## review derivation: particle in a 1D box

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Index Inference Rule Input latex Feeds latex Output latex step validity dimension check unit check notes
20 change variable X to Y
1. 9988949211; locally 1231131:
$$(\sin(x))^2 = \frac{1 - \cos(2 x)}{2}$$
$$\sin^{2}{\left(pdg_{1464} \right)} = \frac{1}{2} - \frac{\cos{\left(2 pdg_{1464} \right)}}{2}$$
1. 0009484724:
$$\frac{n \pi}{W}x$$
$$\frac{pdg_{1592} pdg_{3141} pdg_{4037}}{pdg_{2523}}$$
2. 0004934845:
$$x$$
$$pdg_{1464}$$
1. 7575738420; locally 0100404:
$$\left(\sin\left(\frac{n \pi}{W}x\right) \right)^2 = \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2}$$
$$\sin^{2}{\left(\frac{pdg_{1592} pdg_{3141} pdg_{4037}}{pdg_{2523}} \right)} = \frac{1}{2} - \frac{\cos{\left(\frac{2 pdg_{1464} pdg_{1592} pdg_{3141}}{pdg_{2523}} \right)}}{2}$$
LHS diff is sin(pdg1464)**2 - sin(pdg1592*pdg3141*pdg4037/pdg2523)**2 RHS diff is -cos(2*pdg1464)/2 + cos(2*pdg1464*pdg1592*pdg3141/pdg2523)/2 9988949211:
7575738420:
9988949211:
7575738420:
32 square root both sides
1. 8485867742; locally 1029384:
$$\frac{2}{W} = a^2$$
$$\frac{2}{pdg_{2523}} = pdg_{9139}^{2}$$
1. 9485747245; locally 9394857:
$$\sqrt{\frac{2}{W}} = a$$
$$\sqrt{2} \sqrt{\frac{1}{pdg_{2523}}} = pdg_{9139}$$
2. 9485747246; locally 9394858:
$$-\sqrt{\frac{2}{W}} = a$$
$$- \sqrt{2} \sqrt{\frac{1}{pdg_{2523}}} = pdg_{9139}$$
no check performed 8485867742:
9485747245:
9485747246:
8485867742:
9485747245:
9485747246:
2 declare guess solution
1. 5727578862; locally 7572748:
$$\frac{d^2}{dx^2} \psi(x) = -k^2 \psi(x)$$
$$pdg_{9199}$$
1. 8582885111; locally 7572118:
$$\psi(x) = a \sin(kx) + b \cos(kx)$$
$$\operatorname{pdg}_{9489}{\left(pdg_{4037} \right)} = pdg_{1939} \cos{\left(pdg_{4037} pdg_{5321} \right)} + pdg_{9139} \sin{\left(pdg_{4037} pdg_{5321} \right)}$$
no validation is available for declarations 5727578862:
8582885111:
5727578862:
8582885111:
33 substitute LHS of expr 1 into expr 2
1. 9485747245; locally 9394857:
$$\sqrt{\frac{2}{W}} = a$$
$$\sqrt{2} \sqrt{\frac{1}{pdg_{2523}}} = pdg_{9139}$$
2. 2944838499; locally 3452131:
$$\psi(x) = a \sin(\frac{n \pi}{W} x)$$
$$\operatorname{pdg}_{9489}{\left(pdg_{1464} \right)} = pdg_{9139} \sin{\left(\frac{pdg_{1592} pdg_{3141} pdg_{4037}}{pdg_{2523}} \right)}$$
1. 9393939991; locally 8474766:
$$\psi(x) = -\sqrt{\frac{2}{W}} \sin\left(\frac{n \pi}{W} x\right)$$
$$\operatorname{pdg}_{9489}{\left(pdg_{1464} \right)} = - \sqrt{2} \sqrt{\frac{1}{pdg_{2523}}} \sin{\left(\frac{pdg_{1464} pdg_{1592} pdg_{3141}}{pdg_{2523}} \right)}$$
LHS diff is 0 RHS diff is pdg9139*sin(pdg1592*pdg3141*pdg4037/pdg2523) + sqrt(2)*sqrt(1/pdg2523)*sin(pdg1464*pdg1592*pdg3141/pdg2523) 9485747245:
2944838499:
9393939991:
9485747245:
2944838499:
9393939991:
27 change variable X to Y
1. 5857434758; locally 0021030:
$$\int a dx = a x$$
$$\int pdg_{9139}\, dpdg_{1464} = pdg_{1464} pdg_{9139}$$
1. 0002929944:
$$1/2$$
$$\frac{1}{2}$$
2. 0004948585:
$$a$$
$$pdg_{9139}$$
1. 8575746378; locally 9339495:
$$\int \frac{1}{2} dx = \frac{1}{2} x$$
$$\int \frac{1}{2}\, dpdg_{1464} = \frac{pdg_{1464}}{2}$$
LHS diff is pdg1464*(pdg9139 - 1/2) RHS diff is pdg1464*(pdg9139 - 1/2) 5857434758:
8575746378:
5857434758:
8575746378:
38 simplify
1. 8575748999; locally 2838288:
$$\frac{d^2}{dx^2} \left(a \sin(k x) + b \cos(k x) \right) = -k^2 \left(a \sin(kx) + b \cos(kx) \right)$$
$$\frac{d^{2} \left(pdg_{1939} \cos{\left(pdg_{1464} pdg_{5321} \right)} + pdg_{9139} \sin{\left(pdg_{1464} pdg_{5321} \right)}\right)}{pdg_{9199}^{2}} = - pdg_{5321}^{2} \left(pdg_{1939} \cos{\left(pdg_{1464} pdg_{5321} \right)} + pdg_{9139} \sin{\left(pdg_{1464} pdg_{5321} \right)}\right)$$
1. 8485757728; locally 8474762:
$$a \frac{d^2}{dx^2}\sin(kx) + b \frac{d^2}{dx^2}\cos(k x) = -a k^2 \sin(kx) + -b k^2 \cos(kx)$$
$$pdg_{9199}$$
Nothing to split 8575748999:
8485757728:
8575748999:
8485757728:
26 declare identity
1. 5857434758; locally 0021030:
$$\int a dx = a x$$
$$\int pdg_{9139}\, dpdg_{1464} = pdg_{1464} pdg_{9139}$$
no validation is available for declarations 5857434758:
5857434758:
24 declare identity
1. 0948572140; locally 3992939:
$$\int \cos(a x) dx = \frac{1}{a}\sin(a x)$$
$$\int \cos{\left(pdg_{1464} pdg_{9139} \right)}\, dpdg_{9199} = \frac{\sin{\left(pdg_{1464} pdg_{9139} \right)}}{pdg_{9139}}$$
no validation is available for declarations 0948572140:
0948572140:
28 substitute LHS of expr 1 into expr 2
1. 8575746378; locally 9339495:
$$\int \frac{1}{2} dx = \frac{1}{2} x$$
$$\int \frac{1}{2}\, dpdg_{1464} = \frac{pdg_{1464}}{2}$$
2. 1202310110; locally 0203020:
$$\frac{1}{a^2} = \int_0^W \frac{1}{2} dx - \frac{1}{2} \int_0^W \cos\left(2\frac{n \pi}{W}x\right) dx$$
$$\frac{1}{pdg_{9139}^{2}} = \int\limits_{0}^{pdg_{2523}} \left(\frac{pdg_{9199}}{2} - \frac{\int\limits_{0}^{pdg_{2523}} \cos{\left(\frac{2 pdg_{1592} pdg_{3141} pdg_{4037}}{pdg_{2523}} \right)}\, dpdg_{4037}}{2}\right)\, dpdg_{4037}$$
1. 1202312210; locally 8584733:
$$\frac{1}{a^2} = \frac{1}{2}W - \frac{1}{2} \int_0^W \cos\left(2\frac{n \pi}{W}x\right) dx$$
$$\frac{1}{pdg_{9139}^{2}} = \frac{pdg_{2523}}{2} - \frac{\int\limits_{0}^{pdg_{2523}} \cos{\left(\frac{2 pdg_{1592} pdg_{3141} pdg_{4037}}{pdg_{2523}} \right)}\, dpdg_{4037}}{2}$$
LHS diff is 0 RHS diff is Piecewise((pdg2523*(2*pdg1592*pdg3141*pdg9199 - 2*pdg1592*pdg3141 - pdg2523*sin(2*pdg1592*pdg3141) + sin(2*pdg1592*pdg3141))/(4*pdg1592*pdg3141), Ne(pdg1592*pdg3141/pdg2523, 0)), (pdg2523*(-pdg2523 + pdg9199)/2, True)) 8575746378:
1202310110:
1202312210:
8575746378:
1202310110:
1202312210:
34 substitute LHS of expr 1 into expr 2
1. 9485747246; locally 9394858:
$$-\sqrt{\frac{2}{W}} = a$$
$$- \sqrt{2} \sqrt{\frac{1}{pdg_{2523}}} = pdg_{9139}$$
2. 2944838499; locally 3452131:
$$\psi(x) = a \sin(\frac{n \pi}{W} x)$$
$$\operatorname{pdg}_{9489}{\left(pdg_{1464} \right)} = pdg_{9139} \sin{\left(\frac{pdg_{1592} pdg_{3141} pdg_{4037}}{pdg_{2523}} \right)}$$
1. 9393939992; locally 8474765:
$$\psi(x) = \sqrt{\frac{2}{W}} \sin\left(\frac{n \pi}{W} x\right)$$
$$\operatorname{pdg}_{9489}{\left(pdg_{1464} \right)} = \sqrt{2} \sqrt{\frac{1}{pdg_{2523}}} \sin{\left(\frac{pdg_{1464} pdg_{1592} pdg_{3141}}{pdg_{2523}} \right)}$$
LHS diff is 0 RHS diff is pdg9139*sin(pdg1592*pdg3141*pdg4037/pdg2523) - sqrt(2)*sqrt(1/pdg2523)*sin(pdg1464*pdg1592*pdg3141/pdg2523) 9485747246:
2944838499:
9393939992:
9485747246:
2944838499:
9393939992:
23 expand integrand
1. 9858028950; locally 0495054:
$$\frac{1}{a^2} = \int_0^W \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2} dx$$
$$\frac{1}{pdg_{9139}^{2}} = \int\limits_{0}^{pdg_{2523}} \left(\frac{1}{2} - \frac{\cos{\left(\frac{2 pdg_{1464} pdg_{1592} pdg_{3141}}{pdg_{2523}} \right)}}{2}\right)\, dpdg_{1464}$$
1. 1202310110; locally 0203020:
$$\frac{1}{a^2} = \int_0^W \frac{1}{2} dx - \frac{1}{2} \int_0^W \cos\left(2\frac{n \pi}{W}x\right) dx$$
$$\frac{1}{pdg_{9139}^{2}} = \int\limits_{0}^{pdg_{2523}} \left(\frac{pdg_{9199}}{2} - \frac{\int\limits_{0}^{pdg_{2523}} \cos{\left(\frac{2 pdg_{1592} pdg_{3141} pdg_{4037}}{pdg_{2523}} \right)}\, dpdg_{4037}}{2}\right)\, dpdg_{4037}$$
no check performed 9858028950:
1202310110:
9858028950:
1202310110:
13 normalization condition
1. 1934748140; locally 7575626:
$$\int |\psi(x)|^2 dx = 1$$
$$\int \left|{\operatorname{pdg}_{9489}{\left(pdg_{1464} \right)}}\right|^{2}\, dpdg_{9199} = 1$$
no validation is available for assumptions 1934748140:
1934748140:
17 substitute LHS of expr 1 into expr 2
1. 2944838499; locally 3452131:
$$\psi(x) = a \sin(\frac{n \pi}{W} x)$$
$$\operatorname{pdg}_{9489}{\left(pdg_{1464} \right)} = pdg_{9139} \sin{\left(\frac{pdg_{1592} pdg_{3141} pdg_{4037}}{pdg_{2523}} \right)}$$
2. 4857472413; locally 0595847:
$$1 = \int \psi(x)\psi(x)^* dx$$
$$pdg_{9199}$$
1. 0203024440; locally 0495950:
$$1 = \int_0^W a \sin\left(\frac{n \pi}{W} x\right) \psi(x)^* dx$$
$$1 = \int\limits_{0}^{pdg_{2523}} pdg_{9139} \sin{\left(\frac{pdg_{1464} pdg_{1592} pdg_{3141}}{pdg_{2523}} \right)} \overline{\operatorname{pdg}_{9489}{\left(pdg_{1464} \right)}}\, dpdg_{1464}$$
Nothing to split 2944838499:
4857472413:
0203024440:
2944838499:
4857472413:
0203024440:
25 change variable X to Y
1. 0948572140; locally 3992939:
$$\int \cos(a x) dx = \frac{1}{a}\sin(a x)$$
$$\int \cos{\left(pdg_{1464} pdg_{9139} \right)}\, dpdg_{9199} = \frac{\sin{\left(pdg_{1464} pdg_{9139} \right)}}{pdg_{9139}}$$
1. 0009485858:
$$\frac{2n\pi}{W}$$
$$\frac{2 pdg_{1592} pdg_{3141}}{pdg_{2523}}$$
2. 0004831494:
$$a$$
$$pdg_{9139}$$
1. 7564894985; locally 4948377:
$$\int \cos\left(\frac{2n\pi}{W} x\right) dx = \frac{W}{2n\pi}\sin\left(\frac{2n\pi}{W} x\right)$$
$$\int \cos{\left(\frac{2 pdg_{1592} pdg_{3141} pdg_{4037}}{pdg_{2523}} \right)}\, dpdg_{4037} = \frac{pdg_{2523} \sin{\left(\frac{2 pdg_{1592} pdg_{3141} pdg_{4037}}{pdg_{2523}} \right)}}{2 pdg_{1592} pdg_{3141}}$$
LHS diff is pdg9199*cos(pdg1464*pdg9139) - Piecewise((pdg2523*sin(2*pdg1592*pdg3141*pdg4037/pdg2523)/(2*pdg1592*pdg3141), Ne(pdg1592*pdg3141/pdg2523, 0)), (pdg4037, True)) RHS diff is sin(pdg1464*pdg9139)/pdg9139 - pdg2523*sin(2*pdg1592*pdg3141*pdg4037/pdg2523)/(2*pdg1592*pdg3141) 0948572140:
7564894985:
0948572140:
7564894985:
15 swap LHS with RHS
1. 1934748140; locally 7575626:
$$\int |\psi(x)|^2 dx = 1$$
$$\int \left|{\operatorname{pdg}_{9489}{\left(pdg_{1464} \right)}}\right|^{2}\, dpdg_{9199} = 1$$
1. 8572657110; locally 5577567:
$$1 = \int |\psi(x)|^2 dx$$
$$1 = \int \left|{\operatorname{pdg}_{9489}{\left(pdg_{1464} \right)}}\right|^{2}\, dpdg_{1464}$$
LHS diff is pdg9199*Abs(pdg9489(pdg1464))**2 - Integral(Abs(pdg9489(pdg1464))**2, pdg1464) RHS diff is pdg9199*Abs(pdg9489(pdg1464))**2 - Integral(Abs(pdg9489(pdg1464))**2, pdg1464) 1934748140:
8572657110:
1934748140:
8572657110:
37 substitute RHS of expr 1 into expr 2
1. 5727578862; locally 7572748:
$$\frac{d^2}{dx^2} \psi(x) = -k^2 \psi(x)$$
$$pdg_{9199}$$
2. 8582885111; locally 7572118:
$$\psi(x) = a \sin(kx) + b \cos(kx)$$
$$\operatorname{pdg}_{9489}{\left(pdg_{4037} \right)} = pdg_{1939} \cos{\left(pdg_{4037} pdg_{5321} \right)} + pdg_{9139} \sin{\left(pdg_{4037} pdg_{5321} \right)}$$
1. 8575748999; locally 2838288:
$$\frac{d^2}{dx^2} \left(a \sin(k x) + b \cos(k x) \right) = -k^2 \left(a \sin(kx) + b \cos(kx) \right)$$
$$\frac{d^{2} \left(pdg_{1939} \cos{\left(pdg_{1464} pdg_{5321} \right)} + pdg_{9139} \sin{\left(pdg_{1464} pdg_{5321} \right)}\right)}{pdg_{9199}^{2}} = - pdg_{5321}^{2} \left(pdg_{1939} \cos{\left(pdg_{1464} pdg_{5321} \right)} + pdg_{9139} \sin{\left(pdg_{1464} pdg_{5321} \right)}\right)$$
Nothing to split 5727578862:
8582885111:
8575748999:
5727578862:
8582885111:
8575748999:
18 substitute LHS of expr 1 into expr 2
1. 8849289982; locally 3452132:
$$\psi(x)^* = a \sin(\frac{n \pi}{W} x)$$
$$\overline{\operatorname{pdg}_{9489}{\left(pdg_{4037} \right)}} = pdg_{9139} \sin{\left(\frac{pdg_{1592} pdg_{3141} pdg_{4037}}{pdg_{2523}} \right)}$$
2. 0203024440; locally 0495950:
$$1 = \int_0^W a \sin\left(\frac{n \pi}{W} x\right) \psi(x)^* dx$$
$$1 = \int\limits_{0}^{pdg_{2523}} pdg_{9139} \sin{\left(\frac{pdg_{1464} pdg_{1592} pdg_{3141}}{pdg_{2523}} \right)} \overline{\operatorname{pdg}_{9489}{\left(pdg_{1464} \right)}}\, dpdg_{1464}$$
1. 8889444440; locally 8478550:
$$1 = \int_0^W a^2 \left(\sin\left(\frac{n \pi}{W} x\right) \right)^2 dx$$
$$1 = \int\limits_{0}^{pdg_{2523}} pdg_{9139}^{2} \sin^{2}{\left(\frac{pdg_{1464} pdg_{1592} pdg_{3141}}{pdg_{2523}} \right)}\, dpdg_{1464}$$
LHS diff is 0 RHS diff is pdg9139*(-pdg9139*Piecewise((pdg2523*(pdg1592*pdg3141/2 - sin(pdg1592*pdg3141)*cos(pdg1592*pdg3141)/2)/(pdg1592*pdg3141), Ne(pdg1592*pdg3141/pdg2523, 0)), (0, True)) + Integral(sin(pdg1464*pdg1592*pdg3141/pdg2523)*conjugate(pdg9489(pdg1464)), (pdg1464, 0, pdg2523))) 8849289982:
0203024440:
8889444440:
8849289982:
0203024440:
8889444440:
14 conjugate function X
1. 2944838499; locally 3452131:
$$\psi(x) = a \sin(\frac{n \pi}{W} x)$$
$$\operatorname{pdg}_{9489}{\left(pdg_{1464} \right)} = pdg_{9139} \sin{\left(\frac{pdg_{1592} pdg_{3141} pdg_{4037}}{pdg_{2523}} \right)}$$
1. 0009587738:
$$\psi$$
$$pdg_{9489}$$
1. 8849289982; locally 3452132:
$$\psi(x)^* = a \sin(\frac{n \pi}{W} x)$$
$$\overline{\operatorname{pdg}_{9489}{\left(pdg_{4037} \right)}} = pdg_{9139} \sin{\left(\frac{pdg_{1592} pdg_{3141} pdg_{4037}}{pdg_{2523}} \right)}$$
no check performed 2944838499:
8849289982:
2944838499:
8849289982:
31 multiply both sides by
1. 4857475848; locally 9493949:
$$\frac{1}{a^2} = \frac{W}{2}$$
$$\frac{1}{pdg_{9139}^{2}} = \frac{pdg_{2523}}{2}$$
1. 0009485857:
$$a^2\frac{2}{W}$$
$$\frac{2 pdg_{9139}^{2}}{pdg_{2523}}$$
1. 8485867742; locally 1029384:
$$\frac{2}{W} = a^2$$
$$\frac{2}{pdg_{2523}} = pdg_{9139}^{2}$$
valid 4857475848:
8485867742:
4857475848:
8485867742:
22 divide both sides by
1. 8576785890; locally 9485800:
$$1 = \int_0^W a^2 \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2} dx$$
$$1 = \int\limits_{0}^{pdg_{2523}} pdg_{9139}^{2} \left(\frac{1}{2} - \frac{\cos{\left(\frac{2 pdg_{1464} pdg_{1592} pdg_{3141}}{pdg_{2523}} \right)}}{2}\right)\, dpdg_{1464}$$
1. 0000040490:
$$a^2$$
$$pdg_{9139}^{2}$$
1. 9858028950; locally 0495054:
$$\frac{1}{a^2} = \int_0^W \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2} dx$$
$$\frac{1}{pdg_{9139}^{2}} = \int\limits_{0}^{pdg_{2523}} \left(\frac{1}{2} - \frac{\cos{\left(\frac{2 pdg_{1464} pdg_{1592} pdg_{3141}}{pdg_{2523}} \right)}}{2}\right)\, dpdg_{1464}$$
valid 8576785890:
9858028950:
8576785890:
9858028950:
40 claim LHS equals RHS
1. 8484544728; locally 1214762:
$$-a k^2\sin(k x) + -b k^2\cos(k x) = -a k^2 \sin(kx) + -b k^2 \cos(k x)$$
$$pdg_{4037}$$
Nothing to split 8484544728:
8484544728:
16 expand magnitude to conjugate
1. 8572657110; locally 5577567:
$$1 = \int |\psi(x)|^2 dx$$
$$1 = \int \left|{\operatorname{pdg}_{9489}{\left(pdg_{1464} \right)}}\right|^{2}\, dpdg_{1464}$$
1. 0009458842:
$$\psi(x)$$
$$\operatorname{pdg}_{9489}{\left(pdg_{1464} \right)}$$
1. 4857472413; locally 0595847:
$$1 = \int \psi(x)\psi(x)^* dx$$
$$pdg_{9199}$$
Nothing to split 8572657110:
4857472413:
8572657110:
4857472413:
35 declare final expr
1. 9393939992; locally 8474765:
$$\psi(x) = \sqrt{\frac{2}{W}} \sin\left(\frac{n \pi}{W} x\right)$$
$$\operatorname{pdg}_{9489}{\left(pdg_{1464} \right)} = \sqrt{2} \sqrt{\frac{1}{pdg_{2523}}} \sin{\left(\frac{pdg_{1464} pdg_{1592} pdg_{3141}}{pdg_{2523}} \right)}$$
no validation is available for declarations 9393939992:
9393939992:
30 simplify
1. 0439492440; locally 0405049:
$$\frac{1}{a^2} = \frac{1}{2}W - \frac{1}{2}\left. \frac{W}{2n\pi}\sin\left(\frac{2n\pi}{W} x\right) \right|_0^W$$
$$\frac{1}{pdg_{9139}^{2}} = \frac{pdg_{2523}}{2} - \frac{pdg_{2523} \sin{\left(\frac{2 pdg_{1592} pdg_{3141} pdg_{4037}}{pdg_{2523}} \right)}}{4 pdg_{1592} pdg_{3141}}$$
1. 4857475848; locally 9493949:
$$\frac{1}{a^2} = \frac{W}{2}$$
$$\frac{1}{pdg_{9139}^{2}} = \frac{pdg_{2523}}{2}$$
LHS diff is 0 RHS diff is -pdg2523*sin(2*pdg1592*pdg3141*pdg4037/pdg2523)/(4*pdg1592*pdg3141) 0439492440:
4857475848:
0439492440:
4857475848:
19 declare identity
1. 9988949211; locally 1231131:
$$(\sin(x))^2 = \frac{1 - \cos(2 x)}{2}$$
$$\sin^{2}{\left(pdg_{1464} \right)} = \frac{1}{2} - \frac{\cos{\left(2 pdg_{1464} \right)}}{2}$$
no validation is available for declarations 9988949211:
9988949211:
29 substitute RHS of expr 1 into expr 2
1. 7564894985; locally 4948377:
$$\int \cos\left(\frac{2n\pi}{W} x\right) dx = \frac{W}{2n\pi}\sin\left(\frac{2n\pi}{W} x\right)$$
$$\int \cos{\left(\frac{2 pdg_{1592} pdg_{3141} pdg_{4037}}{pdg_{2523}} \right)}\, dpdg_{4037} = \frac{pdg_{2523} \sin{\left(\frac{2 pdg_{1592} pdg_{3141} pdg_{4037}}{pdg_{2523}} \right)}}{2 pdg_{1592} pdg_{3141}}$$
2. 1202312210; locally 8584733:
$$\frac{1}{a^2} = \frac{1}{2}W - \frac{1}{2} \int_0^W \cos\left(2\frac{n \pi}{W}x\right) dx$$
$$\frac{1}{pdg_{9139}^{2}} = \frac{pdg_{2523}}{2} - \frac{\int\limits_{0}^{pdg_{2523}} \cos{\left(\frac{2 pdg_{1592} pdg_{3141} pdg_{4037}}{pdg_{2523}} \right)}\, dpdg_{4037}}{2}$$
1. 0439492440; locally 0405049:
$$\frac{1}{a^2} = \frac{1}{2}W - \frac{1}{2}\left. \frac{W}{2n\pi}\sin\left(\frac{2n\pi}{W} x\right) \right|_0^W$$
$$\frac{1}{pdg_{9139}^{2}} = \frac{pdg_{2523}}{2} - \frac{pdg_{2523} \sin{\left(\frac{2 pdg_{1592} pdg_{3141} pdg_{4037}}{pdg_{2523}} \right)}}{4 pdg_{1592} pdg_{3141}}$$
LHS diff is 0 RHS diff is -Piecewise((pdg2523*sin(2*pdg1592*pdg3141)/(2*pdg1592*pdg3141), Ne(pdg1592*pdg3141/pdg2523, 0)), (pdg2523, True))/2 + pdg2523*sin(2*pdg1592*pdg3141*pdg4037/pdg2523)/(4*pdg1592*pdg3141) 7564894985:
1202312210:
0439492440:
7564894985:
1202312210:
0439492440:
5 LHS of expr 1 equals LHS of expr 2
1. 9585727710; locally 8577781:
$$\psi(x=0) = 0$$
$$\operatorname{pdg}_{9489}{\left(pdg_{1464} = 0 \right)} = 0$$
2. 8582885111; locally 7572118:
$$\psi(x) = a \sin(kx) + b \cos(kx)$$
$$\operatorname{pdg}_{9489}{\left(pdg_{4037} \right)} = pdg_{1939} \cos{\left(pdg_{4037} pdg_{5321} \right)} + pdg_{9139} \sin{\left(pdg_{4037} pdg_{5321} \right)}$$
1. 8577275751; locally 7547581:
$$0 = a \sin(0) + b\cos(0)$$
$$0 = pdg_{1939}$$
input diff is -pdg9489(pdg4037) + pdg9489(Eq(pdg1464, 0)) diff is 0 diff is -pdg1939*cos(pdg4037*pdg5321) + pdg1939 - pdg9139*sin(pdg4037*pdg5321) 9585727710:
8582885111:
8577275751:
9585727710:
8582885111:
8577275751:
7 substitute RHS of expr 1 into expr 2
1. 1293913110; locally 7572859:
$$0 = b$$
$$0 = pdg_{1939}$$
2. 8582885111; locally 7572118:
$$\psi(x) = a \sin(kx) + b \cos(kx)$$
$$\operatorname{pdg}_{9489}{\left(pdg_{4037} \right)} = pdg_{1939} \cos{\left(pdg_{4037} pdg_{5321} \right)} + pdg_{9139} \sin{\left(pdg_{4037} pdg_{5321} \right)}$$
1. 9059289981; locally 7562671:
$$\psi(x) = a \sin(k x)$$
$$pdg_{1464}$$
Nothing to split 1293913110:
8582885111:
9059289981: no LHS/RHS split
1293913110:
8582885111:
9059289981: N/A
3 boundary condition for expr
1. 5727578862; locally 7572748:
$$\frac{d^2}{dx^2} \psi(x) = -k^2 \psi(x)$$
$$pdg_{9199}$$
1. 9585727710; locally 8577781:
$$\psi(x=0) = 0$$
$$\operatorname{pdg}_{9489}{\left(pdg_{1464} = 0 \right)} = 0$$
no validation is available for assumptions 5727578862:
9585727710:
5727578862:
9585727710:
4 boundary condition for expr
1. 5727578862; locally 7572748:
$$\frac{d^2}{dx^2} \psi(x) = -k^2 \psi(x)$$
$$pdg_{9199}$$
1. 9495857278; locally 8585727:
$$\psi(x=W) = 0$$
$$pdg_{2523}$$
no validation is available for assumptions 5727578862:
9495857278: no LHS/RHS split
5727578862:
9495857278: N/A
10 expr 1 is true under condition expr 2
1. 1020010291; locally 8577672:
$$0 = a \sin(k W)$$
$$0 = pdg_{9139} \sin{\left(pdg_{2523} pdg_{5321} \right)}$$
2. 1857710291; locally 8577711:
$$0 = a \sin(n \pi)$$
$$0 = pdg_{9139} \sin{\left(pdg_{1592} pdg_{3141} \right)}$$
1. 1010923823; locally 9847600:
$$k W = n \pi$$
$$pdg_{2523} pdg_{5321} = pdg_{1592} pdg_{3141}$$
no check performed 1020010291: error for dim with 1020010291
1857710291:
1010923823:
1020010291: N/A
1857710291:
1010923823:
9 declare identity
1. 1857710291; locally 8577711:
$$0 = a \sin(n \pi)$$
$$0 = pdg_{9139} \sin{\left(pdg_{1592} pdg_{3141} \right)}$$
no validation is available for declarations 1857710291:
1857710291:
12 substitute RHS of expr 1 into expr 2
1. 1858772113; locally 9495882:
$$k = \frac{n \pi}{W}$$
$$pdg_{5321} = \frac{pdg_{1592} pdg_{3141}}{pdg_{2523}}$$
2. 9059289981; locally 7562671:
$$\psi(x) = a \sin(k x)$$
$$pdg_{1464}$$
1. 2944838499; locally 3452131:
$$\psi(x) = a \sin(\frac{n \pi}{W} x)$$
$$\operatorname{pdg}_{9489}{\left(pdg_{1464} \right)} = pdg_{9139} \sin{\left(\frac{pdg_{1592} pdg_{3141} pdg_{4037}}{pdg_{2523}} \right)}$$
Nothing to split 1858772113:
9059289981: no LHS/RHS split
2944838499:
1858772113:
9059289981: N/A
2944838499:
6 simplify
1. 8577275751; locally 7547581:
$$0 = a \sin(0) + b\cos(0)$$
$$0 = pdg_{1939}$$
1. 1293913110; locally 7572859:
$$0 = b$$
$$0 = pdg_{1939}$$
valid 8577275751:
1293913110:
8577275751:
1293913110:
1 declare initial expr
1. 5727578862; locally 7572748:
$$\frac{d^2}{dx^2} \psi(x) = -k^2 \psi(x)$$
$$pdg_{9199}$$
no validation is available for declarations 5727578862:
5727578862:
21 substitute RHS of expr 1 into expr 2
1. 7575738420; locally 0100404:
$$\left(\sin\left(\frac{n \pi}{W}x\right) \right)^2 = \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2}$$
$$\sin^{2}{\left(\frac{pdg_{1592} pdg_{3141} pdg_{4037}}{pdg_{2523}} \right)} = \frac{1}{2} - \frac{\cos{\left(\frac{2 pdg_{1464} pdg_{1592} pdg_{3141}}{pdg_{2523}} \right)}}{2}$$
2. 8889444440; locally 8478550:
$$1 = \int_0^W a^2 \left(\sin\left(\frac{n \pi}{W} x\right) \right)^2 dx$$
$$1 = \int\limits_{0}^{pdg_{2523}} pdg_{9139}^{2} \sin^{2}{\left(\frac{pdg_{1464} pdg_{1592} pdg_{3141}}{pdg_{2523}} \right)}\, dpdg_{1464}$$
1. 8576785890; locally 9485800:
$$1 = \int_0^W a^2 \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2} dx$$
$$1 = \int\limits_{0}^{pdg_{2523}} pdg_{9139}^{2} \left(\frac{1}{2} - \frac{\cos{\left(\frac{2 pdg_{1464} pdg_{1592} pdg_{3141}}{pdg_{2523}} \right)}}{2}\right)\, dpdg_{1464}$$
valid 7575738420:
8889444440:
8576785890:
7575738420:
8889444440:
8576785890:
11 divide both sides by
1. 1010923823; locally 9847600:
$$k W = n \pi$$
$$pdg_{2523} pdg_{5321} = pdg_{1592} pdg_{3141}$$
1. 0001334112:
$$W$$
$$pdg_{2523}$$
1. 1858772113; locally 9495882:
$$k = \frac{n \pi}{W}$$
$$pdg_{5321} = \frac{pdg_{1592} pdg_{3141}}{pdg_{2523}}$$
valid 1010923823:
1858772113:
1010923823:
1858772113:
39 simplify
1. 8485757728; locally 8474762:
$$a \frac{d^2}{dx^2}\sin(kx) + b \frac{d^2}{dx^2}\cos(k x) = -a k^2 \sin(kx) + -b k^2 \cos(kx)$$
$$pdg_{9199}$$
1. 8484544728; locally 1214762:
$$-a k^2\sin(k x) + -b k^2\cos(k x) = -a k^2 \sin(kx) + -b k^2 \cos(k x)$$
$$pdg_{4037}$$
Nothing to split 8485757728:
8484544728:
8485757728:
8484544728:
8 LHS of expr 1 equals LHS of expr 2
1. 9495857278; locally 8585727:
$$\psi(x=W) = 0$$
$$pdg_{2523}$$
2. 9059289981; locally 7562671:
$$\psi(x) = a \sin(k x)$$
$$pdg_{1464}$$
1. 1020010291; locally 8577672:
$$0 = a \sin(k W)$$
$$0 = pdg_{9139} \sin{\left(pdg_{2523} pdg_{5321} \right)}$$
Nothing to split 9495857278: no LHS/RHS split
9059289981: no LHS/RHS split
1020010291: error for dim with 1020010291
9495857278: N/A
9059289981: N/A
1020010291: N/A
Physics Derivation Graph: Steps for particle in a 1D box

## Symbols for this derivation

symbol ID category latex scope dimension name value Used in derivations references
4037 variable x
$$x$$
['real']
• length: 1
position
53
9139 variable a
$$a$$
['real'] dimensionless 45
9199 variable dx
$$dx$$
['real']
• length: 1
15
9489 variable \psi
$$\psi$$
complex dimensionless none
• str_note
27
3141 constant \pi
$$\pi$$
['real'] dimensionless pi 3.1415   dimensionless
72
1592 variable n
$$n$$
integer dimensionless index 23
1939 variable b
$$b$$
['real'] dimensionless 20
1464 variable x
$$x$$
['real'] dimensionless 140
5321 variable k
$$k$$
['real']
• length: -1
angular wavenumber
13
2523 variable W
$$W$$
real
• length: 1
width 25
MESSAGE:
• local variable 'all_df' referenced before assignment