Question

How do you model time evolution of the Gaussian packet (quantum mechanics, exercises, wavefunction, schroedinger equation, fourier transform, physics)?

Answer 1

To model the time evolution of a Gaussian packet in quantum mechanics, we start with the wavefunction of the Gaussian packet at an initial time and use the Schrödinger equation to calculate its evolution over time. Here's a step-by-step guide on how to approach this problem:

1. Initial Gaussian Wavefunction: The Gaussian wavefunction is given by:

   ψ(x, t=0) = A * exp[-(x - x₀)² / (4σ²)] * exp[i(k₀x - E₀t) / ℏ]

   Here, A is the normalization constant, x₀ is the initial position of the Gaussian packet, σ is the width (standard deviation) of the Gaussian, k₀ is the initial wave vector, E₀ is the initial energy, t is the initial time, and ℏ is the reduced Planck's constant.

2. Fourier Transform: To determine the momentum-space representation of the Gaussian packet, we perform a Fourier transform of the wavefunction:

   ψ(k, t=0) = ∫ ψ(x, t=0) * exp(-ikx) dx

   The Fourier transform of a Gaussian function is also a Gaussian, and the resulting expression can be obtained by completing the square in the exponent.

3. Energy and Momentum Relations: In quantum mechanics, the energy of a particle is related to its momentum through the dispersion relation:

   E = (ħ²k²) / (2m)

   Here, m is the mass of the particle. Using this relation, we can express the energy in terms of the momentum.

4. Time Evolution: With the initial wavefunction and the momentum-space representation at t=0, we can use the Schrödinger equation to find the time evolution of the Gaussian packet:

   iħ ∂ψ/∂t = - (ħ²/2m) ∂²ψ/∂x²

   This equation is a partial differential equation, and solving it for the Gaussian wavefunction involves applying the appropriate time evolution operator and utilizing the initial conditions.

5. Obtain Time-Dependent Wavefunction: Once the Schrödinger equation is solved, you will have the time-dependent wavefunction ψ(x, t) for the Gaussian packet.

It's worth noting that while this provides an outline of the procedure, the actual calculations and solutions can involve mathematical techniques and approximations depending on the specifics of the problem. It's recommended to consult relevant textbooks or resources for detailed derivations and numerical methods to solve the time evolution of Gaussian wavepackets in quantum mechanics.