Solution for a Single Free Particle: Difference between revisions
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==The General Schrödinger Equation== | ==The General Schrödinger Equation== | ||
The general formulation of the Schrödinger Equation for a time-dependent system of non-relativistic particles in [//en.wikipedia.org/wiki/Bra–ket_notation Bra-Ket notation] is: | |||
<math>i \hbar \frac{d}{d t}\vert\Psi(t)\rangle = \hat H\vert\Psi(t)\rangle</math> | |||
Here <math>i = \sqrt{-1}</math> is the [//en.wikipedia.org/wiki/Imaginary_unit imaginary unit]. <math>\hbar</math> is the [//en.wikipedia.org/wiki/Planck_constant reduced Planck's constant]. <math>\vert\Psi(t)\rangle</math> is the state vector of the quantum system at time <math>t</math>, and <math>\hat H</math> is the [//en.wikipedia.org/wiki/Hamiltonian_(quantum_mechanics) Hamiltonian operator]. For a single particle, the general Schrödinger Equation reduces to a single linear partial differential equation: | |||
<math> | |||
i\hbar\Psi(\vec{r},t) = - \frac{\hbar^2}{2m} \nabla ^2 \Psi(\vec{r},t) + V(\vec{r},t)\Psi(\vec{r},t) | |||
</math> | |||
<math>\Psi(\vec{r},t)</math> becomes the wave function of the particle at position <math>\vec{r}</math> and time <math>t</math>. <math>V(\vec{r},t)</math> is the scalar potential energy of the particle at position <math>\vec{r}</math> and time <math>t</math>. | |||
==References== | ==References== | ||
Revision as of 18:13, 16 April 2022
Claimed by Carlos M. Silva (Spring 2022)
The Schrödinger Equation is a linear partial differential equation that governs the wave function of a quantum mechanical system[1]. Similar to Newton's Laws, the Schrödinger Equation is an equation of motion, meaning that it's capable of describing the time-evolution of a position analog of a system.
The free particle is the name given to the system consisting of a single particle subject to a null or constant potential everywhere in space. It's the simplest system to which the Schrödinger Equation has a solution with physical meaning.
Although the free-particle solution does not have ample practical use in the field of Physics, the methods and conclusions that come from the solution of this system are of great use in a plethora of other quantum systems.
The General Schrödinger Equation
The general formulation of the Schrödinger Equation for a time-dependent system of non-relativistic particles in Bra-Ket notation is:
[math]\displaystyle{ i \hbar \frac{d}{d t}\vert\Psi(t)\rangle = \hat H\vert\Psi(t)\rangle }[/math]
Here [math]\displaystyle{ i = \sqrt{-1} }[/math] is the imaginary unit. [math]\displaystyle{ \hbar }[/math] is the reduced Planck's constant. [math]\displaystyle{ \vert\Psi(t)\rangle }[/math] is the state vector of the quantum system at time [math]\displaystyle{ t }[/math], and [math]\displaystyle{ \hat H }[/math] is the Hamiltonian operator. For a single particle, the general Schrödinger Equation reduces to a single linear partial differential equation:
[math]\displaystyle{ i\hbar\Psi(\vec{r},t) = - \frac{\hbar^2}{2m} \nabla ^2 \Psi(\vec{r},t) + V(\vec{r},t)\Psi(\vec{r},t) }[/math]
[math]\displaystyle{ \Psi(\vec{r},t) }[/math] becomes the wave function of the particle at position [math]\displaystyle{ \vec{r} }[/math] and time [math]\displaystyle{ t }[/math]. [math]\displaystyle{ V(\vec{r},t) }[/math] is the scalar potential energy of the particle at position [math]\displaystyle{ \vec{r} }[/math] and time [math]\displaystyle{ t }[/math].
References
- ↑ Griffiths, David J. (2004). Introduction to Quantum Mechanics (2nd ed.). Prentice Hall. ISBN 978-0-13-111892-8.