function of the laboratory frame ( ψ$_{0}$ ) evaluated at the co- ordinates ξ of the co-moving frame: wave function ˜ ψ is not at all close to the instantaneous ground-state of ˆ H ( t ). Rather, it is the ground-state of the “deformed hamiltonian” ˆ H$_{0}$ [ u ] which is obtained from the initial-time Hamiltonian ˆ H$_{0}$ by a coordinate trans- formation – indeed an elastic deformation. As anticipated in the foregoing discussion the elastic approximation paves the way for a relatively simple cal- culation of the complicated expression that appears in the second line of Eq. (39). Namely, thanks to the fact that ˜ ψ = ψ$_{0}$ is independent of the displacement field we can take the functional derivative of Eq. (38) after taking the average and we arrive at 6 The physical idea behind this approximation is that the time evolution of the wave function in the laboratory frame can be approximated as a continuously evolving elastic deformation of the ground-state wave function. Such a deformation affects all the particles simultane- ously and instantaneously. The burden of describing the time-evolution of the system is entirely placed on the time-dependent geometry (i.e. the time-dependent re- lation between ξ and r ), while the wave function itself remains independent of time. What is lost in this approximation is the fact that in the actual time evolution the system will undergo internal relaxation in order to optimize the correlations between the particles. In other words, the probability of finding the particles in a certain configuration r$_{1}$ , ... r$_{N}$ at time t is not strictly determined by the probability that those particles were initially in the configuration ξ$_{1}$ , ..., ξ$_{N}$ from which r$_{1}$ , ... r$_{N}$ evolve according to Eqs. (22) and (23). However, our approximation should always be valid at sufficiently high frequency, i.e., when the evolution of the geometry is very fast on the scale of the characteristic response times of the system. The equation of motion resulting from the elastic ap- proximation is also strictly valid (and therefore, not an approximation at all) for any one-particle system, be- cause in this case the wave function is completely deter- mined by the displacement field and there is no room for internal relaxation. Finally, our equation of motion is also strictly valid for non-interacting Bose systems in the ground-state (since these systems behave like a single particle), and for non-interacting Fermi systems consist- ing of at most two particles of opposite spins in the same orbital (since these behave like non-interacting Bosons). In all other cases – including the apparently simple case of a non-interacting many-fermion system – the appro- priateness of the elastic approximation must be assessed a posteriori and may depend on the objective of the cal- culation. In general, we can only say that the elastic approximation is expected to work better for collective (many-particle) excitations than for single particle ex- citations, and better for strongly correlated many-body systems (which exhibit bosonic behavior) than for weakly correlated systems. It is important to appreciate the profound difference that exists between the present approximation and an- other common approximation which also entails an in- stantaneous response to a time-dependent field: the adia- batic approximation . In the adiabatic approximation one assumes that the system remains in the instantaneous ground-state of the hamiltonian ˆ H ( t ) – an assumption that is justified only if the time evolution is slow on the scale of the characteristic response time of the system. This is exactly the opposite of the regime of validity of the present approximation. The geometrically distorted where ξ$_{i}$ = r$_{i}$ − u ( r$_{i}$ ) and the last factor on the right hand side is for normalization. This proves that E [ u ] − E [ 0 ] is a positive definite quantity since E [ 0 ] is the ground-state energy of ˆ H$_{0}$ while E [ u ] is the expectation value of ˆ H$_{0}$ in a state that is not the ground-state. of the distorted ground state. Equivalently, E [ u ] can be obtained as the expectation value of the original hamil- tonian ˆ H$_{0}$ in the distorted ground state where E$_{2}$ [ u ] is the second-order term in the expansion of the total energy Putting all together we arrive at the elegant result: is the external potential energy of the distorted ground- state. where We further observe that where ˜ ψ ( ξ$_{1}$ , ..., ξ$_{N}$ , t ) ≃ ψ$_{0}$ ( ξ$_{1}$ , ..., ξ$_{N}$ ) . (41) F µ, $_{1}$( r , t ) = − δE$_{u}$ [ u ] δu$_{µ}$ ( r ) ∣ 1 , (42) E$_{u}$ [ u ] ≡ 〈 ψ$_{0}$ | ˆ H$_{u}$ [ u ] | ψ$_{0}$ 〉 . (43) u · ∇ ∂$_{µ}$V$_{0}$ ( r ) = 1 n$_{0}$ ( r ) δV$_{0}$ [ u ] δu$_{µ}$ ( r ) ∣ 1 , (44) V$_{0}$ [ u ] ≡ ∫ d r V$_{0}$ ( r + u ( r )) n$_{0}$ ( r ) . (45) m∂ 2 t u$_{µ}$ ( r , t ) = − 1 n$_{0}$ ( r ) δE$_{2}$ [ u ] δu$_{µ}$ ( r ) − ∂$_{µ}$V$_{1}$ ( r , t ) . (46) E [ u ] ≡ E$_{u}$ [ u ] + V$_{0}$ [ u ] (47) ψ$_{0}$ [ u ]( r$_{1}$ , ..., r$_{N}$ ) = ψ$_{0}$ ( ξ$_{1}$ , ..., ξ$_{N}$ ) N ∏ i =1 g − 1 / $^{4}$( ξ$_{i}$ ) , (48)