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Review particle in a 1D box

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.
  1. 5727578862
    \(\frac{d^2}{dx^2} \psi(x) = -k^2 \psi(x)\)
 no validation is available for declarations
2
  • 0000111237: 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
3
  • 0000111802: 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
  • 0000111802: 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
5
  • 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}.
  1. 8582885111
    \(\psi(x) = a \sin(kx) + b \cos(kx)\)
  2. 9585727710
    \(\psi(x=0) = 0\)
  1. 8577275751
    \(0 = a \sin(0) + b\cos(0)\)
 input diff is pdg0009489(pdg0004037) - pdg0009489(Eq(pdg0001464, 0)) diff is pdg0001939*cos(pdg0004037*pdg0005321) + pdg0009139*sin(pdg0004037*pdg0005321) diff is -pdg0001939
6
  • 0000111457: 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
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}.
  1. 8582885111
    \(\psi(x) = a \sin(kx) + b \cos(kx)\)
  2. 1293913110
    \(0 = b\)
  1. 9059289981
    \(\psi(x) = a \sin(k x)\)
 Not evaluated due to missing term in SymPy
8
  • 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}.
  1. 9059289981
    \(\psi(x) = a \sin(k x)\)
  2. 9495857278
    \(\psi(x=W) = 0\)
  1. 1020010291
    \(0 = a \sin(k W)\)
 Not evaluated due to missing term in SymPy
9
  • 0000111299: 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
10
  • 0000111698: 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
    \(0 = a \sin(n \pi)\)
  2. 1020010291
    \(0 = a \sin(k W)\)
  1. 1010923823
    \(k W = n \pi\)
 recognized infrule but not yet supported
11
  • 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}.
  1. 1010923823
    \(k W = n \pi\)
  1. 0001334112
    \(W\)
  1. 1858772113
    \(k = \frac{n \pi}{W}\)
 valid
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}.
  1. 9059289981
    \(\psi(x) = a \sin(k x)\)
  2. 1858772113
    \(k = \frac{n \pi}{W}\)
  1. 2944838499
    \(\psi(x) = a \sin(\frac{n \pi}{W} x)\)
 Not evaluated due to missing term in SymPy
13
  • 0000111493: 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
14
  • 0000111996: 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
15
  • 0000111268: 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
    \(1 = \int |\psi(x)|^2 dx\)
 LHS diff is pdg0009199*Abs(pdg0009489(pdg0001464))**2 - Integral(Abs(pdg0009489(pdg0001464))**2, pdg0001464) RHS diff is 0
16
  • 0000111166: 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
    \(1 = \int \psi(x)\psi(x)^* dx\)
17
  • 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}.
  1. 4857472413
    \(1 = \int \psi(x)\psi(x)^* dx\)
  2. 2944838499
    \(\psi(x) = a \sin(\frac{n \pi}{W} x)\)
  1. 0203024440
    \(1 = \int_0^W a \sin\left(\frac{n \pi}{W} x\right) \psi(x)^* dx\)
 Not evaluated due to missing term in SymPy
18
  • 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}.
  1. 0203024440
    \(1 = \int_0^W a \sin\left(\frac{n \pi}{W} x\right) \psi(x)^* dx\)
  2. 8849289982
    \(\psi(x)^* = a \sin(\frac{n \pi}{W} x)\)
  1. 8889444440
    \(1 = \int_0^W a^2 \left(\sin\left(\frac{n \pi}{W} x\right) \right)^2 dx\)
 LHS diff is conjugate(pdg0009489(pdg0004037)) - 1 RHS diff is pdg0009139*(-pdg0009139*Piecewise((pdg0002523*(pdg0001592*pdg0003141/2 - sin(pdg0001592*pdg0003141)*cos(pdg0001592*pdg0003141)/2)/(pdg0001592*pdg0003141), Ne(pdg0001592*pdg0003141/pdg0002523, 0)), (0, True)) + sin(pdg0001592*pdg0003141*pdg0004037/pdg0002523**(pdg0009139*Integral(sin(pdg0001464*pdg0001592*pdg0003141/pdg0002523)*conjugate(pdg0009489(pdg0001464)), (pdg0001464, 0, pdg0002523)))))
19
  • 0000111299: 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
20
  • 0000111886: 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
    \(x\)
  2. 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
21
  • 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}.
  1. 8889444440
    \(1 = \int_0^W a^2 \left(\sin\left(\frac{n \pi}{W} x\right) \right)^2 dx\)
  2. 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
    \(1 = \int_0^W a^2 \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2} dx\)
 LHS diff is -cos(pdg0001592*pdg0003141*pdg0004037/pdg0002523)**2 RHS diff is -pdg0009139**2*Piecewise((pdg0002523*(pdg0001592*pdg0003141/2 - sin(pdg0001592*pdg0003141)*cos(pdg0001592*pdg0003141)/2)/(pdg0001592*pdg0003141), Ne(pdg0001592*pdg0003141/pdg0002523, 0)), (0, True)) + sin(pdg0001464*pdg0001592*pdg0003141/pdg0002523)**2
22
  • 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}.
  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
    \(\frac{1}{a^2} = \int_0^W \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2} dx\)
 valid
23
  • 0000111581: 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
24
  • 0000111299: 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
25
  • 0000111886: 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. 0004831494
    \(a\)
  2. 0009485858
    \(\frac{2n\pi}{W}\)
  1. 7564894985
    \(\int \cos\left(\frac{2n\pi}{W} x\right) dx = \frac{W}{2n\pi}\sin\left(\frac{2n\pi}{W} x\right)\)
 LHS diff is pdg0009199*cos(2*pdg0001464*pdg0001592*pdg0003141/pdg0002523) - Piecewise((pdg0002523*sin(2*pdg0001592*pdg0003141*pdg0004037/pdg0002523)/(2*pdg0001592*pdg0003141), Ne(pdg0001592*pdg0003141/pdg0002523, 0)), (pdg0004037, True)) RHS diff is pdg0002523*(sin(2*pdg0001464*pdg0001592*pdg0003141/pdg0002523) - sin(2*pdg0001592*pdg0003141*pdg0004037/pdg0002523))/(2*pdg0001592*pdg0003141)
26
  • 0000111299: 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
27
  • 0000111886: 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. 0004948585
    \(a\)
  2. 0002929944
    \(1/2\)
  1. 8575746378
    \(\int \frac{1}{2} dx = \frac{1}{2} x\)
 valid
28
  • 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}.
  1. 1202310110
    \(\frac{1}{a^2} = \int_0^W \frac{1}{2} dx - \frac{1}{2} \int_0^W \cos\left(2\frac{n \pi}{W}x\right) dx\)
  2. 8575746378
    \(\int \frac{1}{2} dx = \frac{1}{2} x\)
  1. 1202312210
    \(\frac{1}{a^2} = \frac{1}{2}W - \frac{1}{2} \int_0^W \cos\left(2\frac{n \pi}{W}x\right) dx\)
 LHS diff is pdg0001464/2 - 1/pdg0009139**2 RHS diff is pdg0001464/2 - pdg0002523/2 + Piecewise((pdg0002523*sin(2*pdg0001592*pdg0003141)/(2*pdg0001592*pdg0003141), Ne(pdg0001592*pdg0003141/pdg0002523, 0)), (pdg0002523, True))/2
29
  • 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}.
  1. 1202312210
    \(\frac{1}{a^2} = \frac{1}{2}W - \frac{1}{2} \int_0^W \cos\left(2\frac{n \pi}{W}x\right) dx\)
  2. 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
    \(\frac{1}{a^2} = \frac{1}{2}W - \frac{1}{2}\left. \frac{W}{2n\pi}\sin\left(\frac{2n\pi}{W} x\right) \right|_0^W\)
 LHS diff is Piecewise((pdg0002523*sin(2*pdg0001592*pdg0003141*pdg0004037/pdg0002523)/(2*pdg0001592*pdg0003141), Ne(pdg0001592*pdg0003141/pdg0002523, 0)), (pdg0004037, True)) - 1/pdg0009139**2 RHS diff is -pdg0002523/2 + 3*pdg0002523*sin(2*pdg0001592*pdg0003141*pdg0004037/pdg0002523)/(4*pdg0001592*pdg0003141)
30
  • 0000111457: 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)
31
  • 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}.
  1. 4857475848
    \(\frac{1}{a^2} = \frac{W}{2}\)
  1. 0009485857
    \(a^2\frac{2}{W}\)
  1. 8485867742
    \(\frac{2}{W} = a^2\)
 valid
32
  • 0000111524: 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. 9485747246
    \(-\sqrt{\frac{2}{W}} = a\)
  2. 9485747245
    \(\sqrt{\frac{2}{W}} = a\)
33
  • 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}.
  1. 2944838499
    \(\psi(x) = a \sin(\frac{n \pi}{W} x)\)
  2. 9485747245
    \(\sqrt{\frac{2}{W}} = a\)
  1. 9393939991
    \(\psi(x) = -\sqrt{\frac{2}{W}} \sin\left(\frac{n \pi}{W} x\right)\)
 LHS diff is sqrt(2)*sqrt(1/pdg0002523) - pdg0009489(pdg0001464) RHS diff is pdg0009139 + sqrt(2)*sqrt(1/pdg0002523)*sin(pdg0001464*pdg0001592*pdg0003141/pdg0002523)
34
  • 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}.
  1. 2944838499
    \(\psi(x) = a \sin(\frac{n \pi}{W} x)\)
  2. 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)
35
  • 0000111341: 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
37
  • 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}.
  1. 8582885111
    \(\psi(x) = a \sin(kx) + b \cos(kx)\)
  2. 5727578862
    \(\frac{d^2}{dx^2} \psi(x) = -k^2 \psi(x)\)
  1. 8575748999
    \(\frac{d^2}{dx^2} \left(a \sin(k x) + b \cos(k x) \right) = -k^2 \left(a \sin(kx) + b \cos(kx) \right)\)
 Not evaluated due to missing term in SymPy
38
  • 0000111457: 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
    \(a \frac{d^2}{dx^2}\sin(kx) + b \frac{d^2}{dx^2}\cos(k x) = -a k^2 \sin(kx) + -b k^2 \cos(kx)\)
 Not evaluated due to missing term in SymPy
39
  • 0000111457: 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
    \(-a k^2\sin(k x) + -b k^2\cos(k x) = -a k^2 \sin(kx) + -b k^2 \cos(k x)\)
 Not evaluated due to missing term in SymPy
40
  • 0000111345: 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
    \(-a k^2\sin(k x) + -b k^2\cos(k x) = -a k^2 \sin(kx) + -b k^2 \cos(k x)\)
Not evaluated due to missing term in SymPy

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