Three equal masses lie on a line between two fixed walls

PHYSICS · Multi-part · Fermi University Physics · Mechanics & Dynamical Systems · Condensed Matter, Materials & Soft Matter · Lattice dynamics and thermal properties of solids

Solid State Physics, Ch Phonons: quantisation, occupation and inelastic scattering

Problem

Three equal masses

m

lie on a line between two fixed walls, joined wall to mass to mass to mass to wall by four identical springs of constant

\beta

, and move only along that line with displacements

x_1

x_2

x_3

from equilibrium. ((a)) Write the potential energy

V

in terms of

x_1

x_2

and

x_3

, and show that it contains cross terms such as

x_1x_2

, so the three masses are coupled. ((b)) Define

Q_1=\tfrac12(x_1+\sqrt2\,x_2+x_3)

Q_2=(x_1-x_3)/\sqrt2

and

Q_3=\tfrac12(x_1-\sqrt2\,x_2+x_3)

. Show that the three coefficient patterns are orthonormal, so that the kinetic energy becomes

\tfrac12m(\dot Q_1^2+\dot Q_2^2+\dot Q_3^2)

, and that

V

becomes

\tfrac12\sum_jK_jQ_j^2

with no cross terms. Find

K_1

K_2

and

K_3

in terms of

\beta

. ((c)) For

m=1.0\,\mathrm{kg}

and

\beta=4.0\,\mathrm{N\,m^{-1}}

, find the three normal-mode angular frequencies.

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