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Mathematics for Physics

85 Schrodinger Equations

The family of linear differential equations that is used to describe the wave-particle behavior and probability densities of non-relativistic particles.

Related definitions:

definition (Time Dependent Schrödinger 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.

definition (Time Dependent Schrödinger Equation — Generalized) The coordinate-independent and dimension-independent form of the fundamental law of quantum mechanics. This abstract version represents the general principle that a quantum state changes continuously and unitarily over time, governing the evolution of any quantum system—including those with multiple particles or non-spatial degrees of freedom. The general time-dependent form 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-vector representing the state of the system
  • \(t\) is time
  • \(\hat{H}\) is the Hamiltonian operator representing the total energy of the system.

Note:

  • \(\Psi\) is also written \(|\Psi\rangle\).
  • \(\Psi\) is also called the wave function.
  • the total energy is kinetic energy plus potential energy.

The time-independent version is a specific “eigenvalue equation” used when the potential energy \(V\) does not change with time. It is used to find permitted energy levels and stationary states—states where the probability density remains constant even though the wave function carries a “time-dependent wiggle factor”.

definition (Schrödinger Equation — Time-Independent) An eigenvalue equation used to determine the stationary states and quantized energy levels of a quantum system when the potential energy is independent of time. It determines the spatial part of the wave function \(\psi\) and the allowed energy values \(E\) that the system can possess. The generalized form is:

  • \(\hat{H}\psi = E\psi\)

where

  • \(\hat{H}\) is the Hamiltonian operator representing the total energy of the system.
  • \(\psi\) is the spatial wave function.
  • \(E\) is the energy of the state.

Note:

  • \(\psi\) is lower-case psi.
  • \(\psi\) is an eigenfunction of \(\hat{H}\).
  • \(E\) is the corresponding eigenvalue.

In one dimension, this is explicitly written as:

  • \(-\dfrac{\hbar^2}{2m} \dfrac{d^2\psi}{dx^2} + V\psi = E\psi\).