Time-Dependent Schrödinger Solver
Time-Dependent Schrödinger Solver computes wavefunction evolution in a 1D potential, aiding quantum dynamics and quantum computing studies.
Formulas Used in Time-Dependent Schrödinger Solver
The calculator solves the time-dependent Schrödinger equation numerically:
Time-Dependent Schrödinger Equation:
\\[ i \hbar \frac{\partial \psi(x,t)}{\partial t} = -\frac{\hbar^2}{2m} \frac{\partial^2 \psi(x,t)}{\partial x^2} + V(x) \psi(x,t) \\]Initial Wavefunction (Gaussian):
\\[ \psi(x,0) = \left( \frac{2}{\pi \sigma^2} \right)^{1/4} e^{-(x-x_0)^2 / \sigma^2} e^{i k_0 x} \\]Harmonic Oscillator Potential:
\\[ V(x) = \frac{1}{2} m \omega^2 x^2 \\]Numerical Time Evolution (Euler):
\\[ \psi(x, t + \Delta t) \approx \psi(x, t) – \frac{i \Delta t}{\hbar} \left[ -\frac{\hbar^2}{2m} \frac{\psi(x + \Delta x, t) – 2\psi(x, t) + \psi(x – \Delta x, t)}{\Delta x^2} + V(x) \psi(x, t) \right] \\]Probability Density:
\\[ |\psi(x,t)|^2 \\]Where:
- \\( \psi(x,t) \\): Wavefunction
- \\( m \\): Particle mass (kg)
- \\( x_0 \\): Initial position (nm)
- \\( \sigma \\): Gaussian width (nm)
- \\( k_0 \\): Initial wavevector (nm\\(^{-1}\\))
- \\( \omega \\): Angular frequency (rad/s, 0 for free particle)
- \\( t \\): Evolution time (fs)
- \\( \hbar \\): Reduced Planck’s constant (\\( 1.0545718 \times 10^{-34} \, \text{J·s} \\))
Example Calculations
Example 1: Free Particle Wavepacket Spreading
Input: m = 9.109e-31 kg, x_0 = 0 nm, σ = 1 nm, k_0 = 1 nm\\(^{-1}\\), ω = 0 rad/s, t = 100 fs
Result: Wavepacket spreads; peak probability density decreases, normalization ~1.
Example 2: Harmonic Oscillator, Stationary Gaussian
Input: m = 9.109e-31 kg, x_0 = 0 nm, σ = 1 nm, k_0 = 0 nm\\(^{-1}\\), ω = 1e14 rad/s, t = 100 fs
Result: Oscillatory behavior; probability density remains centered, normalization ~1.
Example 3: Harmonic Oscillator, Moving Wavepacket
Input: m = 9.109e-31 kg, x_0 = 1 nm, σ = 1 nm, k_0 = 1 nm\\(^{-1}\\), ω = 1e14 rad/s, t = 100 fs
Result: Wavepacket oscillates with potential; normalization ~1.