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If I have a state of two equal fermions: [tex]b^\dagger(p_1)b^\dagger(p_2)|0>[/tex] I can construct the general state of two fermions:

[tex]\int d^3p_1 d^3p_2f(p_1,p_2)b^\dagger(p_1)b^\dagger(p_2)|0>[/tex]

where f is the wave function. Now because [tex]\{b^\dagger(p_1),b^\dagger(p_2)\}=0[/tex]

the wave function f mast be antisymmetric.

The question is: if I now consider two different fermions: [tex]b^\dagger(p_1)d^\dagger(p_1)|0>[/tex]

so that the general state is

[tex]\int d^3p_1 d^3p_2f(p_1,p_2)b^\dagger(p_1)d^\dagger(p_2)|0>[/tex]

because

[tex]\{b^\dagger(p_1),d^\dagger(p_2)\}=0[/tex]

remains true, does it mean the wave function of any two fermions will be antisymmetric? I thought it was true only for two identical particles!

Thank you for the answers! :)