Phy5645/Problem 1D sample: Difference between revisions
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:<math>-\frac{\hbar^2}{2m}\frac{d^2\psi(x,y,z)}{dx^2}+\left(X(x)+Y(y)+Z(z)\right)\psi(x,y,z)=E\psi(x,y,z).</math> | :<math>-\frac{\hbar^2}{2m}\frac{d^2\psi(x,y,z)}{dx^2}+\left(X(x)+Y(y)+Z(z)\right)\psi(x,y,z)=E\psi(x,y,z).</math> | ||
Let us assume that <math>\psi</math> has the form, <math>\psi(x,y,z)=\Phi(x) \Delta(y) \Omega (z).</math> Then | Let us assume that <math>\psi</math> has the form, <math>\psi(x,y,z)=\Phi(x) \Delta(y) \Omega (z).\!</math> Then | ||
:<math> | :<math> | ||
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Dividing by <math>\psi(x,y,z),</math> we obtain | Dividing by <math>\psi(x,y,z),\!</math> we obtain | ||
:<math> | :<math> | ||
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</math> | </math> | ||
We may now separate the left-hand side into three parts, each depending on only one of the three coordinates <math>x,\,y,</math> and <math>z.</math> Each of these parts must be equal to a constant. Therefore, | We may now separate the left-hand side into three parts, each depending on only one of the three coordinates <math>x,\,y,</math> and <math>z.\!</math> Each of these parts must be equal to a constant. Therefore, | ||
:<math> | :<math> | ||
-\frac{\hbar^2}{2m}\frac{d^2\Phi(x)}{dx^2} + X(x)\Phi(x) = E_x\Phi(x) </math> | -\frac{\hbar^2}{2m}\frac{d^2\Phi(x)}{dx^2} + X(x)\Phi(x) = E_x\Phi(x) </math> | ||
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-\frac{\hbar^2}{2m}\frac{d^2\Omega(z)}{dz^2} + Z(z)\Omega(z) = E_z\Omega(z), </math> | -\frac{\hbar^2}{2m}\frac{d^2\Omega(z)}{dz^2} + Z(z)\Omega(z) = E_z\Omega(z), </math> | ||
where <math> E_x </math>, <math> E_y </math> and <math> E_z </math> are constants and <math> E = E_x+E_y+E_z.</math> | where <math> E_x,\!</math>, <math> E_y,\!</math> and <math> E_z\!</math> are constants and <math> E = E_x+E_y+E_z.\!</math> | ||
Hence, the three-dimensional problem has been divided into three one-dimensional problems where the total energy <math>E</math> is the sum of the energies <math> E_x </math> | Hence, the three-dimensional problem has been divided into three one-dimensional problems where the total energy <math>E</math> is the sum of the energies <math> E_x,\!</math> <math>E_y,\!</math> and <math>E_z\!</math> in each dimension. | ||
Back to [[Motion in One Dimension]] | Back to [[Motion in One Dimension]] | ||
Revision as of 16:13, 31 July 2013
(Submitted by team 1)
The Schrödinger equation takes the form,
Let us assume that has the form, Then
![{\displaystyle {\begin{aligned}-{\frac {\hbar ^{2}}{2m}}\left[{\frac {d^{2}\Phi (x)}{dx^{2}}}\Delta (y)\Omega (z)+\Phi (x){\frac {d^{2}\Delta (y)}{dy^{2}}}\Omega (z)+\Phi (x)\Delta (y){\frac {d^{2}\Omega (z)}{dz^{2}}}\right]\\+\left[X(x)+Y(y)+Z(z)\right]\Phi (x)\Delta (y)\Omega (z)&=E\Phi (x)\Delta (y)\Omega (z).\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a6ebb1eebcc6ff52d7a5c0c7d9ce717c22cecdaa)
Dividing by we obtain
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{\hbar^2}{2m} \frac{1}{\Phi(x)} \frac{d^2\Phi(x)}{dx^2} + X(x) -\frac{\hbar^2}{2m} \frac{1}{\Delta(y)} \frac{d^2\Delta(y)}{dy^2} + Y(y) -\frac{\hbar^2}{2m} \frac{1}{\Omega(z)} \frac{d^2\Omega(z)}{dz^2} + Z(z) = E }
We may now separate the left-hand side into three parts, each depending on only one of the three coordinates Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x,\,y,} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z.\!} Each of these parts must be equal to a constant. Therefore,
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{\hbar^2}{2m}\frac{d^2\Phi(x)}{dx^2} + X(x)\Phi(x) = E_x\Phi(x) }
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{\hbar^2}{2m}\frac{d^2\Delta(y)}{dy^2} + Y(y)\Delta(y) = E_y\Delta(y) }
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{\hbar^2}{2m}\frac{d^2\Omega(z)}{dz^2} + Z(z)\Omega(z) = E_z\Omega(z), }
where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_x,\!} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_y,\!} and are constants and
Hence, the three-dimensional problem has been divided into three one-dimensional problems where the total energy is the sum of the energies and in each dimension.
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