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A spin-3/2 system can have four values of angular momentum (Sz) when measured along the z-...
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...
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...
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....
Consider the state of a spin-1/2 particle 14) = v1o (31+z) + i] – z)) where...
Consider the state of a spin-1/2 particle 14) = v1o (31+z) + i] – z)) where | z) are the eigenstates of the operator of the spin z-component $z. 1. Show that [V) is properly normalized, i.e. (W14) = 1. 2. Calculate the probability that a measurement of $x = 6x yields 3. Calculate the expectation value (Šx) for the state 14) and its dispersion ASx = V(@z) – ($()2. 4. Assume that the spin is placed in the magnetic...
Parts B, C D, E Rules for Orbital Angular Momentum Constants Periodic Table Part A Learning...
Parts
B, C D, E
Rules for Orbital Angular Momentum Constants Periodic Table Part A Learning Goal How many different values of I are possible for an electron with principal quantum number n Express your answer as an integer To understand and be able to use the ruiles for determining allowable orbital angular momentum states 52 Several numbers are necessary to describe the states available to an electron in the hydrogen atom. The principal quantum number n determines the energy...
3 Angular Momentum and Spherical Harmonics For a quantum mechanical system that is able to rotate...
3 Angular Momentum and Spherical Harmonics For a quantum mechanical system that is able to rotate in 3D, one can always define a set of angular momentum operators J. Jy, J., often collectively written as a vector J. They must satisfy the commutation relations (, ] = ihſ, , Îu] = ihſ, J., ſu] = ihỈy. (1) In a more condensed notation, we may write [1,1]] = Žiheikh, i, j= 1,2,3 k=1 Here we've used the Levi-Civita symbol, defined as...
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...
qm 2019.3 3. The Hamiltonian corresponding to the magnetic interaction of a spin 1/2 particle with...
qm 2019.3
3. The Hamiltonian corresponding to the magnetic interaction of a spin 1/2 particle with charge e and mass m in a magnetic field B is À eB B. Ŝ, m where Ŝ are the spin angular momentum operators. You should make use of expres- sions for the spin operators that are given at the end of the question. (i) Write down the energy eigenvalue equation for this particle in a field directed along the y axis, i.e. B...
[ 2 + 3 + 3 + 5 = 13 marks ] Question 4 (a) For the case of two qubits, briefly explain the main difference between pro...
[ 2 + 3 + 3 + 5 = 13 marks ] Question 4 (a) For the case of two qubits, briefly explain the main difference between product states and entangled states. Provide one example of each. (b) Show that the identity between spin angular momentum operators S+S- = S2- S2 ± hS, holds. Data: S S tiSy, [Se, Sy] = ihS2, [Sy, S2] = ihS, [S2, S = ihSy. (c) An s = 1/2 particle at t =0 is...
SG 3 *** 11- SG 2 Y- SG 1 Trap Trap Figure 1: Schematic illustration of...
SG 3 *** 11- SG 2 Y- SG 1 Trap Trap Figure 1: Schematic illustration of a the stream of atoms described, each individually pre- pared in the superposition V = 1 + , being measured by sequential Stern-Gerlach machines. 2. (20 points) Spin of an electron is a quantum mechanical observable, and as such, there are corresponding operators for electron spin. For example, the operator S, relates to the observable called 2-spin' and has exactly two associated eigenfunctions and...
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...
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...
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....
Consider the state of a spin-1/2 particle 14) = v1o (31+z) + i] – z)) where | z) are the eigenstates of the operator of the spin z-component $z. 1. Show that [V) is properly normalized, i.e. (W14) = 1. 2. Calculate the probability that a measurement of $x = 6x yields 3. Calculate the expectation value (Šx) for the state 14) and its dispersion ASx = V(@z) – ($()2. 4. Assume that the spin is placed in the magnetic...
Parts
B, C D, E
Rules for Orbital Angular Momentum Constants Periodic Table Part A Learning Goal How many different values of I are possible for an electron with principal quantum number n Express your answer as an integer To understand and be able to use the ruiles for determining allowable orbital angular momentum states 52 Several numbers are necessary to describe the states available to an electron in the hydrogen atom. The principal quantum number n determines the energy...
3 Angular Momentum and Spherical Harmonics For a quantum mechanical system that is able to rotate in 3D, one can always define a set of angular momentum operators J. Jy, J., often collectively written as a vector J. They must satisfy the commutation relations (, ] = ihſ, , Îu] = ihſ, J., ſu] = ihỈy. (1) In a more condensed notation, we may write [1,1]] = Žiheikh, i, j= 1,2,3 k=1 Here we've used the Levi-Civita symbol, defined as...
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...
qm 2019.3
3. The Hamiltonian corresponding to the magnetic interaction of a spin 1/2 particle with charge e and mass m in a magnetic field B is À eB B. Ŝ, m where Ŝ are the spin angular momentum operators. You should make use of expres- sions for the spin operators that are given at the end of the question. (i) Write down the energy eigenvalue equation for this particle in a field directed along the y axis, i.e. B...
[ 2 + 3 + 3 + 5 = 13 marks ] Question 4 (a) For the case of two qubits, briefly explain the main difference between product states and entangled states. Provide one example of each. (b) Show that the identity between spin angular momentum operators S+S- = S2- S2 ± hS, holds. Data: S S tiSy, [Se, Sy] = ihS2, [Sy, S2] = ihS, [S2, S = ihSy. (c) An s = 1/2 particle at t =0 is...
SG 3 *** 11- SG 2 Y- SG 1 Trap Trap Figure 1: Schematic illustration of a the stream of atoms described, each individually pre- pared in the superposition V = 1 + , being measured by sequential Stern-Gerlach machines. 2. (20 points) Spin of an electron is a quantum mechanical observable, and as such, there are corresponding operators for electron spin. For example, the operator S, relates to the observable called 2-spin' and has exactly two associated eigenfunctions and...
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