Homework Answers
possible
states for identicle particles are (2,2) (2,-2) (2,0) (1,1) (1,-1)
(1,0) (0,0) (2,1) (2,-1)
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Exercise 8.3 (a) Write down all possible states of two nonidentical particles of spin 1 (i.e.,...
For a system of two particles with spin 3/2 and 1/2 write out all the possible...
For a system of two particles with spin 3/2 and 1/2 write out all the possible |j1m1> x |j2m2> states. how many states should exist?
Problem2 Two possible wave functions for two spin 1 /2 particles with Sz = 0 are Apply the operator S+ to both states as many times as needed to find the largest possible value for m and hence det...
Problem2 Two possible wave functions for two spin 1 /2 particles with Sz = 0 are Apply the operator S+ to both states as many times as needed to find the largest possible value for m and hence determine the value of S2 for each state
Problem2 Two possible wave functions for two spin 1 /2 particles with Sz = 0 are Apply the operator S+ to both states as many times as needed to find the largest possible value...
System A consists of two spin-1/2 particles, and has a four-dimensional Hilbert space. 1. Write down...
System A consists of two spin-1/2 particles, and has a four-dimensional Hilbert space. 1. Write down a basis for the Hilbert space of two spin-1/2 particles. 2. Calculate the matrix of the angular momentum operator, Sfot = (ŜA, ŠA, ŜA) for system A, in the basis of question 4A.1, and express them in this basis. 3. Calculate the square of the total angular momentum of system A , Spotl?, and express this operator in the basis of question 4A.1. 4....
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...
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...
Exercise 1: addition of angular momentum a) Explicitly construct the states of total spin for a system of two spin-z pa...
Exercise 1: addition of angular momentum a) Explicitly construct the states of total spin for a system of two spin-z particles b) Use the table (given below) to verify the Clebsch-Gordan coefficients c) Construct the 1, 12;1, m) 1, 1;0, 0) state explicitly and by using the table Table 1: Clebsch-Gordan coefficients (1,mi, m2ljm) m2 =- ji 2 131-m+ Table 2: Clebsch-Gordan coefficients (j1,1;m, m2jm) m20 1-m+1)01+m+1 2j1101+ j11 (21 +1) (21+2) 71 231 (ว่า+ (31+ (1-m)1-m+1)12
Exercise 1: addition...
(14 points) Write Slater determinants for all possible spin states of the first excited state of...
(14 points) Write Slater determinants for all possible spin states of the first excited state of He (one electron in the 1s orbital, the other in the 2s orbital). Indicate the S (sum of all e spins) and Ms (vector sum of e spins) values of the corresponding wavefunctions. Evaluate the Slater determinants to obtain the total wavefunctions we wrote in class. Hint: The wavefunctions with σ (1,2) and σ (1,2) require two Slater determinants to correctly represent them.
If the spin angular momenta of two spin-1 particles are added, the possible m valnes for...
If the spin angular momenta of two spin-1 particles are added, the possible m valnes for the z-component of the total spin angular momentum (such that Š ) m)are a) m=-1,0, or 1 b) m= c) m=2.0, or-2 d) m--2-3/2,-1,-1/2, 0, 1/2, 1, 3/2, or2 e) m 2, 1, or f) none of the above 2,1, 0,1, or 2
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
Exercise 1: addition of angular momentum a) Explicitly construct the states of total spin for a...
Exercise 1: addition of angular momentum a) Explicitly construct the states of total spin for a system of two spin-^ particles b) Use the table (given below) to verify the Clebsch-Gordan coefficients c) Construct the , 12;l, m)- 1,1;0,0) state explicitly and by using the table Table 1: Clebsch-Gordan coefficients (ji, 1, m2ljm) m2 =- Clebsch-Gordan coefficients 〈J1, 1; m1, m2|JM》 Table 2: 1 +1 71 2j1i+ 1-m)(i-m+1 2j1 (2j1+1 2j1 (2j1+ 11 (21+
Problem2 Two possible wave functions for two spin 1 /2 particles with Sz = 0 are Apply the operator S+ to both states as many times as needed to find the largest possible value for m and hence determine the value of S2 for each state
Problem2 Two possible wave functions for two spin 1 /2 particles with Sz = 0 are Apply the operator S+ to both states as many times as needed to find the largest possible value...
System A consists of two spin-1/2 particles, and has a four-dimensional Hilbert space. 1. Write down a basis for the Hilbert space of two spin-1/2 particles. 2. Calculate the matrix of the angular momentum operator, Sfot = (ŜA, ŠA, ŜA) for system A, in the basis of question 4A.1, and express them in this basis. 3. Calculate the square of the total angular momentum of system A , Spotl?, and express this operator in the basis of question 4A.1. 4....
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...
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...
Exercise 1: addition of angular momentum a) Explicitly construct the states of total spin for a system of two spin-z particles b) Use the table (given below) to verify the Clebsch-Gordan coefficients c) Construct the 1, 12;1, m) 1, 1;0, 0) state explicitly and by using the table Table 1: Clebsch-Gordan coefficients (1,mi, m2ljm) m2 =- ji 2 131-m+ Table 2: Clebsch-Gordan coefficients (j1,1;m, m2jm) m20 1-m+1)01+m+1 2j1101+ j11 (21 +1) (21+2) 71 231 (ว่า+ (31+ (1-m)1-m+1)12
Exercise 1: addition...
(14 points) Write Slater determinants for all possible spin states of the first excited state of He (one electron in the 1s orbital, the other in the 2s orbital). Indicate the S (sum of all e spins) and Ms (vector sum of e spins) values of the corresponding wavefunctions. Evaluate the Slater determinants to obtain the total wavefunctions we wrote in class. Hint: The wavefunctions with σ (1,2) and σ (1,2) require two Slater determinants to correctly represent them.
If the spin angular momenta of two spin-1 particles are added, the possible m valnes for the z-component of the total spin angular momentum (such that Š ) m)are a) m=-1,0, or 1 b) m= c) m=2.0, or-2 d) m--2-3/2,-1,-1/2, 0, 1/2, 1, 3/2, or2 e) m 2, 1, or f) none of the above 2,1, 0,1, or 2
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
Exercise 1: addition of angular momentum a) Explicitly construct the states of total spin for a system of two spin-^ particles b) Use the table (given below) to verify the Clebsch-Gordan coefficients c) Construct the , 12;l, m)- 1,1;0,0) state explicitly and by using the table Table 1: Clebsch-Gordan coefficients (ji, 1, m2ljm) m2 =- Clebsch-Gordan coefficients 〈J1, 1; m1, m2|JM》 Table 2: 1 +1 71 2j1i+ 1-m)(i-m+1 2j1 (2j1+1 2j1 (2j1+ 11 (21+
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