where we have used in the second line the fact that the norm of always remains equal to unity. This expression for S is similar to the initial form (7), but without the real part.
2-b. Stationarity
Suppose now has an arbitrary time dependence between t0 and t1, while keeping its norm constant, as imposed by (6); the functional then takes a certain value S, a priori different from zero. Let us see under which conditions S will be stationary when changes by an infinitely small amount :
For what follows, it will be convenient to assume that the variation is free; we therefore have to ensure that the norm of remains constant, equal to unity3. We introduce Lagrange multipliers (Appendix V) λ(t) to control the square of the norm at every time between t0 and t1, and we look for the stationarity of a function where the sum of constraints has been added. This sum introduces an integral, and we the function in question is:
(12)
where λ(t) is a real function of the time t.
The variation of to first order is obtained by inserting (11) in (10). It yields the sum of a first term containing the ket and of another containing the bra :
which yields a variation of ; in this second variation, the term in becomes , whereas the term in becomes . Now, if the functional is stationary in the vicinity of , the two variations and are necessarily zero, as are also and . In those combinations, only terms in appear for the first one, and in for the second; consequently they must both be zero. As a result, we can write the stationarity conditions with respect to variations of the bra and the ket separately.
Let us write for example that , which means the right-hand side of the second line in (13) must be zero. As the time evolution between t0 and t1 of the bra is arbitrary, this condition imposes this bra multiplies a zero-value ket, at all times. Consequently, the ket must obey the equation:
which is none other than the Schrödinger equation associated with the Hamiltonian H(t) + λ(t).
Actually, λ(t) simply introduces a change in the origin of the energies and this only modifies the total phase4 of the state vector , which has no physical effect. Without loss of generality, this Lagrange factor may therefore be ignored, and we can set:
(16)
A necessary condition5 for the stationarity of S is that obey the Schrödinger equation (8) – or be physically equivalent (i.e. equal to within a global time-dependent phase factor) to a solution of this equation. Conversely, assume is a solution of the Schrödinger equation, and give this ket a variation as in (11). It is then obvious from the second line of (13)
term. The final result is then:
