91影视

An expression has been given as $0<\mathrm{x}<1,0<\mathrm{y}<1$.

A linear partial differential equation that determines the wave function of a quantum-mechanical system is known as the Schr枚dinger equation.

The Schrodinger equation is $\mathbf{-}\frac{{\mathbf{h}}^{\mathbf{2}}}{\mathbf{2}\mathbf{m}}{\mathbf{鈭�}}^{\mathbf{2}}\mathbf{蠄}\mathbf{}\mathbf{+}\mathbf{痴蠄}\mathbf{=}\mathbf{i}\mathit{h}\frac{\mathit{鈭�}}{\mathit{鈭�}\mathit{t}}\mathit{}\mathit{蠄}$.

Use the Schrodinger equation.

$\mathbf{-}\frac{{\mathbf{h}}^{\mathbf{2}}}{\mathbf{2}\mathbf{m}}{\mathbf{鈭�}}^{\mathbf{2}}\mathbf{蠄}\mathbf{}\mathbf{+}\mathbf{痴蠄}\mathbf{=}\mathbf{i}\mathit{h}\frac{\mathit{鈭�}}{\mathit{鈭�}\mathit{t}}\mathit{}\mathit{蠄}$

Assume a solution of the form mentioned below.

$\mathrm{蠄}\left(\mathrm{x},\mathrm{y},\mathrm{t}\right)=\mathrm{蠄}(\mathrm{x},\mathrm{y})\mathrm{T}\left(\mathrm{t}\right)$

The equation can be separated into a time equation with the solution given below.

$\mathrm{T}\left(\mathrm{t}\right)={\mathrm{e}}^{-\frac{\mathrm{iEt}}{h}}$

It can be separated into a space equation (with V = 0 )

$-\frac{{\mathrm{h}}^2}{2\mathrm{m}}{鈭�}^2\mathrm{蠄}=\mathrm{贰蠄}$

Put a solution of the form given below.

$\mathrm{蠄}=\mathrm{X}\left(\mathrm{x}\right)\mathrm{Y}\left(\mathrm{y}\right)$

Then divide by$\mathrm{X}\left(\mathrm{x}\right)\mathrm{Y}\left(\mathrm{y}\right)$ .

$$\begin{gathered} \frac{1}{\mathrm{x}}\frac{d^{2}X}{d{x}^2}+\frac{1}{\mathrm{Y}}\frac{{\mathrm{d}}^2\mathrm{Y}}{{\mathrm{dy}}^2}=-\frac{2mE}{h^2}=-k_x^2-k_y^2 \\ \left(\frac{1}{x}\frac{d^{2}X}{d{x}^2}+k_x^2\right)+\left(\frac{1}{Y}\frac{{\mathrm{d}}^2}{{\mathrm{dy}}^2}+k_y^2\right)=0 \end{gathered}$$

The solutions are a trigonometric solutions.
$$\begin{gathered} \mathrm{X}\left(\mathrm{x}\right)=\left\{\begin{array}{l}\cos \left({\mathrm{k}}_{\mathrm{x}}\mathrm{x}\right)\\ \sin \left({\mathrm{k}}_{\mathrm{x}}\mathrm{x}\right)\end{array}\right. \\ \mathrm{Y}\left(\mathrm{y}\right)=\left\{\begin{array}{l}\cos \left({\mathrm{k}}_{\mathrm{y}}\mathrm{y}\right)\\ \sin \left({\mathrm{k}}_{\mathrm{y}}\mathrm{y}\right)\end{array}\right. \end{gathered}$$

On the boundary$蠄=0$ .
$$\begin{gathered} \mathrm{蠄}(0,\mathrm{y},\mathrm{t})=0 \\ \mathrm{蠄}(\mathrm{l},\mathrm{y},\mathrm{t})=0 \\ \mathrm{蠄}(\mathrm{x},0,\mathrm{t})=0 \\ \mathrm{蠄}(\mathrm{x},\mathrm{I},\mathrm{t})=0 \end{gathered}$$

$$\begin{gathered} \mathrm{x}\left(0\right)=0 \\ 鈬�0=\mathrm{Csin}\left({\mathrm{k}}_{\mathrm{x}}0\right)+\mathrm{D}\cos \left({\mathrm{k}}_{\mathrm{x}}0\right) \end{gathered}$$
$\mathrm{D}=0$
$$\begin{gathered} \mathrm{X}\left(\mathrm{I}\right)=0 \\ 鈬�0=\sin \left({\mathrm{k}}_{\mathrm{x}}\mathrm{I}\right) \end{gathered}$$
$$\begin{gathered} {\mathrm{k}}_{\mathrm{x}}\mathrm{I}=\mathrm{苍蟺} \\ {\mathrm{k}}_{\mathrm{x}}=\frac{\mathrm{苍蟺}}{\mathrm{I}} \end{gathered}$$

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in file: /var/www/html/integration/lib/com/wiris/plugin/impl/HttpImpl.class.php line 68
#0 /var/www/html/integration/lib/php/Boot.class.php(769): com_wiris_plugin_impl_HttpImpl_1(Object(com_wiris_plugin_impl_HttpImpl), NULL, 'http://www.wiri...', 'Http Error #500')
#1 /var/www/html/integration/lib/haxe/Http.class.php(532): _hx_lambda->execute('Http Error #500')
#2 /var/www/html/integration/lib/php/Boot.class.php(769): haxe_Http_5(true, Object(com_wiris_plugin_impl_HttpImpl), Object(com_wiris_plugin_impl_HttpImpl), Array, Object(haxe_io_BytesOutput), true, 'Http Error #500')
#3 /var/www/html/integration/lib/com/wiris/plugin/impl/HttpImpl.class.php(30): _hx_lambda->execute('Http Error #500')
#4 /var/www/html/integration/lib/haxe/Http.class.php(444): com_wiris_plugin_impl_HttpImpl->onError('Http Error #500')
#5 /var/www/html/integration/lib/haxe/Http.class.php(458): haxe_Http->customRequest(true, Object(haxe_io_BytesOutput), Object(sys_net_Socket), NULL)
#6 /var/www/html/integration/lib/com/wiris/plugin/impl/HttpImpl.class.php(43): haxe_Http->request(true)
#7 /var/www/html/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(268): com_wiris_plugin_impl_HttpImpl->request(true)
#8 /var/www/html/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(307): com_wiris_plugin_impl_RenderImpl->showImage('7b27ed52a9b086c...', NULL, Object(PhpParamsProvider))
#9 /var/www/html/integration/createimage.php(17): com_wiris_plugin_impl_RenderImpl->createImage('
uncaught exception: Http Error #500

in file: /var/www/html/integration/lib/com/wiris/plugin/impl/HttpImpl.class.php line 68
#0 /var/www/html/integration/lib/php/Boot.class.php(769): com_wiris_plugin_impl_HttpImpl_1(Object(com_wiris_plugin_impl_HttpImpl), NULL, 'http://www.wiri...', 'Http Error #500')
#1 /var/www/html/integration/lib/haxe/Http.class.php(532): _hx_lambda->execute('Http Error #500')
#2 /var/www/html/integration/lib/php/Boot.class.php(769): haxe_Http_5(true, Object(com_wiris_plugin_impl_HttpImpl), Object(com_wiris_plugin_impl_HttpImpl), Array, Object(haxe_io_BytesOutput), true, 'Http Error #500')
#3 /var/www/html/integration/lib/com/wiris/plugin/impl/HttpImpl.class.php(30): _hx_lambda->execute('Http Error #500')
#4 /var/www/html/integration/lib/haxe/Http.class.php(444): com_wiris_plugin_impl_HttpImpl->onError('Http Error #500')
#5 /var/www/html/integration/lib/haxe/Http.class.php(458): haxe_Http->customRequest(true, Object(haxe_io_BytesOutput), Object(sys_net_Socket), NULL)
#6 /var/www/html/integration/lib/com/wiris/plugin/impl/HttpImpl.class.php(43): haxe_Http->request(true)
#7 /var/www/html/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(268): com_wiris_plugin_impl_HttpImpl->request(true)
#8 /var/www/html/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(307): com_wiris_plugin_impl_RenderImpl->showImage('7b27ed52a9b086c...', NULL, Object(PhpParamsProvider))
#9 /var/www/html/integration/createimage.php(17): com_wiris_plugin_impl_RenderImpl->createImage('

$鈬�0=\mathrm{E}\sin \left({\mathrm{k}}_{\mathrm{y}}0\right)+\mathrm{F}\cos \left({\mathrm{k}}_{\mathrm{y}}0\right)$

$\mathrm{Y}\left(\mathrm{I}\right)=0$
$0=\sin \left({\mathrm{k}}_{\mathrm{y}}\mathrm{l}\right)$
${\mathrm{k}}_{\mathrm{y}}\mathrm{l}=\mathrm{尘蟺}$
${\mathrm{k}}_{\mathrm{y}}=\frac{\mathrm{尘蟺}}{\mathrm{l}}$

Calculate the energy.

$$\begin{gathered} {\mathrm{n}}_{\mathrm{x}}^2+{\mathrm{n}}_{\mathrm{y}}^2=\frac{2\mathrm{mE}}{{\mathrm{h}}^2} \\ ={\left(\frac{\mathrm{苍蟺}}{\mathrm{I}}\right)}^2+{\left(\frac{\mathrm{尘蟺}}{\mathrm{l}}\right)}^2 \\ =\frac{{\mathrm{蟺}}^2}{{\mathrm{l}}^2}({\mathrm{n}}^2+{\mathrm{m}}^2) \end{gathered}$$

$\mathrm{E}={\mathrm{h}}^2{\mathrm{蟺}}^2\frac{({\mathrm{n}}^2+{\mathrm{m}}^2)}{2{\mathrm{ml}}^2}$

It can be seen that the energies are a degenerate. n and m can vary to get the same value.

Write the solution.
$$\begin{gathered} {\mathrm{蠄}}_{\mathrm{N}}=\sin \left(\frac{\mathrm{苍蟺}}{\mathrm{l}}\mathrm{x}\right)\sin \left(\frac{\mathrm{尘蟺}}{\mathrm{l}}\mathrm{y}\right){\mathrm{e}}^{-\frac{{\mathrm{iE}}_{{\mathrm{N}}^{\mathrm{t}}}}{h}} \\ {\mathrm{E}}_{\mathrm{N}}=h^{\mathit{2}}{蟺}^{\mathit{2}}\frac{\mathit{(}{\mathit{n}}^{\mathit{2}}\mathit{+}{\mathit{m}}^{\mathit{2}}\mathit{)}}{\mathit{2}{\mathit{ml}}^{\mathit{2}}} \end{gathered}$$
$\mathrm{蠄}(\mathrm{x},\mathrm{y},\mathrm{t})=\underset{\mathrm{n}=1}{\overset{鈭�}{鈭�}}\underset{\mathrm{m}=1}{\overset{鈭�}{鈭�}}\sin \left(\frac{\mathrm{苍蟺}}{\mathrm{I}}\mathrm{x}\right)\sin \left(\frac{\mathrm{尘蟺}}{\mathrm{l}}\mathrm{y}\right){\mathrm{e}}^{-\mathrm{i}h{蟺}^{\mathit{2}}\frac{\mathit{(}{\mathit{n}}^{\mathit{2}}\mathit{+}{\mathit{m}}^{\mathit{2}}\mathit{)}}{\mathit{2}{\mathit{ml}}^{\mathit{2}}}\mathrm{t}}$
Hence, the final solution is$\mathrm{蠄}(\mathrm{x},\mathrm{y},\mathrm{t})=\underset{\mathrm{n}=1}{\overset{鈭�}{鈭�}}\underset{\mathrm{m}=1}{\overset{鈭�}{鈭�}}\sin \left(\frac{\mathrm{苍蟺}}{\mathrm{I}}\mathrm{x}\right)\sin \left(\frac{\mathrm{尘蟺}}{\mathrm{l}}\mathrm{y}\right){\mathrm{e}}^{-\mathrm{i}h{蟺}^{\mathit{2}}\frac{\mathit{(}{n}^{\mathit{2}}\mathit{+}{m}^{\mathit{2}}\mathit{)}}{\mathit{2}m{l}^{\mathit{2}}}\mathrm{t}}$