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Mathematics for Physics

91 Time Dependent Schrodinger Equation 1-Dimensional

The linear partial differential equation that is used to calculate the spatial and temporal evolution of a particle’s wavefunction along a single axis.

definition (Time Dependent Schrodinger Equation 1-Dimensional) The specific case of the fundamental law of quantum mechanics tailored for a single particle of mass \(m\) constrained to move in one dimension along the \(x\)-axis. It serves as a direct quantum analog to Newton’s second law, determining the particle’s wave function \(\Psi(x, t)\) for all future time given the potential energy \(V(x, t)\) and initial conditions. It is expressed as:

  • \(i\hbar \dfrac{\partial \Psi}{\partial t} = -\dfrac{\hbar^2}{2m} \dfrac{\partial^2 \Psi}{\partial x^2} + V\Psi\)

where

  • \(i\) is the imaginary unit
  • \(\hbar\) is the reduced Planck constant
  • \(\Psi\) is the wave function, which depends on position \(x\) and time \(t\)
  • \(m\) is the mass of the particle
  • \(V\) is the potential energy function
  • \(x\) is the position coordinate
  • \(t\) is time.