Quantum Engi Q and A_Part_6

Quantum Engi Q and A_Part_6 - 16 CHAPTER 2. MATHEMATICAL...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 16 CHAPTER 2. MATHEMATICAL PREREQUISITES To get rid of the change of sign, you can add a factor i to the operator, since the i adds a compensating minus sign when you bring it inside the complex conjugate: (bigg f vextendsingle vextendsingle vextendsingle vextendsingle i d d x g )bigg = − integraldisplay b a parenleftBigg − i d f d x parenrightBigg ∗ g d x = (bigg i d d x f vextendsingle vextendsingle vextendsingle vextendsingle g )bigg This makes id / d x a Hermitian operator. 2.6.7 Solution herm-g Question: Show that if A is a Hermitian operator, then so is A 2 . As a result, under the conditions of the previous question, − d 2 / d x 2 is a Hermitian operator too. (And so is just d 2 / d x 2 , of course, but − d 2 / d x 2 is the one with the positive eigenvalues, the squares of the eigenvalues of id / d x .) Answer: To show that A 2 is Hermitian, just move the two operators A to the other side of the inner product one by one. As far as the eigenvalues are concerned, each application of A to one of its eigenfunctions multiplies by the eigenvalue, so two applications of A multiplies by the square eigenvalue. 2.6.8 Solution herm-h Question: A complete set of orthonormal eigenfunctions of − d 2 / d x 2 on the interval 0 ≤ x ≤ π that are zero at the end points is the infinite set of functions sin( x ) radicalBig π/ 2 , sin(2 x ) radicalBig π/ 2 , sin(3 x ) radicalBig π/ 2 , sin(4 x ) radicalBig π/ 2 , . . . Check that these functions are indeed zero at x = 0 and x = π , that they are indeed orthonor- mal, and that they are eigenfunctions of − d 2 / d x 2 with the positive real eigenvalues 1 , 4 , 9 , 16 , . . ....
View Full Document

This note was uploaded on 11/13/2011 for the course PHY 4458 taught by Professor Garvin during the Fall '11 term at University of Florida.

Page1 / 3

Quantum Engi Q and A_Part_6 - 16 CHAPTER 2. MATHEMATICAL...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online