Introduction To Quantum Mechanics
Introduction To Quantum Mechanics
3rd Edition
ISBN: 9781107189638
Author: Griffiths, David J., Schroeter, Darrell F.
Publisher: Cambridge University Press
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Chapter 4.4, Problem 4.33P
To determine

The matrix Sr representing the component of spin angular momentum along an arbitrary direction r^. And the eigenvalues and normalized eigenspinors of Sr.

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3.20 Construct the matrix representations of the operators J, and J, for a spin 1 system, in the J. basis, spanned by the kets |+) = [1,1), 10) = 1,0), and -) = 1,-1). Use these matrices to find the three analogous eigenstates for each of the two operators J, and J, in terms of +), 10), and -).
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?
1.4 Show that the distance of closest approach d, in Rutherford scattering leading to an angle of deflection 0, is given by d=(1+ cosec 0,2). where p is detined in Fig. 1.6. (Use the conservation of energy and angular momentum.]
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