The probability p; of occupying an available state j is: Pj 9je-BEj Z where g; is the degeneracy of the state and 1/ẞ=kBT. Derive an expression for the temperature I in terms of the probabilities of occupation of two different states i and k.

36.7. If 2 = .09 and initially N = 10, what is the probability that N = 12 one year later? What is the probability that N= 12 two years later? Use the equation below to estimate the probability of N=10 (No+j-1)(No+j-2)...N 0が j! |

3) Four bones of eqyual sizes are to Contain distingulshable nine particles. Find the probability of macyostate j (3,401) and ii(3,3,3). the

The most probable thermal distribution for the states of a system of particles is defined by having the probability of the th state, P₁, proportional to... the number of particles, N Oits Boltzmann factor O volume O pressure O the number of moles, n O temperature ►

a) Make a diagram showing how many distinct ways (how many microstates, the multiplicity) there are of putting q = 2 indistinguishable objects in N = 3 boxes. Assuming that all microstates are equally probable, what is the probability that both objects are in the left-most box? What is the correct formula for the mulitiplicity as a function of N and q? b) Make a diagram showing how many distinct ways (the multiplicity) there are of putting q = 2 distinguishable objects in N= 3 boxes. Assuming that all microstates are equally probable, what is the probability that both objects are in the left-most box? Label the two objects R and G. What is the correct formula for the mulitiplicity as a function of N and q? Below are the diagrams, started for you. Complete the diagrams. distinguishable indistinguishable RG •. !R !G

a. Calculate the minimum uncertainty in the position of an electron in meters, if its velocity has been measured to within 2.34e-06 m/s. For the mass of an electron use 9.10938356 × 10-31 kg and for huse 1.0545718 x 10-34 Js. m b. 44 = 141² = ² represents a: O the pitchfork operator O probability density O numerov program O wave function c. The sum of all the little bits of probability over possible measurements that can be made (i.e. over all space): S4²4 = 1 is called: O the quantum condition O the normalization condition O the human condition O the eignen condition

Answer in 90 minutes please.

Parts 1,2 and 3 have been answers in a previous so 3-6 is what I need to be solved.

a) Storting of desity of Steltes from the general formula 272 Show that g(w) dw = 3 x Vw?dw for phonons 272 83 under the assumption that the phonons are ac custic, meaning U = 3. doesn't depend on w or b)n a solid of N atoms, there are 3N vibrati- onal modes (phonons) with the highelt frequ- eicy wp Wp Using 9(w) dw =3N find wD, and find characteistie temperature Kwo kB

Check Your Understanding If a=3+4i , what is the product a* a?