For simplicity, suppose the particles are bosons, and let \[\hat{\psi}(\mathbf{r}) = \sum_\mu \varphi_\mu(\mathbf{r}) \, \hat{a}_\mu, \quad\;\; \hat{\psi}^\dagger(\mathbf{r}) = \sum_\mu \varphi_\mu^*(...For simplicity, suppose the particles are bosons, and let ˆψ(r)=∑μφμ(r)ˆaμ,ˆψ†(r)=∑μφ∗μ(r)ˆa†μ. Using the aforementioned wavefunction properties, we can derive the inverse relations \[\hat{a}_\mu = \int d^dr \; \varphi_\mu^*(\mathbf{r}) \, \hat{\psi}(\mathbf{r}), \quad\;\; \hat{a}_\mu^\dagger = \int d^dr \; \varphi_\mu(\mathbf{r}) \, \hat{\psi}^\dagger(\m…
We have surreptitiously introduced the basic elements of non-relativistic quantum field theory. Consider again the Heisenberg model, where we described a lattice of spins with nearestneighbour interac...We have surreptitiously introduced the basic elements of non-relativistic quantum field theory. Consider again the Heisenberg model, where we described a lattice of spins with nearestneighbour interactions. If we take the limit of the lattice constant a→0 we end up with a continuum of creation and annihilation operators for each point in space. This is a field.
