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Ch. 20.4 Time Evaluation of WavePacket

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Time Evaluation of WavePacket

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 Mathematical Problem Formulation

                                        
        

    
Consider a one-dimensional quantum system described by the time-dependent Schrödinger equation.
	    

        \[
        i\hbar \frac{\partial \psi(x,t)}{\partial t} = \left( -\frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2} + V(x) \right) \psi(x,t)
        \]
    

	
Time evolution of a wave packet moving in free space. A Gaussian wave packet is defined as:


    

        $$\psi(x, t=0) = A \cdot e^{-\frac{(x - x_0)^2}{2 \sigma^2}} \cdot e^{i k_0 x}$$
    

	    
The energy of the wave packet is defined as:

    

        $$E = \frac{\hbar^2 k_0^2}{2m} + \frac{\hbar^2}{2m \sigma^2}$$
    

	
Analyze Time evolution of the wave packet by Crank-Nicolson Method 

                                    

                                    
                                                                    

                            

                                                    

                    

                

                            
            
            
            
            
            

                

                    

                         Theoretical Principles & Computational Physics Guide
                    

                    

                        Time Evaluation of WavePacket: Quantum Mechanical Simulation & Wave Mechanics                    

                    

                         Dr. Alorika Chatterjee & Dr. Anirban Shaw
                         Science & Physics Learners
                         Python 3 / NumPy / SciPy
                    

                


                
                

                    

                         1. Core Principles & Theoretical Framework
                    

                    

                        
                    

                        
 Wavefunction Dynamics: Simulates quantum state wavefunctions $\psi(\mathbf{r}, t)$ governed by the Schrödinger equation.

                        
 Energy Quantization: Physical boundary conditions restrict allowable solutions to discrete bound-state energy eigenvalues $E_n$.

                        
 Quantum Tunneling: Models penetration and reflection of wavepackets through classically forbidden potential energy barriers.

                    
                    

                


                
                

                    

                         2. Mathematical Formulation & Algorithmic Pipeline
                    

                    

                        
                    
Time-Dependent and Time-Independent Schrödinger Equations:

                    
$$ i\hbar \frac{\partial\psi}{\partial t} = \hat{H}\psi = \left(-\frac{\hbar^2}{2m}\nabla^2 + V(\mathbf{r})\right)\psi, \quad \hat{H}\psi_n = E_n \psi_n $$
                    

                


                
                

                    

                         3. Numerical Mechanics & Implementation Logic
                    

                    

                        
                    
    
                        
  • Spatial Discretization: Represent wavefunction on a discrete coordinate grid with spacing $\Delta x$.
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  • Unitary Propagation: Apply Cayley/Crank-Nicolson or split-operator stepping to conserve total probability $\int |\psi|^2 dx = 1$.
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  • Expectation Values: Evaluate physical observables via inner product contractions $\langle \hat{A} \rangle = \int \psi^* \hat{A} \psi dx$.
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                         4. Relevant Computational Practice Exercises
                    

                    

                                                    

                                Exercise 1
                                

                                    Simulate Time Evaluation of WavePacket and calculate the first 3 quantized energy eigenvalues.                                

                                
                            

                                                    

                                Exercise 2
                                

                                    Verify normalization condition $\int_{-\infty}^\infty |\psi(x)|^2 dx = 1.0$ across the simulation grid.                                

                                
                            

                                                    

                                Exercise 3
                                

                                    Plot the transmission and reflection coefficients as a function of incident particle kinetic energy.                                

                                
                            

                                            

                


                
                

                     Standard Literature & References:
                    
    
                                                    
  • David J. Griffiths, "Introduction to Quantum Mechanics", Cambridge University Press (2018).
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  • Claude Cohen-Tannoudji et al., "Quantum Mechanics", Wiley (2019).
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