Elements of Dirac Notation

Transcription

1 Elemets of Dirac Notatio Frak Rioux I the early days of quatum theory, P. A. M. (Paul Adria Maurice) Dirac created a powerful ad cocise formalism for it which is ow referred to as Dirac otatio or bra-ket (bracket ) otatio. Two major mathematical traditios emerged i quatum mechaics: Heiseberg s matrix mechaics ad chrödiger s wave mechaics. These distictly differet computatioal approaches to quatum theory are formally equivalet, each with its particular stregths i certai applicatios. Heiseberg s variatio, as its ame suggests, is based matrix ad vector algebra, while chrödiger s approach requires itegral ad differetial calculus. Dirac s otatio ca be used i a first step i which the quatum mechaical calculatio is described or set up. After this is doe, oe chooses either matrix or wave mechaics to complete the calculatio, depedig o which method is computatioally the most expediet. Kets, Bras, ad Bra-Ket Pairs I Dirac s otatio what is kow is put i a ket,. o, for example, p expresses the fact that a particle has mometum p. It could also be more explicit: p =, the particle has mometum equal to ; x =.3, the particle has positio.3. Ψ represets a system i the state Q ad is therefore called the state vector. The ket ca also be iterpreted as the iitial state i some trasitio or evet. The bra represets the fial state or the laguage i which you wish to express the cotet of the ket. For example, x =.5 Ψ is the probability amplitude that a particle i state Q will be foud at positio x =.5. I covetioal otatio we write this as Q(x=.5), the value of the fuctio Q at x =.5. The absolute square of the probability amplitude,

2 x =.5 Ψ, is the probability desity that a particle i state Q will be foud at x =.5. Thus, we see that a bra-ket pair ca represet a evet, the result of a experimet. I quatum mechaics a experimet cosists of two sequetial observatios - oe that establishes the iitial state (ket) ad oe that establishes the fial state (bra). If we write x Ψ we are expressig Q i coordiate space without beig explicit about the actual value of x. x =.5 Ψ is a umber, but the more geeral expressio x Ψ is a mathematical fuctio, a mathematical fuctio of x, or we could say a mathematical algorithm for geeratig all possible values of x Ψ, the probability amplitude that a system i state Ψ has positio x. For the groud state of the well-kow particle-i-a-box of uit dimesio x Ψ = Ψ ( x) = si( π x). However, if we wish to express Q i mometum space we would write, [ ] π p Ψ = Ψ ( p) = π exp( ip) + p. How oe fids this latter expressio will be discussed later. The major poit here is that there is more tha oe laguage i which to express Ψ. The most commo laguage for chemists is coordiate space (x, y, ad z, or r,, ad N, etc.), but we shall see that mometum space offers a equally importat view of the state fuctio. It is importat to recogize that x Ψ ad p Ψ are formally equivalet ad cotai the same physical iformatio about the state of the system. Oe of the teets of quatum mechaics is that if you kow Ψ you kow everythig there is to kow about the system, ad if, i particular, you kow x Ψ you ca calculate all of the properties of the system ad trasform x Ψ, if you wish, ito ay other appropriate laguage such as mometum space. A bra-ket pair ca also be thought of as a vector projectio - the projectio of the cotet of the ket oto the cotet of the bra, or the shadow the ket casts o the bra. For example,

3 ΦΨ is the projectio of the state Q oto the state M. It is the amplitude (probability amplitude) that a system i state Q will be subsequetly foud i state M. It is also what we have come to call a overlap itegral. The state vector Ψ ca be a complex fuctio (that is have the form, a + ib, or exp(-ipx), for example, where i = ). Give the relatio of amplitudes to probabilities metioed above, it is ecessary that ΨΨ, the projectio of Q oto itself is real. This Ψ = Ψ * * Ψ Ψ Ψ = a+ ib requires that, where is the complex cojugate of. o if Ψ = a ib ΨΨ = a + b the, which yields, a real umber. Ket-Bra Products - Projectio Operators Havig examied kets, bras, ad bra-ket pairs, it is ow appropriate to study projectio operators which are ket-bra products. Take the specific example of i i Ψ i i Ψ. operatig o the state vector, which is This operatio reveals the cotributio of i to Ψ, or the legth of the shadow that Ψ casts o i. We are all familiar with the simple two-dimesioal vector space i which a arbitrary vector ca be expressed as a liear combiatio of the uit vectors (basis vectors, basis states, etc) i the mutually orthogoal x- ad y-directios. We label these basis vectors i ad j. For the two-dimesioal case the projectio operator which tells how i ad j cotribute to a v i i + j j v = i i v + j j v arbitrary vector is:. I other words,. This 3

4 i i + j j i i + j j = meas, of course, that is the idetity operator:. This is also called the completeess coditio ad is true if i ad j spa the space uder cosideratio. For discrete basis states the completeess coditio is : =. For cotiuous basis states, such as positio, the completeess coditio is: x xdx=. If Ψ is ormalized (has uit legth) the ΨΨ =. We ca use Dirac s otatio to express this i coordiate space as follows. ΨΨ = Ψx x Ψ dx = Ψ * ( x) Ψ( x) dx I other words itegratio of Q(x)*Q(x) over all values of x will yield if Q(x) is ormalized. Note how the cotiuous completeess relatio has bee iserted i the bra-ket pair o the left. Ay vertical bar ca be replaced by the discrete or cotiuous form of the completeess relatio. The same procedure is followed i the evaluatio of the overlap itegral, ΦΨ, referred to earlier. ΦΨ = Φx x Ψ dx = Φ * ( x) Ψ( x) dx Now that a basis set has bee chose, the overlap itegral ca be evaluated i coordiate space by traditioal mathematical methods. The Liear uperpositio The aalysis above ca be approached i a less direct but still revealig way by writig Ψ ad Φ as liear superpositios i the eigestates of the positio operator as is show 4

5 below. Ψ = x x Ψ dx Φ = Φ x' x' dx ' Combiig these as a bra-ket pair yields, ΦΨ = Φx ' x' x x Ψ dx' dx = Φx x Ψdx The xndisappears because the positio eigestates are a orthogoal basis set ad x' x = 0 uless xn = x i which case it equals. Ψ = Ψ is a liear superpositio i the discrete (rather tha cotiuous) basis set. A specific example of this type of superpositio is easy to demostrate usig matrix mechaics. For example, = The vector o the left represets spi-up i the x-directio, while the vectors o the right side are spi-up ad spi-dow i the z-directio, respectively. This ca also be expressed symbolically i Dirac otatio as = + xu zd. It is easy to show that all three vector states are ormalized, ad that ad zd form a orthoormal basis set. I other words, =, ad. = zd zd = zd 0 It caot be stressed too strogly that a liear superpositio is ot a mixture. For example, whe the system is i the state xu every measuremet of the x-directio spi yields 5

6 the same result: spi-up. However, measuremet of the z-directio spi yields spi-up 50% of the time ad spi-dow 50% of the time. The system has a well-defied value for the spi i the x- directio, but a idetermiate spi i the z-directio. It is easy to calculate the probabilities for the z-directio spi measuremets: ad The reaso xu = zd xu =. xu zd caot be iterpreted as a mixture of ad is because ad are liear superpositios of ad : ad xu xd xu xd zd = + ; =. zd xu xd xu, Thus, if is a mixture of ad it would yield a idefiite measuremet of the spi i the x-directio i spite of the fact that it s a eigefuctio of the x-directio spi operator. Just oe more example of the liear superpositio. Cosider a trial wave fuctio for the particle i the oe-dimesioal, oe-bohr box such as, zd 3 ( ) Θ ( x) = 05 x x. Because the eigefuctios for the particle-i-a-box problem form a complete basis set, M(x) ca be writte as a liear combiatio or liear superpositio of these eigefuctios. Φ= Φ= xxφ dx I this otatio Φ is the projectio of M oto the eigestate. This projectio or shadow of M o to ca be writte as c. It is a measure of the cotributio makes to the state Φ. It is also a overlap itegral. Therefore we ca write, Φ = c. Usig Mathcad, for example, it is easy to show that the first te coefficiets i this equatio are:.935, -.35,.035, -.044,.007, -.03,.003, -.005,.00,

7 Operators, Eigevectors, Eigevalues, ad Expectatio Values I matrix mechaics operators are matrices ad states are represeted by vectors. The matrices operate o the vectors to obtai useful physical iformatio about the state of the system. Accordig to quatum theory there is a operator for every physical observable ad a system is either i a state with a well-defied value for that observable or it is ot. The operators associated with spi i the x- ad z-directio are show below i uits of h/4b. x = z 0 = = Whe operates o the result is xu : ˆ =. xu is a ˆx xu x xu xu eigefuctio or eigevector of ˆx with eigevalue (i uits of h/4b). However, xu is ot a eigefuctio of because where xd =. This meas, as ˆz ˆz xu xd = metioed i the previous sectio, that xu does ot have a defiite value for spi i the z- directio. Uder these circumstaces we ca t predict with certaity the outcome of a z-directio spi measuremet, but we ca calculate the average value for a large umber of measuremets. This is called the expectatio value ad i Dirac otatio it is represeted as follows: ˆ = 0. I matrix mechaics it is calculated as follows. xu z xu ( ) 0 0 = 0 7

8 This result is cosistet with the previous discussio which showed that xu is a liear superpositio of ad with eigevalues of + ad -, respectively. I other words, zd half the time the result of the measuremet is + ad the other half -, yieldig a average value of zero. Now we will look at the calculatio for the expectatio value for a system i the state Ψ, which is set up as follows: Ψ xˆ Ψ. To make this calculatio computatioally friedly we expad Ψ i the eigestates of the positio operator. * Ψ xˆ Ψ = Ψ xˆ x x Ψ dx= Ψ xxx Ψ dx=ψ( x) xψ( xdx ) Note the simplificatio that occurs because xx ˆ = xx= xx. The Variatio Method We have had a prelimiary look at the variatio method, a approximate method used whe a exact solutio to chrödiger s equatio is ot available. Usig Φ ( x) = 30x( x) as a trial wave fuctio for the particle-i-the-box problem, we evaluate the expectatio value for the eergy as E = Φ Hˆ Φ. However, employig Dirac s formalism we ca expad M, as oted above, i terms of the eigefuctios of H as follows. Hˆ E E = Φ Hˆ Φ = Φ Hˆ Φ = = si( π x) But, because the states are eigefuctios of the eergy operator, H ˆ. Thus, the eergy expressio becomes, E = Φ E Φ = c E 8

9 E π where =. Because M is ot a eigefuctio of H ˆ, the eergy operator, this system does ot have a well-defied eergy ad all we ca do is calculate the average value for may experimetal measuremets of the eergy. Each idividual eergy measuremet will yield oe of the eigevalues of the eergy operator, E, ad the c tells us the probability of this result beig achieved. Usig Mathcad it is easy to show that c =.9987, c 3 =.004, c 5 =.00006, ad c 7 = All other coefficiets are zero or vaishigly small. These results say that if we make a eergy measuremet o a system i the state represeted by M there is a 99.87% chace we will get 4.935, a 0.4% chace we will get 9.739, ad so o. We might say the that the state M is a liear combiatio of the first four odd eigefuctios, with the first eigefuctio makig by far the biggest cotributio. The variatioal theorem says that o matter how hard you try i costructig trial wave fuctios you ca t do better tha the true groud state value for the eergy, ad this equatio captures that importat priciple. The oly way M ca give the correct result for the groud state of the particle i the box, for example, is if c =, or if M is the eigefuctio itself. If this is ot true, the c < ad the other values of c are o-zero ad the eergy has to be greater tha E. Takig aother look at the last two equatios reveals that a measuremet operator ca = always be writte as a projectio operator ivolvig its eigestates, Hˆ E. Mometum Operator i Coordiate pace Wave-particle duality is at the heart of quatum mechaics. A particle with wavelegth 8 has wave fuctio (u-ormalized) x x λ = exp iπ. However, accordig to debroglie s wave equatio λ the particle s mometum is p = h/8. Therefore the mometum wave fuctio of the particle i x p exp ipx = coordiate space is. ħ I mometum space the followig eigevalue equatio holds: ˆp p = p p. Operatig o the 9

10 mometum eigefuctio with the mometum operator i mometum space returs the mometum eigevalue times the origial mometum eigefuctio. I other words, i its ow space the mometum operator is a multiplicative operator (the same is true of the positio operator i coordiate space). To obtai the mometum operator i coordiate space this expressio ca be projected oto coordiate space by operatig o the left by x. ipx ħ d ipx ħ d x pˆ p = p x p = pexp = exp = x p ħ idx ħ idx Comparig the first ad last terms reveals that x pˆ = ħ d i dx x d i dx ipx ħ ad that is the mometum operator i coordiate space. px = exp is the ħ positio wave fuctio i mometum space. Usig the method outlied above it is easy to show d i dp that the positio operator i mometum space is ħ. Fourier Trasform Quatum chemists work maily i positio (x,y,z) space because they are iterested i electro desities, how the electros are distributed i space i atoms ad molecules. However, quatum mechaics has a equivalet formulatio i mometum space. It aswers the questio of what does the distributio of electro velocities look like? The two formulatios are equivalet, that is, they cotai the same iformatio, ad are coected formally through a Fourier trasform. The Dirac otatio shows this coectio very clearly. p Ψ = px xψ dx tartig from the left we have the amplitude that a system i state Q has positio x. The, if it has positio x, the amplitude that it has mometum p. We the sum over all values of x to fid all the 0

11 ways a system i the state Q ca have mometum p. As a particular example we ca chose the particle-i-a-box problem with eigefuctios x Ψ oted above. It is easy to show that the mometum eigestates i positio space i atomic uits (see previous sectio) are x p p x = exp( ipx). This, of course, meas that the complex cojugate is = exp ( ipx). Therefore, the Fourier traform of Q(x) ito mometum space is p Ψ =Ψ ( p) = exp( ipx) si( π x) dx π 0 This itegral ca be evaluated aalytically ad yields the followig mometum space wave fuctios for the particle-i-a-box. cos( π ) exp( ip) p Ψ =Ψ ( p) = π π p A graphical display of the mometum distributio fuctio, Q*(p)Q(p), for several states is show below.

12 ummary ad Refereces J. L. Marti (see refereces below) has idetified four virtues of Dirac otatio.. It is cocise. There are a small umber of basic elemets to Dirac s otatio: bras, kets, bra-ket pairs, ket-bra products, ad the completeess relatio (cotiuous ad discreet). With these few buildig blocks you ca costruct all of quatum theory.. It is flexible. You ca use it to say the same thig i several ways; traslate with ease from oe laguage to aother. Perhaps the isight that the Dirac otatio offers to the Fourier trasform is the best example of this virtue. 3. It is geeral. It is a sytax for describig what you wat to do without committig yourself to a particular computatioal approach. I other words, you use it to set up a problem ad the choose the most expeditious way to execute the calculatio. 4. While it is ot exactly the idustry stadard, it should be for the reasos listed i -3 above. It is widely used, so if you wat to read the literature i quatum chemistry ad physics, you eed to lear Dirac otatio. I additio most of the best quatum textbooks i chemistry ad physics use it. I would like to add a 5 th virtue. 5. Oce you get the hag of it you will fid that it is simple to use ad very elighteig. It facilitates the uderstadig of all the fudametal quatum cocepts. The followig texts have bee used i preparig this tutorial: Chester, M. Primer of Quatum Mechaics; Krieger Publishig Co.:Malabar, FL, 99. Das, A.; Melissios, A. C. Quatum Mechaics: A Moder Itroductio; Gordo ad Breach ciece Publishers: New York, 986. Feyma, R. P.; Leighto, R. B.; ads, M. The Feyma Lectures o Physics, Vol.3; Addiso-Wesley: Readig, 965. Marti, J. L. Basic Quatum Mechaics; Claredo Press, Oxford, 98.