# Is the field wavefunction sum or product of the wavefunctions of individual modes?

I understand that each mode has its own wavefunction. I would like to know, how do I calculate the wavefunction of a quantum field? Should I multiply or sum the wavefunctions of the individual modes?

Not sure what you mean by "wavefunction of a quantum field". Quantum fields involve an infinity of oscillator operators acting on the Fock vacuum. If you (over-)simplified the field as a 2 mode system, with subscripts 1 and 2, $$\phi = c_1 a^\dagger _1 + c_1^* a _1 + c_2 a^\dagger _2 + c_2^* a _2,$$ then the implied tensor-product structures would generate product Fock states $$a_1^{\dagger k} a_2^{\dagger j} |0,0\rangle,$$ where I denote the empty vacuum with the vanishing occupation number of each mode, so a symmetrize tensor product of 1-2-states. If you absolutely had to consider space wave functions, you'd have a conventional product of wavefunctions, $$\langle x_1| \langle x_2|n_1,n_2\rangle = \psi (x_1) \psi(x_2),$$ (real) Hermite functions whose arguments $$x_1$$ and $$x_2$$ have absolutely nothing to do with our spacetime, and merely keep track of the decoupled mode indices in notional spaces.