Quantum Mechancs for Scentsts and Engneers Davd Mller
Types of lnear operators
Types of lnear operators Blnear expanson of operators
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
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
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
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
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,
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
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
Types of lnear operators The dentty operator
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 1 0 0 ˆ 0 1 0 I 0 0 1 Iˆ
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
Identty operator The statement Iˆ s trval f s the bass used to represent the space Then 1 1 0 0 so that 1 1 1 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0
Identty operator Smlarly so 2 2 0 0 0 0 1 0 0 0 0 Iˆ 3 3 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 1
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
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
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
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
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
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
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
Types of lnear operators Inverse and untary operators
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
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
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 11 12 13 11 21 31 21 22 23 12 22 32 31 32 33 13 23 33 u u u u u u u u u u u u u u u u u u
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
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