Suppose you have a "box" in which each particle may occupy any of 10 single-particle states. For simplicity, assume that each of these states has energy zero. What is the partition function if the box contains two identical fermions? A. 144. В. 87. С. 45. D. 0.
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- Imagine a particle that can be in only two states, with energies 0.05 eV, and 0 eV. This particle is in equilibrium with a reservoir at 300 K. Calculate the partition function for this particle. A. 0.353. В. 1.145. C. 13.464. D. 212.774. O A. O B. c. C. D.Statistical Mechanics. We have a system of N bosons with zero spin. Each boson can have two states of energies 0 and E. Let μ be the chemical potential of the system and suppose that N >> 1. a) Determine the temperature T so that the mean population of the ground state is twice that of the excited state of energy E. Express kT only in terms of N and E. b) What would be the corresponding temperature T′ if the particles obeyed Boltzmann statistics? Compare both results and discuss them physically.First consider some simple electronic partition functions: a. Consider a two-level system of N particles separated by an energy of hv. i. Derive expressions for ē, E, and P, as a function of T. P, is the probability that the system is in the higher energy level. ii. What are the limiting values for each of these at T = 0 and kT » hv. iii. For a level spacing 200 cm what is T when Ē = Nhv. iv. What is P, at the T found in part iii?
- 4. Consider two indistinguishable spin-1/2 fermions in a one-dimensional infinite square well of length L. Construct the state vector and determine the energy for the ground-state of the two-particle system assuming the particles are non-interacting. i. ii. Determine the first excited state energy of the two-particle system and give all its state vectors, again assuming the particles are non-interacting. What is the degeneracy of the first excited state? ji. Suppose the particles interact via the potential V(x, x2) = k(x1 – x2)² where k is positive and small. Discuss quantitatively how the ground and first-excited state energies differ from the case for non-interacting particles. 5. A one-electron atom has atomic number Z, mass number M and a spherical nucleus of radius ry. Assume electric charge +Ze is uniformly distributed throughout the volume of the nucleus. Ignoring spin, use first order non-degenerate perturbation theory and the hydrogenic wave functions adapted to the…C. For a particle of mass 9.10938356×10-31 kg scooting back and forth on a wire of length 13×10-10 m, compute it's energy in the n = 9 state. Use: π = 3.14159265359 and h = 1.0545718×10-34 Js. Eg J. D. What is the rule for the number of nodes for a particle-in-a-box state as a function of its quantum number? Number of nodes = On ΟΙ O n-1 On--1In how many different configurations can four particles be distributed over a set of evenly spaced energy levels (AE=ɛ) such that the total energy is 4ɛ? Select one: а. 8 b. 11 С. 5 d. 3
- 1. Cannot be! Imagine a certain kind of particle, such that each single-particle state can be occupied by at most M particles, with M > 1. For M = 1 and M = ∞, we recover the usual fermion and boson cases. We will focus on the case when the particles do not interact with each other, and the single particle quantum states i have energies e¿. Assume that the system is in equilibrium at temperature T and chemical potential μ. (a) (b) (c) above. Calculate the appropriate partition function for the system in the conditions discussed Compute, as a function of temperature and chemical potential, the average occupation number (n) for state i, and find simplified expressions in the low and high temperature limits. For M1, does the system have the analog of a Fermi energy, i.e. an energy at which the occupation number is discontinuous at T = 0?7c.2. Evaluate for the total energy & at the following k-directions and evaluate at the BZ boundary. a. (1,0,0) b. (1,1,0) c. (1,1,1)2. Two particles, one of spin 2 and another of spin ½, are at rest in a box. The spin state of the entire system is 12). Identify the possible values that would be obtained from the measurements of Sand $2) and the probability of obtaining each value.
- Show that the equation in the picture is a fundamental equation by answering the following: a. Show that it is a homogenous first order function. b. Show that it has an exact differentiaI. c. Find the 3 equations-of-state. d. Find P=P(T, V, N)6. An electron in hydrogen atom is in initial state p(r, 0) = A(2µ100 + ¡Þ210 + 4Þ21–1 – 2i4211) where ynim are the eigenfunctions of the hydrogen atom a. Determine the constant A b. What is the probability of finding the electron in the first excited state? c. Write the state p(r, t) at time t, using energy eigenvalues as En = d. Find the expectation value of L in the state (r,t e. Find the expectation values of Lx and Ly in the state (r, t f. If measurement of L, led to the value -ħ what will be results of measurement of energy and the square of total orbital momentum immediately afterwards and what are their probabilities? hw n21. Derive the density of states as a function of energy for a purely two-dimensional electron system. Show the result in a graph and compare qualitatively with that for 0, 1 and 3 dimensions (1 page maximum).