# Superconducting order parameter seems to depend on the choice of the ground state wave function!

As it is described in this question the BCS wave function has definite phase but indefinite electron number. Therefore the superconducting order parameter $$\langle \phi| c_{k \uparrow}c_{-k \downarrow} |\phi\rangle$$ can be nonzero. However one can construct a wave function with definite particle number. In that case the order parameter $$\langle N|c_{k \uparrow}c_{-k \downarrow} |N\rangle$$ will be zero!

Why does the order parameter depend on our choice of the wave function?

• The value of order parameters always depend on the choice of the wavefunction. Why would you think this is an issue? – Norbert Schuch Mar 30 at 7:36
• Compared to e.g ferromagnet I assume the order parameter should have a definite value! – richard Mar 30 at 7:51
• For the Ising ferromagnet you can also write states with zero magnetization. – Norbert Schuch Mar 30 at 13:43
• @NorbertSchuch Yes but for $T<T_c$ magnetization (order parameter) must be nonzero, right? – richard Mar 30 at 15:12
• No, there can are always (well, almost) states with zero order parameter, take e.g. |all up>+|all down>. The point is that (i) random states have a non-zero order parameter, and more importantly (ii) there is long-range order (long-range correlations), which are independent of the ground state chosen, and which are closely linked to the maximum value of the order parameter. – Norbert Schuch Mar 30 at 23:12