Static Linear Response of the Schrodinger Equation

We now consider the effect of small changes in the external field on the expectation values of physical observables. This is exactly what is studied in most experimental situations where one switches on and off an external field and studies how the system reacts to this. We will here study a more specific case in which we look at static changes in the external potential and their accompanying changes in the ground state expectation values. By investigating this problem we will learn how to take 

Suppose that we have solved the following ground state problem  

Let us now calculate the functional derivative 8(9/8v at a given potential v. According to our definition in the previous section we have to calculate the quantity 

To evaluate this limit we have to calculate 0[v + e 8v] which we will do using static perturbation theory. We therefore make a slight change 

We will solve this equation to first order in e with the condition I P(O)) = I XT (]). We note that the solution of equation (75) is not unique, because if l e)) is a solution then also k (e)) = eme] lf/(e)) is a solution, where 0(e) is an arbitrary function of e. If we choose 0(0) = 0 then l ( e)) also satisfies the condition I 3A0)) = I The arbitrariness of the phase factor obviously does not affect the value of any expectation value, i.e., 

---