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SaFaRJaCk

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## Homework Statement

Show that the antisymmetry of the two nucleon wave function in an oscillator model implies that L + S + T = odd. Secondly would this condition change if one worked in a more general single particle model?

T = isospin

S = intrinsic spin

L = orbital angular momentum

## The Attempt at a Solution

Using the Slater determinant gives us the antisymmetric wave function for a two nucleon system such that.

[tex]\psi(1,2) = 1/\sqrt(2)*[\varphi_{j1 m1} (1)\varphi_{j2 m2} (2) - \varphi_{j2 m2} (1)\varphi_{j1 m1} (2)][/tex]

Which can be rephrased as...

[tex]\psi(1,2) = <j_1 m_1 j_2 m_2|JM>\varphi_{j1 m1} (1)\varphi_{j2 m2} (2) + <j_2 m_2 j_1 m_1|JM> \varphi_{j2 m2} (1)\varphi_{j1 m1} (2)[/tex]

Using the transformation property of the Clebsch Gordon co-efficients we can substitute the below into the above.

[tex]<j_2 m_2 j_1 m_1|JM> = (-1)^{j_1+j_2-J}<j_1 m_1 j_2 m_2|JM> [/tex]

[tex]\psi(1,2) = <j_1 m_1 j_2 m_2|JM>\varphi_{j1 m1} (1)\varphi_{j2 m2} (2) + (-1)^{j_1+j_2-J<j_1 m_1 j_2 m_2|JM>\varphi_{j2 m2} (1)\varphi_{j1 m1} (2)[/tex]

According to the clebsch gordons of the two nucleon problem we should find the following.

[tex]\psi(1,2) = 1/\sqrt(2)*[1-(-1)^{j_1+j_2-J}]\varphi_{j1 m1} (1)\varphi_{j2 m2} (2)[/tex]

Rephrasing in terms of L & S...

[tex]\psi(1,2) = 1/\sqrt(2)*[1-(-1)^{l_1+s_1+l_2+s_2-L-S}]\varphi_{j1 m1} (1)\varphi_{j2 m2} (2)[/tex]

We know for a two fermion system the spin's will each be a half thus S = 1, we also know that for a two nucleon system isospin will be a half for each nucleon therefore T = 1;

So the above equation now reads...

[tex]\psi(1,2) = 1/\sqrt(2)*[1-(-1)^{l_1+l_2-L}]\varphi_{j1 m1} (1)\varphi_{j2 m2} (2)[/tex]

Two things to note here from the above argument.

If [tex] l_1 +l_2 - L = even [/tex] then the wavefunction = 0 however if it = odd then the wave function is non-zero and therefore possible.

L + T + S = L + 2.

I would like someone to please help me figure out how L can be odd in order for L + T + S to make sense /Or possibly point out a flaw in my logic/ better way to do this problem.

This is my first time posting so I'm not sure if I've got this in the right section. Please feel free to move it.

Thanks

Darryl Fleming