Consider a spin system of 1,000,000 spins where each spin is equally likely to be up...
Consider a spin system of 1,000,000 spins where each spin is equally likely to be up (u) or down (d).
The probability that exactly 500,000 spins are up is
a. greater than 50%
b. exactly equal to 50%
c. less than 50%
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Consider a spin system of 1,000,000 spins where each spin is
equally likely to be up...
3.3 One spin in thermal contact with a large spin system Generalize the preceding problem by...
3.3 One spin in thermal contact with a large spin system Generalize the preceding problem by considering the case where the system A' consists of some arbitrarily large number N of spins J, each having magnetic moment plo. The system A consists again of a single spin { with mag- netic moment Mo. Both A and A' are located in the same magnetic field B and are placed in contact with each other so that they are free to exchange...
2. Interacting Spins (5 points each part, 30 points total). Two spins, each of which can...
2. Interacting Spins (5 points each part, 30 points total). Two spins, each of which can be in one of two states, up or down, are in equilibrium with a heat reservoir at temperature t. They interact as follows: When the two spins point in the same direction, their interaction energy is – J, and when they point in opposite directions, their interaction energy is J. The spins also each have a magnetic moment m and are subject to a...
A proton has tWo posslble spins States, spin up With energies e+ and Spin down with...
A proton has tWo posslble spins States, spin up With energies e+ and Spin down with energies e e+ is positive and e. is negative. The spin partition function for a collection of N non-interacting spins is kTexp Show All Your Work! a) Derive the expression for b) As the temperature, T, of the system decreases, the average energy decreases. Why is that? (E)-- In(QCN, V, β)] Cv aP эт
Consider one dimensional lattice of N particles having a spin of 1 /2 with an associated magnetic...
Consider one dimensional lattice of N particles having a spin of 1 /2 with an associated magnetic moment μ The spins are kept in a magnetic field with magnetic induction B along the z direction. The spin can point either up, t, or down, , relative to the z axis. The energy of particle with spin down is e B and that of particle with spin up is ε--B. We assume that the system is isolated from. its environment so...
Will rate!! The outcome of a spin on the wheel is a discrete random variable. Consider...
Will rate!!
The outcome of a spin on the wheel is a discrete random variable. Consider X the dollar amount spun on the wheel, where Bankrupt and Lose a Turn s0, and Free Play - $500. There are 24 wedges on the wheel PMF for Spln Outcome 8 0 300 350 400 450 500 550 600 700 800 900 5000 The following table provides the probability mass function for X. Round values to 3 decimal places. XSO 300 $35o $400...
Consider a couple who gave birth to 4 children (assume a boy is equally likely as...
Consider a couple who gave birth to 4 children (assume a boy is equally likely as a girl) The events space for the experiment S={GGGG,GGGB,GGBG,GGBB,GBGG,GBGB.GBBG GBBB,BGGG BGGB,BGBG,BGBB,BBGG,BBGB,BBBG BBBB } Find the probability of the event E1 of no more than 2 girls in the family () Find the probability of event E2 of no less than two boys in the family (i) (ii) Find the probability of E1 n E2 E1 U E2 (iv) Find the probability of Number Number...
1. (30 points). Coupled spins. Spin-1/2 particles A and B evolve under the influence of the follo...
1. (30 points). Coupled spins. Spin-1/2 particles A and B evolve under the influence of the following Hamiltonian (for simplicity takeh-1 so that energies are expressed in frequency units): We work in the uncoupled basis aba) Ib), where a,b E 0,1 and where states 0) (1)) correspond to single spins aligned (antialigned) with the z-direction. As we discussed in lecture, the eigenstates of the Hamiltonian are 100), 111), and 2-1/2 (101) 110)). a. We prepare the initial state t01). Since...
Consider a family with eight children. Assuming that the sex each child is determined independently of the others and that each child is equally likely to be female or male,
Consider a family with eight children. Assuming that the sex each child is determined independently of the others and that each child is equally likely to be female or male, a. What is the probability that exactly four children are female? Hint: Use the binomial distribution. (10 points) b. What is the probability that at most seven children are female? Hint: What is the complement of this event? (10 points)c. Find the conditional probability that the first two children born are female...
11 Consider an assembly of N-4 particles in a system which has equally spaced non degenerate...
11 Consider an assembly of N-4 particles in a system which has equally spaced non degenerate energy levels, U-0.e,2e,3e, The total energy of the system is U 6. a) Assuming the particles are distinguishable, how many distributions of the particles over the energy levels are possible? List all of them in a table showing the number [7] of particles, n, in each energy level U b) To which particle statistics does this scenario correspond? c) How many microstates contribute to...
A simple model for human offspring is that each child is equally likely to be male...
A simple model for human offspring is that each child is equally likely to be male or female. With this model, a three-child family can be thought of as three random determinations of sex, in order. a. What are the 8 possible outcomes? What is the probability of each? b. What outcomes make up the event “the oldest child is a daughter”? What is the event’s probability? c. What outcomes make up the event “the family has one daughter and...
3.3 One spin in thermal contact with a large spin system Generalize the preceding problem by considering the case where the system A' consists of some arbitrarily large number N of spins J, each having magnetic moment plo. The system A consists again of a single spin { with mag- netic moment Mo. Both A and A' are located in the same magnetic field B and are placed in contact with each other so that they are free to exchange...
2. Interacting Spins (5 points each part, 30 points total). Two spins, each of which can be in one of two states, up or down, are in equilibrium with a heat reservoir at temperature t. They interact as follows: When the two spins point in the same direction, their interaction energy is – J, and when they point in opposite directions, their interaction energy is J. The spins also each have a magnetic moment m and are subject to a...
A proton has tWo posslble spins States, spin up With energies e+ and Spin down with energies e e+ is positive and e. is negative. The spin partition function for a collection of N non-interacting spins is kTexp Show All Your Work! a) Derive the expression for b) As the temperature, T, of the system decreases, the average energy decreases. Why is that? (E)-- In(QCN, V, β)] Cv aP эт
Consider one dimensional lattice of N particles having a spin of 1 /2 with an associated magnetic moment μ The spins are kept in a magnetic field with magnetic induction B along the z direction. The spin can point either up, t, or down, , relative to the z axis. The energy of particle with spin down is e B and that of particle with spin up is ε--B. We assume that the system is isolated from. its environment so...
Will rate!!
The outcome of a spin on the wheel is a discrete random variable. Consider X the dollar amount spun on the wheel, where Bankrupt and Lose a Turn s0, and Free Play - $500. There are 24 wedges on the wheel PMF for Spln Outcome 8 0 300 350 400 450 500 550 600 700 800 900 5000 The following table provides the probability mass function for X. Round values to 3 decimal places. XSO 300 $35o $400...
Consider a couple who gave birth to 4 children (assume a boy is equally likely as a girl) The events space for the experiment S={GGGG,GGGB,GGBG,GGBB,GBGG,GBGB.GBBG GBBB,BGGG BGGB,BGBG,BGBB,BBGG,BBGB,BBBG BBBB } Find the probability of the event E1 of no more than 2 girls in the family () Find the probability of event E2 of no less than two boys in the family (i) (ii) Find the probability of E1 n E2 E1 U E2 (iv) Find the probability of Number Number...
1. (30 points). Coupled spins. Spin-1/2 particles A and B evolve under the influence of the following Hamiltonian (for simplicity takeh-1 so that energies are expressed in frequency units): We work in the uncoupled basis aba) Ib), where a,b E 0,1 and where states 0) (1)) correspond to single spins aligned (antialigned) with the z-direction. As we discussed in lecture, the eigenstates of the Hamiltonian are 100), 111), and 2-1/2 (101) 110)). a. We prepare the initial state t01). Since...
11 Consider an assembly of N-4 particles in a system which has equally spaced non degenerate energy levels, U-0.e,2e,3e, The total energy of the system is U 6. a) Assuming the particles are distinguishable, how many distributions of the particles over the energy levels are possible? List all of them in a table showing the number [7] of particles, n, in each energy level U b) To which particle statistics does this scenario correspond? c) How many microstates contribute to...
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