The spin matrices
Answer to Problem 4.62P
The spin matrices
Explanation of Solution
From Equation 4.135, the quantization of
Identifying the states by the value of
The matrix element of
From Equation 4.136, the quantization of
All non-zero elements have row index
Similarly, for
Therefore,
Write the expression to fine the spin matrix
Write the expression to fine the spin matrix
Therefore,
And,
Conclusion:
Thus, the spin matrices
Want to see more full solutions like this?
Chapter 4 Solutions
Introduction To Quantum Mechanics
- Divergence theorem. (a) Use the divergence theorem to prove, v = -478 (7) (2.1) (b) [Problem 1.64, Griffiths] In case you're not persuaded with (a), try replacing r by (r² + e²)2 and watch what happens when ɛ → 0. Specifically, let 1 -V². 4л 1 D(r, ɛ) (2.2) p2 + g2 By taking note of the defining conditions of 8°(7) [(1) at r = 0, its value goes to infinity, (2) for all r + 0, its value is 0, and (3) the integral over all space is 1], demonstrate that 2.2 goes to 8*(F) as ɛ → 0.arrow_forwardnumber 6 a b c pleasearrow_forwardProblem #1 (Problem 5.3 in book). Come up with a function for A (the Helmholtz free energy) and derive the differential form that reveals A as a potential: dA < -SdT – pdV [Eqn 5.20]arrow_forward
- Part B with all steps and workingarrow_forwardProblem 3.27 Sequential measurements. An operator Ä, representing observ- able A, has two normalized eigenstates 1 and 2, with eigenvalues a1 and a2, respectively. Operator B, representing observable B, has two normalized eigen- states ø1 and ø2, with eigenvalues b1 and b2. The eigenstates are related by = (3ø1 + 402)/5, 42 = (401 – 302)/5. (a) Observable A is measured, and the value aj is obtained. What is the state of the system (immediately) after this measurement? (b) If B is now measured, what are the possible results, and what are their probabilities? (c) Right after the measurement of B, A is measured again. What is the proba- bility of getting a¡? (Note that the answer would be quite different if I had told you the outcome of the B measurement.)arrow_forwardplease solvearrow_forward
- Let f(x, y) = x² + 4y² and let C be the line segment from (0, 0) to (2, 2). You are going to compute là Vf. dr two ways: first, using the method learned in section 6.2 for с evaluating line integrals, and second, using the fundamental theorem for line integrals. First way: Vf=( C can be parameterized by r(t) = (t, Then '(t) and ▼ ƒ(r(t)) = { so sv. So 2 = [² = 2 - 1² || ( Vf. dr ▼ f(r(t)). r' (t)dt dt ). > > for 0 ≤ t ≤ 2. ).arrow_forwardFor Problem 8.35, how do I prove, or perhaps verify, what it is they're asking for?arrow_forwardExplain step by step this integrationarrow_forward
- Consider the following unattached spring system. m₁ = 1, C₁ = 14, Ea{ } m₂ = 7 Write the stiffness matrix K = Write the matrix M-¹K Find the eigenvalues and eigenvectors of M-¹K: • Smaller eigenvalue = with eigenvector • Larger eigenvalue = with eigenvector If the spring system oscillates beginning with initial displacement u(0) = [8] and initial velocity u'(0) = u₁(t) = u₂(t) = Find a nonzero initial velocity vector such that the displacement of the masses will be bounded. • u'(0) = | -22 then compute the displacements of the masses at time t. 10arrow_forward1. Consider the 2D motion of a particle of mass u in a central force field with potential V(r). a) Find the r, o polar-coordinate expression of the Lagrangian for this system and write down the corresponding Euler-Lagrange e.o.m.s. b) Note that the angular variable o is cyclic. What is the physical interpretation of the correspond- ing integral of motion? (For the definitions of the italicized terms see this link.) c) Solve for o in terms of this integral of motion and substitute the result into the Euler-Lagrange equation for r. Show that the result can be arranged to look like a purely 1D e.o.m. of the form dVef(r) (1) dr Identify in the process the explicit expression for Vef(r), which will depend among other things on the integral of motion. d) Take now k V (r) = with k > 0 to be an attractive electrostatic/gravitational-type potential. Sketch the profile of the corresponding effective potential function Vef(r). Find the equilibrium solution for the correspond- ing e.o.m. (1). What…arrow_forwardThe Hamiltonian of a one-dimensional harmonic oscillator can be written in natural units (m = hbar= ω = 1) as: (image1) Where ˆa =(ˆx+ipˆ)/√2, and ˆa† =(ˆx−ipˆ)/√2 One of the proper functions is: (image2) Find the two eigenfunctions closest in energy to the function ψa (you don't have to normalize) .arrow_forward
