Trying to derive two functions which are eigenfunctions of the hamiltonian of 2 identical and indistinguisable particles and also eigenfunctions of the 2-particle exchange operator P.(adsbygoogle = window.adsbygoogle || []).push({});

Need some help with my workings I think.

Have particle '1' and particle '2' in a hamiltonian given as:

[tex] \hat{H} = \frac{p_1^2}{2m} + \frac{p_2^2}{2m} +V(x_1,x_2) [/tex]

And for particles 1,2 to be indistinguisable, H must be symmetric w.r.t particle interaction, so

[tex]\hat{H}(1,2)=\hat{H}(2,1)[/tex]

So the time independent S.E can be:

[tex]\hat{H}(1,2)\psi(1,2)=E\psi(1,2) \\

and\\

\hat{H}(2,1)\psi(2,1)=E\psi(2,1)[/tex]

[itex]\psi(2,1)[/itex] is also an eigenfunction of H(1,2) belonging to the same eigenvalue E.

And for a linear hermitian exchange operator P:

[tex]\hat{P}f(1,2)=f(2,1)[/tex]

Therefore, can write:

[tex]\hat{P}[\hat{H}(1,2)\psi(1,2)]=\hat{H}(2,1)\psi(2,1)=\hat{H}(1,2)\psi(2,1) = \hat{H}(1,2)\psi(1,2)[/tex]

Not sure where to go from here....??

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# 2 particle exchange operator P

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