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Eq. \ref{eq:7572748} is an initial equation. \begin{equation} \frac{d^2}{dx^2} \psi(x) = -k^2 \psi(x) \label{eq:7572748} \end{equation} Judicious choice as a guessed solution to Eq. \ref{eq:7572748} is \begin{equation} \psi(x) = a \sin(kx) + b \cos(kx) \label{eq:7572118} \end{equation} A boundary condition for Eq. \ref{eq:7572748} is Eq. \ref{eq:8577781} \begin{equation} \psi(x=0) = 0 \label{eq:8577781} \end{equation} A boundary condition for Eq. \ref{eq:7572748} is Eq. \ref{eq:8585727} \begin{equation} \psi(x=W) = 0 \label{eq:8585727} \end{equation} LHS of Eq. \ref{eq:8577781} is equal to LHS of Eq. \ref{eq:7572118}; yields Eq. \ref{eq:7547581}. \begin{equation} 0 = a \sin(0) + b\cos(0) \label{eq:7547581} \end{equation} Simplify Eq. \ref{eq:7547581}; yields Eq. \ref{eq:7572859}. \begin{equation} 0 = b \label{eq:7572859} \end{equation} Substitute RHS of Eq. \ref{eq:7572859} into Eq. \ref{eq:7572118}; yields Eq. \ref{eq:7562671}. \begin{equation} \psi(x) = a \sin(k x) \label{eq:7562671} \end{equation} LHS of Eq. \ref{eq:8585727} is equal to LHS of Eq. \ref{eq:7562671}; yields Eq. \ref{eq:8577672}. \begin{equation} 0 = a \sin(k W) \label{eq:8577672} \end{equation} Eq. \ref{eq:8577711} is an identity. \begin{equation} 0 = a \sin(n \pi) \label{eq:8577711} \end{equation} Eq. \ref{eq:8577672} is valid when Eq. \ref{eq:8577711} occurs; yields Eq. \ref{eq:9847600}. \begin{equation} k W = n \pi \label{eq:9847600} \end{equation} Divide both sides of Eq. \ref{eq:9847600} by \(W\); yields Eq. \ref{eq:9495882}. \begin{equation} k = \frac{n \pi}{W} \label{eq:9495882} \end{equation} Substitute RHS of Eq. \ref{eq:9495882} into Eq. \ref{eq:7562671}; yields Eq. \ref{eq:3452131}. \begin{equation} \psi(x) = a \sin(\frac{n \pi}{W} x) \label{eq:3452131} \end{equation} Normalization condition is Eq. \ref{eq:7575626}. \begin{equation} \int |\psi(x)|^2 dx = 1 \label{eq:7575626} \end{equation} Conjugate \(\psi\) in Eq. \ref{eq:3452131}; yields Eq. \ref{eq:3452132}. \begin{equation} \psi(x)^* = a \sin(\frac{n \pi}{W} x) \label{eq:3452132} \end{equation} Swap LHS of Eq. \ref{eq:7575626} with RHS; yields Eq. \ref{eq:5577567}. \begin{equation} 1 = \int |\psi(x)|^2 dx \label{eq:5577567} \end{equation} Expand \(\psi(x)\) in Eq. \ref{eq:5577567} with conjugate; yields Eq. \ref{eq:0595847}. \begin{equation} 1 = \int \psi(x)\psi(x)^* dx \label{eq:0595847} \end{equation} Substitute LHS of Eq. \ref{eq:3452131} into Eq. \ref{eq:0595847}; yields Eq. \ref{eq:0495950}. \begin{equation} 1 = \int_0^W a \sin\left(\frac{n \pi}{W} x\right) \psi(x)^* dx \label{eq:0495950} \end{equation} Substitute LHS of Eq. \ref{eq:3452132} into Eq. \ref{eq:0495950}; yields Eq. \ref{eq:8478550}. \begin{equation} 1 = \int_0^W a^2 \left(\sin\left(\frac{n \pi}{W} x\right) \right)^2 dx \label{eq:8478550} \end{equation} Eq. \ref{eq:1231131} is an identity. \begin{equation} (\sin(x))^2 = \frac{1 - \cos(2 x)}{2} \label{eq:1231131} \end{equation} Change variable \(\frac{n \pi}{W}x\) to \(x\) in Eq. \ref{eq:1231131}; yields Eq. \ref{eq:0100404}. \begin{equation} \left(\sin\left(\frac{n \pi}{W}x\right) \right)^2 = \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2} \label{eq:0100404} \end{equation} Substitute RHS of Eq. \ref{eq:0100404} into Eq. \ref{eq:8478550}; yields Eq. \ref{eq:9485800}. \begin{equation} 1 = \int_0^W a^2 \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2} dx \label{eq:9485800} \end{equation} Divide both sides of Eq. \ref{eq:9485800} by \(a^2\); yields Eq. \ref{eq:0495054}. \begin{equation} \frac{1}{a^2} = \int_0^W \frac{1-\cos\left(2\frac{n \pi}{W}x\right)}{2} dx \label{eq:0495054} \end{equation} Expand integrand of Eq. \ref{eq:0495054}; yields Eq. \ref{eq:0203020}. \begin{equation} \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 \label{eq:0203020} \end{equation} Eq. \ref{eq:3992939} is an identity. \begin{equation} \int \cos(a x) dx = \frac{1}{a}\sin(a x) \label{eq:3992939} \end{equation} Change variable \(\frac{2n\pi}{W}\) to \(a\) in Eq. \ref{eq:3992939}; yields Eq. \ref{eq:4948377}. \begin{equation} \int \cos\left(\frac{2n\pi}{W} x\right) dx = \frac{W}{2n\pi}\sin\left(\frac{2n\pi}{W} x\right) \label{eq:4948377} \end{equation} Eq. \ref{eq:0021030} is an identity. \begin{equation} \int a dx = a x \label{eq:0021030} \end{equation} Change variable \(1/2\) to \(a\) in Eq. \ref{eq:0021030}; yields Eq. \ref{eq:9339495}. \begin{equation} \int \frac{1}{2} dx = \frac{1}{2} x \label{eq:9339495} \end{equation} Substitute LHS of Eq. \ref{eq:9339495} into Eq. \ref{eq:0203020}; yields Eq. \ref{eq:8584733}. \begin{equation} \frac{1}{a^2} = \frac{1}{2}W - \frac{1}{2} \int_0^W \cos\left(2\frac{n \pi}{W}x\right) dx \label{eq:8584733} \end{equation} Substitute RHS of Eq. \ref{eq:4948377} into Eq. \ref{eq:8584733}; yields Eq. \ref{eq:0405049}. \begin{equation} \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 \label{eq:0405049} \end{equation} Simplify Eq. \ref{eq:0405049}; yields Eq. \ref{eq:9493949}. \begin{equation} \frac{1}{a^2} = \frac{W}{2} \label{eq:9493949} \end{equation} Multiply both sides of Eq. \ref{eq:9493949} by \(a^2\frac{2}{W}\); yields Eq. \ref{eq:1029384}. \begin{equation} \frac{2}{W} = a^2 \label{eq:1029384} \end{equation} Take the square root of both sides of Eq. \ref{eq:1029384}; yields Eq. \ref{eq:9394857} and Eq. \ref{eq:9394858}. \begin{equation} \sqrt{\frac{2}{W}} = a \label{eq:9394857} \end{equation} \begin{equation} -\sqrt{\frac{2}{W}} = a \label{eq:9394858} \end{equation} Substitute LHS of Eq. \ref{eq:9394857} into Eq. \ref{eq:3452131}; yields Eq. \ref{eq:8474766}. \begin{equation} \psi(x) = -\sqrt{\frac{2}{W}} \sin\left(\frac{n \pi}{W} x\right) \label{eq:8474766} \end{equation} Substitute LHS of Eq. \ref{eq:9394858} into Eq. \ref{eq:3452131}; yields Eq. \ref{eq:8474765}. \begin{equation} \psi(x) = \sqrt{\frac{2}{W}} \sin\left(\frac{n \pi}{W} x\right) \label{eq:8474765} \end{equation} Eq. \ref{eq:8474765} is one of the final equations. Substitute RHS of Eq. \ref{eq:7572748} into Eq. \ref{eq:7572118}; yields Eq. \ref{eq:2838288}. \begin{equation} \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) \label{eq:2838288} \end{equation} Simplify Eq. \ref{eq:2838288}; yields Eq. \ref{eq:8474762}. \begin{equation} 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) \label{eq:8474762} \end{equation} Simplify Eq. \ref{eq:8474762}; yields Eq. \ref{eq:1214762}. \begin{equation} -a k^2\sin(k x) + -b k^2\cos(k x) = -a k^2 \sin(kx) + -b k^2 \cos(k x) \label{eq:1214762} \end{equation} Thus we see that LHS of Eq. \ref{eq:1214762} is equal to RHS.