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Q7M.6 Assume that the 1+z) and -z) states for an elec- tron in a magnetic field...
Consider an electron in a uniform magnetic field along the z direction. A measurement shows that...
Consider an electron in a uniform magnetic field along the z direction. A measurement shows that the spin is along the negative x direction at -0. a. Find the eigenvector describing the initial spin state. 5. 0 -1 b. Write the Hamiltonian as a 2x2 matrix by starting with H =-7S-Band taking the field B in the z- direction. Find the energy eigenvalues and eigenvectors. Solve for | Ψ(t) using these eigenvalues, eigenvectors, and the initial condition from part a....
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...
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...
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...
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...
The behavior of a spin- particle in a uniform magnetic field in the z-direction, , with...
The behavior of a spin-
particle in a uniform magnetic field in the z-direction,
, with the Hamiltonian
You found that the expectation value of the spin vector
undergoes Larmor precession about the z axis. In this sense, we can
view it as an analogue to a rotating coin, choosing the
eigenstate with eigenvalue
to represent heads and the eigenstate with eigenvalue
to represent tails. Under time-evolution in the magnetic field,
these eigenstates will “rotate” between each other.
(a) Suppose...
) cos(0/2) + -2) state is placed in a magnetic field with strength B pointing 4....
) cos(0/2) + -2) state is placed in a magnetic field with strength B pointing 4. Larmor precession: an electron prepared in the V(t 0 sin(0/2)e in the a-direction. Calculate the time evolution of the electron's spin state. In addition calculate the time evolution of (S), S and (S ). (2 points)
Intro to Quantum Mechanics Problem: An electron under the influence of a uniform magnetic field By...
Intro to Quantum Mechanics Problem:
An electron under the influence of a uniform magnetic field By in the y-direction has its spin initially (at 0) pointing in the positive x-direction. That is, it is in an eigenstate of S with eigenvalue +,S h. The Hamiltonian H--μ . B-γ By Sy consists of the interaction of the magnetic dipole moment μ due to spin and the magnetic field B. Show that the probability of finding the electron with its spin pointing...
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...
The energy of a magnetic moment in a magnetic field is . A certain paramagnetic salt contai...
The energy of a magnetic moment in a magnetic field is . A certain paramagnetic salt contains 1025 magnetic moments per m3. Each one has a value , due to the atom's spin. As the spin is 1/2, there only are two possible states and the magnetic moments can be parallel or antiparallel to the field. Each magnetic moment belongs to one distinguishable atom. A 1 cm3 sample of this salt is placed in a electromagnet producing a uniform magnetic...
Consider an electron in a uniform magnetic field along the z direction. A measurement shows that the spin is along the negative x direction at -0. a. Find the eigenvector describing the initial spin state. 5. 0 -1 b. Write the Hamiltonian as a 2x2 matrix by starting with H =-7S-Band taking the field B in the z- direction. Find the energy eigenvalues and eigenvectors. Solve for | Ψ(t) using these eigenvalues, eigenvectors, and the initial condition from part a....
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...
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...
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...
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...
The behavior of a spin-
particle in a uniform magnetic field in the z-direction,
, with the Hamiltonian
You found that the expectation value of the spin vector
undergoes Larmor precession about the z axis. In this sense, we can
view it as an analogue to a rotating coin, choosing the
eigenstate with eigenvalue
to represent heads and the eigenstate with eigenvalue
to represent tails. Under time-evolution in the magnetic field,
these eigenstates will “rotate” between each other.
(a) Suppose...
) cos(0/2) + -2) state is placed in a magnetic field with strength B pointing 4. Larmor precession: an electron prepared in the V(t 0 sin(0/2)e in the a-direction. Calculate the time evolution of the electron's spin state. In addition calculate the time evolution of (S), S and (S ). (2 points)
Intro to Quantum Mechanics Problem:
An electron under the influence of a uniform magnetic field By in the y-direction has its spin initially (at 0) pointing in the positive x-direction. That is, it is in an eigenstate of S with eigenvalue +,S h. The Hamiltonian H--μ . B-γ By Sy consists of the interaction of the magnetic dipole moment μ due to spin and the magnetic field B. Show that the probability of finding the electron with its spin pointing...
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...
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