Introduction To Quantum Mechanics
3rd Edition
ISBN: 9781107189638
Author: Griffiths, David J., Schroeter, Darrell F.
Publisher: Cambridge University Press
expand_more
expand_more
format_list_bulleted
Question
Chapter 4.4, Problem 4.36P
(a)
To determine
The Hamiltonian matrix for an electron at rest in an oscillating magnetic field.
(b)
To determine
The eigen vector
(c)
To determine
The probability of getting
(d)
To determine
The minimum field
Expert Solution & Answer
Trending nowThis is a popular solution!
Students have asked these similar questions
Prove the following commutator identity:
[AB, C] = A[B. C]+[A. C]B.
Consider a composite state of spin j1 = s = 1/2 and angular momentum j2 = l = 2
of an electron. Find all the eigenstates of |j1, j2; j,m〉 as the linear combination of
product states of spin and angular momentum. Give the values of Clebsch-Gordon
coefficients you get from here.
The Hamiltonian of a spin in a constant magnetic field B aligned with the y axis is given by H = aSy, where a is a constant.
a) Use the energies and eigenstates for this case to determine the time evolution psi(t) of the state with initial condition psi(0) = (1/root(2))*matrix(1,1). (Vertical matrix, 2x1!)
b) For your solution from part (a), calculate the expectation values <Sx>, <Sy>, <Sz> as a function of time.
I have attached the image of the orginial question!
Chapter 4 Solutions
Introduction To Quantum Mechanics
Ch. 4.1 - Prob. 4.1PCh. 4.1 - Prob. 4.3PCh. 4.1 - Prob. 4.4PCh. 4.1 - Prob. 4.5PCh. 4.1 - Prob. 4.6PCh. 4.1 - Prob. 4.7PCh. 4.1 - Prob. 4.8PCh. 4.1 - Prob. 4.9PCh. 4.1 - Prob. 4.10PCh. 4.1 - Prob. 4.11P
Ch. 4.2 - Prob. 4.12PCh. 4.2 - Prob. 4.13PCh. 4.2 - Prob. 4.14PCh. 4.2 - Prob. 4.15PCh. 4.2 - Prob. 4.16PCh. 4.2 - Prob. 4.17PCh. 4.2 - Prob. 4.18PCh. 4.2 - Prob. 4.19PCh. 4.2 - Prob. 4.20PCh. 4.3 - Prob. 4.21PCh. 4.3 - Prob. 4.22PCh. 4.3 - Prob. 4.23PCh. 4.3 - Prob. 4.24PCh. 4.3 - Prob. 4.25PCh. 4.3 - Prob. 4.26PCh. 4.3 - Prob. 4.27PCh. 4.4 - Prob. 4.28PCh. 4.4 - Prob. 4.29PCh. 4.4 - Prob. 4.30PCh. 4.4 - Prob. 4.31PCh. 4.4 - Prob. 4.32PCh. 4.4 - Prob. 4.33PCh. 4.4 - Prob. 4.34PCh. 4.4 - Prob. 4.35PCh. 4.4 - Prob. 4.36PCh. 4.4 - Prob. 4.37PCh. 4.4 - Prob. 4.38PCh. 4.4 - Prob. 4.39PCh. 4.4 - Prob. 4.40PCh. 4.4 - Prob. 4.41PCh. 4.5 - Prob. 4.42PCh. 4.5 - Prob. 4.43PCh. 4.5 - Prob. 4.44PCh. 4.5 - Prob. 4.45PCh. 4 - Prob. 4.46PCh. 4 - Prob. 4.47PCh. 4 - Prob. 4.48PCh. 4 - Prob. 4.49PCh. 4 - Prob. 4.50PCh. 4 - Prob. 4.51PCh. 4 - Prob. 4.52PCh. 4 - Prob. 4.53PCh. 4 - Prob. 4.54PCh. 4 - Prob. 4.55PCh. 4 - Prob. 4.56PCh. 4 - Prob. 4.57PCh. 4 - Prob. 4.58PCh. 4 - Prob. 4.59PCh. 4 - Prob. 4.61PCh. 4 - Prob. 4.62PCh. 4 - Prob. 4.63PCh. 4 - Prob. 4.64PCh. 4 - Prob. 4.65PCh. 4 - Prob. 4.66PCh. 4 - Prob. 4.70PCh. 4 - Prob. 4.72PCh. 4 - Prob. 4.73PCh. 4 - Prob. 4.75PCh. 4 - Prob. 4.76P
Knowledge Booster
Similar questions
- Spin/Field Hamiltonian Consider a spin-1/2 particle with a magnetic moment µ = -e/m$ placed in a uniform magnetic field aligned along the z axis. (a) Write the Hamiltonian for this system in matrix form. (b) Verify by explicit matrix calculation that the Hamiltonian does not commute with the spin operators in the r and y directions. Comment on how this affects the expectation values of these operators.arrow_forwardA free electron is at rest in a uniform magnetic field B = Bok, where Bo is a constant. At time t=0, the electron is in a state with spin up in the x-direction: ((Sx) = +½). Find the probability that at later time t=T, the electron will be found in a state with spin down in the x-direction.arrow_forwardThe Hamiltonian of an electron in a constant magnetic fieldB is given by H = µỡ · B. where u is a positive constant and o = (0,,0,,0,) denotes the Pauli matrices. Let %3D 2 = µB /h and I be the 2x2 unit matrix. Then the operator eiHt/h simplifies toarrow_forward
- The operator în · ở measures spin in the direction of unit vector f = (nx, Ny, N₂) nx = sin cosp ny = sinesino nz = cose in spherical polar coordinates, and ở = (x, y, z) for Pauli spin matrices. (a) Determine the two eigenvalues of û.o.arrow_forwardAt time t = 0, an electron and a positron are formed in a state with total spin angular momentum equal to zero, perhaps from the decay of a spinless particle. The particles are situated in a uniform magnetic field B0 in the z direction. If interaction between the electron and the positron may be neglected, show that the spin Hamiltonian of the system may be written as Ĥ = ω0(Ŝ1z - Ŝ2z), where Ŝ1 is the spin operator of the electron, Ŝ2 is the spin operator of the positron, and ω0 is a constant. Show all work and explanations pleasearrow_forwardConsider the Spin Hamiltonian: Ĥ = -AOT where A E R is a non-negative real number. Use as a trial wave function: |&(0)) = cos(0/2) |+z) + sin(0/2) |−z) (a.) What is the exact ground state and exact ground state energy for this Hamiltonian? (b.) Using the trial wave function, calculate E(0), the energy as a function of the parameter 0. (c.) Using the variational principle, estimate the ground state energy. How good is your solution and why?arrow_forward
- = A free electron is at rest in a uniform magnetic field B Bok, where Bo is a constant. At time t=0, the electron is in a state with spin up in the x-direction: ((S) = +2). Find the probability that at later time t=T, the electron will be found in a state with spin down in the x-direction.arrow_forwardThe Hamiltonian of a spin S in a magnetic field B is given by \hspace {3cm} H = -yS · B where y is a positive constant. If a magnetic field of strength B is applied in the z– direction for a spin-half single electron system, taking @ = y B, the eigenvalues of the Hamiltonian are Select one: a. -ho, ħo O b. -hw,0, ¿ħo -ho, ho Ос. O d. -ho, ho O e. -ħw, 0, ħoarrow_forwardHamiltonian of an axially symmetric rotator acting on a state |lm) is given by: H = 21 212 where I, and Iz are the moments of inertia and Ly, Ly and L, are components of the angular momentum operator. a) Find expectation values of H. b) In the case of a rigid rotator (i.e., I, = 1½ = I ), find the energy expression and the corresponding degeneracy relations. c) Calculate the orbital quantum number / and the corresponding energy degeneracy of the rotator where the magnitude of the total angular momentum is v30h.arrow_forward
- Consider a particle of spin s = 3/2. (a) Find the matrices representing the operators S^ x , S^ y ,S^ z , ^ Sx 2 and ^ S y 2 within the basis of ^ S 2 and S^ z (b) Find the energy levels of this particle when its Hamiltonian is given by ^H= ϵ 0 h 2 ( Sx 2−S y 2 )− ϵ 0 h ( S^ Z ) where ϵ 0 is a constant having the dimensions of energy. Are these levels degenerate? (c) If the system was initially in an eigenstate Ψ0=( 1 0 0 0) , find the state of the system at timearrow_forwardThe following Hamiltonian describes spins in a magnetic field: Ĥ = ω0Ŝz.Show the following: the entangled state 1/√2 (|+z⟩1 |+z⟩2 + |−z⟩1 |−z⟩2) gets a relative phase at two times the rate that the independent spins would.arrow_forwardThe Hamiltonian of a system with two states is given by the following expression: H = ħwoox where x = [₁¹] And the eigen vectors for this Hamiltonian are: |+) 調 1-) = √2/24 Whose eigen values are E₁ = = = ħwo and E₂ We initialise the system in the state. (t = 0)) = cos (a)|+) + sin(a)|-) -ħwo respectively. a. Show the initial state |(t = 0)) is correctly normalised. b. For this problem write the full time-dependent wavefunction. C. If the system starts in the state |(t = 0)) = (1+) + |−))/√√2 (i.e., a = π/4), what is the probability it will stay in this state as a function of time?arrow_forward
arrow_back_ios
SEE MORE QUESTIONS
arrow_forward_ios
Recommended textbooks for you
Classical Dynamics of Particles and SystemsPhysicsISBN:9780534408961Author:Stephen T. Thornton, Jerry B. MarionPublisher:Cengage Learning
Classical Dynamics of Particles and Systems
Physics
ISBN:9780534408961
Author:Stephen T. Thornton, Jerry B. Marion
Publisher:Cengage Learning