Difference between field and wavefunction

Ignore spin, polarization, and even Lorentz issues, absorb all superfluous constants, and consider time and one space dimension.

• A one-particle complex wavefunction $psi(x,t)=langle x|psi rangle$ is a spacetime function serving as a probability density amplitude. For the quantum oscillator in coordinate space, it obeys the suitable Schroedinger equation, $(i2partial_t +partial^2_x -x^2)psi =0$. (Well, in a twisted sense, it is some sort of "probability amplitude classical field", as you suggest.)

• A real classical field $phi(x,t)$ is a spacetime function. As you review in classical mechanics texts, a free one represents a superposition of normal modes of an infinity of coupled oscillators on a line (lattice, as their spacing vanishes). Here, however, x represents the equilibrium location of each oscillator, instead, and not the displacement from equilibrium (as it did for the classical parent of the above single oscillator). The Euler-Lagrange equation it satisfies is $(partial_t^2-partial_x^2 + m^2)phi=0$. The decoupled normal modes are more apparent in the Fourier transform, $tilde{phi}(k,t)=int dx ~e^{-ikx} phi(x)$, so $tilde{phi}(k,t)=e^{itsqrt{k^2+m^2}} c_k + e^{-itsqrt{k^2+m^2}} c^*_k$, for numerical coefficients c_k.

• A quantum field $Phi(x,t)$ is a noncommutative operator, linked to the above expression, except with suitable (k-dependent) normalized creation and annihilation operators $a_k^dagger, a_k$ supplanting the above c-numbers c_k and c_k* of the classical field. (x remains a classical parameter!) These normal modes are decoupled from each other (the string Lagrangian has been diagonalized for them), and commute among themselves; they evidently obey the same linear equation as the classical field, since the operator coefficients do not affect it: the wave operators act on the c-number parameters. That is, each normal mode $tilde{phi}_k$ of the above string was “first quantized” as a classical oscillator, but this organized assembly of an infinity of identical such amounts to “second quantization”; also see WP. Since it packages an infinity of oscillators (excitations, identical particles), each configuration state vector is a particular assembly of creation operators acting on the Fock vacuum, while transition amps are vacuum expectation values of strings of such fields. In the first volume of the text by Bjorken and Drell, you may go through the solution of dynamical equations as wavefunctions, and in the second, the very same equations and solutions as fields. As long as you stick to the proper interpretation and use...

• If your life depended on it, you could construct some type of a wavefunction out of a field operator, $langle x| Phi (x',t)|0rangle$, now an almost localized c-number function, but careful treading is necessary to avoid a range of confusions.