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N, where α is an internal parameter, such as spin for electron, for energy level of two-level atoms, or for a system and an environment which are accessible to a quasiparticle. The total wavefunction is written as,   ψ1 (x) Ψ (x) =  ....  . ψn (x) On the other hand, state-tensor or direct-product wavefunction is often associated with many-body problems, such as a combined system of spin- 12 electron and electromagnetic-field photons or lattice vibrational-mode phonons, where the wavefunction for each part must be specified simultaneously.

This can be illustrated by simulating the measurement in the limit of zero error, as in classical mechanics, using the position operator Q whose eigenvalue is q and abstract eigenket |q . In this limiting case, the corresponding eigenstate in Hilbert space is the Dirac-delta function in the coordinate representation given by |x, q = x |q = δ (x − q) centered at position q. Thus, we Q |x, q = q |x, q . Since the set of position eigenfunctions, {|q }, forms a complete basis states, we can expand the momentum eigenfunction in terms of the position eigenfunctions of all q values.

R. The position of the κth atom in the lth cell is Xl,κ = Xl + xκ . A basic assumption which we shall make meantime is that the change in potential energy when the atoms are given small displacements u(l, κ) is ∆Φ(2) = 1 2 uα (l, κ)Φll ,κκ ,αβ uβ (l , κ ), ll ,κκ ,αβ where α, β, denote components in a Cartesian coordinate system. We have Φll ,κκ ,αβ = ∂2Φ ∂uα (l, κ)∂uβ (l , κ ) which is the analogue of the spring constant. 5in Lattice Vibrations in Crystalline Solids: Phonons swp0002 37 displaced in the β-direction an amount uβ (l , κ ).