TRUE GROUND

Many (probably most) electrical circuits have one side of the power supply attached to the ground.  

These stray radio waves are sometimes called electromagnetic fields, or EMFs. (The old meaning of EMF, still used occasionally, is described on 

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Xi,, which can be varied. The quantity is then a funetion of these parameters (X, X2,. ..). For eaeh set of parameter values, the corresponding value of (Ai, /I2,. ..) is always greater than or equal to the true ground-state energy Eq. The value of X, X2,. ..) closest to Eq is obtained, therefore, by minimizing with respeet to eaeh of these parameters. Selecting a sufficiently large number of parameters in a well-chosen analytieal form for the trial funetion 0 yields an approximation very close to Eq. 

Stated in still other words this means that for any trial density p(r) - which satisfies the necessary boundary conditions such as p( ) - 0, J p( ) dr = N, and which is associated with some external potential Vext - the energy obtained from the functional given in equation (4-6) represents an upper bound to the true ground state energy E0. E0 results if and only if the exact ground state density is inserted into equation (4-8). The proof of the inequality (4-11) is simple since it makes use of the variational principle established for wave functions as detailed in Chapter 1. We recall that any trial density p(r) defines its own Hamiltonian H and hence its own wave function. This wave function can now be taken as the trial wave function for the Hamiltonian generated from the true external potential Vext. Thus, we arrive at... 

Let us summarize what we have shown so far. First, all properties of a system defined by an external potential Vext are determined by the ground state density. In particular the ground state energy associated with a density p is available through the functional J P(f )VNedr + I)ikIp] Second, this functional attains its minimum value with respect to all allowed densities if and only if the input density is the true ground state density, i. e for p(r) = p0(r). Of course, the applicability of this variational recipe is limited to the ground... 

In practice, using trial wave functions with one or more variable parameters, the true ground state may be approached by minimizing the expectation value of the energy with respect to these parameters. 

The overlap of ip with the true ground state eigenfunction ipo is greater than or equal to 1 — e that is, the spatial distribution of the trial wave function is a very good approximation to the true wave function, and... 

We shall now assume that an adequate approximation to the true ground state 0O> is given by the BO ground state, - The ground state energy, Fo, is taken to be zero. Thus,... 

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