Sometimes, I see this called the Dirac Hamiltonian: $$\hat{H} =\vec{\mathbf{\alpha}}\cdot\hat{\vec{p}} + \mathbf{\beta}m ;$$ but, if we take the Dirac Lagrangian density, obtain the corresponding Hamiltonian density via Legendre transforming and integrate, we get --- if I'm not mistaken --- $$\hat{H} = \iiint_{\mathbb{R}^3}\hat{\bar\Psi}(x)\, (-\mathrm{i} \gamma^i \partial_i + m)\hat\Psi(x)\,\mathrm{d}^3{x}.$$
If both expressions are equal, this would mean $$ \mathbf{\beta}\hat{1} = \iiint_{\mathbb{R}^3} \hat{\bar\Psi}(x)\,\hat\Psi(x)\,\mathrm{d}^3{x},\hspace{1em}\mathbf{\alpha}^i\hat{p}_i=-\mathrm{i}\iiint_{\mathbb{R}^3}\hat{\bar\Psi}(x)\gamma^i\partial_i\hat\Psi(x)\,\mathrm{d}^3x.$$ Could this be? Or am I missing something, and these two Hamiltonians aren't the same object?