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, Problem 4.59P
To determine
The expectation value of the product of the given components of the spin
Expert Solution & Answer
Want to see the full answer?
Check out a sample textbook solution
Students have asked these similar questions
By using the joint eigenstates of eigenstates {|s, m¿}} of Ŝ² and Ŝ, as basis,
Find the matrix representation and Dirac form of Ŝx, Ŝy, Ŝz, Ŝ+, §ž and
(a)
3
S, for a particle of spin s =
2°
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.
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 time
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
- In terms of the totally antisymmetric E-symbol (Levi-Civita tensor) with €123 = +1, the vector product can be written as (A x B) i = tijk Aj Bk, where i, j, k = 1, 2, 3 and summation over repeated indices (here j and k) is implied. i) ii) iii) iv) For general vectors A and B, using (2) prove the following relations: a) A x B=-B x A b) (A x B) A = (A x B) - B = 0. The Levi-Civita symbol is related to the Kronecker delta. Prove the following very useful formula €ijk€ilm = 8j18km - Sjm³ki. (2) Prove the formula (3) €imn€jmn = 2dij. Assuming that (3) is true (and using antisymmetry of the E-symbol), prove the relation A x (B x C) = (AC) B- (AB) Carrow_forwardDerive Eq.(7.13) by using set of equations (7.2 a,b,c and d).arrow_forwardThe spin component along y is measured and found to be h-bar/2. Write down the spinor immediately after the measurement, in (a)the basis of eigenstates of sy, (b) the usual basis. The measurement in part (1) occurred at time t = 0. The particle subsequently moves in an applied field such that H = Ωsz. What is the spinor, in the usual basis, at time t?arrow_forward
- Problem 3: Two-level system and density matrice Suppose a 2 x 2 matrix X (not necessarily Hermitian or unitary) is written as X = a000 + a.σ, where ao and ak, k = 1, 2, 3, are numbers, 0o = 1 is the identity matrix and o are the Pauli matrices. (a) How are ao and a related to tr(X) and tr(OX)? Obtain ao and ak in terms of the matrix elements Xij. Assume that ao, ak ER such that X is Hermitian and could be interpreted as a Hamiltonian, what are the eigenvalues of X?arrow_forwardThe 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!arrow_forwardConsider a system of two particles, one with spin 1 and the other with spin 1/2. The particles interact with one another via the Hamiltonian H = Eo [S(1). S(2)] + E₁ [ S:(1) + S. (2)] ħ " where Eo and E₁ are constants with units of energy, S is the spin operator and the labels (1) and (2) indicate the particle on which it operates. (a) Find the energy spectrum of this system, i.e. the eigenvalues of the Hamiltonian H and identify the ground state and first excited state. For numerical purposes assume Eo = 0.2eV and E₁ = 0.1eV. (b) If the two particle system is known to be in a configuration such that the total spin is 1/2, what values a mea- surement of S% of the spin 1 particle would produce and what are the probabilities associated with each one of those values?arrow_forward
- Consider the 2D harmonic oscillator Hamiltonian: 87 - 12/24 (13² + 8² ) + 2 ²³² (1² + 8² ) Ĥ mw² 2m Unless otherwise specified, we will work in the eigenstates that satisfy: Ĥ|nz, ny) = Enz,ny |nx, ny) x, with Eng,ny = ħw(nx + Ny + 1). (a.) Some energy levels are degenerate. For example, E 2ħw can be achieve with (nx, ny) = (1, 0); (0, 1). This energy level has a degeneracy D(2ħw) = 2. What is the degeneracy of energy level E (where N is a positive integer)? = Nhw (b.) Consider the state (0)) = √ (12,0) + 2 |1, 1) + (0,2)). (c.) Calculate (Ĥ), (px), (py), and (âŷ) for the state above. = What is (t)) at a later time t > 0?arrow_forwardConsider a rectangular surface of length L and width W in the xy plane with its center at the origin: Which of the following are valid expressions for the area vector of this surface? Check all that apply. O (0,0, LW) O (W, L, 0) O (0,0, -LW) O (LW, LW, 0) O (0, LW, 0) O (L, W, 0)arrow_forwardStarting with the equation of motion of a three-dimensional isotropic harmonic ocillator dp. = -kr, dt (i = 1,2,3), deduce the conservation equation dA = 0, dt where 1 P.P, +kr,r,. 2m (Note that we will use the notations r,, r2, r, and a, y, z interchangeably, and similarly for the components of p.)arrow_forward
- Prove the following commutator identity: [AB, C] = A[B. C]+[A. C]B.arrow_forwardShow that if a matrix is orthogonal and its determinant is +1, then each element of the matrix is equal to its own cofactor. Hint: Use (6.13) and the definition of an orthogonal matrix.arrow_forwardWrite the matrices which produce a rotation θ about the x axis, or that rotation combined with a reflection through the (y,z) plane. [Compare (7.18) and (7.19) for rotation about the z axis.]arrow_forward
arrow_back_ios
SEE MORE QUESTIONS
arrow_forward_ios
Recommended textbooks for you
College PhysicsPhysicsISBN:9781305952300Author:Raymond A. Serway, Chris VuillePublisher:Cengage Learning
University Physics (14th Edition)PhysicsISBN:9780133969290Author:Hugh D. Young, Roger A. FreedmanPublisher:PEARSONIntroduction To Quantum MechanicsPhysicsISBN:9781107189638Author:Griffiths, David J., Schroeter, Darrell F.Publisher:Cambridge University Press
Physics for Scientists and EngineersPhysicsISBN:9781337553278Author:Raymond A. Serway, John W. JewettPublisher:Cengage Learning
Lecture- Tutorials for Introductory AstronomyPhysicsISBN:9780321820464Author:Edward E. Prather, Tim P. Slater, Jeff P. Adams, Gina BrissendenPublisher:Addison-Wesley
College Physics: A Strategic Approach (4th Editio...PhysicsISBN:9780134609034Author:Randall D. Knight (Professor Emeritus), Brian Jones, Stuart FieldPublisher:PEARSON
College Physics
Physics
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Cengage Learning
University Physics (14th Edition)
Physics
ISBN:9780133969290
Author:Hugh D. Young, Roger A. Freedman
Publisher:PEARSON
Introduction To Quantum Mechanics
Physics
ISBN:9781107189638
Author:Griffiths, David J., Schroeter, Darrell F.
Publisher:Cambridge University Press
Physics for Scientists and Engineers
Physics
ISBN:9781337553278
Author:Raymond A. Serway, John W. Jewett
Publisher:Cengage Learning
Lecture- Tutorials for Introductory Astronomy
Physics
ISBN:9780321820464
Author:Edward E. Prather, Tim P. Slater, Jeff P. Adams, Gina Brissenden
Publisher:Addison-Wesley
College Physics: A Strategic Approach (4th Editio...
Physics
ISBN:9780134609034
Author:Randall D. Knight (Professor Emeritus), Brian Jones, Stuart Field
Publisher:PEARSON