Читать книгу Quantum Mechanics, Volume 3 - Claude Cohen-Tannoudji - Страница 140

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 :

(11)

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 :

(13)

We now imagine another variation for the ket:

(14)

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:

(15)

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) that is zero. As for , an integration by parts over time shows that it is the complex conjugate of , and therefore also equal to zero. The functional S is thus stationary in the vicinity of any exact solution of the Schrödinger equation.

Suppose we choose any variational family of normalized kets , but which now includes a ket for which S is stationary. A simple example is the case where is a family that contains the exact solution of the Schrödinger equation; according to what we just saw, this exact solution will make S stationary, and conversely, the ket that makes S stationary is necessarily . In this case, imposing the variation of S to be zero allows identifying, inside the family , the exact solution we are looking for. If we now change the family continuously from to , in general will no longer contain the exact solution of the Schrödinger equation. We can however follow the modifications at all times of the values of the ket . Starting from an exact solution of the equation, this ket progressively changes, but, by continuity, will stay in the vicinity of this exact solution if stays close to . This is why annulling the variation of S in the family is a way of identifying a member of that family whose evolution remains close to that of a solution of the Schrödinger equation. This is the method we will follow, using the Fock states as a particular variational family.