Example of a variational calculation

We will demonstrate the variational derivation of single-particle states in the case of the Hartree approximation, where the energy is given by Eq. (2.10), starting with the many-body wavefunction of Eq. (2.9). We assume that this state is a stationary state of the system, so that any variation in the wavefunction will give a zero variation in the energy (this is equivalent to the statement that the derivative of a function at an extremum is zero). We can take the variation in the wavefunction to be of the form I, subject to the constraint that ( / = 1, which can be taken into account by 

Notice that the variations of the bra and the ket of (pi are considered to be independent of each other this is allowed because the wavefunctions are complex quantities, so varying the bra and the ket independently is equivalent to varying the real and 

Since this has to be true for any variation we conclude that 

---