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Mathematics for Physics

143 Schrodinger Equations

A family of linear differential equations that is used to describe the evolution and the allowed energies of a non-relativistic quantum system, where a non-relativistic system is a system whose speeds are much smaller than the speed of light.

The generalized time-dependent Schrödinger equation. The state of any quantum system evolves according to the Hamiltonian. This principle is used to predict the state at a later time from the state now.

The generalized time-dependent Schrödinger equation is

\[ i\hbar\dfrac{\partial\Psi}{\partial t} = \hat{H}\Psi \]

where

  • \(i\) is the imaginary unit.
  • \(\hbar\) is the reduced Planck constant.
  • \(\Psi\) is the state.
  • \(t\) is time.
  • \(\hat{H}\) is the Hamiltonian operator.

The one-dimensional time-dependent equation. For one particle in one dimension the Hamiltonian is kinetic energy plus potential energy. This principle is used to write an explicit partial differential equation for \(\Psi(x,t)\).

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

  • \(\Psi\) is the wavefunction.
  • \(m\) is the mass.
  • \(V\) is the potential energy.
  • \(x\) is the position.
  • \(t\) is time.

The time-independent eigenvalue equation. When \(V\) does not depend on time, stationary states satisfy an eigenvalue equation. A stationary state is a state whose probability density does not change with time. This principle is used to find the allowed energies.

The time-independent Schrödinger equation is

\[ \hat{H}\psi = E\psi \]

where

  • \(\hat{H}\) is the Hamiltonian operator.
  • \(\psi\) is the spatial wavefunction.
  • \(E\) is the energy eigenvalue.

Its one-dimensional form. In one dimension that eigenvalue equation is a second-order ordinary differential equation. This principle is used to solve wells, barriers, and the harmonic oscillator.

The one-dimensional time-independent Schrödinger equation is

\[ -\dfrac{\hbar^{2}}{2m}\dfrac{d^{2}\psi}{dx^{2}} + V\psi = E\psi \]

where

  • \(\psi\) is the spatial wavefunction.
  • \(V\) is the potential energy.
  • \(E\) is the energy.
  • \(m\) is the mass.
  • \(x\) is the position.

Note: \(\Psi\) is also written \(|\Psi\rangle\).

143.1 References

  1. Knight, R. D. Physics for Scientists and Engineers: A Strategic Approach with Modern Physics. Pearson, 2023. — Schrödinger equations as the non-relativistic quantum dynamical law.
  2. Griffiths, D. J. Introduction to Quantum Mechanics. Cambridge University Press, 2018. §1.2, §2.1 — time-dependent and time-independent Schrödinger equations.