I've got several questions regarding the so called second quantization of the Schrödinger equation.
My professor introduced the field operators for the Schrödinger field by simply stating them as follows:
$$
\hat\psi (\vec{r},\xi)=\sum\limits _i \psi_i(\vec{r},\xi) \hat a_i\\
\hat\psi^\dagger (\vec{r},\xi)=\sum\limits _i \psi_i^\star (\vec{r},\xi)\hat a^\dagger_i
$$
Where $\psi_i(\vec{r},\xi)$ are the time independent one particle wave functions and $\hat a_i,\, \hat a^\dagger_i$ the corresponding creation and annihilation operators.
Is there a way to explain, why one does this? If I understood correctly what I've been taught so far, in QFT one must find some way to quantize the fields obeying the field equation in question. I do, however, not quite understand why in this particular case it is done like this.
Shouldn't the $\psi_i(\vec{r},\xi)$ be the time dependent one particle wave functions? Because I thought the field operators for a system in a box look like this:
$$
\hat\psi (\vec{r},\xi)\sim\int\text{d}^3k\ \exp(i\omega_k t-i\vec k\vec x) \hat a_k
$$
My professor then proceeded to prove the (anti)commutation relations between the field operators, postulating the corresponding relations between the fermionic or bosonic creation and annihilation operators:
$$
\left[\hat\psi (\vec{r},\xi);\hat\psi^\dagger (\vec{r}',\xi')\right]_\pm=
\left[\sum\limits_i\psi_i (\vec{r},\xi)\hat a_i;\sum\limits_j\psi^\star_j (\vec{r}',\xi')\hat a^\dagger_j\right]_\pm
=\sum\limits_i\psi_i (\vec{r},\xi)\psi^\star_i (\vec{r}',\xi')\\
=\delta(\vec{r}-\vec{r}')\delta _{\xi,\xi'}
$$
Here I do not understand the last step. Is that conclusion possible? And shouldn't or couldn't one postulate the commutation relations between the field operators and arrive at the relations for the creation and annihilation operators?
Best Answer
Is this not simply the closure relation $\sum_{i}\psi _{i}(\mathbf{r},\xi )\psi _{i}^{\ast }(\mathbf{r}\prime ,\xi \prime )=<\mathbf{r,\xi }|{\{}\sum_{i}|\psi _{i}><\psi _{i}|{\}}|% \mathbf{r}\prime ,\xi \prime >=<\mathbf{r,\xi }|\mathbf{r% }\prime ,\xi \prime >$?