Quantum Mechanics for Scientists and Engineers

Transcription

1 Quantu Mechancs or Scentsts and Engneers Sangn K Advanced Coputatonal Electroagnetcs Lab redkd@yonse.ac.kr Nov. 4 th, 26

2 Outlne Quantu Mechancs or Scentsts and Engneers Blnear expanson o lnear operators The dentty operator Inverse and Untary operators 2

3 Blnear expanson o lnear operators Expand unctons n a bass set n n n ( x) c ( x) or ( x) c ( x) n n n By actng wth a specc operator  g Expand g and on the bass set g d g Fro our atrx representaton o c d Ac we know that c So, d A 3

4 Blnear expanson o lnear operators Expand unctons n a bass set d Substtutng back nto c Reeber that s sply a nuber g A, A g d So, swtched ultplcatve order g, A, A, A 4

5 Blnear expanson o lnear operators Expand unctons n a bass set Ths or, A s reerred to as a blnear expanson o the operator on the bass and s analogous to the lnear expanson o a vector on a bass  Though the Drac notaton s ore general g x) A ( x), * ( x ) ( ˆ dx A ( x ) 5

6 Outer product Blnear expanson or An expresson o the or, A Contans an outer product o two vectors An nner product expresson o the or g Coplex nuber An outer product expresson o the or g Matrx 6

7 Outer product The specc suaton, s actually, then, a su o atrces A In the atrx the eleent n the th row and the th colun s, another's are 7

8 The dentty operator The dentty operator Î In atrx or, the dentty operator s In bra-ket or The dentty operator can be wrtten Iˆ where the or a coplete bass or the space 8

9 The dentty operator Proo For an arbtrary uncton, we know So And, the ultplcaton each sde o Iˆ Iˆ By usng,. So, Iˆ 9

10 The dentty operator The stateent Slarly, we can obtan I ˆ So that ˆ I

11 Proo that the trace s ndependent o the bass Consder the su, S o the dagonal eleents o an operator on soe coplete orthonoral bass S A ˆ And, suppose soe other coplete orthonoral bass S Iˆ In, we can nsert an dentty operator ust beore S A ˆ  IA ˆˆ Iˆ Â

12 Proo that the trace s ndependent o the bass Rearrangng S IA ˆˆ 2

13 Proo that the trace s ndependent o the bass So, wth now Usng the Iˆ S Iˆ S Hence the trace o an operator the su o the dagonal eleents s ndependent o the bass used to represent the operator 3

14 Inverse and proecton operator Inverse operator  For an operator operatng on an arbtrary uncton The nverse operator ˆ A Usng the nverse operator ˆ A ˆ A Iˆ Proecton operator For exaple, Pˆ In general has no nverse because t proects all nput vectors onto only one axs n the space the one correspondng to the specc vector 4

15 Untary operator Untary operator Uˆ ˆ One or whch U U That s, ts nverse s ts Hertan adont ˆ The Hertan adont s ored by relectng on a -45 degree lne and takng the coplex conugate 5

16 Untary operator Conservaton o length or untary operators For two atrces  and Bˆ That s, the Hertan adont o the product s the lpped round product o the Hertan adonts Consder the untary operator Uˆ Usng the operator Uˆ new old old and vectors g Uˆ new g old gold g Uˆ Then, So, new g old The untary operaton does not change the nner product The length o a vector s not changed by a untary operator 6

17 Thank You or Your Attenton, Do You Have Any Questons?