Quantization of elastic waves. Let N particles of mass M be connected by a spring of force constant C and length a. To fix the boundary conditions, let the particles form a circular ring. Consider that the transverse displacements of the particles are out of plane of the ring. The displacement of the particle s is q, and its momentum is pç. Given that the Hamiltonian of the system is 72 1 H = Σ{2MP² + 2 C(qs+1 − 9₁)²} s=1 Show that the energy eigenvalues are En = (n + 2) hw, where n = 0,1,2,3,... Hint: You may use the transformation and the corresponding inverse transformation of the particle's coordinates a. and momentum n. to the phonon coordinates 0₁ and
Quantization of elastic waves. Let N particles of mass M be connected by a spring of force constant C and length a. To fix the boundary conditions, let the particles form a circular ring. Consider that the transverse displacements of the particles are out of plane of the ring. The displacement of the particle s is q, and its momentum is pç. Given that the Hamiltonian of the system is 72 1 H = Σ{2MP² + 2 C(qs+1 − 9₁)²} s=1 Show that the energy eigenvalues are En = (n + 2) hw, where n = 0,1,2,3,... Hint: You may use the transformation and the corresponding inverse transformation of the particle's coordinates a. and momentum n. to the phonon coordinates 0₁ and
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Quantization of elastic waves. Let N particles of mass M be connected by a spring of
force constant C and length a. To fix the boundary conditions, let the particles form a
circular ring. Consider that the transverse displacements of the particles are out of
plane of the ring. The displacement of the particle sis q, and its momentum is ps. Given
that the Hamiltonian of the system is
n
H = Σ{2MP² + 2 C(9s +1 - 95) ²}
s=1
Show that the energy eigenvalues are
€n = (n + 2²2 ) hw
where n = 0,1,2,3,...
Hint: You may use the transformation and the corresponding inverse transformation
of the particle's coordinates q, and momentum p, to the phonon coordinates Qk and
momentum Pk. That is,
= N- Σ Q₁ exp(iksa), QK = N-²9, exp(-iksa)
9s =
9s
Ps = Nz ΣP₁ exp(-iksa),
S
P₁ = N-Ps exp(iksa)
S
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