\ref{eq:#4}.
  1. 9988949211
    \((\sin(x))^2=\frac{1 - \cos(2 x)}{2}\)
  1. 0004934845
    \(x\)
  1. 0009484724
    \(\frac{n \pi}{W}x\)
  1. 7575738420
    \(\left(\sin\left(\frac{n \pi}{W}x\right) \right)^2=\frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2}\)
LHS diff is 0 RHS diff is cos(2*pdg0001464*pdg0001592*pdg0003141/pdg0002523)/2 - cos(2*pdg0001592*pdg0003141*pdg0004037/pdg0002523)/2
32
  • ID: 111524; square root both sides
  • number of inputs: 1; feeds: 0; outputs: 2
  • Take the square root of both sides of Eq.~\\ref{eq:#1}; yields Eq.~\\ref{eq:#2} and Eq.~\\ref{eq:#3}.
  1. 8485867742
    \(\frac{2}{W}=a^2\)
  1. 9485747245
    \(\sqrt{\frac{2}{W}}=a\)
  1. 9485747246
    \(-\sqrt{\frac{2}{W}}=a\)
recognized infrule but not yet supported
2
  • ID: 111237; declare guess solution
  • number of inputs: 1; feeds: 0; outputs: 1
  • Judicious choice as a guessed solution to Eq.~\\ref{eq:#1} is Eq.~\\ref{eq:#2},
  1. 5727578862
    \(\frac{d^2}{dx^2} \psi(x)=-k^2 \psi(x)\)
  1. 8582885111
    \(\psi(x)=a \sin(kx) + b \cos(kx)\)
no validation is available for declarations
33
  • ID: 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. 9485747245
    \(\sqrt{\frac{2}{W}}=a\)
  1. 2944838499
    \(\psi(x)=a \sin(\frac{n \pi}{W} x)\)
  1. 9393939991
    \(\psi(x)=-\sqrt{\frac{2}{W}} \sin\left(\frac{n \pi}{W} x\right)\)
LHS diff is 0 RHS diff is pdg0009139*sin(pdg0001592*pdg0003141*pdg0004037/pdg0002523) + sqrt(2)*sqrt(1/pdg0002523)*sin(pdg0001464*pdg0001592*pdg0003141/pdg0002523)
27
  • ID: 111886; change variable X to Y
  • number of inputs: 1; feeds: 2; outputs: 1
  • Change variable $#1$ to $#2$ in Eq.~\\ref{eq:#3}; yields Eq.~\\ref{eq:#4}.
  1. 5857434758
    \(\int a dx=a x\)
  1. 0002929944
    \(1/2\)
  1. 0004948585
    \(a\)
  1. 8575746378
    \(\int \frac{1}{2} dx=\frac{1}{2} x\)
Type Tuple cannot be instantiated; use tuple() instead
38
  • ID: 111457; simplify
  • number of inputs: 1; feeds: 0; outputs: 1
  • Simplify Eq.~\\ref{eq:#1}; yields Eq.~\\ref{eq:#2}.
  1. 8575748999
    \(\frac{d^2}{dx^2} \left(a \sin(k x) + b \cos(k x) \right)=-k^2 \left(a \sin(kx) + b \cos(kx) \right)\)
  1. 8485757728
    \(a \frac{d^2}{dx^2}\sin(kx) + b \frac{d^2}{dx^2}\cos(k x)=-a k^2 \sin(kx) + -b k^2 \cos(kx)\)
invalid syntax (<string>, line 0)
26
  • ID: 111299; declare identity
  • number of inputs: 0; feeds: 0; outputs: 1
  • Eq.~\\ref{eq:#1} is an identity.
  1. 5857434758
    \(\int a dx=a x\)
no validation is available for declarations
24
  • ID: 111299; declare identity
  • number of inputs: 0; feeds: 0; outputs: 1
  • Eq.~\\ref{eq:#1} is an identity.
  1. 0948572140
    \(\int \cos(a x) dx=\frac{1}{a}\sin(a x)\)
no validation is available for declarations
28
  • ID: 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. 8575746378
    \(\int \frac{1}{2} dx=\frac{1}{2} x\)
  1. 1202310110
    \(\frac{1}{a^2}=\int_0^W \frac{1}{2} dx - \frac{1}{2} \int_0^W \cos\left(2\frac{n \pi}{W}x\right) dx\)
  1. 1202312210
    \(\frac{1}{a^2}=\frac{1}{2}W - \frac{1}{2} \int_0^W \cos\left(2\frac{n \pi}{W}x\right) dx\)
Type Tuple cannot be instantiated; use tuple() instead
34
  • ID: 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. 2944838499
    \(\psi(x)=a \sin(\frac{n \pi}{W} x)\)
  1. 9485747246
    \(-\sqrt{\frac{2}{W}}=a\)
  1. 9393939992
    \(\psi(x)=\sqrt{\frac{2}{W}} \sin\left(\frac{n \pi}{W} x\right)\)
LHS diff is -sqrt(2)*sqrt(1/pdg0002523) - pdg0009489(pdg0001464) RHS diff is pdg0009139 - sqrt(2)*sqrt(1/pdg0002523)*sin(pdg0001464*pdg0001592*pdg0003141/pdg0002523)
23
  • ID: 111581; expand integrand
  • number of inputs: 1; feeds: 0; outputs: 1
  • Expand integrand of Eq.~\\ref{eq:#1}; yields Eq.~\\ref{eq:#2}.
  1. 9858028950
    \(\frac{1}{a^2}=\int_0^W \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2} dx\)
  1. 1202310110
    \(\frac{1}{a^2}=\int_0^W \frac{1}{2} dx - \frac{1}{2} \int_0^W \cos\left(2\frac{n \pi}{W}x\right) dx\)
recognized infrule but not yet supported
13
  • ID: 111493; normalization condition
  • number of inputs: 0; feeds: 0; outputs: 1
  • Normalization condition is Eq.~\\ref{eq:#1}.
  1. 1934748140
    \(\int |\psi(x)|^2 dx=1\)
no validation is available for assumptions
17
  • ID: 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. 4857472413
    \(1=\int \psi(x)\psi(x)^* dx\)
  1. 2944838499
    \(\psi(x)=a \sin(\frac{n \pi}{W} x)\)
  1. 0203024440
    \(1=\int_0^W a \sin\left(\frac{n \pi}{W} x\right) \psi(x)^* dx\)
invalid syntax (<string>, line 0)
25
  • ID: 111886; change variable X to Y
  • number of inputs: 1; feeds: 2; outputs: 1
  • Change variable $#1$ to $#2$ in Eq.~\\ref{eq:#3}; yields Eq.~\\ref{eq:#4}.
  1. 0948572140
    \(\int \cos(a x) dx=\frac{1}{a}\sin(a x)\)
  1. 0009485858
    \(\frac{2n\pi}{W}\)
  1. 0004831494
    \(a\)
  1. 7564894985
    \(\int \cos\left(\frac{2n\pi}{W} x\right) dx=\frac{W}{2n\pi}\sin\left(\frac{2n\pi}{W} x\right)\)
Type Tuple cannot be instantiated; use tuple() instead
15
  • ID: 111268; swap LHS with RHS
  • number of inputs: 1; feeds: 0; outputs: 1
  • Swap LHS of Eq.~\\ref{eq:#1} with RHS; yields Eq.~\\ref{eq:#2}.
  1. 1934748140
    \(\int |\psi(x)|^2 dx=1\)
  1. 8572657110
    \(1=\int |\psi(x)|^2 dx\)
Type Tuple cannot be instantiated; use tuple() instead
37
  • ID: 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. 8582885111
    \(\psi(x)=a \sin(kx) + b \cos(kx)\)
  1. 5727578862
    \(\frac{d^2}{dx^2} \psi(x)=-k^2 \psi(x)\)
  1. 8575748999
    \(\frac{d^2}{dx^2} \left(a \sin(k x) + b \cos(k x) \right)=-k^2 \left(a \sin(kx) + b \cos(kx) \right)\)
invalid syntax (<string>, line 0)
18
  • ID: 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. 0203024440
    \(1=\int_0^W a \sin\left(\frac{n \pi}{W} x\right) \psi(x)^* dx\)
  1. 8849289982
    \(\psi(x)^*=a \sin(\frac{n \pi}{W} x)\)
  1. 8889444440
    \(1=\int_0^W a^2 \left(\sin\left(\frac{n \pi}{W} x\right) \right)^2 dx\)
Type Tuple cannot be instantiated; use tuple() instead
14
  • ID: 111996; conjugate function X
  • number of inputs: 1; feeds: 1; outputs: 1
  • Conjugate $#1$ in Eq.~\\ref{eq:#2}; yields Eq.~\\ref{eq:#3}.
  1. 2944838499
    \(\psi(x)=a \sin(\frac{n \pi}{W} x)\)
  1. 0009587738
    \(\psi\)
  1. 8849289982
    \(\psi(x)^*=a \sin(\frac{n \pi}{W} x)\)
recognized infrule but not yet supported
31
  • ID: 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. 4857475848
    \(\frac{1}{a^2}=\frac{W}{2}\)
  1. 0009485857
    \(a^2\frac{2}{W}\)
  1. 8485867742
    \(\frac{2}{W}=a^2\)
valid
22
  • ID: 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. 8576785890
    \(1=\int_0^W a^2 \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2} dx\)
  1. 0000040490
    \(a^2\)
  1. 9858028950
    \(\frac{1}{a^2}=\int_0^W \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2} dx\)
Type Tuple cannot be instantiated; use tuple() instead
40
  • ID: 111345; claim LHS equals RHS
  • number of inputs: 1; feeds: 0; outputs: 0
  • Thus we see that LHS of Eq.~\\ref{eq:#1} is equal to RHS.
  1. 8484544728
    \(-a k^2\sin(k x) + -b k^2\cos(k x)=-a k^2 \sin(kx) + -b k^2 \cos(k x)\)
invalid syntax (<string>, line 0)
16
  • ID: 111166; expand magnitude to conjugate
  • number of inputs: 1; feeds: 1; outputs: 1
  • Expand $#1$ in Eq.~\\ref{eq:#2} with conjugate; yields Eq.~\\ref{eq:#3}.
  1. 8572657110
    \(1=\int |\psi(x)|^2 dx\)
  1. 0009458842
    \(\psi(x)\)
  1. 4857472413
    \(1=\int \psi(x)\psi(x)^* dx\)
recognized infrule but not yet supported
35
  • ID: 111341; declare final expression
  • number of inputs: 1; feeds: 0; outputs: 0
  • Eq.~\\ref{eq:#1} is one of the final equations.
  1. 9393939992
    \(\psi(x)=\sqrt{\frac{2}{W}} \sin\left(\frac{n \pi}{W} x\right)\)
no validation is available for declarations
30
  • ID: 111457; simplify
  • number of inputs: 1; feeds: 0; outputs: 1
  • Simplify Eq.~\\ref{eq:#1}; yields Eq.~\\ref{eq:#2}.
  1. 0439492440
    \(\frac{1}{a^2}=\frac{1}{2}W - \frac{1}{2}\left. \frac{W}{2n\pi}\sin\left(\frac{2n\pi}{W} x\right) \right|_0^W\)
  1. 4857475848
    \(\frac{1}{a^2}=\frac{W}{2}\)
LHS diff is 0 RHS diff is -pdg0002523*sin(2*pdg0001592*pdg0003141*pdg0004037/pdg0002523)/(4*pdg0001592*pdg0003141)
19
  • ID: 111299; declare identity
  • number of inputs: 0; feeds: 0; outputs: 1
  • Eq.~\\ref{eq:#1} is an identity.
  1. 9988949211
    \((\sin(x))^2=\frac{1 - \cos(2 x)}{2}\)
no validation is available for declarations
29
  • ID: 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. 1202312210
    \(\frac{1}{a^2}=\frac{1}{2}W - \frac{1}{2} \int_0^W \cos\left(2\frac{n \pi}{W}x\right) dx\)
  1. 7564894985
    \(\int \cos\left(\frac{2n\pi}{W} x\right) dx=\frac{W}{2n\pi}\sin\left(\frac{2n\pi}{W} x\right)\)
  1. 0439492440
    \(\frac{1}{a^2}=\frac{1}{2}W - \frac{1}{2}\left. \frac{W}{2n\pi}\sin\left(\frac{2n\pi}{W} x\right) \right|_0^W\)
Type Tuple cannot be instantiated; use tuple() instead
5
  • ID: 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. 9585727710
    \(\psi(x=0)=0\)
  1. 8582885111
    \(\psi(x)=a \sin(kx) + b \cos(kx)\)
  1. 8577275751
    \(0=a \sin(0) + b\cos(0)\)
input diff is -pdg0009489(pdg0004037) + pdg0009489(Eq(pdg0001464, 0)) diff is 0 diff is pdg0001939*cos(pdg0004037*pdg0005321) - pdg0001939 + pdg0009139*sin(pdg0004037*pdg0005321)
7
  • ID: 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. 1293913110
    \(0=b\)
  1. 8582885111
    \(\psi(x)=a \sin(kx) + b \cos(kx)\)
  1. 9059289981
    \(\psi(x)=a \sin(k x)\)
invalid syntax (<string>, line 0)
3
  • ID: 111802; boundary condition for expression
  • number of inputs: 1; feeds: 0; outputs: 1
  • A boundary condition for Eq.~\\ref{eq:#1} is Eq.~\\ref{eq:#2}
  1. 5727578862
    \(\frac{d^2}{dx^2} \psi(x)=-k^2 \psi(x)\)
  1. 9585727710
    \(\psi(x=0)=0\)
no validation is available for assumptions
4
  • ID: 111802; boundary condition for expression
  • number of inputs: 1; feeds: 0; outputs: 1
  • A boundary condition for Eq.~\\ref{eq:#1} is Eq.~\\ref{eq:#2}
  1. 5727578862
    \(\frac{d^2}{dx^2} \psi(x)=-k^2 \psi(x)\)
  1. 9495857278
    \(\psi(x=W)=0\)
no validation is available for assumptions
10
  • ID: 111698; expr 1 is true under condition expr 2
  • number of inputs: 2; feeds: 0; outputs: 1
  • Eq.~\\ref{eq:#1} is valid when Eq.~\\ref{eq:#2} occurs; yields Eq.~\\ref{eq:#3}.
  1. 1857710291
    \(0=a \sin(n \pi)\)
  1. 1020010291
    \(0=a \sin(k W)\)
  1. 1010923823
    \(k W=n \pi\)
recognized infrule but not yet supported
9
  • ID: 111299; declare identity
  • number of inputs: 0; feeds: 0; outputs: 1
  • Eq.~\\ref{eq:#1} is an identity.
  1. 1857710291
    \(0=a \sin(n \pi)\)
no validation is available for declarations
12
  • ID: 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. 9059289981
    \(\psi(x)=a \sin(k x)\)
  1. 1858772113
    \(k=\frac{n \pi}{W}\)
  1. 2944838499
    \(\psi(x)=a \sin(\frac{n \pi}{W} x)\)
invalid syntax (<string>, line 0)
6
  • ID: 111457; simplify
  • number of inputs: 1; feeds: 0; outputs: 1
  • Simplify Eq.~\\ref{eq:#1}; yields Eq.~\\ref{eq:#2}.
  1. 8577275751
    \(0=a \sin(0) + b\cos(0)\)
  1. 1293913110
    \(0=b\)
valid
1
  • ID: 111981; declare initial expression
  • number of inputs: 0; feeds: 0; outputs: 1
  • Eq.~\\ref{eq:#1} is an initial equation.
  1. 5727578862
    \(\frac{d^2}{dx^2} \psi(x)=-k^2 \psi(x)\)
no validation is available for declarations
21
  • ID: 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. 8889444440
    \(1=\int_0^W a^2 \left(\sin\left(\frac{n \pi}{W} x\right) \right)^2 dx\)
  1. 7575738420
    \(\left(\sin\left(\frac{n \pi}{W}x\right) \right)^2=\frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2}\)
  1. 8576785890
    \(1=\int_0^W a^2 \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2} dx\)
Type Tuple cannot be instantiated; use tuple() instead
11
  • ID: 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. 1010923823
    \(k W=n \pi\)
  1. 0001334112
    \(W\)
  1. 1858772113
    \(k=\frac{n \pi}{W}\)
valid
39
  • ID: 111457; simplify
  • number of inputs: 1; feeds: 0; outputs: 1
  • Simplify Eq.~\\ref{eq:#1}; yields Eq.~\\ref{eq:#2}.
  1. 8485757728
    \(a \frac{d^2}{dx^2}\sin(kx) + b \frac{d^2}{dx^2}\cos(k x)=-a k^2 \sin(kx) + -b k^2 \cos(kx)\)
  1. 8484544728
    \(-a k^2\sin(k x) + -b k^2\cos(k x)=-a k^2 \sin(kx) + -b k^2 \cos(k x)\)
invalid syntax (<string>, line 0)
8
  • ID: 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. 9495857278
    \(\psi(x=W)=0\)
  1. 9059289981
    \(\psi(x)=a \sin(k x)\)
  1. 1020010291
    \(0=a \sin(k W)\)
invalid syntax (<string>, line 0)


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