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Module 4.1 1D Time-Independent Schroedinger Equation (TISE) - QHO Shooting Method

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Simulation 1

1D Time-Independent Schroedinger Equation (TISE) - QHO Shooting Method

Python 3.11 with Matplotlib Slider Ready

Interactive Physics Ready

Click "Run Code" to launch simulation and interactive parameter sliders.


                                

                            


                            
                                                        

                                

                                                                            

                                            
 Mathematical Problem Formulation

                                            
Solve 1-dimensional time-independent Schroedinger Equation (TISE) for Quantum Harmonic Oscillator (QHO) using shooting algorithm and Numerov method

                                        

                                    
                                                                            

                                            
 Theoretical Background & Explanation

                                            

    

        

             Quantum Harmonic Oscillator (QHO) - Boundary Value Problem
        

        

            The Quantum Harmonic Oscillator represents a foundational model in quantum mechanics, describing vibrating diatomic molecules, lattice vibrations (phonons), and quantum field fluctuations around stable equilibrium positions.
        

    


    

        1. One-Dimensional Time-Independent Schrödinger Equation (TISE)
    

    

        The stationary states $\psi(x)$ of a non-relativistic quantum particle of mass $m$ confined within a potential well $V(x)$ are governed by the time-independent Schrödinger equation:
    

    

        $$-\frac{\hbar^2}{2m} \frac{d^2\psi(x)}{dx^2} + V(x)\psi(x) = E \psi(x)$$
    

    

        Rearranging into standard second-order ordinary differential equation (ODE) form:
    

    

        $$\frac{d^2\psi(x)}{dx^2} = \frac{2m}{\hbar^2} \left[ V(x) - E \right] \psi(x)$$
    


    

        2. Parabolic Harmonic Potential & Analytical Eigenvalues
    

    

        For a harmonic oscillator with angular frequency $\omega$, the parabolic restoring potential is:
    

    

        $$V(x) = \frac{1}{2} m \omega^2 x^2$$
    

    

        Analytical solution yields discrete, equally-spaced energy eigenvalues:
    

    

        $$E_n = \left(n + \frac{1}{2}\right) \hbar \omega, \quad n = 0, 1, 2, 3, \dots$$
    

    

        In dimensionless natural units ($m = 1, \hbar = 1, \omega = 1$):
        
• Ground state ($n = 0$): $E_0 = 0.5$ (Zero-point energy)
        
• 1st Excited state ($n = 1$): $E_1 = 1.5$ (Odd parity, single central node)
        
• 2nd Excited state ($n = 2$): $E_2 = 2.5$ (Even parity, two symmetric nodes)
        
• 3rd Excited state ($n = 3$): $E_3 = 3.5$ (Odd parity, three nodes)
        
• 4th Excited state ($n = 4$): $E_4 = 4.5$ (Even parity, four nodes)
    


    

        3. The Shooting Algorithm & Boundary Conditions
    

    

        Because bound-state wave functions must be square-integrable, the physical boundary conditions require $\psi(x) \to 0$ as $x \to \pm \infty$. On a finite computational domain $[-L, L]$ with $L = 6$:
    

    

        $$\psi(-L) = 0, \quad \psi(+L) = 0$$
    

    

        The shooting algorithm converts the 2nd-order boundary value problem into a system of two coupled 1st-order differential equations:
    

    

        $$\frac{dy_1}{dx} = y_2, \qquad \frac{dy_2}{dx} = \frac{2m}{\hbar^2} \left[ V(x) - E \right] y_1$$
    

    

        Starting from $x = -L$ with initial vector $u = [y_1(-L), y_2(-L)] = [0, 0.1]$, the numerical integrator (scipy.integrate.odeint) integrates across the domain to $x = +L$ for a trial test energy $E$.
    


    

        4. Wave Function Normalization & Interactive Slider
    

    

        The probability of finding the quantum particle anywhere along the $x$-axis is identically unity:
    

    

        $$\int_{-\infty}^{\infty} |\psi(x)|^2 dx = 1 \implies \psi_{\text{norm}}(x) = \frac{\psi(x)}{\sqrt{\int_{-L}^{L} \psi^2(x) dx}}$$
    

    

        In the simulation code, the trapezoidal quadrature rule (scipy.integrate.trapezoid) computes the normalization integral dynamically.
        By adjusting the Energy Slider ($E \in [0, 10]$), observe how non-eigenvalues diverge rapidly at $x = +L$, whereas true quantized energies ($E = 0.5, 1.5, 2.5, 3.5, 4.5, \dots$) smoothly decay to zero at both boundaries, displaying exact quantum nodes!