Mathematical Review for Theoretical Chemistry

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1 Mthemticl Review for Theoreticl Chemistry Rndl Hllford Deprtment of Chemistry Midwestern Stte University Introduction: This review of the bsic mthemticl techniques tht re generlly necessry to understnd the mteril in n upper level undergrdute chemistry courses is not intended to be complete, but is review of the minimum tools tht will be required to solve problems in theoreticl chemistry. Chemistry is fscinting subject s it dels with the nture of the physicl world, but there is requirement for modest level of skill in using mthemtics. Consider this brief guide to chemistry mthemtics, nd llow this to help you build some fcility in pplied mthemtics. For some this will be review, but very commonly using the clculus in this pplied wy genertes significnt proficiency for those who hve not been eposed to higher clculus. Clculus: Essentil underlying opertions in the clculus re differentition nd integrtion. Consider functions of one vrible such s y = f(), nd ssume the functions re well behved -mening the derivtives eist nd the functions re single vlued for given vlue of.. Derivtives The derivtive of the function f() t the point is defined by: = ' = lim = lim h ( ) ( ) df f( ) f( ) f h f ( ) f ( ) h

2 This definition gives the instntneous slope of line tngent to curve. the derivtive cn lso be written s f () (f prime of ). The bove eqution llows derivtion of the generl formul for polynomils: df ( ) n '( ) ( ) = f = n for f = n In generl, if given function in the denomintor, write the epression s negtive eponent. This will simplify finding the derivtive. Emple, d d ( ) ( ) ( = + = where the function /f() is treted s f() - nd therefore n = - nd nf() n- = (-)f() -. This emple utilizes the chn rule. The chin rule pplies when one function is function inside nother, e.g. g(f()). ( ( ( ))) = ' ( ) d g f g f f ( ) '( ) First, tke the derivtive with respect to g() treting the whole of f() s the vrible, then tke the derivtive with respect to f(): α ( ) d e = α e α This is the derivtive of Gussin function with respect to. Note tht here g(f()) = e f() nd f() = -. The derivtive of n eponentil is itself n eponentil (d/)e = e. )

3 Derivtives To Memorize d d e n = n = e n d e d d Specil Derivtive Reltionships = e cos = sin sin = cos d log = Chin Rule: d[ F( u( ))] df du = du Rtio Rule: du dv v u d( u/ v) = v Product rule duv ( ) du dv = v+ u Differentil Equtions Equtions which involve derivtives of functions giving rise to solution (or fmily of solutions) for function re referred to s differentil equtions. A set of boundry conditions must eist, either s function of the physicl system, or be soluble for them. A differentil sttement hs physicl reltionship to rte. 3

4 Ordinry Differentil Equtions A differentil sttement with derivtives with respect to only one vrible, or epressions with seprble terms with derivtives of only one vrible re referred to s ordinry differentils. dv Newton s second lw in vector form is F = m. The ccelertion is written s dt, where v dr is the velocity vector, or s, where r is the displcement vector. Thus, dt differentil cn be written for ny mechnicl system where motion is the result of force. The force of mechnicl system is given by the first derivtive of the potentil energy function V ( ) F = where V() is the potentil energy function. The ccelertion is the second derivtive with respect to time d = dt nd since the form of the potentil is known, the derivtive gives the force d F( ) = m dt An importnt clssicl nd quntum mechnicl model is the hrmonic oscilltor. This models the oscilltions of spring systems nd the vibrtions of molecules. The potentil function is given by V = m ω where ω is the oscilltor frequency. The force is determined from the differentil form to give F( ) = mω The negtive sign indictes tht the force cts in opposition to the motion long nd is referred to s the restoring force. The results re set equl to the F() epression bove d m = mω dt nd setting equl to zero d + ω = dt gives second order (highest eponentil is ) homogeneous (seprble nd equls zero, not some function of t) liner (eponent on is one) differentil eqution. 4

5 . Properties of Derivtives The Totl differentil of function tht depends upon more thn one other vrible cn be written s: Z = f (, y) dz f = y f + y dy hold const. hold const. This is convenient for depicting reltionships between vribles in severl dimensions. The prtil derivtive symbol,, replces the stndrd d when it necessry to hold one or more vribles constnt while tking the derivtive of function with respect to nother vrible. Subscripts on the prentheticl terms note the vrible(s) held constnt. The totl differentil for the function depicted in the Grph is: dy dy = For functions tht represent physicl systems responding to more thn one vrible t time, the totl differentil ccounts for the vribles, nd describes the chnge in the 5

6 system. For the y z grph below, pth is shown tht cn be described by the totl differentil for z: z dz = y z dy + y this implies tht generl form of differentil cn be written: dz = i z qi dq where the q i re the vribles in the system. Emple: From physics it is known tht for quntity of n idel gs, the pressure responds to temperture nd volume chnges: i RT P = V m ; Find the totl differentil for dp Write prtil term for ech vrible, holding the others constnt: P dp = V m T dv P + T V m dt P Vm T = RTV m ; P T vm = RV m 6

7 R RT the function then is: dp = dt dv m V V Independent nd Dependnt Vribles For multivrite totl differentils, functions my contin both dependnt nd independent vribles. f (,y,w)= nd g (,z,w)= contin four vribles. m m If w = + y + z then let z = + y w So tht, y re independent: y nd if, z re independent: w This hs physicl interprettion: z y ( ). The differentil then is: + y + ( + y ) ( + z + z ). The differentil is: ( ) = = w = z w w does not chnge in -y plne but does in the z = y Eigenfunctions Equtions tht hve n opertor multiplied by some function cn return vlue nd the function gin. This process produces n eigen eqution, where the vlue is n eigenvlue. de e = d The function, e, is multiplied by the opertor nd returns the function multiplied by vlue,. Not ll mth functions cn behve in n eigenvlue eqution. One emple is the log function. Cycle (cyclic)rule in Differentil Equtions Totl differentils in thermodynmics my be independent or relted s result of ssocited stte vribles. Determintion of which cse is true, nd how mny eist, is 7

8 ccomplished vi the cyclic rule. This lso indictes the function is rel, i.e., is on the surfce of gph: z z dz = + dy : y y if the chnges in vribles re restricted so tht z is unchnged, then dz = nd z z = + dy y y divide through by ( y) : z z z = + y y y z y multiply through by yields: z y z = z y z y This result indictes tht the prtil differentil sttements re relted. A different representtion of the cyclic rue llows the determintion of missing prtil terms. P P V For the epression : = find the prtil term T V T, Vn, Tn, Pn V P Tn,. This representtion is written s rtio: V P T = T V Vn, P T V V Moving the prtil terms to the other side leves : = P, T, P, Pn, Tn, Vn Pn Tn Ectness Tests If some differentil form is ect, the coefficients must be the derivtive of the function. For some set of functions du = M(,y) + N(,y) dy then: 8

9 u u M(, y) = ; N(, y) = y y u u The Euler reciprocity reltion = gurntees tht: y y for n ect differentil. N M = y y Emple: 9 3 Show tht du = y + + dy y y Is ect. 9 9 y + = y y y = y y y Since both derivtives re the sme, the differentil is ect. The Chin Rule If z = z(u,,y) but cn be epressed s function of u,v nd y, then z z = y uv, uv, yuv, If function A(B,C) is written where B nd C re lso functions of the vribles C nd D so tht B(D,E) nd C(D,E) then: A A D A E = + B C DE BC ED B. C Notice tht D nd E in the first nd second term cn lso be result of division of totl derivtive for da by db t constnt C. Reciprocity P nr T If = then = T V P Vn, Vn, V nr More detils on the use of derivtives re in the definitions nd theorems section. 3. Integrtion The integrl is the sum of smll res under curve, s the res pproch n infinitely smll dimension: 9

10 I n = lim f( i ) h h i Where f() is the height nd h is the width of smll re centered on the point i. In the limit of very smll h, this converts to the integrl: I = f ( ) Integrtion is the ntiderivtive tht is, it is the inverse opertion of derivtive. If derivtive is known, the integrl cn be deduced. e e = + c is n indefinite integrl, where c is the integrtion constnt. The integrting constnt is vlue determined by the physicl ppliction of the integrnd to specific problem. In cses where the problem eists within definite limits, the definite integrl cn be written s b b I = cos = sin = sin b sin. The difference of the function vlues t nd b result in the loss of the integrting constnt (it is the sme for both function vlues nd subtrcts out) nd the nswer is simply the numericl difference of the function t nd b. Integrtion for physicl processes: summtion of smll steps The epnsion of n idel gs ginst n eternl force produces grph of given re. However, there is no functionl dependence between the two vribles s they cn chnge independently of ech other. The work done by gs under these conditions does depend on both the eternl pressure on the gs nd the volume chnge by the gs: w = P ( V ) et V which is just the re under the curve. By mesuring the re, we cn determine the work without the need for considertion of ll of the other prmeters in the eperiment. In less strightforwrd cses, where severl vribles re chnging, mesuring the re is problemtic without the clculus, nd the evlution of integrls describing the comple re chnges re necessry step in the solution of physicl problems. When problem involves severl vribles tht re relted to chnge in one vrible during process, n nlyticl solution is not vilble unless the function relting the vribles

11 over the pth is known. Allowing only one vrible to chnge nd integrting the pth for this vrible cn be ccomplished by removing constnt vrible from the integrl in some cses. For instnce, cusing the eternl pressure to remin constnt nd removing it from the integrl provides n nlyticl solution: V = V A P et dv becomes A = P et dv which is Pet (V V ). V V If the system is reversible, then the eternl pressure nd the internl pressure become equl, nd the temperture vrible is introduced. The resulting integrl looks like: V A = V Pgs ( T, V ) dv nd direct evlution is not possible. Integrls to Memorize n+ n = + c n + = ln + c = + c ( ) n n n e e = + c sin( ) = cos( ) + c cos( ) = sin( ) + c 4. Methods of Integrtion When more complicted integrls result from nlysis of problem, integrl tbles cn provide templtes for the solution of the integrls. Mny times the integrl you hve determined is necessry to solve the problem does not seem similr to ny in the tbles. This requires mnipultion to re-rrnge the integrl to stndrd form.

12 . Brek integrls into smll steps: f ( ) = f ( ) + f ( ) = b f ( ) + f ( ) + f ( ) b. Chnge vrible (dummy vrible) The results of n integrtion is independent of the identity of the vrible over which integrtion is crried out, nd cn be treted s dummy vrible, u, such tht it s derivtive llows substitution in the integrnd. b f ( ) = f( udu ) = = u b Chnge of vribles If u=k, then =u/k: = du = du du k b kb u F( ) = F( ) du k k = u= k 3. Reversing Limits Chnging the limits chnges the sign of the integrl. 4. Integrtion by Prts An importnt method of integrtion stems from the identity: d(uv) = udv + vdu nd leds to the integrl: udv = uv vdu This is the formul presented in most clculus books.

13 A useful emple to consider is the rel prt of Fourier trnsform. In generl: i t ω ( ω) () F = e f t d π t this cn be solved for the eponentil function f(t) = e -t/τ to yield two terms, one rel nd one imginry. Solving for the rel nd imginry prts seprtely using the identity: Results in iωt e = cos t i ( ω ) sin ( ωt) i F t f t dt t π π ( ω) = cos( ω ) ( ) sin ( ω ) ( ) f t dt From this form it is cler tht the rel nd imginry prts re seprted prior to crrying out integrtion. π t / The rel prt is ( ) cos ωt e Apply the integrtion by prts t this point. τ dt Let u = cos(ωt) nd dv = e -t/τ dt. Then v = τe -t/τ nd du = ωsin(ωt)dt. Using the formul for integrtion by prts (nd ignoring the fctor of /π) : t/ τ t/ τ cos( ω ) = τ cos( ω ) ω sin( ω ) te dt te t te t / τ dt The first term on the right hnd side evlutes to τ. The sme integrtion by prts is performed for the second integrl on the right hnd side. t/ τ t/ τ sin( ω ) = τsin( ω ) + ωτcos( ω ) te dt te te t / τ dt The first term on the right hnd side evlutes to zero. Substituting the second integrl into the first : 3

14 t/ τ t/ τ cos( ωte ) dt= τ ω τ cos( ωte ) dt nd solving for the integrl term we find: Another emple: ω cos( ωte ) t / τ dt τ = + ω τ Eponentil Function e - The re under the curve is: e = To obtin this nswer use the substitution u = - nd du = -. Therefore = -du /. The integrl cn be written: u u edu e = = = ( ) In generl eponentil integrls multiplied by polynomil cn be solved using integrtion by prts. For emple, the integrl 4

15 e cn be solved by substitution u = nd dv = e -. Then du = nd v = (-/)e - to obtin the integrl. In generl so tht in the present cse: udv = uv vdu e = e + e = Notice tht the first (uv) term is zero nd the second term contins the integrl of e - tht ws solved bove. By this technique ny polynomil cn be solved yielding the generl formul n e = n! n+ 5.. Even-oddness If n integrnd is even over the limits [f()=f(-)] then: And if odd [f()=-f(-)]: f ( ) = f( ) f( ) = 5. Multiple Integrtion Multiple dimensionl spces require multiple integrtions nd re written over sets of coordintes or vribles. The multiple integrl is evluted step-wise, holding ll but one vrible constnt. A two dimensionl integrl looks like: b c d f (, y) dy 5

16 If the function f(,y) is in the form of f(,y) = g()h(y) then the function vribles re seprble, nd the integrl becomes the product of two integrls. b d b d f (, y) dy = g( ) h( y) dy c c Frequently, this is not possible nd evlution of multiple integrtions becomes more comple. Chnging vribles in Multiple Integrtion Commonly, limits on multiple integrls must be converted form crtesin coordintes to polr coordintes. Recll tht = r cos θ nd y = r sin θ with rnges of r nd θ π. The function must be epressed in terms of r nd θ, nd the re or volume element is chnged from dy to J drdθ, where J is the Jcobin determinnt of the coordinte trnsform: r θ J = The two dimensionl determinnt is given by b d bc y y c d = r θ After solving the four prtil derivtives nd inserting them, the solution gives ( cos θ sin θ) J = r + since cos θ + sin θ = then J = r The volume (or re for two dimensionl system) chnge is dy rdrdθ. The units remin the sme. In the three dimensionl cse, the vrible chnge is = r sinθ cosφ; y = r sinθ sinφ; z = r cosθ with rnges of r ; θ π; φ π. The determinnt cn be tken from clculus tet. The chnge in the volume element is dydz r dr sinθdθdφ. 6. Vector Algebr Vectors re noted in type s bold letter,, nd in written form s. Since vectors re directionl, tht is they re combintion of mgnitude (sclr) nd direction, there is nottion set of subscripts nd opertors tht give direction to ech component of vector sclr. i i = i j = k = i+ y j+ zk nd unit vectors multiply: j j = j k =i k k = k i = j The i nottion ( i ht ) is the unit vector in the direction. 6

17 Sum of Vectors + b= + b i+ + b j+ + b k ( ) ( ) ( ) y y z z Represents the sum of vectors nd b Multipliction by constnt distributes the constnt throughout s sclr multiplier. α = α i+ α j+ α k y z The sclr vlue of vector cn be found s the norm of the vector. = + + ( ) y z The unit vector for vector is found s the rtio of the vector to the norm: = = unit vlue; this is prllel to The sclr product of vectors (dot product) is found from the product of the norms of the vectors nd the cosine of the ngle between them: b = bcosθ And the mgnitude of the vector is =. For n orthogonl set of vectors cos θ is zero, therefore the orthogonl vectors hve zero dot product The cross product of vectors (vector product) requires unit vector, n, tht is orthogonl to both vectors. b= b sinθ n θ π Comple lgebr includes epressions hving rel nd imginry prt, typiclly denoted by z= + iy. The comple portion contins i, symbol representing -. The modulus of comple number is ( ) ( ) Z = rel z + imginry z = + y 7

18 And stisfies : Z = ZZ Z ( is the comple conjugte of Z) (Z = + iy then Z * = iy) Comple crtesin coordintes cn be epressed s polr coordintes such tht Z = r(cosθ + i sinθ), where θ is the rgument of the epression. Vlues cn be obtined for r, sinθ, nd cosθ : r y = + ; cos θ = ; sinθ = + y + y y 7. Series Epnsions Mny pproimtions to physicl processes for which no ect solution eists cn be found by substituting terms from series epnsion into n empiricl or nlyticl function. Severl series which should be kept in mind re listed here. e n 3 = = n!!! 3! n= n n ( ) sin = = +... (n + )! 3! 5! 7! n= n n 4 6 ( ) cos = = ( n)!! 4! 6! n= An ppliction to quntum wve mechnics is relted to the sin nd cosine functions. Using series of epnsions provides simplifiction nd very fundmentl reltionship used in wide vriety of pplictions. 8

19 e z n 3 z z z z = = n!!! 3! n= iθ let z = e, where θ is rel nd z is, therefore, imginry. if = iθ 3 ( iθ) ( iθ) iθ iθ e = !! 3! i e iθ θ iθ θ iθ = + + +!! 3! 4! 5! seprte terms bsed on lternting rel nd imiginry prts e iθ θ θ θ iθ iθ iθ iθ = i +...! 4! 6!! 3! 5! 7! cos θ i sinθ therefore : e iθ = cosθ + isin θ Euler's Formul In quntum mechnics it is necessry to constrin systems of wvefunctions to rel positive vlues. Utilizing comple conjugtes ensures this is true. If Z = r e iθ, then Z * i = r e θ = r, rel vlue sinc r = Z. Then ZZ * = Z. For non-zero vlues of r we hve: iθ iθ re Z Z = r e = sotht Z = r Z If wvefunction requires shift by some ngle then Euler s formul cn be used to ensure only integer vlues eist for the shift: imθ f φ = f φ + π where f( φ) = e ( ) ( ) then : e = e = e e e imθ im( θ+ π) imθ πim cncelling e iπ m = imθ : using Euler's formul: = cos( πm) + isin( πm) = cos πm = nd sin πm = cosφ = if φ=, ± π, ± 4 π... sin φ= if φ=, ± π, ± π,... so tht m =, ±, ±, ± 3,... m = integer vlues only 9

20 Delt functions A delt function is n infinitely nrrow, infinitely high function whose re is normlized to one. This is little difficult to understnd unless we give model for delt function. One model is the squre function. Imgine function whose vlue is / over the rnge - / < < / nd outside these vlues. This function is plotted below for three vlues of. The re under ech rectngle is the sme becuse the bse length is nd the height is /. Since (/)() = the re is lwys one. Imgine we tke this function in the limit tht goes to zero. The function becomes nrrower nd tller, but lwys hs n re of one. In the limit tht goes to zero this is delt function. Since the delt function δ( - ). Note tht since the delt function is centered bout in this instnce the vlue of is. In this cse our delt function is δ(). In spectroscopy, delt function is often used to represent mtching condition. If we imgine the energy levels of molecule s being etremely nrrow then the energy of n incident photon must mtch ectly to the energy difference in order for trnsition to occur. The Einstein reltion sttes tht E - E = ΔE = hν. One wy to write this for infinitely nrrow level widths is to use delt function δ(δe - hν). Properties. The delt function is the eigenfunction of the position opertor. For free prticle we cn operte with the position opertor (ht). The eigenvlue eqution is:

21 ( ) = δ ( ) ˆ δ o o o The eigenvlue is the ctul position of the prticle. The delt function (δ) specifies tht of ll the possible vlues only is non-zero.. The integrl properties of delt function re s follows:. the integrl over δ( - ) is equl to the function evluted t. f ( ) δ ( ) = f( ) b. The re under the delt function is one. δ ( ) = 3. The vlue of delt function is zero everywhere ecept where the rgument is zero. δ ( ) = for δ ( ) = for 4. A chnge of rgument by fctor results in multipliction by the inverse of the fctor. δ( k) = δ( ) k To see this consider the bove rectngulr function. The delt function is / over the limits -/ < < /. Thus, the height is / nd the bse is. If we multiply the height by k then it becomes k/. This mens tht we should multiply the bse by k. In other words since δ ( ) =

22 we require tht Thus, k δ ( k) = k δ( k) = δ( ) In spectroscopy we mke use of this (non-intuitive) property of the delt function to mke the eqution: ( + ) = ( E+ E ) δ ω ω δ A finl point is tht ny of the lineshpe functions re representtion of delt function. A Gussin, Lorentzin or sin()/ function re ll delt functions in the limit tht their width goes to zero. In fct δ ( ) is lso representtion for delt function. ik = e π 8. Some importnt mthemticl definitions nd theorems : Vector Opertor used like d, hs no mening unless function or sclr is present ˆ i ˆ j kˆ = + + y z Grdient ( ): Directionl derivtive vector ˆ φ grd i ˆ φ j kˆ φ φ = φ = + + y z the vector is perpendiculr to the surfce of Φ (= constnt t some point,,y,z) Divergence ( i φ ): Flu sclr Dot product φ φ φz y z y = + + i φ

23 If φ represents the velocity (chnge of position per unit time, chnge in density, etc.) then div is the chnge per unit volume (the flu). Useful for E-M theory, liquid mechnics, nd etc. Curl ( φ ): Rottion (ngulr velocity) Vector Fundmentl to Stoke s theorem φ i j k φ φ φ φ φ φ y z z y y z φ φ φ z y z y = i + j + k = div grd ( φ = φ ) Lplcin y z φ = φ = φ y z φ y φz + + Importnt spects of Lplcin mthemtics in the physicl sciences re: φ = φ = φ = Lplce's eqution φ The wve eqution t φ diffusion eqution (of het conduction) t Green s Theorem (line integrls) F I The double integrl of the function HG K J Qy (, ) where n re enclosed by the function Q(,y) nd P(,y) hs continuous first prtil differentils, is equl to the line integrl. F HG zz Q A I P dy = P + Qdy ykj za The line integrl is counterclockwise rottion of the totl re. Stokes Theorem (open surfce reltionship to the line integrl) 3

24 z zz Vdr = b g ndσ curve bounding σ surfce σ Butterfly net: the rim is the curve bounding the surfce mde of the net. Surfces treted with this theorem must be two sided in order to know the sense of the norml vector ( not Moebius surfce). Mtrices Definition of mtri A mtri is two-dimensionl rry. The dimensions of mtri re given by its number of rows n nd columns m. We designte mtri by bold letter. The mtri A is A. The dimensions of re n m (n by m). The elements of re ij. The inde i is the row inde nd the inde j is the column inde. We represent A s: Specil cses A 3 = A squre mtri hs n = m. A digonl mtri D hs vlues only long the digonl d, d, d 33, etc. nd zero elsewhere. D d... = d... d33... The unit mtri or identity mtri I hs ones long the digonl. I... = A column vector is mtri of dimension n. 4

25 A row vector is mtri of dimension n. r r = r r... 3 r = ( r r r... 3 ) Mtri ddition nd subtrction Two mtrices cn be dded or subtrcted only if they hve the sme dimesions. The result of dding mtrices A nd B is obtined by dding their corresponding elements. A + B = C is obtined by dding ij + b ij = c ij for ll i nd j. A + B= C 3... b b b3... c c c b b b3... = c c c b3 b3 b33... c3 c3 c33... Mtri multipliction In order to multiply two mtrices AB = C the number of columns in the A mtri must be the sme s the number of rows in the B mtri. If A is k l mtri nd B is l m mtri, then C hs dimensions k m. The prescription for mtri multipliction is: c l = b ij ir rj r= The first row elements of the A mtri re multiplied by the first column elements of the B mtri to give the first element of the C mtri. AB = C 3... b b b3... c c c b b b3... = c c c b3 b3 b33... c3 c3 c33... Note tht the bove eqution indictes the sum of the products of the elements c = b + b + 3 b 3 + until the row nd column re finished. The c element is obtined in similr fshion: 5

26 3... b b b3... c c c b b b3... = c c c b b b... c c c This process is continued until ll of the elements of the C mtri hve been obtined. The order of opertion is importnt. This cn be simply illustrted with the following importnt emple. The product of row mtri nd column mtri is sclr. b 3... b= b + b + 3b b3... ( ) The product of n column mtri nd n row mtri is n n mtri. b b b 3 b ( 3... ) = b b b 3... b 3 b 3 b 3 b b... Mtri Trnspose The trnspose of mtri is obtined by interchnging the rows nd columns of mtri T A=..., A = Mtri Inverse The inverse of mtri is defined such tht AA - = A - A = I where I is the identity mtri. The determinnt of mtri The determinnt of mtri is number obtined by summing n! products of n n mtri. This is best illustrted by strting with mtri. The determinnt of mtri is: 6

27 = Det For lrger dimension mtrices, the determinnt cn be obtined by reduction to lowerorder determinnts (minors). This is illustrted for 3 3 mtri Det 3 = Det Det + 3 Det The determinnt is obtined s bove for ech of the minors. The determinnt of A is obtined by summing cross ny row s follows: Mtri solution of liner equtions n i= ( ) A = i+ j ij ij A Mtrices re importnt tools for the solution of liner equtions. The generl form of set of simultneous liner equtions is A = b. This illustrted below: 3... b... = b b Here the i re the unknowns nd ij re the liner coefficients. To find the ith coefficient i we replce tht column of the A mtri by the b vector. b 3... A = b b The unknown is given by the rtio of the determinnts, i = ( i ) ( ) Det A Det A 7

28 Brket nottion In quntum spectroscopy it is commonly necessry to solve integrls of the type Ψ * m AΨ dτ where the Ψ m nd Ψ n re wvefunctions, A (ht) is n opertor, nd the volume element dτ represents the integrl over ll spce. The significnce of this epression is tht it gives the verge of the physicl quntity corresponding to opertor A for the wvefunctions in question. There re severl equivlent wys of writing this integrl. The bove integrl cn be rewritten using the following bbrevitions: * Ψ A Ψ dτ = Ψ A Ψ = m A n m n m n Keep in mind tht ll of these epression men integrtion over ll spce. These epressions re referred to s mtri elements. This is becuse often there re mny different wvefunctions for different electronic, vibrtionl, or rottionl sttes of molecule nd the opertor my represent the trnsitions between them. This is often the cse in spectroscopy. For emple, if the opertor A is the dipole opertor then the bove mtri element represents trnsition dipoles between sttes m nd n. In the sme mnner the overlp of two sttes cn be epressed s Ψ Ψ dτ = Ψ Ψ = m n * m n m n Another wy to view the bove terms is tht they represent the projection of one stte onto nother. The shorthnd nottion with ngle brckets is result of P.A.M. Dirc. This is known s brket nottion. One importnt spect of the shorthnd nottion is tht the br m implies tht the comple conjugte of Ψ m * is represented. The ket n represents Ψ n. Note tht since: * then : Ψ Ψ dτ = Ψ Ψ dτ * m n m n mn * = n nm There is no prticulr significnce to the choice of symbol for the wvefunction. Vriously Ψ, ψ, φ, χ, or the ket n is used to represent wvefunction. The verge properties nd probbilities re required to be rel. In other words Ψ*Ψ must be rel. Wvefunctions tht stisfy this condition re clled Hermitin. In order to solve mtrices (groups of vectors or sptil functions, etc.) tht contin imginry nd rel prts, performing opertions requires them to be Hermitin. A squre mtri is Hermitin if the comple mtri trnspose is equl to the originl mtri: * ( A ) T = A 8