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## Main Question or Discussion Point

**Quantum Field Theory -- variational principle**

In non-relativistic quantum mechanics, the ground state energy (and wavefunction) can be found via the variational principle, where you take a function of the n particle positions and try to minimize the expectation value of that function with the hamiltonian.

In relativistic quantum mechanics (but fixed particle number, like the dirac equation), the same can be said.

Is there something equivalent in quantum field theory?

What exactly would I be varying as there doesn't really seem to be a wavefunction anymore? Could one write the state as a sum of a wavefunction in front of each possible number of particles?

Something like:

[tex] \Psi = \left(

\phi(\mathbf{r}_1)a^{\dagger}(\mathbf{r}_1) +

\phi(\mathbf{r}_1,\mathbf{r}_2)a^{\dagger}(\mathbf{r}_1)a^{\dagger}(\mathbf{r}_2) +

\phi(\mathbf{r}_1,\mathbf{r}_2,\mathbf{r}_3)a^{\dagger}(\mathbf{r}_1)a^{\dagger}(\mathbf{r}_2)a^{\dagger}(\mathbf{r}_3) + ...

\right) |0\rangle [/tex]

Is there an exact solution to the hydrogen atom in quantum field theory? Or do they use the relativistic quantum mechanics solutions are a starting point and do perturbations around that?