- #1

H Psi equal E Psi

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Hi guys!

I'm struggling with the following problem:

Consider two distinguishable (not interacting) particles in a quadratic 2 dimensional potential well. So

##

V(x,y)=\left\{\begin{matrix}

0,\quad\quad-\frac { L }{ 2 } \le \quad x\quad \le \quad \frac { L }{ 2 } \quad and\quad -\frac { L }{ 2 } \le \quad y\quad \le \quad \frac { L }{ 2 } \\ \infty ,\quad \quad \quad rest

\end{matrix}\right.

##

I am now asked to find the normalized wave function in the ground state for two particles within the given potential. I tried to solve the schroedinger equation by means of the method of separation of variables:

##\psi ({ x }_{ 1 },{ x }_{ 2 },{ y }_{ 1 }{ y }_{ 2 })=\alpha ({ x }_{ 1 })\beta ({ x }_{ 2 })\delta ({ y }_{ 1 })\varepsilon ({ y }_{ 2 })##

This was harder then i thought so i didn't quiet got an sensible answer...

The second part of the exercise is to replace the two distinguishable particle with two spin ½ (Not interacting) electrons. Now pauli's principle has to be taken into account. Since i didn't manged to find radial function I am not able to construct the anti symmetric wave function for the electrons:

##\psi ({ x }_{ 1,2 },{ y }_{ 1,2 })=\phi ({ x }_{ 1,2 },{ y }_{ 1,2 })\cdot \frac { 1 }{ \sqrt { 2 } } ((\left| \left \uparrow \downarrow \right> \right) -(\left| \left \downarrow \uparrow \right> \right))##

While ##\phi (x,y)## is the searched function (has to be symmetric).

Sorry for my English ( not my mother tongue )

Thanks and Cheers!

I'm struggling with the following problem:

Consider two distinguishable (not interacting) particles in a quadratic 2 dimensional potential well. So

##

V(x,y)=\left\{\begin{matrix}

0,\quad\quad-\frac { L }{ 2 } \le \quad x\quad \le \quad \frac { L }{ 2 } \quad and\quad -\frac { L }{ 2 } \le \quad y\quad \le \quad \frac { L }{ 2 } \\ \infty ,\quad \quad \quad rest

\end{matrix}\right.

##

I am now asked to find the normalized wave function in the ground state for two particles within the given potential. I tried to solve the schroedinger equation by means of the method of separation of variables:

##\psi ({ x }_{ 1 },{ x }_{ 2 },{ y }_{ 1 }{ y }_{ 2 })=\alpha ({ x }_{ 1 })\beta ({ x }_{ 2 })\delta ({ y }_{ 1 })\varepsilon ({ y }_{ 2 })##

This was harder then i thought so i didn't quiet got an sensible answer...

The second part of the exercise is to replace the two distinguishable particle with two spin ½ (Not interacting) electrons. Now pauli's principle has to be taken into account. Since i didn't manged to find radial function I am not able to construct the anti symmetric wave function for the electrons:

##\psi ({ x }_{ 1,2 },{ y }_{ 1,2 })=\phi ({ x }_{ 1,2 },{ y }_{ 1,2 })\cdot \frac { 1 }{ \sqrt { 2 } } ((\left| \left \uparrow \downarrow \right> \right) -(\left| \left \downarrow \uparrow \right> \right))##

While ##\phi (x,y)## is the searched function (has to be symmetric).

Sorry for my English ( not my mother tongue )

Thanks and Cheers!

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