Section 7_19

This was installed into tensor doc at 2 PM on March 17, 2016. Do not edit here! 7.19 More on basis vectors and matrix elements of R and S Ambiguity In the summary tables of Section 7.18 we see various equations like these, (un)i = δni (en)i = Rni (e'n)i = δni (u'n)i = Rin . (7.19.1) There is a certain ambiguity in writing, for example, (en)i : it is not clearly stated in what basis component i is evaluated! The basis depends on the expansion in which a component appears. For example, from (7.13.10) we write for a generic vector V, V = ΣiVi ui ui V = Vi (7.13.10) and we then know that Vi is a component in the u-basis. Applying this to V = en (an x-space vector) we conclude that en = Σi (en)i ui ui en = (en)i . (7.19.2) Thus, (en)i is in fact a component in the u-basis, not for example the e-basis. We would then write (en)i = (en)(u)i = Rni . (7.19.3) This is the "natural basis" to use for the components of en and for any x-space vector V where we use the natural expansion V = ΣiVi ui shown above. One could however expand en on the en basis to get en = Σi (en)(e)i ei ei en = (en)(e)i = δin . (7.19.4) Then with no ambiguity one could write (en)(u)i = Rni // natural (en)(e)i = δni (7.19.5) Thus, when we write (en)i, we imply that we are using the u-basis for the components. Since this applies to any x-space vector, it is also true for (un)i . What about x'-space vectors? From (7.13.11) we write V' = Σi(V')i e'i e'i V' = (V')i . (7.13.11) Applying this to V' = e'n (an x'-space vector) we conclude that e'n = Σi(e'n)i e'i e'i e'n = (e'n)i = δin . (7.19.6) Thus, (e'n)i is in fact a component in the e'-basis, not for example the u'-basis. We would then write (e'n)i = (e'n)(e')i = δni . (7.19.7) This is the "natural basis" to use for the components of e'n and for any x'-space vector V' where we use the natural expansion V' = Σi(V')i e'i shown above. One could however expand e'n on the u'n basis to get e'n = Σi (e'n)(u')i u'i u'i e'n = (e'n)(u')i = ui en = (en)i = Rni . (7.19.8) Then with no ambiguity one could write (e'n)(e')i = δni // natural (e'n)(u')i = Rni (7.19.9) Thus, when we write (e'n)i, we imply that we are using the e'-basis for the components. Since this applies to any x'-space vector, it is also true for (u'n)i . We summarize the above discussion as follows: Fact: un and en components are by default presented in the u-basis (7.19.10) e'n and u'n components are by default presented in the e'-basis Table of basis vector dot products Since there are four kinds of basis vectors of interest, un, en, u'n and e'n with lower labels. one might imagine there are 16 scalar products of interest, u e u' e' u uu ue uu' ue' e eu ee eu' ee' u' u'u u'e u'u' u'e' e' e'u e'e e'u' e'e' However, since a scalar product only exists within a Hilbert space (x-space or x'-space), the cross space entries in this table make no sense, so we eliminate them to get u e u' e' u uu ue - - e eu ee - - u' - - u'u' u'e' e' - - e'u' e'e' (7.19.11) Since ue = eu and u'e' = e'u', we see that there are only 6 distinct scalar products of interest. Collecting data from the tables (7.13.1) and (7.13.2) we summarize these 6 cases as follows: Table of basis vector dot products: (un)i = ui un = <ui | un > = gin = u'i u'n = < u'i | u'n > (en)i = ui en = <ui | en > = Sin = Rni (en')i = e'i e'n = < e'i | e'n > = g'in = ei en = <ei | en > (un')i = e'i u'n = <e'i | u'n> = Rin = Sni (7.19.12) Since a b = a'b', the first and third lines each contain 2 of the 6 cases. We are careful to use the correct tensor notation gni and g'ni for δni as discussed near equation (7.4.17). Doing this, we may state the following : Fact: In any of the above equation lines, one may raise the label n and/or lower the index i and end up with another valid equation line. (7.19.13) In this manner, from the 4 lines above one may generate 12 more lines of equations. Note that we have slipped in the Dirac notation < | > described in (E.7.4) as an alternate way to write the dot products. As an example of the above Fact, if we raise the label n in the third line we get these valid equations, (en')i = e'i en' = < e'i | en' > = g'in = ei en = <ei | en > and if we then lower index i, we get another set of valid equations, (en')i = e'i en' = < e'i | en' > = g'in = ei en = <ei | en > . To prove the above Fact, one first observes in which space a vector lies, so one knows which metric tensor raises and lowers the index. Then one uses these facts from (7.18.1) and (7.18.3), en = g'ni ei e'n = g'ni e'i un = gni ui u'n = gni u'i en = g'ni ei e'n = g'ni e'i un = gni un u'n = gni u'i . (7.19.14) For example, consider the third line in (7.19.12), (en)i = ui en = <ui | en > = Rni . Apply g'mn to get g'mn (en)i = g'mn ui en = g'mn <ui | en > = g'mn Rni or [g'mn en]i = ui [g'mn en] = <ui | [g'mn] en > = g'mn Rni or (em)i = ui [em] = <ui | em > = Rmi // see (7.5.9) about Rmi which shows that the label n can be raised to get another valid set of equations. Now start with the same third line in (7.19.12) and instead apply gji to get gji(en)i = gjiui en = gji<ui | en > = gjiRni or gji(en)i = [gjiui] en = <gjiui | en > = gjiRni or (en)j = uj en = <uj | en > = Rnj // see (7.5.9) about Rnj which shows that the index i can be lowered to get another valid set of equations. Matrix elements of R and S One can regard the elements Rij and Sij as being certain matrix elements of operators R and S, and this is most easily handled in the Dirac bra-ket notation of (E.7.4). Since V' = RV (vector transformation rule) one sees that a matrix element <a' | R | b> = <a' | R b> must have | b> being an x-space vector, and <a' | being a transposed x'-space vector. We shall now compute matrix elements of R in several basis combinations. As examples of V' = RV we know that, e'n = R en u'n = R un or |e'n> = R |en> |u'n> = R |un> . (7.19.15) We can "close on the left" in various ways. For example, <e'i | e'n> = <e'i | R | en> = g'in <u'i | e'n> = <u'i | R | en> = Sin = Rni (7.19.16) Here we look up the scalar products on the left of each line in (7.19.16) using table (7.19.12), making use of Fact (7.19.13), and adjusting the various index names and the up and down index sense. Now start instead with the second equation of (7.19.15) and close two ways, again looking up the basis vector scalar products in (7.19.12), <e'i |u'n> = <e'i | R | un> = Rin = Sni <u'i |u'n> = <u'i | R | un> = gin . (7.19.17) Here then are four R matrix elements of interest, <e'i | R | en> = g'in <u'i | R | en> = Sin = Rni <e'i | R | un> = Rin = Sni <u'i | R | un> = gin . (7.19.18) For the S matrix elements, start with this rewrite of (7.19.15), en = S e'n // S = R-1 un = S u'n or |en> = S |e'n> |un> = S |u'n> (7.19.19) Matrix elements of S must be of the form <a | S | b'> = <a | S b'>. Again using the table (7.19.12) for scalar product evaluations, close on the left to get <ei |en> = <ei | S | e'n> = g'in <ui |en> = <ui | S | e'n> = Sin = Rni (7.19.20) and similarly <ei |un> = <ei | S | u'n> = Rin = Sni <ui |un> = <ui | S | u'n> = gin . (7.19.21) Here then is a summary of matrix elements of R and S, where it happens that the basis vector label is up on the left side and down on the right side, <e'i | R | en> = g'in <ei | S | e'n> = g'in <u'i | R | en> = Sin = Rni <ui | S | e'n> = Sin = Rni <e'i | R | un> = Rin = Sni <ei | S | u'n> = Rin = Sni <u'i | R | un> = gin <ui | S | u'n> = gin (7.19.22) As usual, g'in = gin = δin but we maintain the true tensor form to allow for up down modifications as per the following claim: Fact: In the above equations, one can raise/lower indices on either or both sides to get new valid equations. (7.19.23) The proof of this claim is the same as the proof of Fact (7.19.13) above using (7.19.14). For example, <u'i | R | un> = gin so gji<u'i | R | un> = gjigin or <u'j | R | un> = gjn Similarly <e'i | R | un> = Rin = Sni <e'i | R | un> = Rin = Sni. Notice in (7.19.22) that the right-side expressions for the left column of equations is exactly the same as for the right column. The reason for this perhaps surprising fact can be explained by first reordering the right column while maintaining the order of the left column <e'i | R | en> = g'in <ui | S | u'n> = gin <u'i | R | en> = Sin = Rni <ei | S | u'n> = Rin = Sni <e'i | R | un> = Rin = Sni <ui | S | e'n> = Sin = Rni <u'i | R | un> = gin <ei | S | e'n> = g'in Now the second column can be obtained from the first column using the earlier rules g'↔ g R ↔ S en ↔ u'n e'n ↔ un en ↔ u'n e'n ↔ un . (7.18.2) Basically one is just changing from the transformation x' = F(x) to the inverse transformation x = F-1(x'). For example, the tangent basis vectors for F are the en while those for F-1 are the u'n, and so on. Interpretation of Rin and Sin Since V' = R V, one can regard the operator R as a mapping R : X → X' (x-space → x'-space). R may be regarded as a "cross tensor" having one foot in each space. As shown in (7.19.22), Rij = <e'i | R | uj> = [ R(e',u)]ij (7.19.24) so one can regard R as an abstract cross-tensor being expanded as follows in the mixed-basis sense of (E.10.4), R = Σij [ R(e',u)]ij (e'i uj) . (7.19.25) As usual, the coefficient (the cross tensor components) can be projected out using (e'a ub) R = Σij [ R(e',u)]ij (e'a ub) (e'i uj) = Σij [ R(e',u)]ij (e'a e'i) (ub uj) = Σij [ R(e',u)]ij δai δbj = [ R(e',u)]ab . (7.19.26) Similarly, since V = S V' one can regard the operator S as a mapping S : X' → X where, from (7.19.22), Sij = <ui | S | e'j> = [ S(u,e')]ij (7.19.27) and then the cross-tensor expansion and projection is given by, S = Σij [ S(u,e')]ij (ui e'j) (ua e'b) S = [ S(u,e')]ab . (7.19.28) A Pitfall Example One knows from (7.19.22) that < u'i | R ua> = < u'i | R | ua> = gia = δia . This is also known from, < u'i | R ua> = < u'i | u'a> = δia . Consider now the following slight of hand wherein we get a different result, < u'i | R ua> = [R ua]i = Rij(ua)j = Rijδaj = Ria wrong! Where is the error being made here? Well, the vector u'a = Rua being in x'-space has a natural e'-type basis as discussed below (7.19.7), but here the component [Rua]i is in the u'-basis. When doing something "unnaturally" one must pay more attention to labels. The correct version of the above is < u'i | R ua> = [R ua](u')i = [R(u',u)]ij (ua)(u)j = [R(u',u)]ij δaj = [R(u',u)]ia = <u'i | R | ua> = gia = δia . (7.19.29) Now consider instead a different example. From (7.19.22) one knows that, < e'i | R ua> = < e'i | R | ua> = Ria . In this case write < e'i | R ua> = [R ua]i = Rijδaj = Ria and there is no problem because e' is the natural basis. In more detail < e'i | R ua> = [R ua](e')i = [R(e',u)]ij (ua)(u)j = [R(e',u)]ij δaj = [R(e',u)]ia = <e'i | R | ua> = Ria (7.19.30) We find that the Dirac notation is useful because it provides a bulletproof formalism for avoiding ambiguities such as that of the previous example. The above equation sequence can be written (implied sum on j as usual) < e'i | R ua> = < e'i | R [1] ua> = < e'i | R [ | uj><uj | ] ua> = < e'i | R | uj><uj | ua> = [R(e',u)]ij (ua)(u)j (7.19.31) where we use the (E.7.4) completeness relation 1 = | uj><uj |. In fact x-space completeness: 1 = | uj><uj| = | uj><uj| = | ej><ej| = | ej><ej| x'-space completeness: 1 = | u'j><u'j| = | u'j><u'j| = | e'j><e'j| = | e'j><e'j| (7.19.32) There are just restatements of the (6.2.8) duality idea that bi bj = < bi | bj > = δij for any basis.