THE SCIENCE AND APPLICATIONS OF ACOUSTICS - H. H. Arnold ...

456 16. Ultrasonicscomplete as a linear combination of the normal modes, i.e.,ξ j = ∑ k(e ik( ja) X k + e −ik( ja) X −k )e iω kt(16.17)The normal modes are orthogonal, so the normalization constants can be obtainedfor any boundary conditions, whether they be clamped, periodic, and so on. Thetotal energy E of the system can be found fromE = m ∑ k|X k | 2 ω 2 kCoupled Quantum ParticlesIn dealing with harmonically coupled particles it is more expeditious to use aLagrangian formulation rather than Newtonian formation of Equation (16.15).The Lagragian for such a set of connected particles isL = 1 ∑( dξ jm j2 dtj) 2− 1 ∑s j (ξ j+1 − ξ j ) 22With the canonical momenta p j = m j (dξ j /dt), the Hamiltonian becomesH = ∑ ( ) dξ jp j − LdtjThe system is quantitized with the commutation relations[ ]ξ j , p j ′ = i¯hδ j, j; (16.18)jwhere δ j, j; represents the Dirac delta function that equals unity when j =j ′ andzero when j ≠ j ′ . We then assume a periodic system and set m j = m and s = s j ,and also make use of Equation (16.17) without the coefficient e −iωkt . We obtainthe HamiltonianH = 1 ∑P k P −k + 1 2m2 m ∑ ωk 2 X k X −kkwhereP k = m dX ( )−kka, ω k = ω 0 sindt2The above Hamiltonian could be used to construct a Schrödinger wave equationfor a field ψ(X), where X represents a point in the 2N-dimensional X k space and( ) ∂ψP k =−i¯h∂ X kAnother approach is to construct the properties of the eigenfuctions and eigenvaluesthrough the use of the commutation relations of Equation (16.18) for x j and p j .