Naive quantization of Schrödinger field

Question

I just started learning QFT and I was wondering if one is able to quantize the Schrödinger field similar to the way one is able to quantize electromagnetic or elastic mechanical wave modes. E.g. phonons are acquired by solving the classical Newton's equation for the crystal lattice in normal coordinates, and then each normal mode is considered as a harmonic oscillator which is able to gain or lose energy in quanta of $\hbar \omega$. Is there a similar way for the Schrödinger field? E.g. by solving the Schrödinger equation for the required potential and then considering each wave mode as a harmonic oscillator. By direct calculation one is able to see that this does not give the correct energies (e.g. this way one doesn't get the $\approx0.5~MeV$ energy of the electron), but maybe with some small correction, it could. Any ideas?

Answer 1

Indeed, the ordinary Schrödinger equation can be second-quantized, yielding non-relativistic QFT.

To do so, one first rewrites the Schrödinger equation as a "classical" field theory:

• the phase space of this field theory is the Hilbert space $\mathcal{K}$ of the first-quantized theory

• with (complex) coordinates $a_k(\psi) = \left\langle e_k \middle| \psi \right\rangle$ for some ONB $\left(e_k\right)_k$ of $\mathcal{K}$

• obeying the Poisson brackets $\left\{ a_k, a^*_l \right\} = -i \delta_kl$

• and with Hamiltonian function(al) $\mathcal{H}(\psi) = \left\langle \psi \middle| H \middle| \psi \right\rangle$, where $H$ denotes the first-quantized Hamiltonian operator (that this Hamiltonian indeed encodes the first-quantized Schrödinger equation as can be checked from $\frac{d}{dt} a_k(\psi) = \{ a_k, \mathcal{H} \}$).

Remarkably, the canonical (second-)quantization of this field theory describes precisely an arbitrary number of undistinguishable (bosonic) particles, which each obey the first-quantized theory:

• the second-quantized Hilbert space is the Fock space $\mathcal{F} = \bigoplus_{N \geq 0} \mathcal{K}^{\otimes N, \text{sym}}$, where $\mathcal{K}^{\otimes N, \text{sym}}$ is the symmetric subspace of $\mathcal{K}^{(1)} \otimes \dots \otimes \mathcal{K}^{(N)}$, which describes $N$ undistinguishable particles

• the quantization of $a_k$, resp. $a^*_l$, is the annihilation operator $\hat{a}_k$ for a particle of "type $e_k$", resp. the creation operation $\hat{a}^+_l$, whose commutator is indeed $\left[ \hat{a}_k, \hat{a}^+_l \right] = \delta_kl$

• the second-quantized Hamiltonian, obtained by quantizing $\mathcal{H}$, is $\hat{H} = \bigoplus_{N \geq 0} \sum_{n=1}^N 1^{(1)} \otimes \dots \otimes H^{(n)} \otimes \dots \otimes 1^{(N)}$ (this is easier to show if the ONB $\left(e_k\right)_k$ is chosen as the eigenbasis of the first-quantized Hamiltonian $H$, in which case $\hat{H} = \sum_k E_k \hat{a}^+_k \hat{a}_k$), meaning that each particle evolves independently, according to the first-quantized Hamiltonian H.

Of course, the Schrödinger equation is non-relativistic, so if $e_k$ is an energy eigenstate for, say, a single electron, the corresponding $E_k$ will correspond to the NR limit $E = \sqrt{m^2 c^4 + p^2 c^2} \approx m c^2 + \frac{p^2}{2m}$. Although the constant term $mc^2$ is often dropped from the 1st quantized theory, one should keep it here if we want the creation/annihilation operator to change the total energy by the correct order of magnitude.

I like this example as it cleanly demonstrates how the 2 aspects of second-quantization tie together, namely:

1. describing an arbitrary number of particles obeying this first-quantized theory;

2. taking a (first-quantized) wave-equation, and thinking of it as a "classical" field theory to be quantized again, hence the name "second-quantization": although the physical interpretation is a priori different, the mathematics of going from the 1st to the 2nd-quantized theory happen to be exactly the same as the ones to go from the classical theory to the 1st-quantized one!

In a typical introduction to QFT, which goes directly to the relativistic case and deals with the second-quantization of the Dirac/Klein-Gordon wave-equations, this remarkable match is somewhat obscured by the lack of a well-defined first-quantized theory (due to the pathologies of these relativistic wave-equations).

In addition, the above holds for any first-quantized theory, so one can see the magic happens already by working out the second-quantization of a simple spin $1/2$: there, the phase space to be (second-)quantized is just 4 dimensional (thus bypassing field theoretic subtleties arising from an infinite-dimensional phase space), with two independent complex variables, each of which can be identified with the $z = x + ip$ variable of an harmonic oscillator. For reference purposes, the more difficult case of 2nd-quantizing the Schrödinger equation of a non-relativistic particle is treated in details eg. in Schiff L.I. "Quantum mechanics" book (sections 45 & 46 of chapter XIII).