a system consitst of N weakly interacting particles, each of which can be in one of two states with energies (e1) or (e2) with (e1)<(e2). Calculate the mean energy, heat capacity of the system and what are thehigh and low end temp. limits of E and Cv(T)?

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1)Three dice with 6 faces each are tossed. Find the followingprobability:
a) to get exactly one "5" and one "6", P(5, 6);
b) to get exactly two "3", P(3, 3);
c) to get exactly two "3" and one "6", P(3, 3, 6);
d) to get "6" in none of the dice, P(no 6).

2)The intrinsic energy of the oscillator of the Problem 1 can takethree values. The state with the energy e1 is not degenerate andconsists of a single quantum state. The second energy stateconsists of three different quantum substates with the same energye2. The third energy state consists of five different quantumsubstates with the same energy e3. A system of N such oscillatorsis in thermodynamic equilibrium at temperature T.
Find
a) probability that the oscillator in one of the three substateswith the intrinsic energy
e2;
b) probability that the oscillator has the energy e2;
c) the average numbers of molecules (N1, N2, N3) in each of theenergy states;

A quantum dot can be thought of as essentially a man-made atom.Consider a 2-dimensional dot (effectively an electron in a squareinfinite well), with a single ground state (with energy Eo = 0.04eV), and a double degenerate first excited state (with energy E1 =0.1eV); the relevant energy levels are shoen below.

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1. At T=300K what is the probability that a dot will be excited?

0.02

0.08

0.16

0.4

0.67

2. What is the probability that the dot will be in the groundstate for very high T, i.e., as T approaches infinity?

1/3

1/2

3. For what temperature will about half of the dots be in theirground state, and half in one of the first excited states ?

T = 100K

T = 1000K

There is no temperature at which this canoccur.

4. What is the entropy of a collection of N such dots at verylow temperatures (T-->0) ?

S=0

S=Nk ln(2)

S=Nk ln(3)

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a system consitst of N weakly interacting particles, each of which can be in one of two states with energies (e1) or (e2) with (e1)<(e2). Calculate the mean energy, heat capacity of the system and what are thehigh and low end temp. limits of E and Cv(T)?

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Physics: Principles with Applications

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