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Mathematics for Physics

150 Time Dependent Schrodinger Equation 1-Dimensional

A linear partial differential equation that is used to compute the wavefunction of one particle moving along a single axis.

The one-dimensional time-dependent Schrödinger equation. For a particle of mass \(m\) on the \(x\) axis, the Schrödinger equation is the quantum analog of Newton’s second law. This principle is used to determine \(\Psi(x,t)\) from \(V(x,t)\) and the initial wavefunction.

The one-dimensional time-dependent Schrödinger equation is

\[ i\hbar\dfrac{\partial\Psi}{\partial t} = -\dfrac{\hbar^{2}}{2m}\dfrac{\partial^{2}\Psi}{\partial x^{2}} + V\Psi \]

where

  • \(i\) is the imaginary unit.
  • \(\hbar\) is the reduced Planck constant.
  • \(\Psi\) is the wavefunction.
  • \(m\) is the mass of the particle.
  • \(V\) is the potential energy.
  • \(x\) is the position.
  • \(t\) is time.

The kinetic-energy operator. The first term on the right is the kinetic-energy operator. This principle is used to identify \(-\dfrac{\hbar^{2}}{2m}\dfrac{\partial^{2}}{\partial x^{2}}\) with \(p^{2}/2m\).

Uniqueness from initial data. Given \(\Psi(x,0)\) and \(V\), the equation determines \(\Psi(x,t)\) at every later time. This principle is used to treat the Schrödinger equation as an initial-value problem.

150.1 References

  1. Hall, B. C. Quantum Theory for Mathematicians. Springer, 2013. — one-dimensional time-dependent Schrödinger equation.
  2. Griffiths, D. J. Introduction to Quantum Mechanics. Cambridge University Press, 2018. §1.2 — \(\Psi(x,t)\) in one dimension.