Quantum Mechanics for Scientists and Engineers. David Miller

Transcription

1 Quantum Mechancs for Scentsts and Engneers Davd Mller

2 Types of lnear operators

3 Types of lnear operators Blnear expanson of operators

4 Blnear expanson of lnear operators We know that we can expand functons n a bass set f x cn n x as n f x c x or n n n What s the equvalent expanson for an operator? We can deduce ths from our matrx representaton Consder an arbtrary functon f, wrtten as the ket f from whch we can calculate a functon g wrtten as the ket g by actng wth a specfc operator  g Aˆ f n

5 Blnear expanson of lnear operators We expand g and f on the bass set g d f c From our matrx representaton of g Aˆ f we know that d A c and, by defnton of the expanson coeffcent we know that c f so d A f

6 Blnear expanson of lnear operators Substtutng d A f back nto g d g A f, Remember that f c s smply a number so we can move t wthn the multplcatve expresson Hence we have g A f A f,, But g Aˆ f and g and f are arbtrary, so Aˆ A, gves

7 Blnear expanson of lnear operators Ths form Aˆ A, s referred to as a blnear expanson of the operator  on the bass and s analogous to the lnear expanson of a vector on a bass Any lnear operator that operates wthn the space can be wrtten ths way

8 Blnear expanson of lnear operators Though the Drac notaton s more general and elegant for functons of a smple varable where g x Af ˆ x1 dx1 we can analogously wrte the blnear expanson n the form Aˆ A x x1,

9 Outer product An expresson of the form Aˆ A, contans an outer product of two vectors An nner product expresson of the form g f results n a sngle, complex number An outer product expresson of the form g f generates a matrx

10 Outer product d1 dc 1 1 dc 1 2 dc 1 3 d 2 dc 2 1 dc 2 2 dc 2 3 g f c1 c2 c 3 d3 dc 3 1 dc 3 2 dc 3 3 The specfc summaton Aˆ A, s actually, then, a sum of matrces In the matrx the element n the th row and the th column s 1 All other elements are zero

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12 Types of lnear operators The dentty operator

13 Identty operator The dentty operator Î s the operator that when t operates on a vector (functon) leaves t unchanged In matrx form, the dentty operator s In bra-ket form the dentty operator can be wrtten where the form a complete bass for the space ˆ I Iˆ

14 Identty operator - proof For an arbtrary functon f c we know cm m so f f Now, wth our proposed form then Iˆ f f But f s ust a number Iˆ and so t can be moved n the product Hence Iˆ f f and hence, usng f f, Î f f f

15 Identty operator The statement Iˆ s trval f s the bass used to represent the space Then so that

16 Identty operator Smlarly so Iˆ

17 Identty operator Note, however, that Iˆ even f the bass beng used s not the set Then some specfc s not a vector wth an th element of 1 and all other elements 0 and the matrx n general has possbly all of ts elements non-zero Nonetheless, the sum of all matrces stll gves the dentty matrx Î We can use any convenent complete bass to wrte Î

18 Identty operator The expresson Iˆ has a smple vector meanng In the expresson f s ust the proecton of f onto the axs so multplyng by f that s, f f gves the vector component of f on the axs Provded the form a complete set addng these components up ust reconstructs f f f

19 Identty matrx n formal proofs Snce the dentty matrx s the dentty matrx no matter what complete orthonormal bass we use to represent t we can use the followng trcks Frst, we nsert the dentty matrx n some bass nto an expresson Then, we rearrange the expresson Then, we fnd an dentty matrx we can take out of the result

20 Proof that the trace s ndependent of the bass Consder the sum, S of the dagonal elements of an operator on some complete orthonormal bass S Aˆ Now suppose we have some other complete orthonormal bass m We can therefore also wrte the dentty operator as Iˆ m m m Â

21 Proof that the trace s ndependent of the bass In S Aˆ we can nsert an dentty operator ust before  whch makes no dfference to the result snce ÎAˆ Aˆ so we have ˆˆ S IA ˆ m m A m

22 Proof that the trace s ndependent of the bass Rearrangng ˆˆ S ˆ IA m m A m reorderng the sums movng the number movng a sum and assocatng recognzng Iˆ m S m Aˆ m m m Aˆ m m ˆ m A m m AI ˆ ˆ m m m

23 Proof that the trace s ndependent of the bass So, wth now the fnal step s to note that so S Aˆ AI ˆˆ m m m Hence the trace of an operator the sum of the dagonal elements s ndependent of the bass used to represent the operator whch s why the trace s a useful operator property AI ˆˆ Aˆ S Aˆ Aˆ m m m

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25 Types of lnear operators Inverse and untary operators

26 Inverse operator For an operator  operatng on an arbtrary functon then the nverse operator, f t exsts 1 s that operator  such that ˆ1 f A Aˆ f Snce the functon f s arbtrary we can therefore dentfy ˆ1 A Aˆ Iˆ Snce the operator can be represented by a matrx fndng the nverse of the operator reduces to fndng the nverse of a matrx f

27 Proecton operator For example, the proecton operator ˆP f f n general has no nverse because t proects all nput vectors onto only one axs n the space the one correspondng to the specfc vector f

28 Untary operators A untary operator, Uˆ, s one for whch 1 Uˆ Uˆ that s, ts nverse s ts Hermtan adont The Hermtan adont s formed by reflectng on a -45 lne and takng the complex conugate u u u u u u u u u u u u u u u u u u

29 Conservaton of length for untary operators Note frst that t can be shown generally that for two matrces  and ˆB that can be multpled ˆ ˆ ˆ ˆ AB B A (ths s easy to prove usng the summaton notaton for matrx or vector multplcaton) That s, the Hermtan adont of the product s the flpped round product of the Hermtan adonts Explctly, for matrx-vector multplcaton A ˆ h h Aˆ

30 Conservaton of length for untary operators Consder the untary operator Uˆ and vectors fold and gold We form two new vectors by operatng wth Uˆ f ˆ new U fold and g ˆ new U gold Then gnew gold U So g ˆ ˆ new fnew gold UU f ˆ 1 old g ˆ old U U fold g ˆ old I fold gold fold The untary operaton does not change the nner product So, n partcular fnew fnew fold fold the length of a vector s not changed by a untary operator

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