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 how this energy influences the time evolution of the wave packet.

Simulating Time Evolution of a Gaussian Wave Packet in a Potential Well

Consider a one-dimensional system where a quantum particle is evolving in time under the influence of a potential barrier. The time evolution of the wave function \( \psi(x, t) \) is governed by the time-dependent Schrödinger equation:

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

take: \( m = 1.0 \) and \( \hbar = 1.0 \)

The initial wave function is a Gaussian wave packet centered at \( x_0 = 100 \) with width \( \sigma = 10 \) and wavenumber \( k_0 = \frac{\pi}{10} \). The corresponding initial wave function is:

\[ \psi(x, 0) = A \exp\left( -\frac{(x - x_0)^2}{2\sigma^2} \right) e^{ik_0 x} \]

where \( A \) is a normalization constant.

The barrier height is given by \( V_0 = 0.1 \).

The simulation is performed over \( N = 1000 \) spatial points and up to \( T = 10,000 \) time steps.

The energy of the wave packet is given by:

\[ E = \frac{\hbar^2}{2m} \left( k_0^2 + \frac{1}{2\sigma^2} \right) \]

Python program simulates the time evolution of the wave packet using finite difference methods, and plot the real and imaginary parts of the wave function \( \psi(x, t) \) as well as the probability density \( |\psi(x,t)|^2 \) at various time steps.