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app.py

@@ -1,529 +1,533 @@
-import streamlit as st
-
-# ==========================================
-# 0. PAGE CONFIGURATION & STYLING
-# ==========================================
-st.set_page_config(
-    page_title="Pili-Pili Quantum Solver | Ahilan Kumaresan",
-    page_icon="🍟",
-    layout="wide",
-    initial_sidebar_state="expanded"
-)
-
-import numpy as np
-import matplotlib.pyplot as plt
-import math
-import time
-
-try:
-    import mediapipe as mp
-    import cv2
-    import plotly.graph_objects as go
-    from plotly.subplots import make_subplots
-except ImportError as e:
-    st.error(f"CRITICAL ERROR: Failed to import required libraries. {e}")
-    st.stop()
-
-# Import physics engine
-try:
-    import functions as f
-except ImportError as e:
-    st.error(f"CRITICAL ERROR: Failed to import physics engine. {e}")
-    st.stop()
-
-
-# ==========================================
-# 0. SESSION STATE (for camera flow)
-# ==========================================
-if 'countdown_finished' not in st.session_state:
-    st.session_state.countdown_finished = False
-if 'V_user_defined' not in st.session_state:
-    st.session_state.V_user_defined = None
-
-# Custom CSS for a professional look
-st.markdown("""
-    <style>
-    .main {
-        background-color: #0e1117;
-    }
-    .stButton>button {
-        width: 100%;
-        border-radius: 5px;
-        height: 3em;
-        background-color: #262730;
-        color: white;
-        border: 1px solid #4b4b4b;
-    }
-    .stButton>button:hover {
-        border-color: #00ADB5;
-        color: #00ADB5;
-    }
-    h1, h2, h3 {
-        color: #00ADB5;
-        font-family: 'Helvetica Neue', sans-serif;
-    }
-    </style>
-    """, unsafe_allow_html=True)
-
-# ==========================================
-# 1. SIDEBAR: PERSONALIZATION & NAV
-# ==========================================
-with st.sidebar:
-    st.title("Quantum Solver 2.0")
-    st.markdown("---")
-    
-    # Navigation
-    page = st.radio("Navigation", ["Simulator", "Benchmarks & Verification", "Theory & Method"])
-    
-    st.markdown("---")
-    
-    # Author Profile
-    st.markdown("### About Moi")
-    st.markdown("""
-    **Ahilan Kumaresan**
-    
-    *Aspiring Mathematical & Computational Physicist*
-    
-    Developing Interative and accurate numerical tools for quantum mechanics.
-    """)
-    
-    st.info("Verified against Analytical Solutions & QMSolve Package.")
-
-# ==========================================
-# 2. HELPER FUNCTIONS (Plotting)
-# ==========================================
-def plot_interactive(E, psi, V, x, nos=5):
-    """
-    Creates a professional interactive Plotly chart for wavefunctions and energy levels.
-    """
-    # Limit states
-    states = min(nos, len(E))
-    
-    # Create subplots: Main plot (Potential + Psi) and Side plot (Energy Levels)
-    fig = make_subplots(
-        rows=1, cols=2, 
-        column_widths=[0.8, 0.2],
-        shared_yaxes=True,
-        horizontal_spacing=0.02,
-        subplot_titles=("Wavefunctions & Potential", "Energy Spectrum")
-    )
-
-    # Scaling factor for wavefunctions
-    if len(E) >= 2:
-        scale = (E[1] - E[0]) * 0.4
-    else:
-        scale = max(E[0] * 0.1, 0.5)
-        
-    max_E = E[states-1] if states > 0 else 10
-    window_height = max_E * 1.5
-
-    # Get x coordinates for internal points (matching psi dimensions)
-    x_internal = x[1:-1]
-    V_internal = V[1:-1]
-    
-    # 1. Plot Potential V(x) - using internal points for better visibility
-    V_clipped = np.clip(V_internal, 0, window_height)
-    
-    fig.add_trace(
-        go.Scatter(
-            x=x_internal.tolist() if hasattr(x_internal, 'tolist') else x_internal, 
-            y=V_clipped.tolist() if hasattr(V_clipped, 'tolist') else V_clipped,
-            mode='lines',
-            name='V(x)',
-            line=dict(color='#FFFFFF', width=2.5),
-            hovertemplate='V(x): %{y:.2f}<extra></extra>'
-        ),
-        row=1, col=1
-    )
-
-    # 2. Plot Wavefunctions (shifted by Energy)
-    colors = ['#00ADB5', '#FF2E63', '#F38181', '#FCE38A', '#EAFFD0', 
-              '#95E1D3', '#FFB6C1', '#DDA0DD', '#87CEEB', '#98FB98']
-    
-    for n in range(states):
-        # Normalize wavefunction amplitude
-        psi_n = psi[:, n]
-        max_amp = np.max(np.abs(psi_n))
-        if max_amp > 1e-9:
-            psi_n = psi_n / max_amp
-        else:
-            psi_n = psi_n
-            
-        # Shift by energy
-        y_shifted = psi_n * scale + E[n]
-        
-        # Hide where potential is infinite
-        y_shifted[V_internal > 1e5] = np.nan
-
-        color = colors[n % len(colors)]
-        
-        # Ensure arrays match in length
-        if len(x_internal) != len(y_shifted):
-            # Fallback: truncate to minimum length
-            min_len = min(len(x_internal), len(y_shifted))
-            x_plot = x_internal[:min_len]
-            y_plot = y_shifted[:min_len]
-        else:
-            x_plot = x_internal
-            y_plot = y_shifted
-        
-        fig.add_trace(
-            go.Scatter(
-                x=x_plot.tolist() if hasattr(x_plot, 'tolist') else x_plot, 
-                y=y_plot.tolist() if hasattr(y_plot, 'tolist') else y_plot,
-                mode='lines',
-                name=f'n={n+1}, E={E[n]:.4f}',
-                line=dict(color=color, width=2),
-                hovertemplate=f'n={n+1}<br>E={E[n]:.4f}<br>x: %{{x:.2f}}<br>ψ: %{{y:.2f}}<extra></extra>'
-            ),
-            row=1, col=1
-        )
-        
-        # Add Energy Level to Side Bar
-        fig.add_trace(
-            go.Scatter(
-                x=[0, 1], y=[E[n], E[n]],
-                mode='lines',
-                line=dict(color=color, width=3),
-                showlegend=False,
-                hovertemplate=f'E_{n+1}={E[n]:.4f}<extra></extra>'
-            ),
-            row=1, col=2
-        )
-
-    # Layout Styling - Enhanced dark mode
-    fig.update_layout(
-        template="plotly_dark",
-        height=600,
-        margin=dict(l=20, r=20, t=50, b=20),
-        legend=dict(
-            orientation="h", 
-            yanchor="bottom",
-            y=1.02,
-            xanchor="right",
-            x=1,
-            font=dict(size=10)
-        ),
-        hovermode="closest",
-        plot_bgcolor='#0e1117',
-        paper_bgcolor='#0e1117',
-        font=dict(color='#FAFAFA')
-    )
-    
-    fig.update_xaxes(
-        title_text="Position (a.u.)", 
-        row=1, col=1,
-        gridcolor='#2a2a2a',
-        showgrid=True
-    )
-    fig.update_xaxes(
-        showticklabels=False, 
-        row=1, col=2,
-        showgrid=False
-    )
-    fig.update_yaxes(
-        title_text="Energy (Hartree)", 
-        range=[0, max_E * 1.2], 
-        row=1, col=1,
-        gridcolor='#2a2a2a',
-        showgrid=True
-    )
-    
-    return fig
-
-# ==========================================
-# 3. HELPER: MediaPipe hand → 1D potential
-# ==========================================
-def process_frame_to_potential(frame):
-    """
-    Takes a BGR frame (OpenCV) and returns:
-      pot_profile: 1D array in [0,1] representing V(x) profile
-      msg: human-friendly label
-    Modes:
-      - 2 hands → Square well (0 inside, 1 outside)
-      - 1 hand → QHO-like parabola
-    """
-    mp_hands = mp.solutions.hands
-    
-    with mp_hands.Hands(max_num_hands=2, min_detection_confidence=0.5) as hands:
-        h, w, _ = frame.shape
-        rgb = cv2.cvtColor(frame, cv2.COLOR_BGR2RGB)
-        res = hands.process(rgb)
-
-        if not res.multi_hand_landmarks:
-            return None, "No Hands Detected, But Cute Smile :)"
-
-        # --- LOGIC: Square Well vs QHO ---
-        
-        # 1. Square Well (2 Hands)
-        if len(res.multi_hand_landmarks) >= 2:
-            INDEX_TIP_ID = 8
-            x_coords = [lm.landmark[INDEX_TIP_ID].x * w for lm in res.multi_hand_landmarks]
-            x_coords.sort()
-            
-            xL_hand, xR_hand = x_coords[0], x_coords[1]
-            well_width = xR_hand - xL_hand
-            
-            center_screen = w / 2
-            centered_L = center_screen - (well_width / 2)
-            centered_R = center_screen + (well_width / 2)
-            
-            x_space = np.linspace(0, w, 400)
-            pot_profile = np.ones_like(x_space)
-            pot_profile[(x_space > centered_L) & (x_space < centered_R)] = 0
-            
-            return pot_profile, "Square Well (Captured)"
-
-        # 2. Harmonic Oscillator (1 Hand)
-        elif len(res.multi_hand_landmarks) == 1:
-            lm = res.multi_hand_landmarks[0]
-            THUMB = lm.landmark[4]
-            INDEX = lm.landmark[8]
-            
-            dx = INDEX.x - THUMB.x
-            dy = INDEX.y - THUMB.y
-            dist = math.sqrt(dx**2 + dy**2)
-            
-            # Map pinch distance → curvature
-            A = np.interp(dist, [0.05, 0.3], [100.0, 1.0]) 
-            
-            x_space = np.linspace(-1, 1, 400)
-            pot_profile = A * (x_space**2)
-            
-            pot_profile = np.clip(pot_profile, 0, 100)
-            pot_profile = pot_profile / 100.0  # normalize 0..1
-            
-            return pot_profile, f"Harmonic Oscillator (k={A:.1f})"
-            
-    return None, "Error"
-
-# ==========================================
-# 4. PAGE: SIMULATOR
-# ==========================================
-if page == "Simulator":
-    st.title("Pili-Pili - Quantum Potential Solver")
-    st.markdown("Show a potential with your hands or select a preset to solve the **Time-Independent Schrödinger Equation**.")
-
-    # Shared grid for all modes
-    L = 50
-    N_GRID = 1000
-    x_full, dx, x_internal = f.make_grid(L, N_GRID)
-
-    V_full_to_solve = None
-    status_msg = ""
-    
-    col1, col2 = st.columns([1, 3])
-
-    with col1:
-        st.subheader("Controls")
-        
-        # Settings
-        potential_mode = st.selectbox(
-            "Potential Type",
-            [
-                "Static Square Well",
-                "Static Harmonic Oscillator",
-                "Double Well",
-                "Hand Gesture (Camera)"
-            ]
-        )
-        
-        nos_user = st.slider("Eigenstates to Plot", 1, 10, 5)
-
-        # ---- STATIC MODES ----
-        if potential_mode == "Static Square Well":
-            width = st.slider("Well Width", 1.0, 20.0, 10.0)
-            V_physics = np.zeros_like(x_internal)
-            V_physics[np.abs(x_internal) > width/2] = 200
-            V_full_to_solve = np.pad(V_physics, (1,1), constant_values=1e10)
-            status_msg = f"Static Square Well (width = {width:.1f})"
-            
-        elif potential_mode == "Static Harmonic Oscillator":
-            k = st.slider("Spring Constant (k)", 0.1, 50.0, 5.0)
-            V_physics = 0.5 * k * x_internal**2
-            # scale a bit so it shows nicely under energies
-            V_physics = V_physics / np.max(V_physics) * 50
-            V_full_to_solve = np.pad(V_physics, (1,1), constant_values=1e10)
-            status_msg = f"Static Harmonic Oscillator (k = {k:.2f})"
-
-        elif potential_mode == "Double Well":
-            sep = st.slider("Separation", 0.5, 5.0, 2.0)
-            depth = st.slider("Depth", 0.1, 5.0, 1.0)
-            V_physics = depth * ((x_internal**2 - sep**2)**2)
-            V_physics = V_physics / np.max(V_physics) * 50
-            V_full_to_solve = np.pad(V_physics, (1,1), constant_values=1e10)
-            status_msg = f"Double Well (sep = {sep:.2f}, depth = {depth:.2f})"
-
-        # ---- HAND-GESTURE / CAMERA MODE ----
-        elif potential_mode == "Hand Gesture (Camera)":
-            st.subheader("Hand Gesture Controls")
-            st.info(
-                "1. Click **'Start Countdown'**. (IGNORE)\n"
-                "2. Get your **two hands** ready for a Square Well, "
-                "or **one-hand pinch** for a Harmonic Oscillator.\n"
-                "3. When you'r ready, use **'Take a snapshot'**."
-            )
-
-
-            st.subheader("Hand Gesture Input")
-
-            img_file = st.camera_input("Take a Snapshot")
-
-            if img_file:
-                file_bytes = np.asarray(bytearray(img_file.read()), dtype=np.uint8)
-                frame = cv2.imdecode(file_bytes, 1)
-                frame = cv2.flip(frame, 1)
-
-                V_raw, msg = process_frame_to_potential(frame)
-
-                if V_raw is not None:
-                    st.success(f"Detected: {msg}")
-                    st.session_state.V_user_defined = V_raw
-
-                    # Map to simulation grid
-                    V_interpolated = np.interp(
-                        np.linspace(0, 1, len(x_internal)),
-                        np.linspace(0, 1, len(V_raw)),
-                        V_raw
-                    )
-                    V_physics = V_interpolated * 200.0
-                    V_full_to_solve = np.pad(V_physics, (1,1), constant_values=1e10)
-                    status_msg = f"Camera Potential: {msg}"
-                else:
-                    st.error(msg)
-
-
-    # --------- RIGHT COLUMN: SOLVE & PLOT ----------
-    with col2:
-        if V_full_to_solve is not None:
-            start_time = time.time()
-            T = f.kinetic_operator(len(x_internal), dx)
-            E, psi = f.solve(T, V_full_to_solve, dx)
-            solve_time = time.time() - start_time
-            
-            if status_msg:
-                st.markdown(f"**Potential:** {status_msg}")
-            st.markdown(f"**Solver Status:** ✅ Converged in {solve_time:.3f} s")
-            
-            fig = plot_interactive(E, psi, V_full_to_solve, x_full, nos=nos_user)
-            st.plotly_chart(fig, use_container_width=True)
-            
-            # Eigenenergies panel
-            st.markdown("### Eigenenergies")
-            cols = st.columns(nos_user)
-            for i in range(nos_user):
-                if i < len(E):
-                    cols[i].metric(f"n={i}", f"{E[i]:.4f} Ha")
-        else:
-            if potential_mode == "Hand Gesture (Camera)":
-                st.info("Follow the instructions on the left to capture a potential from your hands.")
-            else:
-                st.info("Select parameters on the left to generate a potential and solve.")
-
-# ==========================================
-# 5. PAGE: BENCHMARKS
-# ==========================================
-elif page == "Benchmarks & Verification":
-    st.title("🛡️ Verification & Accuracy")
-    st.markdown("""
-    This solver has been rigorously tested against known analytical solutions and external libraries to ensure physical accuracy.
-    """)
-    
-    tab1, tab2, tab3 = st.tabs(["Analytical Benchmarks", "QMSolve Comparison", "Code"])
-    
-    with tab1:
-        st.subheader("1. Infinite Square Well")
-        st.markdown("Particle in a box of length $L=20$. Error < 0.003%.")
-        st.table({
-            "State (n)": [1, 2, 3, 4, 5],
-            "Analytic E": [0.012337, 0.049348, 0.111033, 0.197392, 0.308425],
-            "Numerical E": [0.012337, 0.049348, 0.111032, 0.197389, 0.308419],
-            "% Error": ["0.0001%", "0.0003%", "0.0007%", "0.0013%", "0.0021%"]
-        })
-        
-        st.subheader("2. Harmonic Oscillator")
-        st.markdown("Standard QHO with $k=1$. Error < 0.02%.")
-        st.table({
-            "State (n)": [0, 1, 2, 3, 4],
-            "Analytic E": [0.5, 1.5, 2.5, 3.5, 4.5],
-            "Numerical E": [0.499980, 1.499902, 2.499746, 3.499512, 4.499200],
-            "% Error": ["0.0039%", "0.0065%", "0.0101%", "0.0139%", "0.0178%"]
-        })
-
-    with tab2:
-        st.subheader("Cross-Verification: Double Well Potential")
-        st.markdown("""
-        Comparison with the Python package `QMSolve` for a Double Well potential (no simple analytic solution).
-        **Agreement within 0.25%**.
-        """)
-        
-        col_a, col_b = st.columns(2)
-        with col_a:
-            st.markdown("**Parameters:** $V(x) = 2(x^2 - 1)^2$")
-            st.table({
-                "State (n)": [0, 1, 2, 3, 4],
-                "psi_solve2 (Ha)": [1.400886, 2.092533, 4.455252, 6.917808, 9.872632],
-                "QMSolve (Ha)": [1.402472, 2.097767, 4.466368, 6.936807, 9.900227],
-                "% Difference": ["0.11%", "0.25%", "0.25%", "0.27%", "0.28%"]
-            })
-        with col_b:
-            st.info("Note: QMSolve uses eV units. Results were converted to Hartree (1 Ha ≈ 27.211 eV) for comparison.")
-
-    with tab3:
-        st.subheader("Code Verification")
-        st.code("""
-def kinetic_operator(N, dx, hbar=1, m=1):
-    # 3-point central difference stencil for 2nd derivative
-    main_diagonal = (1/dx**2) * np.diag(-2 * np.ones(N))
-    off_diagonal1 = (1/dx**2) * np.diag(np.ones(N-1), -1)
-    off_diagonal2 = (1/dx**2) * np.diag(np.ones(N-1), 1)
-    D2 = (main_diagonal + off_diagonal1 + off_diagonal2)
-    
-    # Kinetic Energy Operator T = -hbar^2 / 2m * d^2/dx^2
-    T = (-(hbar**2 / (2*m)) * D2)
-    return T
-        """, language="python")
-        st.code("""
-def harmonic(x,k,center=0.0):
-    # A Parabola, setting the global k-value.
-    global Last_k_value
-    Last_k_value = k
-    
-    constant_factor = 1 
-    potential = 0.5*k*(x - center)**2
-    return constant_factor * potential
-""")
-        
-        
-        
-# ==========================================
-# 6. PAGE: THEORY
-# ==========================================
-elif page == "Theory & Method":
-    st.title("📖 Theory & Methodology")
-    
-    st.markdown("### The Time-Independent Schrödinger Equation")
-    st.latex(r" \hat{H}\psi(x) = E\psi(x) ")
-    st.latex(r" \left[ -\frac{\hbar^2}{2m}\frac{d^2}{dx^2} + V(x) \right]\psi(x) = E\psi(x) ")
-    
-    st.markdown("### Numerical Method: Finite Difference")
-    st.markdown(r"""
-    We discretize the spatial domain $x$ into a grid of $N$ points. The second derivative is approximated using the **Central Difference Formula**:
-    """)
-    st.latex(r" \frac{d^2\psi}{dx^2} \approx \frac{\psi_{i+1} - 2\psi_i + \psi_{i-1}}{\Delta x^2} ")
-    
-    st.markdown(r"""
-    This transforms the differential operator into a **Tridiagonal Matrix** equation:
-    """)
-    st.latex(r" \mathbf{H}\mathbf{\psi} = E\mathbf{\psi} ")
-    
-    st.markdown(r"""
-    Where $\mathbf{H}$ is an $N \times N$ matrix. We then use `numpy.linalg.eigh` to solve for the eigenvalues ($E$) and eigenvectors ($\psi$).
-    """)
-    
-    st.markdown("### Implementation Details")
-    st.markdown(r"""
-    - **Grid Size:** Dynamic (default 1000–2000 points)
-    - **Boundary Conditions:** Dirichlet ($ \psi(0) = \psi(L) = 0 $) via infinite walls at grid edges.
-    - **Units:** Hartree Atomic Units ($\hbar=1, m=1$).
-    """)
+import streamlit as st
+
+# ==========================================
+# 0. PAGE CONFIGURATION & STYLING
+# ==========================================
+st.set_page_config(
+    page_title="Pili-Pili Quantum Solver | Ahilan Kumaresan",
+    page_icon="🍟",
+    layout="wide",
+    initial_sidebar_state="expanded"
+)
+
+import numpy as np
+import matplotlib.pyplot as plt
+import math
+import time
+
+try:
+    import mediapipe as mp
+    import cv2
+    import plotly.graph_objects as go
+    from plotly.subplots import make_subplots
+except ImportError as e:
+    st.error(f"CRITICAL ERROR: Failed to import required libraries. {e}")
+    st.stop()
+
+# Import physics engine
+try:
+    import functions as f
+except ImportError as e:
+    st.error(f"CRITICAL ERROR: Failed to import physics engine. {e}")
+    st.stop()
+
+
+# ==========================================
+# 0. SESSION STATE (for camera flow)
+# ==========================================
+if 'countdown_finished' not in st.session_state:
+    st.session_state.countdown_finished = False
+if 'V_user_defined' not in st.session_state:
+    st.session_state.V_user_defined = None
+
+# Custom CSS for a professional look
+st.markdown("""
+    <style>
+    .main {
+        background-color: #0e1117;
+    }
+    .stButton>button {
+        width: 100%;
+        border-radius: 5px;
+        height: 3em;
+        background-color: #262730;
+        color: white;
+        border: 1px solid #4b4b4b;
+    }
+    .stButton>button:hover {
+        border-color: #00ADB5;
+        color: #00ADB5;
+    }
+    h1, h2, h3 {
+        color: #00ADB5;
+        font-family: 'Helvetica Neue', sans-serif;
+    }
+    </style>
+    """, unsafe_allow_html=True)
+
+# ==========================================
+# 1. SIDEBAR: PERSONALIZATION & NAV
+# ==========================================
+with st.sidebar:
+    st.title("Quantum Solver 2.0")
+    st.caption("v2.1 - HF Fix")
+    st.markdown("---")
+    
+    # Navigation
+    page = st.radio("Navigation", ["Simulator", "Benchmarks & Verification", "Theory & Method"])
+    
+    st.markdown("---")
+    
+    # Author Profile
+    st.markdown("### About Moi")
+    st.markdown("""
+    **Ahilan Kumaresan**
+    
+    *Aspiring Mathematical & Computational Physicist*
+    
+    Developing Interative and accurate numerical tools for quantum mechanics.
+    """)
+    
+    st.info("Verified against Analytical Solutions & QMSolve Package.")
+
+# ==========================================
+# 2. HELPER FUNCTIONS (Plotting)
+# ==========================================
+def plot_interactive(E, psi, V, x, nos=5):
+    """
+    Creates a professional interactive Plotly chart for wavefunctions and energy levels.
+    """
+    # Limit states
+    states = min(nos, len(E))
+    
+    # Create subplots: Main plot (Potential + Psi) and Side plot (Energy Levels)
+    fig = make_subplots(
+        rows=1, cols=2, 
+        column_widths=[0.8, 0.2],
+        shared_yaxes=True,
+        horizontal_spacing=0.02,
+        subplot_titles=("Wavefunctions & Potential", "Energy Spectrum")
+    )
+
+    # Scaling factor for wavefunctions
+    if len(E) >= 2:
+        scale = (E[1] - E[0]) * 0.4
+    else:
+        scale = max(E[0] * 0.1, 0.5)
+        
+    max_E = E[states-1] if states > 0 else 10
+    window_height = max_E * 1.5
+
+    # Get x coordinates for internal points (matching psi dimensions)
+    x_internal = x[1:-1]
+    V_internal = V[1:-1]
+    
+    # 1. Plot Potential V(x) - using internal points for better visibility
+    V_clipped = np.clip(V_internal, 0, window_height)
+    
+    fig.add_trace(
+        go.Scatter(
+            x=x_internal.tolist() if hasattr(x_internal, 'tolist') else x_internal, 
+            y=V_clipped.tolist() if hasattr(V_clipped, 'tolist') else V_clipped,
+            mode='lines',
+            name='V(x)',
+            line=dict(color='#FFFFFF', width=2.5),
+            hovertemplate='V(x): %{y:.2f}<extra></extra>'
+        ),
+        row=1, col=1
+    )
+
+    # 2. Plot Wavefunctions (shifted by Energy)
+    colors = ['#00ADB5', '#FF2E63', '#F38181', '#FCE38A', '#EAFFD0', 
+              '#95E1D3', '#FFB6C1', '#DDA0DD', '#87CEEB', '#98FB98']
+    
+    for n in range(states):
+        # Normalize wavefunction amplitude
+        psi_n = psi[:, n]
+        max_amp = np.max(np.abs(psi_n))
+        if max_amp > 1e-9:
+            psi_n = psi_n / max_amp
+        else:
+            psi_n = psi_n
+            
+        # Shift by energy
+        y_shifted = psi_n * scale + E[n]
+        
+        # Hide where potential is infinite
+        y_shifted[V_internal > 1e5] = np.nan
+
+        color = colors[n % len(colors)]
+        
+        # Ensure arrays match in length
+        if len(x_internal) != len(y_shifted):
+            # Fallback: truncate to minimum length
+            min_len = min(len(x_internal), len(y_shifted))
+            x_plot = x_internal[:min_len]
+            y_plot = y_shifted[:min_len]
+        else:
+            x_plot = x_internal
+            y_plot = y_shifted
+        
+        fig.add_trace(
+            go.Scatter(
+                x=x_plot.tolist() if hasattr(x_plot, 'tolist') else x_plot, 
+                y=y_plot.tolist() if hasattr(y_plot, 'tolist') else y_plot,
+                mode='lines',
+                name=f'n={n+1}, E={E[n]:.4f}',
+                line=dict(color=color, width=2),
+                hovertemplate=f'n={n+1}<br>E={E[n]:.4f}<br>x: %{{x:.2f}}<br>ψ: %{{y:.2f}}<extra></extra>'
+            ),
+            row=1, col=1
+        )
+        
+        # Add Energy Level to Side Bar
+        fig.add_trace(
+            go.Scatter(
+                x=[0, 1], y=[E[n], E[n]],
+                mode='lines',
+                line=dict(color=color, width=3),
+                showlegend=False,
+                hovertemplate=f'E_{n+1}={E[n]:.4f}<extra></extra>'
+            ),
+            row=1, col=2
+        )
+
+    # Layout Styling - Enhanced dark mode
+    fig.update_layout(
+        template="plotly_dark",
+        height=600,
+        margin=dict(l=20, r=20, t=50, b=20),
+        legend=dict(
+            orientation="h", 
+            yanchor="bottom",
+            y=1.02,
+            xanchor="right",
+            x=1,
+            font=dict(size=10)
+        ),
+        hovermode="closest",
+        plot_bgcolor='#0e1117',
+        paper_bgcolor='#0e1117',
+        font=dict(color='#FAFAFA')
+    )
+    
+    fig.update_xaxes(
+        title_text="Position (a.u.)", 
+        row=1, col=1,
+        gridcolor='#2a2a2a',
+        showgrid=True
+    )
+    fig.update_xaxes(
+        showticklabels=False, 
+        row=1, col=2,
+        showgrid=False
+    )
+    fig.update_yaxes(
+        title_text="Energy (Hartree)", 
+        range=[0, max_E * 1.2], 
+        row=1, col=1,
+        gridcolor='#2a2a2a',
+        showgrid=True
+    )
+    
+    return fig
+
+# ==========================================
+# 3. HELPER: MediaPipe hand → 1D potential
+# ==========================================
+def process_frame_to_potential(frame):
+    """
+    Takes a BGR frame (OpenCV) and returns:
+      pot_profile: 1D array in [0,1] representing V(x) profile
+      msg: human-friendly label
+    Modes:
+      - 2 hands → Square well (0 inside, 1 outside)
+      - 1 hand → QHO-like parabola
+    """
+    try:
+        mp_hands = mp.solutions.hands
+        with mp_hands.Hands(max_num_hands=2, min_detection_confidence=0.5) as hands:
+            h, w, _ = frame.shape
+            rgb = cv2.cvtColor(frame, cv2.COLOR_BGR2RGB)
+            res = hands.process(rgb)
+
+            if not res.multi_hand_landmarks:
+                return None, "No Hands Detected, But Cute Smile :)"
+
+            # --- LOGIC: Square Well vs QHO ---
+            
+            # 1. Square Well (2 Hands)
+            if len(res.multi_hand_landmarks) >= 2:
+                INDEX_TIP_ID = 8
+                x_coords = [lm.landmark[INDEX_TIP_ID].x * w for lm in res.multi_hand_landmarks]
+                x_coords.sort()
+                
+                xL_hand, xR_hand = x_coords[0], x_coords[1]
+                well_width = xR_hand - xL_hand
+                
+                center_screen = w / 2
+                centered_L = center_screen - (well_width / 2)
+                centered_R = center_screen + (well_width / 2)
+                
+                x_space = np.linspace(0, w, 400)
+                pot_profile = np.ones_like(x_space)
+                pot_profile[(x_space > centered_L) & (x_space < centered_R)] = 0
+                
+                return pot_profile, "Square Well (Captured)"
+
+            # 2. Harmonic Oscillator (1 Hand)
+            elif len(res.multi_hand_landmarks) == 1:
+                lm = res.multi_hand_landmarks[0]
+                THUMB = lm.landmark[4]
+                INDEX = lm.landmark[8]
+                
+                dx = INDEX.x - THUMB.x
+                dy = INDEX.y - THUMB.y
+                dist = math.sqrt(dx**2 + dy**2)
+                
+                # Map pinch distance → curvature
+                A = np.interp(dist, [0.05, 0.3], [100.0, 1.0]) 
+                
+                x_space = np.linspace(-1, 1, 400)
+                pot_profile = A * (x_space**2)
+                
+                pot_profile = np.clip(pot_profile, 0, 100)
+                pot_profile = pot_profile / 100.0  # normalize 0..1
+                
+                return pot_profile, f"Harmonic Oscillator (k={A:.1f})"
+                
+    except Exception as e:
+        return None, f"MediaPipe Error: {e}"
+            
+    return None, "Error"
+
+# ==========================================
+# 4. PAGE: SIMULATOR
+# ==========================================
+if page == "Simulator":
+    st.title("Pili-Pili - Quantum Potential Solver")
+    st.markdown("Show a potential with your hands or select a preset to solve the **Time-Independent Schrödinger Equation**.")
+
+    # Shared grid for all modes
+    L = 50
+    N_GRID = 1000
+    x_full, dx, x_internal = f.make_grid(L, N_GRID)
+
+    V_full_to_solve = None
+    status_msg = ""
+    
+    col1, col2 = st.columns([1, 3])
+
+    with col1:
+        st.subheader("Controls")
+        
+        # Settings
+        potential_mode = st.selectbox(
+            "Potential Type",
+            [
+                "Static Square Well",
+                "Static Harmonic Oscillator",
+                "Double Well",
+                "Hand Gesture (Camera)"
+            ]
+        )
+        
+        nos_user = st.slider("Eigenstates to Plot", 1, 10, 5)
+
+        # ---- STATIC MODES ----
+        if potential_mode == "Static Square Well":
+            width = st.slider("Well Width", 1.0, 20.0, 10.0)
+            V_physics = np.zeros_like(x_internal)
+            V_physics[np.abs(x_internal) > width/2] = 200
+            V_full_to_solve = np.pad(V_physics, (1,1), constant_values=1e10)
+            status_msg = f"Static Square Well (width = {width:.1f})"
+            
+        elif potential_mode == "Static Harmonic Oscillator":
+            k = st.slider("Spring Constant (k)", 0.1, 50.0, 5.0)
+            V_physics = 0.5 * k * x_internal**2
+            # scale a bit so it shows nicely under energies
+            V_physics = V_physics / np.max(V_physics) * 50
+            V_full_to_solve = np.pad(V_physics, (1,1), constant_values=1e10)
+            status_msg = f"Static Harmonic Oscillator (k = {k:.2f})"
+
+        elif potential_mode == "Double Well":
+            sep = st.slider("Separation", 0.5, 5.0, 2.0)
+            depth = st.slider("Depth", 0.1, 5.0, 1.0)
+            V_physics = depth * ((x_internal**2 - sep**2)**2)
+            V_physics = V_physics / np.max(V_physics) * 50
+            V_full_to_solve = np.pad(V_physics, (1,1), constant_values=1e10)
+            status_msg = f"Double Well (sep = {sep:.2f}, depth = {depth:.2f})"
+
+        # ---- HAND-GESTURE / CAMERA MODE ----
+        elif potential_mode == "Hand Gesture (Camera)":
+            st.subheader("Hand Gesture Controls")
+            st.info(
+                "1. Click **'Start Countdown'**. (IGNORE)\n"
+                "2. Get your **two hands** ready for a Square Well, "
+                "or **one-hand pinch** for a Harmonic Oscillator.\n"
+                "3. When you'r ready, use **'Take a snapshot'**."
+            )
+
+
+            st.subheader("Hand Gesture Input")
+
+            img_file = st.camera_input("Take a Snapshot")
+
+            if img_file:
+                file_bytes = np.asarray(bytearray(img_file.read()), dtype=np.uint8)
+                frame = cv2.imdecode(file_bytes, 1)
+                frame = cv2.flip(frame, 1)
+
+                V_raw, msg = process_frame_to_potential(frame)
+
+                if V_raw is not None:
+                    st.success(f"Detected: {msg}")
+                    st.session_state.V_user_defined = V_raw
+
+                    # Map to simulation grid
+                    V_interpolated = np.interp(
+                        np.linspace(0, 1, len(x_internal)),
+                        np.linspace(0, 1, len(V_raw)),
+                        V_raw
+                    )
+                    V_physics = V_interpolated * 200.0
+                    V_full_to_solve = np.pad(V_physics, (1,1), constant_values=1e10)
+                    status_msg = f"Camera Potential: {msg}"
+                else:
+                    st.error(msg)
+
+
+    # --------- RIGHT COLUMN: SOLVE & PLOT ----------
+    with col2:
+        if V_full_to_solve is not None:
+            start_time = time.time()
+            T = f.kinetic_operator(len(x_internal), dx)
+            E, psi = f.solve(T, V_full_to_solve, dx)
+            solve_time = time.time() - start_time
+            
+            if status_msg:
+                st.markdown(f"**Potential:** {status_msg}")
+            st.markdown(f"**Solver Status:** ✅ Converged in {solve_time:.3f} s")
+            
+            fig = plot_interactive(E, psi, V_full_to_solve, x_full, nos=nos_user)
+            st.plotly_chart(fig, use_container_width=True)
+            
+            # Eigenenergies panel
+            st.markdown("### Eigenenergies")
+            cols = st.columns(nos_user)
+            for i in range(nos_user):
+                if i < len(E):
+                    cols[i].metric(f"n={i}", f"{E[i]:.4f} Ha")
+        else:
+            if potential_mode == "Hand Gesture (Camera)":
+                st.info("Follow the instructions on the left to capture a potential from your hands.")
+            else:
+                st.info("Select parameters on the left to generate a potential and solve.")
+
+# ==========================================
+# 5. PAGE: BENCHMARKS
+# ==========================================
+elif page == "Benchmarks & Verification":
+    st.title("🛡️ Verification & Accuracy")
+    st.markdown("""
+    This solver has been rigorously tested against known analytical solutions and external libraries to ensure physical accuracy.
+    """)
+    
+    tab1, tab2, tab3 = st.tabs(["Analytical Benchmarks", "QMSolve Comparison", "Code"])
+    
+    with tab1:
+        st.subheader("1. Infinite Square Well")
+        st.markdown("Particle in a box of length $L=20$. Error < 0.003%.")
+        st.table({
+            "State (n)": [1, 2, 3, 4, 5],
+            "Analytic E": [0.012337, 0.049348, 0.111033, 0.197392, 0.308425],
+            "Numerical E": [0.012337, 0.049348, 0.111032, 0.197389, 0.308419],
+            "% Error": ["0.0001%", "0.0003%", "0.0007%", "0.0013%", "0.0021%"]
+        })
+        
+        st.subheader("2. Harmonic Oscillator")
+        st.markdown("Standard QHO with $k=1$. Error < 0.02%.")
+        st.table({
+            "State (n)": [0, 1, 2, 3, 4],
+            "Analytic E": [0.5, 1.5, 2.5, 3.5, 4.5],
+            "Numerical E": [0.499980, 1.499902, 2.499746, 3.499512, 4.499200],
+            "% Error": ["0.0039%", "0.0065%", "0.0101%", "0.0139%", "0.0178%"]
+        })
+
+    with tab2:
+        st.subheader("Cross-Verification: Double Well Potential")
+        st.markdown("""
+        Comparison with the Python package `QMSolve` for a Double Well potential (no simple analytic solution).
+        **Agreement within 0.25%**.
+        """)
+        
+        col_a, col_b = st.columns(2)
+        with col_a:
+            st.markdown("**Parameters:** $V(x) = 2(x^2 - 1)^2$")
+            st.table({
+                "State (n)": [0, 1, 2, 3, 4],
+                "psi_solve2 (Ha)": [1.400886, 2.092533, 4.455252, 6.917808, 9.872632],
+                "QMSolve (Ha)": [1.402472, 2.097767, 4.466368, 6.936807, 9.900227],
+                "% Difference": ["0.11%", "0.25%", "0.25%", "0.27%", "0.28%"]
+            })
+        with col_b:
+            st.info("Note: QMSolve uses eV units. Results were converted to Hartree (1 Ha ≈ 27.211 eV) for comparison.")
+
+    with tab3:
+        st.subheader("Code Verification")
+        st.code("""
+def kinetic_operator(N, dx, hbar=1, m=1):
+    # 3-point central difference stencil for 2nd derivative
+    main_diagonal = (1/dx**2) * np.diag(-2 * np.ones(N))
+    off_diagonal1 = (1/dx**2) * np.diag(np.ones(N-1), -1)
+    off_diagonal2 = (1/dx**2) * np.diag(np.ones(N-1), 1)
+    D2 = (main_diagonal + off_diagonal1 + off_diagonal2)
+    
+    # Kinetic Energy Operator T = -hbar^2 / 2m * d^2/dx^2
+    T = (-(hbar**2 / (2*m)) * D2)
+    return T
+        """, language="python")
+        st.code("""
+def harmonic(x,k,center=0.0):
+    # A Parabola, setting the global k-value.
+    global Last_k_value
+    Last_k_value = k
+    
+    constant_factor = 1 
+    potential = 0.5*k*(x - center)**2
+    return constant_factor * potential
+""")
+        
+        
+        
+# ==========================================
+# 6. PAGE: THEORY
+# ==========================================
+elif page == "Theory & Method":
+    st.title("📖 Theory & Methodology")
+    
+    st.markdown("### The Time-Independent Schrödinger Equation")
+    st.latex(r" \hat{H}\psi(x) = E\psi(x) ")
+    st.latex(r" \left[ -\frac{\hbar^2}{2m}\frac{d^2}{dx^2} + V(x) \right]\psi(x) = E\psi(x) ")
+    
+    st.markdown("### Numerical Method: Finite Difference")
+    st.markdown(r"""
+    We discretize the spatial domain $x$ into a grid of $N$ points. The second derivative is approximated using the **Central Difference Formula**:
+    """)
+    st.latex(r" \frac{d^2\psi}{dx^2} \approx \frac{\psi_{i+1} - 2\psi_i + \psi_{i-1}}{\Delta x^2} ")
+    
+    st.markdown(r"""
+    This transforms the differential operator into a **Tridiagonal Matrix** equation:
+    """)
+    st.latex(r" \mathbf{H}\mathbf{\psi} = E\mathbf{\psi} ")
+    
+    st.markdown(r"""
+    Where $\mathbf{H}$ is an $N \times N$ matrix. We then use `numpy.linalg.eigh` to solve for the eigenvalues ($E$) and eigenvectors ($\psi$).
+    """)
+    
+    st.markdown("### Implementation Details")
+    st.markdown(r"""
+    - **Grid Size:** Dynamic (default 1000–2000 points)
+    - **Boundary Conditions:** Dirichlet ($ \psi(0) = \psi(L) = 0 $) via infinite walls at grid edges.
+    - **Units:** Hartree Atomic Units ($\hbar=1, m=1$).
+    """)