Problem 3.9Homework Answers
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Problem 3.9
3.9. Calculate deltaSx, and DeltaSy for an eigenstate of SZ for a spin-1/2 particle....
for spin 1 particle ,construct Sx,Sy,Sz and S^2 matrix
for spin 1 particle ,construct Sx,Sy,Sz and S^2 matrix
(7) Consider a spin-1/2 particle that is in the eigenstate of S,. The sys- tem is...
(7) Consider a spin-1/2 particle that is in the eigenstate of S,. The sys- tem is rotated by an angle 0 = T/4 about z axis. What is. (a) State of the system after rotation?? (b) What are the expectation values of S2, S, and S̟ after rotation ??
Problem 111.3. A spin 1/2 particle interacts with a nnagnetic field B = Boe through the...
Problem 111.3. A spin 1/2 particle interacts with a nnagnetic field B = Boe through the Pauli interaction H-μσ. B where μ is the magnetic moment. The Pauli spin matrices are İ-(Oz,@yMwwhere the σί are T0 1 0-il The eigenstates for d, are the spinors 0 (a) (3 pts.) Suppose that at time t-0 the particle is in an eigenstate Xx corresponding to spin pointing along the positive z-axis. Find the eigenstatexz in terms of α and β. (b) (7...
(2.1) (20 points) A spin 1/2 particle is in an eigenstate of Sy with eigenvalue h/2...
(2.1) (20 points) A spin 1/2 particle is in an eigenstate of Sy with eigenvalue h/2 at the initial time t = 0. At that time, it is placed in a magnetic induction B = B2, and it is then allowed to precess in that induction for the time T. Then, at that instant T, B is instantaneously rotated from the z to the y direction, becoming B = Bį. After another identical time interval T occurs, a measurement of...
[3] A spin-1/2 particle is in the state IW) 1/311) +i2/3|). (a) A measurement is made of the x component of the spi...
[3] A spin-1/2 particle is in the state IW) 1/311) +i2/3|). (a) A measurement is made of the x component of the spin. What is the probability that the spin will be in the +z direction? (b) Suppose a measurement is made of the spin in the z direction and it is found that the particle has m,#1/2. what is the state after the measurement? (c) Now a second measurement is made immediately after to determine the spin in the...
1. In this problem, we are going to look at a three-level system. A spin-1 particld is placed in ...
1. In this problem, we are going to look at a three-level system. A spin-1 particld is placed in a constant magnetic field along the a-direction with strength B,. The spin-1 particle İs initialized in a z-eigenstate with positive eigenvalue h, ie, the i 1,m 1) state. What is the probability to find the negative eigenvalue the spin along the z axis as a function of time? Assume that the spin-1 particle has inagnetic moment 2 × μιι, i.e. that...
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...
Assignment 7 Problem 1. [100pt] The Hamiltonian for a spin-1/2 particle with charge te in an...
Assignment 7 Problem 1. [100pt] The Hamiltonian for a spin-1/2 particle with charge te in an external magnetic field is ges.B 2mc Calculate the operator ds/dt if B = Bu, what is S2(t) in matrix form?
1. Consider a particle of mass m in an infinite square well with potential energy 0...
1. Consider a particle of mass m in an infinite square well with potential energy 0 for 0 Sz S a oo otherwise V (x) For simplicity, we may take the 'universe' here to be the region of 0 S z S a, which is where the wave function is nontrivial. Consequently, we may express stationary state n as where En is the associated mechanical energy. It can be shown that () a/2 and (p:)0 for stationary state n. (a)...
(L43*) Spin can be represented by matrices. Show that all three spin matrices l 0 2 0 -1 0),"2=2 1 have eigenval...
(L43*) Spin can be represented by matrices. Show that all three spin matrices l 0 2 0 -1 0),"2=2 1 have eigenvalues of +1/2h and -1/2h. Calculate the corresponding eigenfunctions which we will denote as α-and β-eigenfunctions corresponding to spin l/2 particles. Show that Sj can be determined by the commutation of the other two matrices sn and sm, n, maj. Prove that the (2×2) matrix sz-s' +ss+s, commutes with all spin matrices, ie. s2s,-sis-. Calculate the eigenvalues of s2....
(7) Consider a spin-1/2 particle that is in the eigenstate of S,. The sys- tem is rotated by an angle 0 = T/4 about z axis. What is. (a) State of the system after rotation?? (b) What are the expectation values of S2, S, and S̟ after rotation ??
Problem 111.3. A spin 1/2 particle interacts with a nnagnetic field B = Boe through the Pauli interaction H-μσ. B where μ is the magnetic moment. The Pauli spin matrices are İ-(Oz,@yMwwhere the σί are T0 1 0-il The eigenstates for d, are the spinors 0 (a) (3 pts.) Suppose that at time t-0 the particle is in an eigenstate Xx corresponding to spin pointing along the positive z-axis. Find the eigenstatexz in terms of α and β. (b) (7...
(2.1) (20 points) A spin 1/2 particle is in an eigenstate of Sy with eigenvalue h/2 at the initial time t = 0. At that time, it is placed in a magnetic induction B = B2, and it is then allowed to precess in that induction for the time T. Then, at that instant T, B is instantaneously rotated from the z to the y direction, becoming B = Bį. After another identical time interval T occurs, a measurement of...
[3] A spin-1/2 particle is in the state IW) 1/311) +i2/3|). (a) A measurement is made of the x component of the spin. What is the probability that the spin will be in the +z direction? (b) Suppose a measurement is made of the spin in the z direction and it is found that the particle has m,#1/2. what is the state after the measurement? (c) Now a second measurement is made immediately after to determine the spin in the...
1. In this problem, we are going to look at a three-level system. A spin-1 particld is placed in a constant magnetic field along the a-direction with strength B,. The spin-1 particle İs initialized in a z-eigenstate with positive eigenvalue h, ie, the i 1,m 1) state. What is the probability to find the negative eigenvalue the spin along the z axis as a function of time? Assume that the spin-1 particle has inagnetic moment 2 × μιι, i.e. that...
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...
Assignment 7 Problem 1. [100pt] The Hamiltonian for a spin-1/2 particle with charge te in an external magnetic field is ges.B 2mc Calculate the operator ds/dt if B = Bu, what is S2(t) in matrix form?
1. Consider a particle of mass m in an infinite square well with potential energy 0 for 0 Sz S a oo otherwise V (x) For simplicity, we may take the 'universe' here to be the region of 0 S z S a, which is where the wave function is nontrivial. Consequently, we may express stationary state n as where En is the associated mechanical energy. It can be shown that () a/2 and (p:)0 for stationary state n. (a)...
(L43*) Spin can be represented by matrices. Show that all three spin matrices l 0 2 0 -1 0),"2=2 1 have eigenvalues of +1/2h and -1/2h. Calculate the corresponding eigenfunctions which we will denote as α-and β-eigenfunctions corresponding to spin l/2 particles. Show that Sj can be determined by the commutation of the other two matrices sn and sm, n, maj. Prove that the (2×2) matrix sz-s' +ss+s, commutes with all spin matrices, ie. s2s,-sis-. Calculate the eigenvalues of s2....
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