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diff --git a/Assignment_3/Problem_3/Project.toml b/Assignment_3/Problem_3/Project.toml
index 9023662..8c32031 100644
--- a/Assignment_3/Problem_3/Project.toml
+++ b/Assignment_3/Problem_3/Project.toml
@@ -1,3 +1,6 @@
+name = "SchrödingerSolver"
+uuid = "44c901a1-9596-43aa-b2cd-6fc2ac4d38fa"
+
 [deps]
 LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
-Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80"
+Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80"
\ No newline at end of file
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diff --git a/Assignment_3/Problem_3/problem_3.tex b/Assignment_3/Problem_3/problem_3.tex
index 06f99dd..c5643b9 100644
--- a/Assignment_3/Problem_3/problem_3.tex
+++ b/Assignment_3/Problem_3/problem_3.tex
@@ -1,3 +1,4 @@
+\textit{长海今天人好多啊}
 \subsection{题目描述}
 Solve the 1D Schrödinger equation with the potential (i) \( V(x) = x^2 \); (ii) \( V(x) = x^4 - x^2 \) with the variational approach using a \textbf{Gaussian basis} (either fixed widths or fixed centers)
 \[
@@ -6,7 +7,308 @@ \subsection{题目描述}
 Consider the three lowest energy eigenstates.
 
 \subsection{程序描述}
+本题要求用原版高斯基线性组合，通过变分原理找能级。高斯基并不相互正交，故需要计算重叠积分 $S_{ij}$：
+\[
+    S_{ij} = \left(\frac{v_i v_j}{v_i + v_j}\right)^{1/2} \exp{\left(-\frac{v_i v_j (s_i - s_j)^2}{v_i + v_j}\right)},
+\]
+接着计算动能积分 $T_{ij}$（此解答采用自然单位制 $\hbar = m = 1 $）：
+\[
+    T_{ij} = \left(\frac{v_i^{3/2} v_j^{3/2}}{\sqrt{\pi} (v_i + v_j)^{5/2}}\right) \left[(v_i + v_j) - 2 v_i v_j (s_i - s_j)^2\right] \exp{\left(-\frac{v_i v_j (s_i - s_j)^2}{v_i + v_j}\right)}.
+\]
+而对于势能积分 $V_{ij}$，通过广义本征值问题 $\mathbf{H} \mathbf{c} = E \mathbf{S} \mathbf{c}$ 可以找到能级 $E$ 和对应的波函数 $\psi = \mathbf{c} \phi$。计算的难点在于，对于一般的势能 $V(x)$，高斯基下的积分
+\[
+    V_{ij} = \int \phi_i(x) V(x) \phi_j(x) \, dx
+\]
+没有解析解，需要数值积分来求解。幸运的是，本题所求的势能 $V(x)$ 是多项式形式，使用 Mathematica\textsuperscript{\textregistered} 可以方便地求解 $x^n$ 对应的势能积分。然而，高阶积分的解析表达式会变得非常复杂。GPT 提醒我，两个高斯基的乘积仍是一个新的高斯函数，因此可利用该特性来简化多项式势能的积分，即通过高斯分布的矩来处理。两个高斯函数的乘积为
+\[
+\phi_i(x) \phi_j(x) = \left(\frac{2 v_i v_j}{\pi^2}\right)^{1/4} \exp\left(-v_i (x - s_i)^2 - v_j (x - s_j)^2\right).
+\]
+将 $v_i (x - s_i)^2 + v_j (x - s_j)^2$ 展开为：
+\[
+(v_i + v_j) \left( x - \frac{v_i s_i + v_j s_j}{v_i + v_j} \right)^2 + \frac{v_i v_j (s_i - s_j)^2}{v_i + v_j}.
+\]
+因此，$\phi_i(x) \phi_j(x)$ 可以写成一个新的高斯函数的形式：
+\[
+\phi_i(x) \phi_j(x) = \left(\frac{2v_i v_j}{\pi (v_i + v_j)}\right)^{1/2} \exp\left(-\frac{v_i v_j (s_i - s_j)^2}{v_i + v_j}\right) \exp\left(- (v_i + v_j) \left(x - \mu\right)^2 \right),
+\]
+其中新的高斯分布的中心 $\mu$ 和方差 $\sigma^2$ 为：
+\[
+\mu = \frac{v_i s_i + v_j s_j}{v_i + v_j}, \quad \sigma^2 = \frac{1}{2(v_i + v_j)}.
+\]
+于是对于多项式形式的势能 $V(x) = x^n$，我们可以将势能积分转化为新高斯分布的矩问题。高斯分布 $G(x; \mu, \sigma^2)$ 的 $n$ 阶矩 $M_n = \mathbb{E}[x^n]$ 满足递推关系
+\[
+M_n(\mu, \sigma^2) = \mu M_{n-1}(\mu, \sigma^2) + (n-1) \sigma^2 M_{n-2}(\mu, \sigma^2),
+\]
+其中
+\[
+M_0 = 1, \quad M_1 = \mu.
+\]
+因此，前几个矩的具体值为
+\[
+M_2 = \mu^2 + \sigma^2,
+\]
+\[
+M_3 = \mu^3 + 3 \mu \sigma^2,
+\]
+\[
+M_4 = \mu^4 + 6 \mu^2 \sigma^2 + 3 \sigma^4.
+\]
+综上，势能积分可以表示为
+\[
+V_{ij} = S_{ij} \cdot M_n(\mu, \sigma^2),
+\]
+其中 $S_{ij}$ 是重叠积分，保证新高斯分布的归一化，$M_n$ 是高斯分布的 $n$ 阶矩。通过递推公式可以更方便地计算多项式势能下的高斯积分，\href{https://github.com/bud-primordium/Computational-Physics-Fall-2024/blob/main/Assignment_3/Problem_3/src/methods.jl#L106-L117}{源代码}中实际使用\texttt{Switch}语句处理了$n\in [1,4]$的情形。最后的本征值求解借助了\textit{Julia} \ \texttt{LinearAlgebra}库中的\texttt{eigen}函数，并采用梯形法计算最后波函数的归一化系数。
+
+本题子目录结构如下
+\begin{multicols}{2}
+	\begin{verbatim}
+        |-- Documenter.html
+        |-- figs
+        |-- problem_3.tex
+        `-- src
+            |-- documenter_output
+            |-- integrals.nb
+            |-- interaction.jl
+            |-- main.jl
+            |-- methods.jl
+            `-- utils.jl
+    \end{verbatim}
+\end{multicols}
+\noindent \textbf{助教老师}审阅源代码时，可借助\texttt{Documenter.html}便捷查看\texttt{Documenter}生成的\href{https://bud-primordium.github.io/Computational-Physics-Fall-2024/Assignment_3/Problem_3/src/documenter_output/build/index.html}{注释文档}。在\texttt{src}目录下，运行\ccmd{julia main.jl}即可运行程序（需安装\texttt{LinearAlgebra}与\texttt{Plots}包），\texttt{main.jl}是主程序入口点，其逻辑结构在伪代码 \ \ref{alg:interactive_entry} \ 中有详细说明；\texttt{methods.jl}负责算法实现，包括重叠积分，动能积分，势能积分与求解Schrödinger方程等，逻辑结构在伪代码 \ \ref{alg:integral_functions},\ref{alg:matrix_solver} \ 中有详细说明；\texttt{interaction.jl}负责交互功能，包括展示主菜单，根据不同选项处理用户输入输出等；\texttt{utils.jl}包含一些通用的工具函数，如梯形法，波函数归一化，异常处理等。目录下还准备了\texttt{integrals.nb}，是用于验证高斯积分的 Mathematica\textsuperscript{\textregistered} 笔记本文件，最终源代码采用前述的多级矩计算方法，比直接录入Mathematica\textsuperscript{\textregistered}结果更加高效简洁且具备拓展性\footnote{猛然想起钟阳老师说DeepMD\textsuperscript{\textregistered}就是因为内置了许多积分查找的优化，速度提升不少。}。
+
+程序主菜单提供了四个选项，分别对应使用：自定义参数求解题目要求的两种势阱、使用默认参数一次求解两种势阱，以及自定义四次以内的多项式势能进行求解。自定义参数包括高斯基数量、宽度$v$、中心$s$（在子模式下可选择变动其中之一，为另一个指定范围），输出能级数，自定义势阱系数等。用户还可以选择是否绘制波函数图像，但归一化处理较为耗时，故内置了多线程处理不同能级绘制，但依赖于用户授权，故建议至少使用\ccmd{julia -t 4 main.jl}运行，可显著提升绘图速度。输出结果与绘图详见\ref{sec:problem_3_example}节所示。
+
+\textit{碎碎念：我多次尝试使用\texttt{PackageCompiler}打包编译本程序，奈何依赖屡屡出错，也没有合适的压缩方法，更不用谈交叉编译；尝试过使用\texttt{Pluto}交互式改造，发布至\textit{Binder}，但新语法过多，中道崩殂。尽管我非常欣赏Julia超前的各项设计理念，风格优美在一众科学计算语言中当属清流，但生态实在还是不够配套，继续加油吧！}
+
 
 \subsection{伪代码}
+\begin{algorithm}[H]
+    \caption{Interactive Entry Program for Solving the 1D Schrödinger Equation}
+    \label{alg:interactive_entry}
+
+    % Function and Data Format Settings
+    \SetKwFunction{DisplayMenu}{DisplayMenu}
+    \SetKwFunction{GetUserChoice}{GetUserChoice}
+    \SetKwFunction{GetParameters}{GetParameters}
+    \SetKwFunction{BuildMatrices}{BuildMatrices}
+    \SetKwFunction{SolveSchrodinger}{SolveSchrodinger}
+    \SetKwFunction{PrintEnergies}{PrintEnergies}
+    \SetKwFunction{AskUserToPlot}{AskUserToPlot}
+    \SetKwFunction{ComputeWavefunction}{ComputeWavefunction}
+    \SetKwFunction{NormalizeWavefunction}{NormalizeWavefunction}
+    \SetKwFunction{PlotWavefunctions}{PlotWavefunctions}
+    \SetKwFunction{Range}{Range}
+    \SetKwFunction{InitializeArray}{InitializeArray}
+    \SetKwFunction{Print}{Print}
+
+    \SetKwData{Choice}{choice}
+    \SetKwData{Potential}{potential}
+    \SetKwData{Params}{params}
+    \SetKwData{PotentialName}{potential\_name}
+    \SetKwData{NumLevels}{num\_levels}
+    \SetKwData{XVals}{x\_vals}
+    \SetKwData{Wavefunctions}{wavefunctions}
+    \SetKwData{N}{N}
+    \SetKwData{v}{v}
+    \SetKwData{s}{s}
+    \SetKwData{Energies}{energies}
+    \SetKwData{States}{states}
+    \SetKwData{PotentialList}{potential\_list}
+    \SetKwData{ParamsList}{params\_list}
+    \SetKwData{NameList}{name\_list}
+    \SetKwData{HMatrix}{\textbf{H}}
+    \SetKwData{SMatrix}{\textbf{S}}
+
+    \KwIn{\ \Choice (str): User selection for potential type.\\
+    \qquad \qquad \N (int): Number of basis functions.\\
+    \qquad \qquad \v (Array[float]): Fixed or variable basis widths.\\
+    \qquad \qquad \s (Array[float]): Basis center positions.\\
+    \qquad \qquad \NumLevels (int) [default=3]: Number of energy levels to compute.}
+
+    \KwOut{\Energies (Array), optional plots}
+
+    \While{\textbf{True}}{
+        \DisplayMenu()\tcp*[r]{Display menu options to the user}
+        \Choice $\gets$ \GetUserChoice()\tcp*[r]{Get user's menu selection}
+
+        \If{\Choice \textbf{equals} 'q' \textbf{or} \Choice \textbf{equals} 'Q'}{
+            \Print("Program exited.")\tcp*[r]{Exit message}
+            \textbf{break}\tcp*[r]{Terminate the loop}
+        }
+
+        (\N, \v, \s, \PotentialList, \ParamsList, \NameList, \NumLevels) $\gets$ \GetParameters(\Choice)\;
+        \tcp{Option 1: Custom parameters for $V(x) = x^2$}
+        \tcp{Option 2: Custom parameters for $V(x) = x^4 - x^2$}
+        \tcp{Option 3: Use default parameters for both problems}
+        \tcp{Option 4: Custom polynomial potential up to degree 4}
+
+        \For{$i \gets 1$ \KwTo \texttt{Length}(\PotentialList)}{
+            (\Potential, \Params, \PotentialName) $\gets$ (\PotentialList$[i]$, \ParamsList$[i]$, \NameList$[i]$)\;
+            \tcp{Get potential function and its parameters}
+
+            (\HMatrix, \SMatrix) $\gets$ \BuildMatrices(\N, \v, \s, \Potential, \Params)\tcp*[r]{Construct \HMatrix and \SMatrix matrices}
+
+            (\Energies, \States) $\gets$ \SolveSchrodinger(\HMatrix, \SMatrix, \NumLevels)\tcp*[r]{Solve the eigenvalue problem}
+
+            \PrintEnergies(\Energies, \PotentialName, \NumLevels)\tcp*[r]{Output energy levels}
+
+            \If{\AskUserToPlot(\PotentialName)}{
+                \XVals $\gets$ \Range($-5$, $5$, $200$)\;
+                \Wavefunctions $\gets$ \InitializeArray(\NumLevels)\tcp*[r]{Prepare storage for wavefunctions}
+
+                \For{$n \gets 1$ \KwTo \NumLevels}{
+                    $\psi_n$ $\gets$ \ComputeWavefunction(\XVals, \States$[:, n]$, \v, \s, \N)\tcp*[r]{Compute wavefunction $\psi_n$}
+
+                    \Wavefunctions$[n]$ $\gets$ \NormalizeWavefunction(\XVals, $\psi_n$)
+                    \tcp*[r]{Using \textbf{trapezoidal method}}
+                }
+                \PlotWavefunctions(\XVals, \Wavefunctions, \NumLevels, \PotentialName, \Params)
+            }
+        }
+    }
+\end{algorithm}
+
+\begin{algorithm}[H]
+    \caption{Integral Functions for Gaussian Basis}
+    \label{alg:integral_functions}
+
+    % Function and Data Format Settings
+    \SetKwProg{Fn}{Function}{:}{end}
+    \SetKwFunction{OverlapIntegral}{OverlapIntegral}
+    \SetKwFunction{KineticIntegral}{KineticIntegral}
+    \SetKwFunction{PotentialIntegral}{PotentialIntegral}
+    \SetKwFunction{ComputeMoment}{ComputeMoment}
+
+    % Define Variables
+    \SetKwData{Vp}{$v_p$}
+    \SetKwData{Exponent}{$exponent$}
+    \SetKwData{Prefactor}{$prefactor$}
+    \SetKwData{Sij}{$S_{ij}$}
+    \SetKwData{Tij}{$T_{ij}$}
+    \SetKwData{Vij}{$V_{ij}$}
+    \SetKwData{Mu}{$\mu$}
+    \SetKwData{Sigma}{$\sigma^2$}
+    \SetKwData{Mn}{$M_n$}
+
+    % Mathematical Upright e
+    \newcommand{\me}{\mathrm{e}}
+
+    % Algorithms
+
+    \Fn{\OverlapIntegral{$v_1$, $s_1$, $v_2$, $s_2$}}{
+        \Vp $\gets v_1 + v_2$\tcp*[r]{Combined width parameter}
+        \Exponent, \Prefactor $\gets - \dfrac{v_1 v_2 (s_1 - s_2)^2}{\Vp},\ \dfrac{\sqrt{v_1 v_2}}{\sqrt{\pi \Vp}}$\tcp*[r]{Exponent and prefactor}
+        \Sij $\gets$ \Prefactor $\times$ $\me^{\Exponent}$\tcp*[r]{Compute overlap integral}
+        \Return \Sij\;
+    }
+
+    \Fn{\KineticIntegral{$v_1$, $s_1$, $v_2$, $s_2$}}{
+        \Vp $\gets v_1 + v_2$\tcp*[r]{Combined width parameter}
+        \Exponent, \Prefactor $\gets - \dfrac{v_1 v_2 (s_1 - s_2)^2}{\Vp},\ \dfrac{v_1^{1.5} v_2^{1.5}}{\sqrt{\pi} \Vp^{2.5}}$\tcp*[r]{Exponent and prefactor}
+        $numerator \gets \Vp - 2 v_1 v_2 (s_1 - s_2)^2$\tcp*[r]{Numerator in kinetic integral}
+        \Tij $\gets$ \Prefactor $\times$ $numerator$ $\times$ $\me^{\Exponent}$\tcp*[r]{Compute kinetic energy integral}
+        \Return \Tij\;
+    }
+
+    \Fn{\PotentialIntegral{$v_1$, $s_1$, $v_2$, $s_2$, $n$}}{
+        \Vp $\gets v_1 + v_2$\tcp*[r]{Combined width parameter}
+        \Exponent $\gets - \dfrac{v_1 v_2 (s_1 - s_2)^2}{\Vp}$\;
+        \Sij $\gets \dfrac{\sqrt{v_1 v_2}}{\sqrt{\pi \Vp}} \times \me^{\Exponent}$\tcp*[r]{Compute overlap integral}
+        \Mu $\gets \dfrac{v_1 s_1 + v_2 s_2}{\Vp}$\tcp*[r]{Mean of combined Gaussian}
+        \Sigma $\gets \dfrac{1}{2 \Vp}$\tcp*[r]{Variance of combined Gaussian}
+        \Mn $\gets$ \ComputeMoment{$n$, \Mu, \Sigma}\tcp*[r]{Compute the $n$-th moment}
+        \Vij $\gets$ \Sij $\times$ \Mn\tcp*[r]{Compute potential energy integral}
+        \Return \Vij\;
+    }
+
+    \Fn{\ComputeMoment{$n$, \Mu, \Sigma}}{
+        \eIf{$n = 0$}{
+            \Return $1$\tcp*[r]{Compute $M_n$ using moments of the normal distribution}
+        }{
+            \Return $\sum_{k=0}^{n} \binom{n}{k} \mu^{n-k} \sigma^{k} \cdot (n-k-1)!!$\tcp*[r]{General formula for $M_n$}
+        }
+    }
+
+\end{algorithm}
+
+\begin{algorithm}[H]
+    \caption{Matrix Construction and Schrödinger Equation Solver}
+    \label{alg:matrix_solver}
+
+    % Function and Data Format Settings
+    \SetKwProg{Fn}{Function}{:}{end}
+    \SetKwFunction{BuildMatrices}{BuildMatrices}
+    \SetKwFunction{SolveSchrodinger}{SolveSchrodinger}
+    \SetKwFunction{OverlapIntegral}{OverlapIntegral}
+    \SetKwFunction{KineticIntegral}{KineticIntegral}
+    \SetKwFunction{PotentialFunction}{PotentialFunction}
+    \SetKwFunction{GeneralizedEigenSolve}{GeneralizedEigenSolve}
+
+    % Define Variables
+    \SetKwData{HMatrix}{$\mathbf{H}$}
+    \SetKwData{SMatrix}{$\mathbf{S}$}
+    \SetKwData{Energies}{$energies$}
+    \SetKwData{States}{$states$}
+    \SetKwData{NumLevels}{$num\_levels$}
+    \SetKwData{Idx}{$idx$}
+
+    % Algorithms
+
+    \Fn{\BuildMatrices{$N$, $v$, $s$, PotentialFunction, PotentialParams}}{
+        Initialize \HMatrix\ and \SMatrix\ as $N \times N$ zero matrices\;
+        \For{$i \gets 1$ \KwTo $N$}{
+            \For{$j \gets i$ \KwTo $N$}{
+                $S_{ij} \gets$ \OverlapIntegral{$v[i]$, $s[i]$, $v[j]$, $s[j]$}\;
+                $T_{ij} \gets$ \KineticIntegral{$v[i]$, $s[i]$, $v[j]$, $s[j]$}\;
+                $V_{ij} \gets$ \PotentialFunction($v[i]$, $s[i]$, $v[j]$, $s[j]$, PotentialParams)\;
+                \HMatrix$[i, j] \gets T_{ij} + V_{ij}$\;
+                \SMatrix$[i, j] \gets S_{ij}$\;
+                \If{$i \ne j$}{
+                    \HMatrix$[j, i],$ \SMatrix$[j, i] \gets$ \HMatrix$[i, j],$ \SMatrix$[i, j]$\tcp*[r]{Copy symmetric elements}
+                }
+            }
+        }
+        \Return \HMatrix, \SMatrix\;
+    }
+
+    \Fn{\SolveSchrodinger{\HMatrix, \SMatrix, \NumLevels}}{
+        $(E\_values, E\_vectors) \gets$ \GeneralizedEigenSolve{\HMatrix, \SMatrix}\tcp*[r]{Solve $\mathbf{H} \mathbf{c} = E \mathbf{S} \mathbf{c}$}
+        \Idx $\gets$ Indices that sort $E\_values$ in ascending order\;
+        \Energies $\gets E\_values[\Idx[1..\NumLevels]]$\tcp*[r]{Select lowest \NumLevels\ energies}
+        \States $\gets E\_vectors[:, \Idx[1..\NumLevels]]$\tcp*[r]{Corresponding eigenvectors}
+        \Return \Energies, \States\;
+    }
+
+\end{algorithm}
+
 
 \subsection{结果示例}
+\label{sec:problem_3_example}
+\begin{figure}[H]
+	\centering
+	\includegraphics[width=0.8\textwidth]{Problem_3/figs/choice_3.png}
+    \hspace{5pt}
+    \includegraphics[width=0.5\textwidth]{Problem_3/figs/plot_1.png}
+    \hspace{5pt}
+    \includegraphics[width=0.5\textwidth]{Problem_3/figs/plot_2.png}
+	\caption{选项3：使用默认参数求解$V=x^2$与$V=x^4-x^2$及其前$3$个能级}
+\end{figure}
+
+\begin{figure}[H]
+	\centering
+	\includegraphics[width=0.75\textwidth]{Problem_3/figs/choice_4.png}
+    \hspace{5pt}
+    \includegraphics[width=0.75\textwidth]{Problem_3/figs/plot_3.png}
+	\caption{选项4：求解自定义势阱$V=\frac{1}{2}x^2$及其前$8$个能级}
+	\caption*{\footnotesize{\textbf{注：} 在自然单位制下，与解析解$E_n = (n+\frac{1}{2})\hbar \omega$数值一致}}
+\end{figure}
+
+\begin{figure}[H]
+	\centering
+	\includegraphics[width=0.8\textwidth]{Problem_3/figs/choice_4_2.png}
+    \hspace{5pt}
+    \includegraphics[width=0.8\textwidth]{Problem_3/figs/plot_4.png}
+	\caption{选项4：求解自定义势阱$V=(x+1)^2$及其前$3$个能级}
+	\caption*{\footnotesize{\textbf{注：} 能级与势阱$V=x^2$一致，波函数相较势阱$V=x^2$平移}}
+\end{figure}
+
diff --git a/Assignment_3/Problem_3/src/build.jl b/Assignment_3/Problem_3/src/build.jl
new file mode 100644
index 0000000..58a6009
--- /dev/null
+++ b/Assignment_3/Problem_3/src/build.jl
@@ -0,0 +1,10 @@
+using Pkg
+Pkg.precompile()
+using PackageCompiler
+
+# 生成通用 64 位 CPU 架构的应用
+create_app("./project", "SchrodingerSolver";
+    cpu_target="x86-64", force=true)
+
+# 构建可执行文件
+build_executable("SchrodingerSolver")
diff --git a/Assignment_3/Problem_3/src/project/Manifest.toml b/Assignment_3/Problem_3/src/project/Manifest.toml
new file mode 100644
index 0000000..811845c
--- /dev/null
+++ b/Assignment_3/Problem_3/src/project/Manifest.toml
@@ -0,0 +1,1083 @@
+# This file is machine-generated - editing it directly is not advised
+
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diff --git a/Assignment_3/Problem_3/src/project/Project.toml b/Assignment_3/Problem_3/src/project/Project.toml
new file mode 100644
index 0000000..70bd388
--- /dev/null
+++ b/Assignment_3/Problem_3/src/project/Project.toml
@@ -0,0 +1,6 @@
+name = "SchrodingerSolver"
+uuid = "44c901a1-9596-43aa-b2cd-6fc2ac4d38fa"
+
+[deps]
+LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
+Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80"
diff --git a/Assignment_3/Problem_3/src/project/src/SchrodingerSolver.jl b/Assignment_3/Problem_3/src/project/src/SchrodingerSolver.jl
new file mode 100644
index 0000000..747f71e
--- /dev/null
+++ b/Assignment_3/Problem_3/src/project/src/SchrodingerSolver.jl
@@ -0,0 +1,95 @@
+# main.jl
+# Main script for solving the 1D Schrödinger equation using Gaussian basis functions
+# with additional utility modules.
+#
+# Author: Gilbert Young
+# Date: 2024/10/08
+#
+# This script provides the user interface to select different potential functions,
+# solve the Schrödinger equation, and visualize the results.
+module SchrödingerSolver
+
+include("methods.jl")
+include("utils.jl")
+include("interaction.jl")
+
+using .Methods
+using .Utils
+using .Interaction
+
+using LinearAlgebra
+using Plots
+using Base.Threads
+
+function main()
+    print("Number of threads available: ", Threads.nthreads())
+    while true
+        display_menu()
+        # Prompt for main menu selection
+        print("Enter your choice: ")
+        choice = readline()
+
+        if choice in ["1", "2", "3", "4"]
+            N, v, s, potential_gaussian_integral_list, potential_params_list, potential_name_list, num_levels = get_parameters(choice)
+            if N == 0
+                continue
+            end
+
+            # Loop over potential functions and corresponding parameters/names
+            for i in eachindex(potential_gaussian_integral_list)
+                potential_gaussian_integral = potential_gaussian_integral_list[i]
+                potential_params = potential_params_list[i]
+                potential_name = potential_name_list[i]
+
+                # Build Hamiltonian and overlap matrices
+                H, S = build_matrices(N, v, s, potential_gaussian_integral, potential_params)
+
+                # Solve the Schrödinger equation
+                energies, states = solve_schrodinger(H, S, num_levels)
+
+                # Output results
+                println("\nLowest $(num_levels) energy levels for $potential_name:")
+                for (j, E) in enumerate(energies)
+                    println("Energy Level $(j): E = $(E)")
+                end
+
+                # Ask whether to plot
+                print("Do you want to plot the wave functions for $potential_name? (y/n) ")
+                plot_choice = readline()
+                if plot_choice in ["y", "Y"]
+                    x_vals = range(-5, 5, length=200)
+
+                    # Parallel computation of wave functions
+                    wavefunctions = Vector{Vector{Float64}}(undef, num_levels)
+
+                    @threads for n in 1:num_levels
+                        ψ_n = zeros(Float64, length(x_vals))
+
+                        # Construct the wave function as a linear combination of Gaussian basis functions
+                        for k in 1:N
+                            ψ_n .+= states[k, n] * sqrt(v[k] / π) .* exp.(-v[k] .* (x_vals .- s[k]) .^ 2)
+                        end
+
+                        # Normalize the wave function using the utility function
+                        wavefunctions[n] = normalize_wavefunction(x_vals, ψ_n)
+                    end
+
+                    # Plot the precomputed wave functions
+                    plot_wavefunctions(x_vals, wavefunctions, num_levels, potential_name, potential_params)
+                end
+            end
+        elseif choice in ["q", "Q"]
+            print("Program exited.")
+            break
+        else
+            print("Invalid selection, please try again.\n")
+        end
+    end
+end
+
+# Run the main function if the script is executed directly
+if abspath(PROGRAM_FILE) == @__FILE__
+    main()
+end
+
+end # module SchrödingerSolver
diff --git a/Assignment_3/Problem_3/src/project/src/interaction.jl b/Assignment_3/Problem_3/src/project/src/interaction.jl
new file mode 100644
index 0000000..d9b4eaa
--- /dev/null
+++ b/Assignment_3/Problem_3/src/project/src/interaction.jl
@@ -0,0 +1,75 @@
+# interaction.jl
+# Interaction functions for plotting and displaying the Schrödinger equation solver results.
+#
+# Author: Gilbert Young
+# Date: 2024/10/08
+
+module Interaction
+
+using Plots
+using ..Utils
+
+export display_menu, plot_wavefunctions
+
+@doc raw"""
+    display_menu()
+
+Displays the main menu options to the user.
+"""
+function display_menu()
+    println("\nMain Menu:")
+    println("1. Custom parameters for V(x) = x^2")
+    println("2. Custom parameters for V(x) = x^4 - x^2")
+    println("3. Use default parameters to compute both problems")
+    println("4. Custom polynomial potential function up to degree 4")
+    println("q. Quit")
+end
+
+@doc raw"""
+    plot_wavefunctions(
+        x_vals::AbstractVector{Float64},
+        wavefunctions::Vector{Vector{Float64}},
+        num_levels::Int,
+        potential_name::String,
+        potential_params::Vector{Float64}
+    )
+
+Plots the wave functions.
+
+# Arguments
+- `x_vals`: Values of `x`.
+- `wavefunctions`: Precomputed and normalized wave functions.
+- `num_levels`: Number of energy levels to plot.
+- `potential_name`: The name of the potential function.
+- `potential_params`: Coefficients for the polynomial potential function.
+"""
+function plot_wavefunctions(
+    x_vals::AbstractVector{Float64},
+    wavefunctions::Vector{Vector{Float64}},
+    num_levels::Int,
+    potential_name::String,
+    potential_params::Vector{Float64}
+)
+    # Initialize the primary plot for wave functions on the left y-axis
+    p = plot(title="Wave Functions and Potential\n($potential_name)", xlabel="x", ylabel="Ψ(x)", legend=:topleft)
+
+    # Batch plot wave functions
+    for n in 1:num_levels
+        plot!(p, x_vals, wavefunctions[n], label="Ψ$n")
+    end
+
+    # Define the potential function based on the given parameters
+    V_function = x -> sum(potential_params[i] * x^(4 - (i - 1)) for i in eachindex(potential_params))
+
+    # Calculate the potential values for plotting
+    V_vals = [V_function(x) for x in x_vals]
+
+    # Use `twinx()` to add a second y-axis for the potential
+    p_right = twinx()
+    plot!(p_right, x_vals, V_vals, label="Potential V(x)", linewidth=2, linestyle=:solid, color=:black, ylabel="V(x)", legend=:topright)
+
+    # Display the final plot with dual y-axes
+    display(p)
+end
+
+end  # module Interaction
diff --git a/Assignment_3/Problem_3/src/project/src/methods.jl b/Assignment_3/Problem_3/src/project/src/methods.jl
new file mode 100644
index 0000000..b8bb3a8
--- /dev/null
+++ b/Assignment_3/Problem_3/src/project/src/methods.jl
@@ -0,0 +1,183 @@
+# methods.jl
+# Methods for solving the 1D Schrödinger equation using Gaussian basis functions
+#
+# Author: Gilbert Young
+# Date: 2024/10/08
+#
+# This module provides functions to compute overlap, kinetic energy, and potential energy integrals
+# for Gaussian basis functions and to solve the 1D Schrödinger equation using the variational method.
+
+module Methods
+
+using LinearAlgebra
+using Base.Threads  # For parallel computing
+
+export overlap_integral, kinetic_integral, potential_integral_xn
+export build_matrices, solve_schrodinger
+
+@doc raw"""
+    overlap_integral(v1, s1, v2, s2)
+
+Computes the overlap integral between two Gaussian basis functions with parameters `(v1, s1)` and `(v2, s2)`.
+
+The overlap integral is given by:
+
+```math
+S_{ij} = \frac{\sqrt{v_1 v_2}}{\sqrt{\pi (v_1 + v_2)}} e^{- \frac{v_1 v_2 (s_1 - s_2)^2}{v_1 + v_2}}
+```
+
+# Arguments
+
+- `v1`: Width parameter of the first Gaussian basis function.
+- `s1`: Center of the first Gaussian basis function.
+- `v2`: Width parameter of the second Gaussian basis function.
+- `s2`: Center of the second Gaussian basis function.
+
+# Returns
+
+- `S_ij`: The overlap integral value.
+"""
+function overlap_integral(v1::Real, s1::Real, v2::Real, s2::Real)::Float64
+    vp = v1 + v2
+    exponent = -((v1 * v2 * (s1 - s2)^2) / vp)
+    prefactor = sqrt(v1 * v2) / (sqrt(π) * sqrt(vp))
+    S_ij = prefactor * exp(exponent)
+    return S_ij
+end
+
+@doc raw"""
+    kinetic_integral(v1, s1, v2, s2)
+
+Computes the kinetic energy integral between two Gaussian basis functions.
+
+The kinetic energy integral is given by:
+
+```math
+T_{ij} = \frac{v_1^{3/2} v_2^{3/2} \left( v_1 + v_2 - 2 v_1 v_2 (s_1 - s_2)^2 \right)}{\sqrt{\pi} (v_1 + v_2)^{5/2}} e^{- \frac{v_1 v_2 (s_1 - s_2)^2}{v_1 + v_2}}
+```
+
+# Arguments
+
+- `v1`: Width parameter of the first Gaussian basis function.
+- `s1`: Center of the first Gaussian basis function.
+- `v2`: Width parameter of the second Gaussian basis function.
+- `s2`: Center of the second Gaussian basis function.
+
+# Returns
+
+- `T_ij`: The kinetic energy integral value.
+"""
+function kinetic_integral(v1::Real, s1::Real, v2::Real, s2::Real)::Float64
+    vp = v1 + v2
+    exponent = -((v1 * v2 * (s1 - s2)^2) / vp)
+    numerator = v1 + v2 - 2 * v1 * v2 * (s1 - s2)^2
+    prefactor = v1^(1.5) * v2^(1.5) / (sqrt(π) * vp^(2.5))
+    T_ij = prefactor * numerator * exp(exponent)
+    return T_ij
+end
+
+@doc raw"""
+    potential_integral_xn(v1, s1, v2, s2, n)
+
+Computes the potential energy integral for \( V(x) = x^n \) between two Gaussian basis functions.
+
+# Arguments
+
+- `v1`, `s1`: Parameters of the first Gaussian basis function.
+- `v2`, `s2`: Parameters of the second Gaussian basis function.
+- `n`: The power of x in the potential function (integer from 0 to 4).
+
+# Returns
+
+- `V_ij`: The potential energy integral value.
+"""
+function potential_integral_xn(v1::Real, s1::Real, v2::Real, s2::Real, n::Int)::Float64
+    if n < 0 || n > 4
+        error("n should be an integer between 0 and 4")
+    end
+
+    vp = v1 + v2
+    exponent = -((v1 * v2 * (s1 - s2)^2) / vp)
+    S_ij = sqrt(v1 * v2) / (sqrt(π) * sqrt(vp)) * exp(exponent)
+
+    μ = (v1 * s1 + v2 * s2) / vp
+    σ2 = 1 / (2 * (v1 + v2))
+
+    # Compute the nth moment M_n(μ, σ2)
+    if n == 0
+        moment = 1.0
+    elseif n == 1
+        moment = μ
+    elseif n == 2
+        moment = μ^2 + σ2
+    elseif n == 3
+        moment = μ^3 + 3 * μ * σ2
+    elseif n == 4
+        moment = μ^4 + 6 * μ^2 * σ2 + 3 * σ2^2
+    end
+
+    V_ij = S_ij * moment
+    return V_ij
+end
+
+@doc raw"""
+    build_matrices(N, v, s, potential_gaussian_integral, potential_params)
+
+Builds the Hamiltonian matrix `H` and overlap matrix `S` for the variational method.
+
+# Arguments
+
+- `N`: Number of basis functions.
+- `v`: Width parameters of the Gaussian basis functions.
+- `s`: Centers of the Gaussian basis functions.
+- `potential_gaussian_integral`: Function to compute the potential energy integral.
+- `potential_params`: Additional parameters for the potential function.
+
+# Returns
+
+- `(H, S)`: The Hamiltonian and overlap matrices.
+"""
+function build_matrices(N::Int, v::Vector{Float64}, s::Vector{Float64}, potential_gaussian_integral, potential_params)
+    H = zeros(N, N)
+    S = zeros(N, N)
+    @threads for i in 1:N
+        for j in i:N  # Utilize symmetry, compute only upper triangle
+            S_ij = overlap_integral(v[i], s[i], v[j], s[j])
+            T_ij = kinetic_integral(v[i], s[i], v[j], s[j])
+            V_ij = potential_gaussian_integral(v[i], s[i], v[j], s[j], potential_params...)
+            H_ij = T_ij + V_ij
+            H[i, j] = H_ij
+            S[i, j] = S_ij
+            if i != j
+                H[j, i] = H_ij
+                S[j, i] = S_ij
+            end
+        end
+    end
+    return H, S
+end
+
+@doc raw"""
+    solve_schrodinger(H, S, num_levels)
+
+Solves the generalized eigenvalue problem for the Hamiltonian `H` and overlap matrix `S`.
+
+# Arguments
+
+- `H`: Hamiltonian matrix.
+- `S`: Overlap matrix.
+- `num_levels`: Number of energy levels to compute.
+
+# Returns
+
+- `(energies, states)`: The lowest `num_levels` eigenvalues and eigenvectors.
+"""
+function solve_schrodinger(H::Matrix{Float64}, S::Matrix{Float64}, num_levels::Int)
+    E_values, E_vectors = eigen(H, S)
+    idx = sortperm(real(E_values))
+    energies = real(E_values[idx[1:num_levels]])
+    states = E_vectors[:, idx[1:num_levels]]
+    return energies, states
+end
+
+end  # module Methods
\ No newline at end of file
diff --git a/Assignment_3/Problem_3/src/project/src/utils.jl b/Assignment_3/Problem_3/src/project/src/utils.jl
new file mode 100644
index 0000000..27ae1b5
--- /dev/null
+++ b/Assignment_3/Problem_3/src/project/src/utils.jl
@@ -0,0 +1,241 @@
+# utils.jl
+# Utility functions for the Schrödinger equation solver.
+#
+# Author: Gilbert Young
+# Date: 2024/10/08
+
+module Utils
+
+using LinearAlgebra
+using ..Methods
+
+export check_positive_v, trapezoidal_integration, get_parameters, get_potential_function, read_number, read_int, normalize_wavefunction
+
+@doc raw"""
+    check_positive_v(v)
+
+Checks if `v` is positive.
+
+# Arguments
+- `v`: Value to check.
+
+# Throws
+- `Error` if `v <= 0`.
+"""
+function check_positive_v(v::Float64)
+    if v <= 0
+        error("Error: v must be greater than 0. Please enter a positive value for v.")
+    end
+end
+
+@doc raw"""
+    trapezoidal_integration(x, y)
+
+Performs trapezoidal integration of `y` with respect to `x`.
+
+# Arguments
+- `x`: Vector of x values.
+- `y`: Vector of y values.
+
+# Returns
+- `integral`: The result of the integration.
+"""
+function trapezoidal_integration(x::AbstractVector{Float64}, y::AbstractVector{Float64})
+    integral = sum((y[1:end-1] + y[2:end]) .* diff(x)) / 2
+    return integral
+end
+
+@doc raw"""
+    read_number(prompt::String, default::Float64)
+
+Reads a floating-point number from the user with a prompt. If the user presses enter without input, returns the default value.
+
+# Arguments
+- `prompt`: The prompt message to display.
+- `default`: The default value to return if no input is provided.
+
+# Returns
+- The user-entered number or the default value.
+"""
+function read_number(prompt::String, default::Float64)
+    print("$prompt [Default: $default] ")
+    input = readline()
+    if isempty(input)
+        return default
+    else
+        try
+            return parse(Float64, input)
+        catch
+            println("\nInvalid input. Using default value: $default")
+            return default
+        end
+    end
+end
+
+@doc raw"""
+    read_int(prompt::String, default::Int)
+
+Reads an integer from the user with a prompt. If the user presses enter without input, returns the default value.
+
+# Arguments
+- `prompt`: The prompt message to display.
+- `default`: The default value to return if no input is provided.
+
+# Returns
+- The user-entered integer or the default value.
+"""
+function read_int(prompt::String, default::Int)
+    print("$prompt [Default: $default] ")
+    input = readline()
+    if isempty(input)
+        return default
+    else
+        try
+            return parse(Int, input)
+        catch
+            println("\nInvalid input. Using default value: $default")
+            return default
+        end
+    end
+end
+
+@doc raw"""
+    normalize_wavefunction(x_vals::AbstractVector{Float64}, ψ::AbstractVector{Float64})::Vector{Float64}
+
+Normalizes the wave function `ψ` based on the integration over `x_vals`.
+
+# Arguments
+- `x_vals`: Vector of x values.
+- `ψ`: Wave function to normalize.
+
+# Returns
+- Normalized wave function.
+"""
+function normalize_wavefunction(x_vals::AbstractVector{Float64}, ψ::AbstractVector{Float64})::Vector{Float64}
+    norm_factor = sqrt(trapezoidal_integration(x_vals, ψ .^ 2))
+    return ψ / norm_factor
+end
+
+@doc raw"""
+    get_parameters(choice::String)
+
+Retrieves parameters based on the user's selection.
+
+# Arguments
+- `choice`: User selection for the potential type.
+
+# Returns
+- `(N, v, s, potential_gaussian_integral, potential_params, potential_name, num_levels)`: Parameters for solving the Schrödinger equation.
+"""
+function get_parameters(choice::String)
+    num_levels_default = 3
+    if choice in ["1", "2", "4"]
+        N = read_int("Enter the number of basis functions N:", 40)
+
+        println("Select parameter setting:")
+        println("1. Fixed v, varying s")
+        println("2. Fixed s, varying v")
+        print("Enter your choice: ")
+        param_choice = readline()
+
+        v, s = Float64[], Float64[]
+        if param_choice == "1"
+            v_value = read_number("Enter the fixed value of v:", 0.5)
+            check_positive_v(v_value)
+            v = fill(v_value, N)
+
+            println("Suggested range for s is from -$(N/4) to $(N/4)")
+            s_start = read_number("Enter the starting value of s:", -N / 4)
+            s_end = read_number("Enter the ending value of s:", N / 4)
+            s = collect(range(s_start, stop=s_end, length=N))
+        elseif param_choice == "2"
+            s_value = read_number("Enter the fixed value of s:", 0.0)
+            s = fill(s_value, N)
+
+            v_start = read_number("Enter the starting value of v:", 0.1)
+            check_positive_v(v_start)
+
+            v_end = read_number("Enter the ending value of v:", 1.0)
+            check_positive_v(v_end)
+
+            v = collect(range(v_start, stop=v_end, length=N))
+        else
+            println("Invalid selection, returning to main menu.")
+            return 0, Float64[], Float64[], nothing, Float64[], "", 0
+        end
+
+        num_levels = read_int("Enter the number of energy levels to compute:", num_levels_default)
+
+        potential_gaussian_integral_list, potential_name_list, potential_params_list = get_potential_function(choice)
+        return N, v, s, potential_gaussian_integral_list, potential_params_list, potential_name_list, num_levels
+    elseif choice == "3"
+        N = 40
+        v = fill(0.5, N)
+        s = collect(range(-10.0, stop=10.0, length=N))
+        num_levels = num_levels_default
+        potential_gaussian_integral_list, potential_name_list, potential_params_list = get_potential_function("3")
+        return N, v, s, potential_gaussian_integral_list, potential_params_list, potential_name_list, num_levels
+    else
+        println("Invalid selection, returning to main menu.")
+        return 0, Float64[], Float64[], nothing, Float64[], "", 0
+    end
+end
+
+@doc raw"""
+    get_potential_function(choice::String)
+
+Selects the potential function based on user's choice.
+
+# Arguments
+- `choice`: User selection for the potential type.
+
+# Returns
+- `[(potential_gaussian_integral, potential_name, potential_params)]`: Potential function and parameters.
+"""
+function get_potential_function(choice::String)
+    if choice == "1"
+        potential_gaussian_integral = (v1, s1, v2, s2, params...) -> potential_integral_xn(v1, s1, v2, s2, 2)
+        potential_name = "V(x) = x^2"
+        potential_params = [0.0, 0.0, 1.0, 0.0, 0.0]  # Coefficients from x^4 to x^0
+        return [potential_gaussian_integral], [potential_name], [potential_params]
+    elseif choice == "2"
+        potential_gaussian_integral = (v1, s1, v2, s2, params...) -> potential_integral_xn(v1, s1, v2, s2, 4) - potential_integral_xn(v1, s1, v2, s2, 2)
+        potential_name = "V(x) = x^4 - x^2"
+        potential_params = [1.0, 0.0, -1.0, 0.0, 0.0]
+        return [potential_gaussian_integral], [potential_name], [potential_params]
+    elseif choice == "3"
+        potential_gaussian_integral_list = [
+            (v1, s1, v2, s2, params...) -> potential_integral_xn(v1, s1, v2, s2, 2),
+            (v1, s1, v2, s2, params...) -> potential_integral_xn(v1, s1, v2, s2, 4) - potential_integral_xn(v1, s1, v2, s2, 2)
+        ]
+        potential_name_list = ["V(x) = x^2", "V(x) = x^4 - x^2"]
+        potential_params_list = [[0.0, 0.0, 1.0, 0.0, 0.0], [1.0, 0.0, -1.0, 0.0, 0.0]]
+        return potential_gaussian_integral_list, potential_name_list, potential_params_list
+    elseif choice == "4"
+        println("Enter the coefficients of the polynomial potential (degree up to 4).")
+        println("Format: c4 c3 c2 c1 c0 (separated by spaces) [Default: 1 0 -1 0 0]")
+        print("Enter coefficients: ")
+        potential_params_input = readline()
+        if isempty(potential_params_input)
+            potential_params = [1.0, 0.0, -1.0, 0.0, 0.0]
+        else
+            try
+                potential_params = parse.(Float64, split(potential_params_input))
+                if length(potential_params) != 5
+                    error("Please enter exactly 5 coefficients.")
+                end
+            catch
+                println("Invalid input. Using default coefficients: 1 0 -1 0 0")
+                potential_params = [1.0, 0.0, -1.0, 0.0, 0.0]
+            end
+        end
+        potential_gaussian_integral = (v1, s1, v2, s2, potential_params...) -> sum(potential_params[i] * potential_integral_xn(v1, s1, v2, s2, 4 - (i - 1)) for i in eachindex(potential_params))
+        terms = [c != 0 ? "$(c)x^$(4 - (i - 1))" : "" for (i, c) in enumerate(potential_params)]
+        potential_name = "V(x) = " * join(filter(!isempty, terms), " + ")
+        return [potential_gaussian_integral], [potential_name], [potential_params]
+    else
+        error("\nInvalid potential function selection.")
+    end
+end
+
+end  # module Utils