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6. Consider an ensemble of N particles, where each particle can be treated as a 3-dimensional isotropic simple harmonic oscillator. The Hamiltonian for this system is given by $$H = \sum_{i=1}^{3N} \frac{1}{2m}p_i^2 + \frac{m\omega^2}{2}q_i^2$$ (a) Show that the Liouville's theorem holds for this system. (b) How does the phase-space volume for the system evolves in time? (c) If the initial distribution of $q_i$ and $p_i$'s for each particle is normal, i.e., $$\rho(q_i, p_i, t = 0) = \frac{1}{2\pi} \exp \left[ -\frac{1}{2} (p_i^2 + m\omega^2 q_i^2) \right],$$ what will be the distribution $\rho(q_i, p_i, t)$ at time t?

          6. Consider an ensemble of N particles, where each particle can be treated as a
3-dimensional isotropic simple harmonic oscillator. The Hamiltonian for this
system is given by
$$H = \sum_{i=1}^{3N} \frac{1}{2m}p_i^2 + \frac{m\omega^2}{2}q_i^2$$
(a) Show that the Liouville's theorem holds for this system.
(b) How does the phase-space volume for the system evolves in time?
(c) If the initial distribution of $q_i$ and $p_i$'s for each particle is normal, i.e.,
$$\rho(q_i, p_i, t = 0) = \frac{1}{2\pi} \exp \left[ -\frac{1}{2} (p_i^2 + m\omega^2 q_i^2) \right],$$
what will be the distribution $\rho(q_i, p_i, t)$ at time t?

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6. Consider an ensemble of N particles, where each particle can be treated as a
3-dimensional isotropic simple harmonic oscillator. The Hamiltonian for this
system is given by

H = ∑i=1^3N(1)/(2m)pi^2 + (mω^2)/(2)qi^2

(a) Show that the Liouville's theorem holds for this system.
(b) How does the phase-space volume for the system evolves in time?
(c) If the initial distribution of qi and pi's for each particle is normal, i.e.,

ρ(qi, pi, t = 0) = (1)/(2π)[ -(1)/(2) (pi^2 + mω^2 qi^2) ],

what will be the distribution ρ(qi, pi, t) at time t?

Added by Virginia V.

University Physics with Modern Physics

Hugh D. Young 14th Edition

Instant Answer

Solved by Expert Sri K

05/13/2022

Step 1

In other words, the volume in phase space occupied by a set of trajectories remains constant as the system evolves in time. Show more…

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Consider an ensemble of N particles, where each particle can be treated as a 3-dimensional isotropic simple harmonic oscillator. The Hamiltonian for this system is given by H=sum_(i=1)^(3N) (1)/(2m)p_(i)^(2)+(momega ^(2))/(2)q_(i)^(2) (a) Show that the Liouville's theorem holds for this system. (b) How does the phase-space volume for the system evolves in time? (c) If the initial distribution of q_(i) and p_(i) 's for each particle is normal, i.e., ho (q_(i),p_(i),t=0)=(1)/(2pi )exp[-(1)/(2)(p_(i)^(2)+momega ^(2)q_(i)^(2))], what will be the distribution ho (q_(i),p_(i),t) at time t ? 6. Consider an ensemble of N particles, where each particle can be treated as a 3-dimensional isotropic simple harmonic oscillator. The Hamiltonian for this system is given by 3N 1 mw * H= (a) Show that the Liouville's theorem holds for this system. (b) How does the phase-space volume for the system evolves in time? (c) If the initial distribution of qi and pi's for each particle is normal, i.e., o(qi,Pit=0) what will be the distribution p(qi, Pi,t) at time t?

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