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Consider an electron in a uniform magnetic field along the z direction. A measurement shows that...
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...
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...
Q7M.6 Assume that the 1+z) and -z) states for an elec- tron in a magnetic field...
Q7M.6 Assume that the 1+z) and -z) states for an elec- tron in a magnetic field are energy eigenvectors with ener- gies E and 0, respectively, and assume that the electron's state at t=0 is 14(0) = (Q7.25) Find the probability that we will determine this electron's spin to be in the +x direction at time t.
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...
4. (15 pts Consider the following direction fields IV VI (5 pts)Which of the direction fields corresponds to the system x -Ax, where A is a 2x2 matrix with eigenvalues λ,--1 and λ2-2 and correspondin...
4. (15 pts Consider the following direction fields IV VI (5 pts)Which of the direction fields corresponds to the system x -Ax, where A is a 2x2 matrix with eigenvalues λ,--1 and λ2-2 and corresponding eigenvectors vand v- 1? a. is a 2x2 matrix with repeated eigenvalue λ = 0 with defect 1 (has only one linearly independent eigenvector, not two.) and corresponding eigenvector vi- 13 (5 pts) Which of the direction fields corresponds to the system x -Cx, where...
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...
For anelectron that is at rest in a uniform magnetic field B- Boe, the hamiltonian is...
For anelectron that is at rest in a uniform magnetic field B- Boe, the hamiltonian is where w eBo/m. Note that Bo is the field's constant magde, 1.6021766208 x 10-19 C is the elementary charge, and m ~ 9.10938356 × 10-31 kg is the electron's mass. (a) Find the eigenvalues and a set of normalized eigenvectors for H (b) If the electron starts out in the state ?(0)-( 1 then what is ?(t)?
1. (50 pts) Consider the spin degree of freedom of an electron under an external magnetic...
1. (50 pts) Consider the spin degree of freedom of an electron under an external magnetic field in the r-direction. The spin is initially (at time t 0) in the z-direction. (a) Write down the Hamiltonian for the electron spin. (Do you remember the elementary magnetic moment of electron, the Bohr magneton μΒ, in terms of electron mass me and the elementary charge e?) b) Write down the Schrödinger equation for the spin (c) Describe the motion of spin direction
2. (25 points). Rabi oscillations. Consider a spin-1/2 particle in a magnetic field B - Bo2 such ...
2. (25 points). Rabi oscillations. Consider a spin-1/2 particle in a magnetic field B - Bo2 such that the spin eigenstates are split in energy by hwo (let's label the ground state |0) and the excited state |1)). The Hamiltonian for the system is written as hwo Zeeman - _ here and below. ơng,z are the usual Pauli matrices. A second, oscillating field is applied in the transverse plane, giving rise to a time-dependent term in the Hamiltoniain hw Rabi-...
6. Given the spin Hamiltonian of an electron in a magnetic field B-Bok, H--y5.B, find the...
6. Given the spin Hamiltonian of an electron in a magnetic field B-Bok, H--y5.B, find the time evolution Unitary operator given as U(6) - exp(-()/N). Given that the initial quantum state is le (O) >= 0) find the state after a time t given as (10 points) () >= U(t)|(0) >
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...
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...
Q7M.6 Assume that the 1+z) and -z) states for an elec- tron in a magnetic field are energy eigenvectors with ener- gies E and 0, respectively, and assume that the electron's state at t=0 is 14(0) = (Q7.25) Find the probability that we will determine this electron's spin to be in the +x direction at time t.
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...
4. (15 pts Consider the following direction fields IV VI (5 pts)Which of the direction fields corresponds to the system x -Ax, where A is a 2x2 matrix with eigenvalues λ,--1 and λ2-2 and corresponding eigenvectors vand v- 1? a. is a 2x2 matrix with repeated eigenvalue λ = 0 with defect 1 (has only one linearly independent eigenvector, not two.) and corresponding eigenvector vi- 13 (5 pts) Which of the direction fields corresponds to the system x -Cx, 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...
For anelectron that is at rest in a uniform magnetic field B- Boe, the hamiltonian is where w eBo/m. Note that Bo is the field's constant magde, 1.6021766208 x 10-19 C is the elementary charge, and m ~ 9.10938356 × 10-31 kg is the electron's mass. (a) Find the eigenvalues and a set of normalized eigenvectors for H (b) If the electron starts out in the state ?(0)-( 1 then what is ?(t)?
1. (50 pts) Consider the spin degree of freedom of an electron under an external magnetic field in the r-direction. The spin is initially (at time t 0) in the z-direction. (a) Write down the Hamiltonian for the electron spin. (Do you remember the elementary magnetic moment of electron, the Bohr magneton μΒ, in terms of electron mass me and the elementary charge e?) b) Write down the Schrödinger equation for the spin (c) Describe the motion of spin direction
2. (25 points). Rabi oscillations. Consider a spin-1/2 particle in a magnetic field B - Bo2 such that the spin eigenstates are split in energy by hwo (let's label the ground state |0) and the excited state |1)). The Hamiltonian for the system is written as hwo Zeeman - _ here and below. ơng,z are the usual Pauli matrices. A second, oscillating field is applied in the transverse plane, giving rise to a time-dependent term in the Hamiltoniain hw Rabi-...
6. Given the spin Hamiltonian of an electron in a magnetic field B-Bok, H--y5.B, find the time evolution Unitary operator given as U(6) - exp(-()/N). Given that the initial quantum state is le (O) >= 0) find the state after a time t given as (10 points) () >= U(t)|(0) >
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