Qal QuantP1S - Revision history

Oborn at 03:50, 19 August 2007

2007-08-19T03:50:40Z

Oborn at 03:49, 19 August 2007

2007-08-19T03:49:13Z

Oborn at 03:48, 19 August 2007

2007-08-19T03:48:03Z

Oborn at 03:45, 19 August 2007

2007-08-19T03:45:11Z

Oborn at 03:37, 19 August 2007

2007-08-19T03:37:51Z

Oborn at 03:37, 19 August 2007

2007-08-19T03:37:24Z

Oborn at 03:36, 19 August 2007

2007-08-19T03:36:37Z

Oborn at 03:36, 19 August 2007

2007-08-19T03:36:08Z

Oborn at 03:30, 19 August 2007

2007-08-19T03:30:15Z

Oborn at 21:00, 17 August 2007

2007-08-17T21:00:37Z

New page

Solution:

* <math> [- \frac{\hbar^2}{2M}\Delta^2 + V]W(x,y,z)=E W(x,y,z) </math> 

In our case, using separation of variables, we will get 3 differential equations for x, y and z. W(x,y,z)=w(x)w(y)w(z) 

<math>\frac{d^2 w(x)}{dx^2} - m^2 w(x) = 0</math>(1) 
The same will be for y and z. 

Solution of equation (1) is following 
<math>w(x) = A\sin(mx)+B\cos(mx)</math> 
<math>w(y) = C\sin(ky)+D\cos(ky)</math> 
<math>w(z) = E\sin(qz)+F\cos(qz)</math> 

* Applying B. C. at x=y=z=0 wave function should be zero, that means B=D=F=0. We have 

<math>w(x) = A\sin(mx) </math> 
<math>w(y) = C\sin(ky) </math> 
<math>w(z) = E\sin(qz) </math> 

Also, w(a)=0 which gives <math>A\sin(ma)=0, m=\frac{\pi n_x}{a}</math>. For y component <math>C\sin(kb)=0, k=\frac{\pi n_y}{b}</math> and for z <math>E\sin(qc)=0, q=\frac{\pi n_z}{c}</math> 

A, C and E are normalization constants 

<math>\frac{1}{A^2}=\int\sin^2 (\pi nx/a) dx = \frac{a}{2} </math>, limits are from 0 to a. 

The eigenfunction for each component will be 

<math>w(x) = \sqrt{\frac{2}{a}} \sin(\pi n_x x/a)</math> 
<math>w(y) = \sqrt{\frac{2}{b}} \sin(\pi n_y y/b)</math> 
<math>w(z) = \sqrt{\frac{2}{c}} \sin(\pi n_z z/c)</math> 

The eigenenergies 

<math>E_{n_x} = \frac{\pi^2 \hbar^2 {n_x}^2}{2Ma^2}</math>, <math>E_{n_y} = \frac{\pi^2 \hbar^2 {n_y}^2}{2Mb^2}</math>, <math>E_{n_z} = \frac{\pi^2 \hbar^2 {n_z}^2}{2Mc^2}</math> 
Total energy is sum of these energies. 

* <math>E = \frac{\pi^2 \hbar^2 n^2}{2M^2 a^2}</math>, where <math>n^2=n_x ^2 + n_y ^2 + n_z ^2</math>, n=1,2,3...

← Older revision		Revision as of 03:50, 19 August 2007
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	<math>\lambda_{1,2} = \frac{(w_1 + w_2)+/- \sqrt{(w_1 + w_2)^2 - 4(w_1 w_2 - v^2)}}{2}</math>		<math>\lambda_{1,2} = \frac{(w_1 + w_2)+/- \sqrt{(w_1 + w_2)^2 - 4(w_1 w_2 - v^2)}}{2}</math>
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		+
		+	Dr. Forest: We have not had Perturbation Theory.

← Older revision		Revision as of 03:49, 19 August 2007
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	<math>(w_1 - \lambda)(w_2 - \lambda) - v^2=0</math><br>		<math>(w_1 - \lambda)(w_2 - \lambda) - v^2=0</math><br>

−	<math>\lambda_{1,2} = \frac{(w_1 + w_2)+/- \sqrt{j}}{1}</math>	+	<math>\lambda_{1,2} = \frac{(w_1 + w_2)+/- \sqrt{(w_1 + w_2)^2 - 4(w_1 w_2 - v^2)}}{2}</math>

← Older revision		Revision as of 03:48, 19 August 2007
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	<math>(w_1 - \lambda)(w_2 - \lambda) - v^2=0</math><br>		<math>(w_1 - \lambda)(w_2 - \lambda) - v^2=0</math><br>
		+
		+	<math>\lambda_{1,2} = \frac{(w_1 + w_2)+/- \sqrt{j}}{1}</math>

← Older revision		Revision as of 03:45, 19 August 2007
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	<math>H=\begin{pmatrix}(w_1-\lambda)&v\\v&(w_2-\lambda)\end{pmatrix}</math><br>		<math>H=\begin{pmatrix}(w_1-\lambda)&v\\v&(w_2-\lambda)\end{pmatrix}</math><br>
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		+	<math>(w_1 - \lambda)(w_2 - \lambda) - v^2=0</math><br>

← Older revision		Revision as of 03:37, 19 August 2007
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	a.) <math>H=\begin{pmatrix}w_1&v\\v&w_2\end{pmatrix}</math><br>		a.) <math>H=\begin{pmatrix}w_1&v\\v&w_2\end{pmatrix}</math><br>

−	<math>H=\begin{pmatrix}(w_1-\lambda)&v\\v&w_2\end{pmatrix}</math><br>	+	<math>H=\begin{pmatrix}(w_1-\lambda)&v\\v&(w_2-\lambda)\end{pmatrix}</math><br>

← Older revision		Revision as of 03:37, 19 August 2007
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−	a.) <math>H=\begin{pmatrix}w_1&v\\v&w_2\end{pmatrix}</math>	+	a.) <math>H=\begin{pmatrix}w_1&v\\v&w_2\end{pmatrix}</math><br>
		+
		+	<math>H=\begin{pmatrix}(w_1-\lambda)&v\\v&w_2\end{pmatrix}</math><br>

← Older revision		Revision as of 03:36, 19 August 2007
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−	a.) <math>H=\begin{pmatrix}w_1&v\\v&w_2</math>	+	a.) <math>H=\begin{pmatrix}w_1&v\\v&w_2\end{pmatrix}</math>

← Older revision		Revision as of 03:36, 19 August 2007
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	* <math>E = \frac{\pi^2 \hbar^2 n^2}{2M^2 a^2}</math>, where <math>n^2=n_x ^2 + n_y ^2 + n_z ^2</math>, n=1,2,3...		* <math>E = \frac{\pi^2 \hbar^2 n^2}{2M^2 a^2}</math>, where <math>n^2=n_x ^2 + n_y ^2 + n_z ^2</math>, n=1,2,3...
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		+
		+

	2.)Solution:<br>		2.)Solution:<br>
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		+
		+	a.) <math>H=\begin{pmatrix}w_1&v\\v&w_2</math>

← Older revision		Revision as of 03:30, 19 August 2007
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	* <math>E = \frac{\pi^2 \hbar^2 n^2}{2M^2 a^2}</math>, where <math>n^2=n_x ^2 + n_y ^2 + n_z ^2</math>, n=1,2,3...		* <math>E = \frac{\pi^2 \hbar^2 n^2}{2M^2 a^2}</math>, where <math>n^2=n_x ^2 + n_y ^2 + n_z ^2</math>, n=1,2,3...
		+
		+	2.)Solution:<br>