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 in the positive z-direction at a later time t is P(Ss+ h)-,S [1 + sin ( γ By t )].](http://img.homeworklib.com/questions/9d927730-a0a8-11eb-93c2-bf03a2bad9c6.png?x-oss-process=image/resize,w_560)
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Intro to Quantum Mechanics Problem:
An electron under the influence of a uniform magnetic field By...
Problem 7.49 A hydrogen atom is placed in a uniform magnetic field Bo Bo (the Hamiltonian can be ...
Problem 7.49
Problem 7.49 A hydrogen atom is placed in a uniform magnetic field Bo Bo (the Hamiltonian can be written as in Equation 4.230). Use the Feynman-Hellman theorem (Problem 7.38) to show that a En (7.114) where the electron's magnetic dipole moment10 (orbital plus spin) is Yo l-mechanical + γ S . μ The mechanical angular momentum is defined in Equation 4.231 a volume V and at 0 K (when they're all in the ground state) is41 Note: From...
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...
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....
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
Hello, I need help with a problem for my Quantum Mechanics class. Please explain as if...
Hello, I need help with a problem for my Quantum Mechanics class. Please explain as if I am learning for the first time. I want to be able to understand and do problems like this on my own. Thank you in advance for your help! The infinite square well has solutions that are very familiar to us from previous physics classes. However, in this class we learn that a quantum state of the system can be in a superposition state...
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) >
points An electron enters a region of a uniform magnetic field with velocity V- 5.0 10...
points An electron enters a region of a uniform magnetic field with velocity V- 5.0 10 m/s in sy direction. The magnitude of the field is 10 T and is in ydirection What is the magnetic force matitude and direction) exerted on the electron from the field? 2.4-12 N in the direction 2.4e-12 N in the ty direction 2.de-12 N in the y direction The force is zero 2.4-12 N in the direction
9. An electron moving with non-relativistic velocity v in an electric field E experiences a magnetic...
9. An electron moving with non-relativistic velocity v in an electric field E experiences a magnetic fieldB given by: v x (-V(r)) v x E B=- where (r) is the electric potential. This magnetic field interacts with the magnetic moment u of the electron given by -S, =n me where S is the electron spin. Assuming non-relativistic mechanics, show that the Hamiltonian representing this effect (spin-orbit coupling) for a spherically-symmetric electric potential is: 1 dφ(r) S.L ΔΗ [6] r dr...
An electron of magnetic moment mu = gamma S is placed in a magnetic field B = (-omega x/ gamma, - omega y/gamma,- omega z/gamma). Show that the evolution operator of the spin is U(t,0) = exp(-imut).
An electron of magnetic moment υ = γ S is placed in a magnetic field B = (-ωx/γ, -ωy/γ, -ωz/γ). Show that the evolution operator of the spin is U(t,0) = exp(-iυt)
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...
Problem 7.49
Problem 7.49 A hydrogen atom is placed in a uniform magnetic field Bo Bo (the Hamiltonian can be written as in Equation 4.230). Use the Feynman-Hellman theorem (Problem 7.38) to show that a En (7.114) where the electron's magnetic dipole moment10 (orbital plus spin) is Yo l-mechanical + γ S . μ The mechanical angular momentum is defined in Equation 4.231 a volume V and at 0 K (when they're all in the ground state) is41 Note: From...
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...
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....
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
Hello, I need help with a problem for my Quantum Mechanics class. Please explain as if I am learning for the first time. I want to be able to understand and do problems like this on my own. Thank you in advance for your help! The infinite square well has solutions that are very familiar to us from previous physics classes. However, in this class we learn that a quantum state of the system can be in a superposition state...
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) >
points An electron enters a region of a uniform magnetic field with velocity V- 5.0 10 m/s in sy direction. The magnitude of the field is 10 T and is in ydirection What is the magnetic force matitude and direction) exerted on the electron from the field? 2.4-12 N in the direction 2.4e-12 N in the ty direction 2.de-12 N in the y direction The force is zero 2.4-12 N in the direction
9. An electron moving with non-relativistic velocity v in an electric field E experiences a magnetic fieldB given by: v x (-V(r)) v x E B=- where (r) is the electric potential. This magnetic field interacts with the magnetic moment u of the electron given by -S, =n me where S is the electron spin. Assuming non-relativistic mechanics, show that the Hamiltonian representing this effect (spin-orbit coupling) for a spherically-symmetric electric potential is: 1 dφ(r) S.L ΔΗ [6] r dr...
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...
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