Homework Answers
Total different combinations = 2*2*2*2*2 = 25 =32
Combinations with 4 having spin up and other having spin down = 5C4 = 5
Probability that four have spin up while other have spin down
= 5/32 = 0.15625
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12. Consider five spin-1/2 elementary particles (distinguishable and with no external fields present). What is the...
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...
4. When an external magnetic field B is applied, a "spin-1" ion has 3 magnetic states...
4. When an external magnetic field B is applied, a "spin-1" ion has 3 magnetic states with energies given Em=aBm, m=-1,0,1, where a is a constant of order a few times the Bohr magneton up = en/(2m). (Note: the notation here is quite different from that of Kittel & Kroemer who use "m" for the elementary magnetic moment which we have denoted a. In our terminology, m=ms is an integer quantum number: m=-1 labels the "spin down" state, m=0 labels...
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...
I guessed on the answer, but could be wrong... Question 3 Consider two particles zipping around...
I guessed on the answer, but
could be wrong...
Question 3 Consider two particles zipping around in a plane (i.e., in 2 dimensions), one with speed v, the other with speed 2v. How many different velocities are consistent with these speeds? OA There are half as many v that give a speed of 2v OB The numbers are the same. ° C There are twice as many u that give a speed of 2v O DThere are four times as...
2. Addition of Angular Momentum a) (8pts) Given two spin 1/2 particles, what are the four possibilities for their spin configuration? Put your answer in terms of states such as | 11). where the f...
2. Addition of Angular Momentum a) (8pts) Given two spin 1/2 particles, what are the four possibilities for their spin configuration? Put your answer in terms of states such as | 11). where the first arrow denotes the z-component of the particle's spin. Identify the m values for each state. b)(7pts) If you apply the lowering operator to a state you get Apply the two-state lowering operator S--S(,) +S(), where sti) acts on the first state and S acts on...
A spin-1 particle interacts with an external magnetic field B = B. The interaction Hamiltonian for the system is H = gB-S, where S-Si + Sỳ + SE is the spin operator. (Ignore all degrees of freedom ot...
A spin-1 particle interacts with an external magnetic field B = B. The interaction Hamiltonian for the system is H = gB-S, where S-Si + Sỳ + SE is the spin operator. (Ignore all degrees of freedom other than spin.) (a) Find the spin matrices in the basis of the S. S eigenstates, |s, m)) . (Hint: Use the ladder operators, S -S, iS, and S_-S-iS,, and show first that s_ | 1,0-ћ /2 | 1.-1)) . Then use these...
2) If I flipped a coin five times and it landed heads up each time, what...
2) If I flipped a coin five times and it landed heads up each time, what is the probability that it will land heads up on the sixth time? 3) If a solar panel manufacturing process has a 3% defect rate, and a random sample of 10 panels is selected, what is the probability that none of the panels in the sample will be defective? 4) You are designing an experiment to determine the effect of temperature, stirring rate, and...
Question 9 Consider a quantum system comprising two indistinguishable particles which can occupy only three individual-particle...
Question 9 Consider a quantum system comprising two indistinguishable particles which can occupy only three individual-particle energy levels, with energies 81 0, 82 2 and E3 38.The system is in thermal equilibrium at temperature T. (a) Suppose the particles which can occupy an energy level. are spinless, and there is no limit to the number of particles (i) How many states do you expect this system to have? Justify your answer (ii) Make a table showing, for each state of...
Solve 12-14 12. Consider the molecule shown below. OH OH f. How many stereoisomers are possible?...
Solve 12-14
12. Consider the molecule shown below. OH OH f. How many stereoisomers are possible? g. Draw each of these stereoisomers. h. Identify pairs of enantiomers and pairs of diastereomers. i. Build a model of the stereoisomer that has both OH groups pointing out. Also, build a model of the mirror image of this molecule. j. Do your two molecules represent enantiomers? Are they chiral? Fischer Projections 13. Build a model of 1-bromo-1-chloroethane. a. Draw a3D representation of this...
2. Consider the permutations a (123)(45) and b (2543) in the symmetric group S (a) Compute the co...
2. Consider the permutations a (123)(45) and b (2543) in the symmetric group S (a) Compute the conjugate permutation ca using (i) the definition a-b ab (b) What is the order of a? How many permutations have the same shape as a; that is, (x x x)(x x). (c) What is the subgroup H of all permutations in Ss that commute with the permutation a? d) Using the result of the previous part, or otherwise, find 5 other permutations bi,...
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...
4. When an external magnetic field B is applied, a "spin-1" ion has 3 magnetic states with energies given Em=aBm, m=-1,0,1, where a is a constant of order a few times the Bohr magneton up = en/(2m). (Note: the notation here is quite different from that of Kittel & Kroemer who use "m" for the elementary magnetic moment which we have denoted a. In our terminology, m=ms is an integer quantum number: m=-1 labels the "spin down" state, m=0 labels...
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...
I guessed on the answer, but
could be wrong...
Question 3 Consider two particles zipping around in a plane (i.e., in 2 dimensions), one with speed v, the other with speed 2v. How many different velocities are consistent with these speeds? OA There are half as many v that give a speed of 2v OB The numbers are the same. ° C There are twice as many u that give a speed of 2v O DThere are four times as...
2. Addition of Angular Momentum a) (8pts) Given two spin 1/2 particles, what are the four possibilities for their spin configuration? Put your answer in terms of states such as | 11). where the first arrow denotes the z-component of the particle's spin. Identify the m values for each state. b)(7pts) If you apply the lowering operator to a state you get Apply the two-state lowering operator S--S(,) +S(), where sti) acts on the first state and S acts on...
A spin-1 particle interacts with an external magnetic field B = B. The interaction Hamiltonian for the system is H = gB-S, where S-Si + Sỳ + SE is the spin operator. (Ignore all degrees of freedom other than spin.) (a) Find the spin matrices in the basis of the S. S eigenstates, |s, m)) . (Hint: Use the ladder operators, S -S, iS, and S_-S-iS,, and show first that s_ | 1,0-ћ /2 | 1.-1)) . Then use these...
2) If I flipped a coin five times and it landed heads up each time, what is the probability that it will land heads up on the sixth time? 3) If a solar panel manufacturing process has a 3% defect rate, and a random sample of 10 panels is selected, what is the probability that none of the panels in the sample will be defective? 4) You are designing an experiment to determine the effect of temperature, stirring rate, and...
Question 9 Consider a quantum system comprising two indistinguishable particles which can occupy only three individual-particle energy levels, with energies 81 0, 82 2 and E3 38.The system is in thermal equilibrium at temperature T. (a) Suppose the particles which can occupy an energy level. are spinless, and there is no limit to the number of particles (i) How many states do you expect this system to have? Justify your answer (ii) Make a table showing, for each state of...
Solve 12-14
12. Consider the molecule shown below. OH OH f. How many stereoisomers are possible? g. Draw each of these stereoisomers. h. Identify pairs of enantiomers and pairs of diastereomers. i. Build a model of the stereoisomer that has both OH groups pointing out. Also, build a model of the mirror image of this molecule. j. Do your two molecules represent enantiomers? Are they chiral? Fischer Projections 13. Build a model of 1-bromo-1-chloroethane. a. Draw a3D representation of this...
2. Consider the permutations a (123)(45) and b (2543) in the symmetric group S (a) Compute the conjugate permutation ca using (i) the definition a-b ab (b) What is the order of a? How many permutations have the same shape as a; that is, (x x x)(x x). (c) What is the subgroup H of all permutations in Ss that commute with the permutation a? d) Using the result of the previous part, or otherwise, find 5 other permutations bi,...
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