Schrödinger equation

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The Schrödinger equation is an equation that is fundamental to quantum theory. The time independent Schrödinger equation looks like

\hat{H} \psi = E \psi .

The \hat{H} is called the Hamiltonianw:Hamiltonian_mechanics. \psi \,\; denote the wavefunctions.

[edit] How to construct the Schrödinger equation for a system

The operator \hat{H} is called the Hamiltonian. It contains two parts:

  1. The kinetic energy, denoted by the operator \hat{T}
  2. The potential energy, denoted by the operator \hat{U}

These two operators represent the principal types of energies in any physical system. Putting these two parts together, we can write this as \hat{H} = \hat{T} + \hat{U}. We can think of the Hamiltonian as a mathematical object that encodes how the energies in a system can be distributed.

[edit] Kinetic energy operator

In classical mechanics, we know that the kinetic energy of a moving particle is given by

E = \frac{1}{2} mv^2 = \frac{1}{2} m \vec{v} \cdot \vec{v}

in which m corresponds to the mass of the particle, and \vec{v} is its velocity. Noting that classically the linear momentum \vec{p} is given by \vec{p} = m \vec{v}, so the kinetic energy can also be written as

E = \frac{p^2}{2m}.

In order to get the quantum mechanical version of this, we substitute all occurrences of the momentum \vec{p} with

\vec{p} \to -i \hbar \nabla,

where \nabla is the vector analogue of the partial differential operator \frac{\partial}{\partial x}.

Hence, the kinetic energy operator is given by

\hat{T} = - \frac{\hbar^2}{2m} \nabla^2 .

[edit] How to solve the Schrödinger equation

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