Physics 324, Fall 2002 Dirac Notatio These otes were produced by David Kapla for Phys. 324 i Autum 2001. 1 Vectors 1.1 Ier product Recall from liear algebra: we ca represet a vector V as a colum vector; the V =(V T ) is a row vector, ad the ier product (aother ame for dot product) betwee two vectors is writte as A B = A 1 B 1 + A 2 B 2 +. (1) I covetioal vector otatio, the above is just A B. Note that the ier product of a vector with itself is positive defiite; we ca defie the orm of a vector to be V = V V, (2) which is a o-egative real umber. (I covetioal vector otatio, this is V, which is the legth of V ). 1.2 Basis vectors We ca expad a vector i a set of basis vectors {ê i }, provided the set is complete, which meas that the basis vectors spa the whole vector space. The basis is called orthoormal if they satisfy ê iê j = δ ij (orthoormality), (3) ad a orthoormal basis is complete if they satisfy ê i ê i = I (completeess), (4) i where I is the uit matrix. (ote that a colum vector ê i times a row vector ê i is a square matrix, followig the usual defiitio of matrix multiplicatio). Assumig we have a complete orthoormal basis, we ca write V = IV = i ê i ê iv i V i ê i, V i (ê iv ). (5) The V i are complex umbers; we say that V i are the compoets of V i the {ê i } basis. 1
1.3 Eigevectors as basis vectors Sometimes it is coveiet to choose as basis vectors the eigevectors of a particular matrix. I quatum mechaics, measurable quatities correspod to hermitia operators; so here we will look at hermitia matrices. A hermitia matrix is oe satisfyig M = M (M T ) (hermitia). (6) This just meas that the compoets of a hermitia matrix satisfy M ij = M ji. We say that the vector v is a eigevector of the matrix M if it satisfies Mv = λ v, (7) where λ is a umber called a eigevalue of M. If M is hermitia, the eigevalues λ are all real, ad the eigevectors may be take to be orthoormal: v m v = δ m. (8) So we ca take the v to be our basis vectors, ad write a arbitrary vector A i this basis as A = A v. (9) where the A are i geeral complex umbers. This is a coveiet choice if we wish to kow what is the actio of the hermitia matrix M whe it multiplies the vector A: MA = A Mv = A λ v. (10) 2 Dirac otatio for vectors Now let us itroduce Dirac otatio for vectors. We simply rewrite all the equatios i the above sectio i terms of bras ad kets. We replace V V, V V, A B A B. (11) Suppose we have basis vector i, aalogous to the ê i, which form a complete orthoormal set: i j = δ ij (orthoormality) i i i = 1 (completeess), (12) where 1 is the idetity operator; it has the property 1 ψ = ψ for ay ψ. The ay vector V may be expaded i this basis as V = 1 V = i i i V i V i i, V i i V. (13) 2
Note that V i = V i. As before, we ca use the eigevectors of a hermitia operator for our basis vectors. Matrices become operators i this laguage, M ˆM. The the eigevalue equatio becomes ˆM = λ, (14) where the λ are real ad we ca take the kets to be orthoormal: m = δ m. The we ca write ˆM V = ˆM V = V ˆM = V λ. (15) 3 Dirac otatio for quatum mechaics Fuctios ca be cosidered to be vectors i a ifiite dimesioal space, provided that they are ormalizable. I quatum mechaics, wave fuctios ca be thought of as vectors i this space. We will deote a quatum state as ψ. This state is ormalized if we make it have uit orm: ψ ψ =1. Measurable quatities, such as positio, mometum, eergy, agular mometum, spi, etc are all associated with operators which ca act o ψ. IfÔ is a operator correspodig to some measurable quatity, its expectatio value is give by Ô = ψ Ô ψ. Note that sice A Ô B = B Ô A, (check this i the case of fiite legth vectors ad matrices!) it follows that if your measurable quatity is real (ad they always are!) the Ô = Ô implies that ψ Ô ψ = ψ Ô ψ, or that Ô = Ô. Coclusio: Measurable quatities are associated with hermitia operators. I order to compute expectatio values for give quatum states, it is ofte useful to choose a coveiet basis. I will discuss three commo bases that are ofte used: the positio eigestate basis, the mometum eigestate basis, ad the eergy eigestate basis. 3.1 Positio eigestate basis: x The positio operator ˆx =ˆx is a hermitia operator, ad we ca use its eigevectors as a orthoormal basis. The state x is defied to be the eigestate of ˆx with eigevalue x: ˆx x = x x. (16) What is ew here is that the eigevalues x are ot discrete, ad so we use the Dirac δ-fuctio for ormalizatio: x x = δ(x x ) (orthoormality). (17) 3
The states x form a complete basis for our space ad dx x x = ˆ1 (completeess) (18) Note that the sum over discrete basis vectors i eq. (12) has bee replaced by a itegral. Also, the uit operator ˆ1 has replaced the uit matrix I. Therefore we ca expad our state ψ i terms of the x basis vectors: ψ = ˆ1 ψ = dx x x ψ = dx ψ(x) x, (19) where we have defied ψ(x) x ψ. This ψ(x) is othig but our familiar wavefuctio. I the preset laguage, ψ(x) are the coordiates of the our state ψ i the x basis. Note that i eq. (19) we iserted the uit operator i the guise of a itegral over x x this techique is very powerful, ad is called isertig a complete set of states. If we have ormalized ψ so that ψ ψ = 1, it follows that 1= ψ ψ = ψ ˆ1 ψ = dx ψ x x ψ = dx ψ (x)ψ(x). (20) This is the usual ormalizatio coditio we have see. Computig ˆx for our state is particularly easy i the x basis, sice x is a eigestate of the operator ˆx : ˆx = ψ ˆx ψ = dx ψ ˆx x x ψ = dx x ψ x x ψ = dx x ψ (x)ψ(x). (21) I the ext sectio I will discuss measurig ˆp, usig the positio eigestate basis. 3.2 Mometum eigestate basis: p Aother useful basis is formed by the eigestates of the mometum operator ˆp: ˆp p = p p, p p = δ(p p ), p p = ˆ1. (22) We ca expad ψ i the mometum basis as ψ = p p ψ = φ(p) p, φ(p) p ψ. (23) We see that φ(p) is just the wavefuctio i the mometum basis. As i the above sectio, we ca easily compute expectatio values such as ˆp usig this basis. 4
It is iterestig to ask how we ca traslate betwee the x ad the p bases. For this, we eed to kow the quatity x p. We ca get this by kowig that ψ(x) ad φ(p) are Fourier trasforms of each other: ψ(x) = φ(p)e ipx/ h. (24) We ca rewrite this as ψ(x) = x ψ = p ψ e ipx/ h. (25) Isertig a complete set of states o the left had side of the above equatio we get x p p ψ = p ψ e ipx/ h, (26) implyig that x p = eipx/ h. (27) Usig this result we ca also compute (here y is a eigevector of ˆx with eigevalue y): y ˆp x = y ˆp p p x = p y p p x = p = i x eip(y x)/ h 2π eip(y x)/ h = i h δ(x y). (28) x Therefore if we kow ψ(x) but ot φ(p), we ca still compute ˆp as ˆp = ψ ˆp ψ = dy dx ψ y y ˆp x x ψ = dy dx ψ (y) (i h x ) δ(x y) ψ(x). (29) Itegratig by parts with respect to x (igorig the boudary terms at x = ±, which vaish) we get ( ˆp = = dy dx ψ (y)δ(x y) i h ) ψ(x) x ( = dx ψ (x) i h ) ψ(x) (30) x So we see that i the x represetatio, ˆp i h x. 5
3.3 Eergy eigestates: Fially, aother operator of iterest is the Hamiltoia ow, the Hamiltoias we have see take the form Ĥ which gives the eergy. Up to Ĥ = ˆp2 + V (ˆx). (31) 2m The time depedet Schrodiger equatio ca be writte as i h ψ,t = Ĥ ψ,t. (32) t You ca thik of ψ,t as a vector movig aroud i our vector space as a fuctio of time, ad the above equatio govers how it moves. Sice ˆp ad ˆx are hermitia, the Ĥ is also hermitia, provided that the potetial V (x) is a real fuctio. Therefore we ca use eigestates of Ĥ as basis vectors: Ĥ = E, m = δ m, = ˆ1. (33) Note that this eigevalue equatio is simply the time-idepedet Schrödiger equatio, ad that sice Ĥ is hermitia, the eigevalues E are real umbers. (Here I have assumed that the eergy eigevalues E are discrete; this is correct for boud states but ot scatterig states. For states with cotiuous eigevalue, replace the δ m by δ(m ), ad the by d.) The we ca expad ψ,t i this basis, with time-depedet coefficiets: ψ,t = c (t). (34) Pluggig this ito the Schrödiger equatio eq. (32) we fid i h t ψ,t = i h dc (t) = dt Ĥc (t) = E c (t) (35) Sice the form a orthoormal basis, it is easy to show that the above equatio implies i h dc (t) = E c (t) dt = c (t) =e iet/ h c (0). (36) Therefore the solutio for ψ,t is ψ,t = e iet/ h c (0). (37) So all we eed to kow is what are c (0) (the iitial coditios at t = 0), ad the solutios, E to the time-idepedet Schrödiger equatio, eq. (33). Note that ψ,t = e iet/ h c (0). (38) 6
If ψ,t is ormalized, it follows that 1= ψ,t ψ,t = c m (t)c (t) m = m c (t) 2 = c (0) 2. (39) Note that i the coordiate basis, where ˆx x,ˆp i hd/dx. x = u (x), (40) where [ h2 d 2 ] 2m dx + V (x) u 2 (x) =E u (x). (41) Solvig this sort of equatio for differet potetials V (x) (ad geeraliztios i 3 dimesios) what we will be doig for the rest of this quarter. 3.4 Commets Why are these bases we have discussed ecessarily differet from each other? For example, ca t we fid a basis i which both ˆp ad ˆx are simple? No. If we had a state x, p which was simultaeously a eigestate of ˆx (with eigevalue x) ad ˆp (with eigevalue p) it would follow that [ˆx, ˆp] x, p = ˆxˆp x, p ˆpˆx x, p = pˆx x, p xˆp x, p = (px xp) x, p = 0 (42) sice x ad p are ordiary umbers, ad commute both with each other ad with operators ˆp ad ˆx. But we kow that [ˆx, ˆp] =i hˆ1 (43) (we do t ormally write the ˆ1) which is icosistet with the previous result, ad so there caot be such as state as x, p. The result is that a state ψ caot simultaeously be a eigestate of two operators that do ot commute. For the hydroge atom, we will fid that we ca fid basis sates which are simultaeously eigestates of Ĥ, ˆL ˆL, ad ˆL z, where L refers to the agular mometum vector. We will discuss a ormalized basis of agular mometum states later i the course. 7