Introduction To Quantum Mechanics
3rd Edition
ISBN: 9781107189638
Author: Griffiths, David J., Schroeter, Darrell F.
Publisher: Cambridge University Press
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Chapter 6.4, Problem 6.9P
(a)
To determine
Prove that
(b)
To determine
Prove that
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(a) Show that for a Hermitian bounded linear operator Ĥ : H → H, all of its eigen-
values are real and the eigenvectors corresponding to different eigenvalues are
orthogonal. Hint: start by calculating (ø|H|ø) for an eigenstate lø).
Check if the following operators with the corresponding functions could
form an eigen value equations or not (where Bis a constant value)
No.
function
Оperator
3
2
3
sin(ßx)
sin(Bx)
d
dx
4
sin(ßx)
dx
if A and B are Hermitian operators, show that from (AB+BA), (AB-BA) which one Hermitian and non Hermitian?
Chapter 6 Solutions
Introduction To Quantum Mechanics
Ch. 6.1 - Prob. 6.1PCh. 6.2 - Prob. 6.2PCh. 6.2 - Prob. 6.3PCh. 6.2 - Prob. 6.4PCh. 6.2 - Prob. 6.5PCh. 6.2 - Prob. 6.7PCh. 6.4 - Prob. 6.8PCh. 6.4 - Prob. 6.9PCh. 6.4 - Prob. 6.10PCh. 6.4 - Prob. 6.11P
Ch. 6.4 - Prob. 6.12PCh. 6.4 - Prob. 6.13PCh. 6.5 - Prob. 6.14PCh. 6.5 - Prob. 6.15PCh. 6.5 - Prob. 6.16PCh. 6.5 - Prob. 6.17PCh. 6.6 - Prob. 6.18PCh. 6.6 - Prob. 6.19PCh. 6.7 - Prob. 6.20PCh. 6.7 - Prob. 6.21PCh. 6.7 - Prob. 6.22PCh. 6.7 - Prob. 6.23PCh. 6.7 - Prob. 6.25PCh. 6.8 - Prob. 6.26PCh. 6.8 - Prob. 6.27PCh. 6.8 - Prob. 6.28PCh. 6.8 - Prob. 6.30PCh. 6 - Prob. 6.31PCh. 6 - Prob. 6.32PCh. 6 - Prob. 6.34PCh. 6 - Prob. 6.35PCh. 6 - Prob. 6.36PCh. 6 - Prob. 6.37P
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- Show that the quadrupole term in the multipole expansion can be written as 1 1 Vquad (r): 4neo r3 ij=1 where Note that Qiy is a rank-two tensor, so it is possible to express it as a matrix. Also show that Qij is traceless.arrow_forwardSuppose that you have three vectors: fi (x) = 1, f2 (x) = x quadratic functions with the inner product: (f; | f;) = [° fi (x) f; (x) e¯ªdx. Suppose we have a derivative operator D = . What is (f3 |D| f2) =? 1, and f3 (x) =; (x² – 4x + 2), that make up an orthonormal basis spanning the real vector space of - dx 1 5/8 -3/4arrow_forwardWhat are the normalized eigenfunctions of the operator 'Px = -iħ? Under what conditions are these eigenfunctions orthonormal, i. e. yi (x) y;(x)dx= d} where Vi is the i-th eigenfunction of the previous operator and * = { ° 0 sii+ j 1 sii= j It is the Kronecker delta.arrow_forward
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