9. Complex Numbers. 1. Numbers revisited. 2. Imaginary number i: General form of complex numbers. 3. Manipulation of complex numbers
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1 9. Comple Numbers. Numbers revsted. Imagnar number : General form of comple numbers 3. Manpulaton of comple numbers 4. The Argand dagram 5. The polar form for comple numbers
2 9.. Numbers revsted We saw earler that the soluton to the quadratc equaton of the form a b c 0 can eld up to two real roots, dependng on the values of a, b, and c: b b 4ac wth b 4ac 0 a We can represent the soluton graphcall n terms of where the functon a b c cuts the -as, where = 0. When the dscrmnant s negatve, we see graphcall that the plot of the quadratc functon does not cut the -as at all. Logc would dctate: no real solutons. However, ths problem can be crcumvented b smpl etendng the number sstem to nclude so-called comple numbers, whch ncorporate as a legtmate number. Then, f 0, we can wrte ' where ' 0. In ths case, the soluton of the quadratc equaton becomes: b ', a comple number! a 6 For nstance: 5 0 Ths concept can be shockng. However, we ll see that comple numbers, and ther treatment, are straghtforward. Importantl, we fnd that the allow us to tackle real problems n chemstr n a wa that would otherwse be mpossble.
3 9.. Imagnar number : General form of comple numbers We defne the magnar number. In general, an magnar number s defned as an real number multpled b, for nstance: 8 8,, Note that:,, 4, 3 5. The sum of a real number and an magnar number s termed a comple number. It takes the general form: where and are real numbers, termed the real and magnar parts of, respectvel. The ensemble of comple numbers s desgnated b C. o f 0 and 0 then s an magnar number. o If 0 and 0 then s a real number. Back to our prevous eample: 5 0, hence the roots to a quadratc equaton wth a negatve dscrmnant are epressed as the sum of a real number () and an magnar number ( or ), that s, a comple number. We defne the two functons Re() and Im() (doman of defnton s C) such as: o Re() =, returns the real part onl, o Im() =, return the magnar part onl.
4 9.3. Manpulaton of comple numbers Straghtforward, as long as we remember that. and Consder the two comple numbers a b, c d: o Addton and subtracton. Carr out the operaton separatel on the real and magnar parts: a c b d a b c d o Multplcaton b a scalar (real number m). Multpl the real and magnar parts b the scalar: b ma mb m m a o Multplcaton. Epand the epresson as a sum of terms and group the real and magnar parts: a bc d ac ac bd ad bc ad bc bd o Note that, n general, the operatons above eld comple numbers as a result. The comple conjugate, *, of the comple number s defned b *. That s, smpl change the sgn of the magnar part ( ; * ). Propertes: o Sum: * (real) o Product: * (real) o Dfference: * (magnar)
5 Dvson of comple numbers:? As above, we need to transform nto a comple number. Trck: multpl and dvde b * (snce * R). * * 9.4. The Argand dagram Useful to represent comple numbers usng an Argand dagram, n whch the real and magnar parts defne a pont (,) n a plane: Locaton of specfed b usng ether o Cartesan coordnates (,) where = Re(), = Im(), o Polar coordnates (r,q) where 0 r and. The polar coordnates r and defne the modulus (or absolute value ) and argument, respectvel, of. From Pthagoras theorem and smple trgonometr: 0 r ; a tan tan Im() Re() r q = +
6 Cauton must be used n determnng ( mples that s postve n frst two quadrants!). Eample: o Although and both have tan, the le n the 4 th and nd quadrants, respectvel. Im() = - + q - 0 q Re() = The polar form of comple numbers Let s consder. From trgonometr, we see that: o = Re() = r cos o = Im() = r sn Consequentl, ma be epressed n terms of r and as: r cos sn Usng the mportant result known as Euler s formula: e cos sn (that can be demonstrated used Maclaurn seres for the sne, cosne and eponental functons), we can epress the polar form for n a more compact and powerful wa: re
7 Let s evaluate the number e usng Euler s formula: e cos sn 0 Hence the etraordnar and elegant result: e, whch rearranges to a sngle relatonshp: e 0 contanng the rratonal numbers e and, the magnar number, as well as the numbers ero and unt. Advantage of polar form: certan manpulatons are made much easer. For nstance: o The modulus of s gven drectl from the product of and *: * r e e r r * o Postve and negatve powers of : n n n n re r e (n,, 3,...) where for a gven value of n, n s seen to be a comple number, wth modulus r n and argument n. Can be also appled to ratonal powers of, where n = p/q ( q 0). o From the epresson above: n n n n r e r cosn snn r We obtan the De Movre theorem: n cos sn n n cos sn cosn snn
8 o De Movre s theorem allows us to epress the square root of a comple number (for nstance, where n = ½): / r / cos sn Comple functons: So far we have been concerned wth the concept of comple numbers. We can etend our dscusson to comple functons. Such a functon comprses a real and an magnar part, takng the general form: f() g() h() Euler s formula s a good eample of a comple functon defned b f( ) cos( ) sn( ), and the eponental functon e s another one (wth modulus ). The comple conjugate of the functon f() s gven b: f()* g() h() Thus f()f()* s a real functon of the form: f()f()* g() h() The above propert of comple functons plas a crucal role n quantum mechancs, where the wave functon of an electron,, whch ma be comple n form, s related to the phscall meanngful probablt denst through the product *. If X s a comple functon, then, * s a real functon.
9 The perodct of the eponental functon: It ma seem odd to thnk of the eponental functon e as perodc (t clearl sn t when the eponent s real!). However, defnng modulus and argument of and, respectvel, we see that the values of the eponental functon le on a crcle of radus r = on an Argand dagram: = e q Im() r = q - 0 Re() - Dfferent values of defne the locaton of comple numbers of modulus unt on the crcumference of the crcle. We can also see that the functon s perodc wth perod : e m m e e The egenvalue problem revsted: The three p orbtals resultng from the soluton of the Schrödnger equaton for the hdrogen atom can be wrtten as: r / N e a0r sn e 0 N e r r / a0 / a0 Ne r sn e where N and N are constants, a 0 s the Bohr radus, r s the dstance of the electron from the nucleus, and ml s labelled b the value of the orentaton quantum number m l. r cos e
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