Consider an ensemble of 16 spin-1/2 particles. There is no magnetic field present.
a) List the possible macrostates of the ensemble.b) Determine the total number of microstates possible.
c) Determine the number of microstates that are associated with each of the macrostates.
Your answer should be a list and you can use a calculator or Excel to compute the
numbers.d) How does the probability of attaining the microstate
compare to the probability of attaining the microstate
?
Explain your answer.e) Determine the probability with which each macrostate could occur.
f) Which macrostate is most probable?
g) What is the probability that a macrostate will contain at least 4 spin up particles?
h) What is the probability that a macrostate will contain at most 5 spin up particles?Consider a system of non-interacting spin-1/2 particles. The energy of a single particle with
spin up is B and that with spin down is B. If the total energy of the system is fixed,
then an accessible microstate is one that gives this total energy. Suppose that the system
consists of six particles and that the total energy of the system is 2B.a) Describe which macrostates are possible in terms of the number of particles with spin
up.
b) List all accessible microstates.c) Determine the probability with which on particular given particle (e.g. the leftmost) will
be in a state with spin up.
d) Determine the probability with which any two given particles will be in states with spin
up.3 Ensembles of spin-1/2 particles in a magnetic field: small number example
The energy of a particle with spin up in a magnetic field with magnitude B oriented along
the positive z axis is B. That for a particle with spin down is B. Consider an ensemble
of eight spin-1/2 particles in this field. Assume that the probability with which a particle is
in the spin up state is 3/4 and the probability that it is in the spin down state is 1/4. Let
N+ represent the number of particles with spin up.a) List all macrostates for the ensemble (in terms of N+), the energy of each macrostate
and the probability with which each occurs.b) Which is the most likely macrostate? With what probability does it occur?
c) Determine the mean for the energy, E (this is for a large collection of such ensembles).
Is there a macrostate which has an energy exactly equal to the mean energy?
d) Determine the standard deviation for the energy, E.e) Suppose that one is given a single copy of ensemble without knowing its actual microstate. One way to assign an energy to it is to use the mean energy. What is the
probability that the ensembles microstate is that whose energy is exactly the mean
energy?It should be clear that the ensemble could be in a different macrostate. Consider the two
other macrostates with energy closest to that of the mean energy.f) List the energies of these macrostates and determine the fractional difference between
the energies of these and the mean energy.g) Determine the probability that the ensemble could have either the mean energy or one
of the two other energies that are closest to the mean energy.
h) Determine the probability that the ensemble could have an energy in the range E E
to E + E.i) Based on the previous results you should be able to make a statement of the form: The
energy of the ensemble is within ??% of the mean with probability =??. Provide such
a statement or statements for this example.The energy of a particle with spin up in a magnetic field with magnitude B oriented along
the positive z axis is B. That for a particle with spin down is B. Consider an ensemble
of N spin-1/2 particles in this field and assume that N is even. Let N+ represent the number
of particles with spin up.The probability with which any given particle will be in the spin
up state is different to that in which it will be in the spin down state. It will emerge that
the spin up state is favored over the spin down state.Denote the probability with which any
particle is in the spin up state by (1 + )/2 where 0 ! ! 1.
a) Show that,
E = BN
2
and that the standard deviation of the energy, .
E = B!(1 2)N.b) Show that the magnitude of the ratio of the standard deviation to the mean approaches
zero as the size of the ensemble increases.In order to illustrate these consider numerical examples for which B = 1/2 and = 1/3.
The aim will be to determine the probability with which the energy of a macrostate is within
the range of a standard deviation of the mean.c) Show that E = N/6 and E = !2N/9. Show that the value of N+ that gives the
macrostate with energy exactly equal to E is N+ = 2N/3. Show that the value of N+
that gives the macrostate with energy exactly equal to E E is N+ = 2N/3!2N/9.
In each of the following you can use a numerical tool (e.g. Excel) to do the computations.These all concern the range of the energies E E to E + E, A measure of this range is the
ratio |E/E|.d) Let N = 21. Determine |E/E|. Determine the probability with which the energy of a
macrostate is in the range E E to E + E.e) Let N = 90. Determine the fractional difference between the mean energy E and the
extremes of the energies in the range E E to E +E. Determine the probability with
which the energy of a macrostate is in the range E E to E + E.f) Let N = 150. Determine the fractional difference between the mean energy E and the
extremes of the energies in the range E E to E +E. Determine the probability with
which the energy of a macrostate is in the range E E to E + E.g) Does the range of energies E E to E + E become more precise, relative to the mean
energy, as N increases?h) Does the probability of falling within the range energies E E to E +E become larger
or smaller as N increases? Does it appear to approach a fixed value?5 Einstein solid statistics: small numbers
Consider an Einstein solid consisting of five particles (these are labeled A, B, C, D and E).
a) Determine the multiplicities for the four lowest energy states.
b) Suppose that there are three energy units. Determine the probability with which one
particle will have all three energy units.c) Suppose that there are three energy units. Determine the probability with which particle
1 has two energy units.6 Einstein solid statistics: large numbers
Consider two Einstein solids, labeled A and B. Solid A has 10 particles and solid B has 10
particles. The exercise will consider various energy units. Calculations can be done using the
OSP program called Einstein solid: temperature. Suppose that initially the energy number for
A is qA = 9 and, for B, qB = 3. In the following do not forget the ! and N/2 terms in teh
energy.a) Determine the multiplicities A and B. Determine the total number of microstates for
the composite system. Determine the energy per particle for each solid.
b) Now suppose that the systems can interact, exchange energy and eventually reach equilibrium. Use the program to determine and describe the most probable state for each
solid. Determine the the energy per particle for each solid.c) In what direction did the energy flow as the systems approached equilibrium?
d) Determine the probability that the energy flows from system A to system B. Determine
the probability that the energy flows from system B to system A. How many times is it
more likely that energy flows from A to B rather than from B to A?
Now suppose that solid A initially has 20 particles and qA = 20 while solid B has 5 particles
and qB = 10.e) Determine the equilibrium state after the solids have interacted. Determine the energy
per particle for each solid.
f) Did the energy flow from the solid with higher total energy to that with lower total
energy?
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