A system of three identical distinguishable particles has energy 3. The single particle can take discrete energies 0, &, 2, 3ɛ and so on. The average number of particles in the energy state & is 1.2 0.9 0.6 0.3
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- How to solve this questionA potential function is shown in the following with incident particles coming from -0 with a total energy E>V2. The constants k are defined as k₁ = 2mE h? h? k₂ = √√2m (E - V₁) h² k3 = √√2m (E - V₂) Assume a special case for which k₂a = 2nπ, n = 1, 2, 3,.... Derive the expression, in terms of the constants, k₁, k2, and k3, for the transmission coefficient. The transmis- sion coefficient is defined as the ratio of the flux of particles in region III to the inci- dent flux in region I. Incident particles E>V₂ I V₁ II V2 III x = 0 x = a3. The classical partition function of a gas of noninteracting indistinguishable particles is written as exp{- N! 2m Z= where N is the number of particles of mass m, r, and p, are the position and the momentum of the ith particle, B = 1/(kpT), and Tis the temperature of the gas. The volume of the gas is V. (a) Find the analytic expression of the partition function of the gas. (b) Obtain the total mean energy E of the gas from the partition function. (c) Obtain the entropy S of the gas from the partition function and the total mean energy. Lexp(-x³xdx = Va Hint:
- One-dimensional harmonic oscillators in equilibrium with a heat bath (a) Calculate the specific heat of the one-dimensional harmonic oscillator as a function of temperature (b) Plot the T -dependence of the mean energy per particle E/N and the specific heat c. Show that E/N → kT at high temperatures for which kT > hw In this limit the energy kT is large in comparison to hw , the separation between energy levels. Hint: expand the exponential function 1 ē = ħw + eBhwFour different particles are trapped in one dimensional wells with infinite potential at the walls and zero potential inside. The masses of the particles and their energy level, n , are given by: 1. mass=m, n=1 2. mass=2m, n=2 3. mass =3m, n=3 4. mass=4m, n=2 All these wells have the same length L. Rank the kinetic energy of the particles in order of size, smallest to largest. Group of answer choices E3 <E1<E2<E4 E1 = E4 < E2 < E3 E4 <E3 <E2 = E1 E1 < E2 = E4 < E3 E1 < E2 = E3 <E424. Consider a modified box potential with V(x) = V₁x, Vi(ar), x a Use the orthogonal trial function = c₁f₁+c₂f₂ with f₁ = √√sin (H) and f2 = √√ √√sin sin (2) to determine the upper bound to ground state energy.