Fund Quantum Mechanics Lect & HW Solutions 34

Fund Quantum Mechanics Lect & HW Solutions 34 - 16...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right 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 ofthe inner product one by one....
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online