Solution for a Single Free Particle - Revision history

Carlosmsilva: /* The Free Particle */

2022-04-17T13:51:11Z

The Free Particle

← Older revision		Revision as of 09:51, 17 April 2022
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	</math>		</math>

	The latter is a form of the [//en.wikipedia.org/wiki/Helmholtz_equation Helmholtz Equation]. There is exists a general solution for an unbounded geometry in [//mathworld.wolfram.com/SphericalCoordinates.html Spherical Coordinates]:		The latter is a form of the [//en.wikipedia.org/wiki/Helmholtz_equation Helmholtz Equation]. There exists a general solution for an unbounded geometry in [//mathworld.wolfram.com/SphericalCoordinates.html Spherical Coordinates]:

	<math>		<math>

2022-04-17T01:31:10Z

General Solution in 1 Dimension

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 <math>
 - \frac {\hbar^2}{2m} \frac{\partial^2\psi(x)}{\partial x^2} = E \psi(x)
 </math>
 ==References==

2022-04-17T00:44:47Z

The Free Particle

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 <math>
 V(\vec{r},t) = 0
 </math>
 ==References==

2022-04-17T00:16:38Z

← Older revision		Revision as of 20:16, 16 April 2022
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	</div>		</div>
	</div>		</div>

			==The Free Particle==

			The free particle is a special case of the Schrödinger Equation where the potential is null (or constant) everywhere in space:

			<math>
			V(\vec{r},t) = 0
			</math>

	==References==		==References==

2022-04-17T00:13:51Z

Derivation of the Time-Independent and Space-Independent Schrödinger Equation

← Older revision		Revision as of 20:13, 16 April 2022
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	<math>		<math>
			\frac{i\hbar\frac{d}{d t}\phi(t)}{\phi(t)} = -\frac{\hbar^2}{2m} \frac{\nabla ^2 \psi(\vec{r})}{\psi(\vec{r})} + V(\vec{r})
			\rightarrow
	- \frac {\hbar ^2}{2m} \nabla^2\psi(\vec{r}) + V(\vec{r})\psi(\vec{r})= E \psi(\vec{r})		- \frac {\hbar ^2}{2m} \nabla^2\psi(\vec{r}) + V(\vec{r})\psi(\vec{r})= E \psi(\vec{r})
	</math>		</math>

2022-04-17T00:13:00Z

Derivation of the Time-Independent and Space-Independent Schrödinger Equation

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 \phi(t) = C_2 e^{C_1 t}
 </math>
 ::

2022-04-17T00:08:28Z

Derivation of the Time-Independent and Space-Independent Schrödinger Equation

← Older revision		Revision as of 20:08, 16 April 2022
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	<math>		<math>
	\frac{d}{dt}\left(\dot{\phi}\right)\phi = \left( \dot{\phi}\right)^2 ~~\rightarrow~~		\frac{d}{dt}\left(\dot{\phi}\right)\phi = \left( \dot{\phi}\right)^2
	\rightarrow		\rightarrow
	\frac{d}{dt}\left(v(\phi)\right)\phi = v^2(\phi)		\frac{d}{dt}\left(v(\phi)\right)\phi = v^2(\phi)

2022-04-17T00:06:03Z

Undo revision 39356 by Carlosmsilva (talk)

← Older revision		Revision as of 20:06, 16 April 2022
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	<div class="toccolours">		<div class="toccolours">
	====Space-Independent Schrödinger Equation for 1 Particle====		====Space-Independent Schrödinger Equation for 1 Particle====
	<math>\frac{d}{d t}\phi(t)= -\frac{iE}{\hbar}\phi(t)</math>		<math>\frac{d}{d t}\phi(t)= -\frac{i}{\hbar}\left(E + V \right)\phi(t)</math>
	</div>		</div>

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	<div class="toccolours mw-collapsible mw-collapsed">		<div class="toccolours mw-collapsible mw-collapsed">

	====Derivation of the Time-Independent and Space-Independent Schrödinger Equation====		====Derivation of the Time-Independent and Space-Independent Schrödinger Equation====
	<div class="mw-collapsible-content">		<div class="mw-collapsible-content">

2022-04-17T00:02:27Z

Space-Independent Schrödinger Equation for 1 Particle

← Older revision		Revision as of 20:02, 16 April 2022
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	<div class="toccolours">		<div class="toccolours">
	====Space-Independent Schrödinger Equation for 1 Particle====		====Space-Independent Schrödinger Equation for 1 Particle====
	<math>\frac{d}{d t}\phi(t)= -\frac{i}{\hbar}~~\left(E + V \right)~~\phi(t)</math>		<math>\frac{d}{d t}\phi(t)= -\frac{iE}{\hbar}\phi(t)</math>
	</div>		</div>

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	<div class="toccolours mw-collapsible mw-collapsed">		<div class="toccolours mw-collapsible mw-collapsed">

	====Derivation of the Time-Independent and Space-Independent Schrödinger Equation====		====Derivation of the Time-Independent and Space-Independent Schrödinger Equation====
	<div class="mw-collapsible-content">		<div class="mw-collapsible-content">

2022-04-16T23:54:14Z

@@ Line 59: / Line 59: @@
 <math>\frac{d}{d t}\phi(t)= -\frac{i}{\hbar}\left(E + V \right)\phi(t)</math>
 </div>
 <div class="toccolours mw-collapsible mw-collapsed">
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 i\hbar\left(
 \frac{\ddot{\phi}}{\phi}
--\frac{\dot{phi}^2}{phi^2}
+-\frac{\dot{\phi}^2}{\phi^2}
 \right) = 0
-<\math>
+</math>
 </div>