Hello everyone
I am an undergraduate physics student looking to become an experimental physicist, and I am looking for some advice as to how to do well in the laboratory.
In a week I will begin a course called "Advanced experimental physics", where my group and I will have to do complex experimental physics (at least compared to the usual basic things done in the undergraduate levels). We will be monitored by my university's top researchers in the field, so we would like to make a good impression.

I am trying to find the expression which is similar to the hamiltonian but it seems that I am missing something here,

$\hat{\phi}(x) = \int \frac{d^3k}{(2 \pi)^3} \, \frac{1}{ \sqrt{2 E_{k}}} \left( \hat{a}_{k} + \hat{a}^{\dagger}_{-k} \right) e^{i k \cdot x}$
$\hat{\pi}(x) = \int \frac{d^3k}{(2 \pi)^3} \, \sqrt{\frac{E_{k}}{2}} \left( \hat{a}_{k} - \hat{a}^{\dagger}_{-k} \right) e^{i k \cdot x}$

$\hat{P}^{i} = -\int d^3x \, \hat{\pi}(x) \, \partial_{i} \hat{\phi}(x)$

what I got is
$P = \frac{1}{2} \int \frac{d^3p}{(2\pi)^3} \, p^i \left( \hat{a}_{p}^{\dagger} \hat{a}_{p} + \hat{a}_{p} …

@Student404Mus In $\frac{1}{2} \int \frac{d^3p}{(2\pi)^3} \, p^i \left( \hat{a}_{p}^{\dagger} \hat{a}_{p} + \hat{a}_{p} \hat{a}_{-p} - \hat{a}_{-p}^{\dagger} \hat{a}_{-p} - \hat{a}_{-p}^{\dagger} \hat{a}_{p}^{\dagger} + [\hat{a}_{p},\hat{a}_{p}^{\dagger}] \right) $ for the first and third term you have
$$
\frac{1}{2} \int \frac{d^3p}{(2\pi)^3} \, \left( p^i \hat{a}_{p}^{\dagger} \hat{a}_{p} - p^i \hat{a}_{-p}^{\dagger} \hat{a}_{-p} \right) = \frac{1}{2} \int \frac{d^3p}{(2\pi)^3} \, \left( p^i \hat{a}_{p}^{\dagger} \hat{a}_{p} + p^i \hat{a}_{p}^{\dagger} \hat{a}_{p} \right) = \int \fra…

Using the normalization definition of the momentum state
$$
|\mathbf{p}\rangle=\sqrt{2 E_{\mathbf{p}}} a_{\mathbf{p}}^{\dagger}|0\rangle
$$

since (before normalization)

$$
\langle\mathbf{p} | \mathbf{q}\rangle=(2 \pi)^{3} \delta^{(3)}(\mathbf{p}-\mathbf{q})
$$

it follows that

$$ \langle\mathbf{p} | \mathbf{q}\rangle= 2 E_{\mathbf{p}}(2 \pi)^{3} \delta^{(3)}(\mathbf{p}-\mathbf{q}) $$

How someone could prove the result

$$ (1)_{1-\text { particle }}=\int \frac{d^{3} p}{(2 \pi)^{3}}|\mathbf{p}\rangle \frac{1}{2 E_{\mathbf{p}}}\langle\mathbf{p}| $$