This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Physics 3220 Quantum Mechanics 1 Fall 2008 Problem Set #13 Due Wednesday, December 10 at 2pm Problem 13.1 : Surveys! (20 points) Please take the following surveys. You will not be graded for accuracy for these surveys, you get credit just for participating. a) http : // www . colorado . edu / sei / surveys / Fall08 / Clicker Phys3220 fa08 post . html b) http : // www . colorado . edu / physics / EducationIssues / baily / SurveyFa08 / MPASFall08Post 3220 . htm Problem 13.2 : Projection operators. (20 points) Consider first a Hilbert space spanned by a basis of orthonormal states { n } labeled by one discrete quantum number, which we will call n . The states  n are the eigenstates of some operator Q , and n labels the various eigenvalues, Q  n = q n  n . (For example, Q could be the Hamiltonian, and n could label the allowed energies.) For each n , we define the projection operator onto the state  n as P n  n n  . (1) Thus there is a different P n for each state  n . a) Demonstrate that P n is Hermitian, and that P 2 n = P n . b) What is the result of acting P n on an arbitrary state  = m c m  m ? Explain why the name projection operator is justified. If there are N distinct values of n , all operators will be N N matrices; what does P n look like as such a matrix? c) In general P n  is not normalized; show that the state P n  /  P n  is properly normalized. d) How are the number  P n  and the state P n  /  P n  related to the result of making a measurement of Q ? Relate them to the postulates of quantum mechanics. Now consider a system where the Hilbert space is labeled by more than one quantum number: the hydrogen atom, with states  n m labeled by n , and m . The projection operator associated to a given value of n now has a sum over all values of the other quantum numbers: P n = =0 m =  n m n m  . (2) 1 In the following consider the hydrogen atom wavefunction  = 1 2  2 1 0 + 2...
View
Full
Document
 Fall '08
 STEVEPOLLOCK
 mechanics

Click to edit the document details