Question

2. (a) Find the entropy of a set of N oscillators of frequency ω as a function of the total quantum number n. Using the multiplicity function, g(N, n) = (N + n − 1)! n!(N − 1)! and make the Stirling approximation logN! ≈ NlogN − N. Replace N − 1 by N. (5 points)

Answer #1

Let's look at N one-dimensional harmonic oscillators that are
indistinguishable from each other in a microcanonical ensemble. If
the mass is m and the frequency is w, use Hamiltonian.
(1) Find the number of states whose energy is between E and E +
ΔE through integration, then approximate the entropy from it.
(2) Find the relationship between energy and temperature and the
relationship between pressure and temperature.

Consider a system composed of a very large number (N
>>1) of indistinguishable atoms,
each of which has only two energy levels 0 and . In thermodynamic
equilibrium, there
are n particles each with energy and the total energy of the
system is E. (10 points)
a) Write and equation for the number of microstates in terms of
N, E, and .
b) Use the Stirling approximation to write an equation for the
entropy S.
E 1 and...

A) Find an approximation of the area of the region R
under the graph of the function f on the interval [0, 2].
Use n = 5 subintervals. Choose the representative points
to be the midpoints of the subintervals.
f(x)=x^2+5
=_____ square units
B) Find an approximation of the area of the region R
under the graph of the function f on the interval [-1, 2].
Use n = 6 subintervals. Choose the representative points
to be the left endpoints...

Let S be the set {(-1)^n +1 - (1/n): all n are natural
numbers}.
1. find the infimum and the supremum of S, and prove that these
are indeed the infimum and supremum.
2. find all the boundary points of the set S. Prove that each of
these numbers is a boundary point.
3. Is the set S closed? Compact? give reasons.
4. Complete the sentence: Any nonempty compact set has a....

Consider a system of N distinguishable atoms, each of which can
be in only one of two states: the lowest energy state with energy
0, and an excited state with energy ɛ > 0. When there are n
atoms in the excited state (and N-1 atoms in the lowest state), the
total energy is U = nɛ.
1. Calculate the entropy S/k = ln(Ω(n)) and find the value of n for
which it is maximum.
2. Find an expression for...

Let n be an odd number and X be a set with n elements. Find the
no. of subsets of X which has even no. of elements. The answer
should be a number depending only on n. (For example, when n = 5,
you need to find the no.of subsets of X which has either 0 elements
or 2 elements or 4 elements)

Part C
The quantum state of a particle can be specified by giving a
complete set of quantum numbers (n,l,
ml,ms). How many different
quantum states are possible if the principal quantum number is
n = 2?
To find the total number of allowed states, first write down the
allowed orbital quantum numbers l, and then write down the
number of allowed values of ml for each orbital quantum
number. Sum these quantities, and then multiply by 2 to account...

For the system given by the difference equation below:
Y(n) – (3/2).Y(n-1) – Y(n-2) = -(5/2). X(n-1)
Find the transfer function H(z). You will need to do this
manually.
Find the poles and zeros of H(z). You can do this manually or
use MATLAB.
Plot the poles and zeros in MATLAB
Is the system stable?
Plot the impulse response of the system using MATLAB
Plot the Step Response of the system using MATLAB
Plot the frequency response of the system...

Find the linear approximation to g(x)=ln(x^2+2x+3) at the
critical point x=-1. Then using the same function, find the
quadratic approximation to g(x) at the critical point x=-1.

Create a frequency table and a histogram for each of the data
sets from Part 2. Below are two links explaining how to use Excel
to create frequency tables and histrograms. Using Excel for Data
and Tables Create a Histogram From a Frequency Table 1) Data Set 1:
23, 26, 50, 39, 55, 64, 50, 10, 42, 35, 51, 42, 43, 56, 64, 48, 48,
57, 27, 37 a. List the data values in order from lowest to highest.
b....

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