Eigenvalues and Eigenfunctions of an Operator

Time-Independent Schrödinger Equation

8.4 Eigenvalues and Eigenfunctions of an Operator

A linear operatorAmay be used to generate a homogeneous equation (eigenvalue equation) in the unknownv, having the form

Av=Av, (8.23)

5In this context the termoperatorhas the following meaning: if an operation brings each function fof a given function space into correspondence with one and only one functionsof the same space, one says that this is obtained through the action of a given operatorÂontofand writess=Âf. Alinearoperator is such thatÂ(c1f1+c2f2)=c1Af1+c2Af2for any pair of functionsf1,f2

and of complex constantsc1,c2([78], Chap. II.11).

6The adjoint operator is the counterpart of the conjugate-transpose matrix in vector algebra.

withAa parameter. Clearly (8.23) admits the solutionv=0 which, however, is of no interest; it is more important to find whether specific values ofAexist (eigenvalues), such that (8.23) admits non-vanishing solutions (eigenfunctions). In general (8.23) must be supplemented with suitable boundary or regularity conditions onv.

The set of the eigenvalues of an operator found from (8.23) is the operator’s spectrum. It may happen that the eigenvalues are distinguished by an index, or a set of indices, that take only discrete values; in this case the spectrum is calleddiscrete.

If, instead, the eigenvalues are distinguished by an index, or a set of indices, that vary continuously, the spectrum iscontinuous. Finally, it ismixedif a combination of discrete and continuous indices occurs.

An eigenvalue issimpleif there is one and only one eigenfuction corresponding to it, while it isdegenerate of ordersif there areslinearly independent eigenfuctions corresponding to it. The order of degeneracy may also be infinite. By way of example, the Schrödinger equation for a free particle in one dimension discussed in Sect.8.2.1 has a continuous spectrum of eigenvaluesE= ¯h²k²/(2m) of indexk, namely,E= E_k. Each eigenvalue is degenerate of order 2 because to eachEthere correspond two linearly-independent eigenfunctions exp (ik x), exp (−ik x), withk=√

2m E/h.¯ Instead, the Schrödinger equation for a particle in a box discussed in Sect.8.2.2 has a discrete spectrum of eigenvaluesE_ngiven by the first relation in (8.6). Each eigenvalue is simple as already indicated in Sect.8.2.2.

Letv⁽¹⁾,. . .,v^(s)be the linearly-independent eigenfunctions belonging to an eigen- valueAdegenerate of orders; then a linear combination of such eigenfunctions is also an eigenfunction belonging toA. In fact, lettingα₁,. . . ,αs be the coefficients of the linear combination, fromAv^(k) =Av^(k)it follows

A s

k=1

αkv^(k)= s

k=1

αkAv^(k) = s

k=1

αkAv^(k) =A s

k=1

αkv^(k). (8.24)

8.4.1 Eigenvalues of Hermitean Operators

A fundamental property of the Hermitean operators is that their eigenvalues are real.

Consider, first, the case where the eigenfunctions are square integrable, so thatv|v is different from zero and finite. To proceed one considers the discrete spectrum, where the eigenvalues areAn. Herenindicates a single index or also a set of indices.

If the eigenvalue is simple, letvn be the eigenfunction belonging to An; if it is degenerate, the same symbolvnis used here to indicate any eigenfunction belonging toAn. Then, two operations are performed: in the first one, the eigenvalue equation Avn = Anvnis scalarly multiplied by vn on the left, while in the second one the conjugate equation (Avn)^∗ = A^∗_nv^∗_n is scalarly multiplied byvn on the right. The operations yield, respectively,

vn|Avn =Anvn|vn, Avn|vn =A^∗_nvn|vn. (8.25) The left hand sides in (8.25) are equal to each other due to the hermiticity ofA; as a consequence,A^∗_n=An, that is,Anis real.

164 8 Time-Independent Schrödinger Equation Another fundamental property of the Hermitean operators is that two eigenfunctions belonging to different eigenvalues are orthogonal to each other. Still considering the discrete spectrum, letA_m,A_nbe two different eigenvalues and letv_m(vn) be an eigenfunction belonging toA_m(An). The two eigenvalues are real as demonstrated earlier. Then, the eigenvalue equationAv_n =A_nv_nis scalarly multiplied byv_mon the left, while the conjugate equation for the other eigenvalue, (Av_m)^∗ =Amv^∗_m, is scalarly multiplied byvnon the right. The operations yield, respectively,

8.4.2 Gram–Schmidt Orthogonalization

When two eigenfunctions belonging to a degenerate eigenvalue are considered, the reasoning that proves their orthogonality through (8.26) is not applicable because A_n = A_m. In fact, linearly-independent eigenfunctions of an operatorAbelong- ing to the same eigenvalue are not mutually orthogonal in general. However, it is possible to form mutually-orthogonal linear combinations of the eigenfunctions. As shown by (8.24), such linear combinations are also eigenfunctions, so their norm is different from zero. The procedure (Gram–Schmidt orthogonalization) is described here with reference to the case of thenth eigenfunction of a discrete spectrum, with a degeneracy of orders. Let the non-orthogonal eigenfunctions bev_n⁽¹⁾,. . .,v^(s)_n , and let u⁽¹⁾_n ,. . .,u_n^(s)be the linear combinations to be found. Then one prescribesu⁽¹⁾_n =v_n⁽¹⁾, u⁽²⁾_n =v⁽²⁾_n +a₂₁u⁽¹⁾_n wherea₂₁is such thatu⁽¹⁾_n |u⁽²⁾_n =0; thus

u⁽¹⁾_n |v⁽²⁾_n +a₂₁u⁽¹⁾_n |u⁽¹⁾_n =0, a₂₁ = −u⁽¹⁾_n |v⁽²⁾_n

u⁽¹⁾n |u⁽¹⁾n . (8.27) The next function is found be lettingu⁽³⁾_n =v⁽³⁾_n +a₃₁u⁽¹⁾_n +a₃₂u⁽²⁾_n , withu⁽¹⁾_n |u⁽³⁾_n = 0,u⁽²⁾_n |u⁽³⁾_n =0, whence

u⁽¹⁾_n |v⁽³⁾_n +a₃₁u⁽¹⁾_n |u⁽¹⁾_n =0, a₃₁ = −u⁽¹⁾_n |v⁽³⁾_n

u⁽¹⁾n |u⁽¹⁾n , (8.28) u⁽²⁾_n |v⁽³⁾_n +a₃₂u⁽²⁾_n |u⁽²⁾_n =0, a₃₂ = −u⁽²⁾_n |v⁽³⁾_n

u⁽²⁾n |u⁽²⁾n . (8.29) Similarly, thekth linear combination is built up recursively from the combinations of indices 1,. . .,k−1:

u^(k)_n =v^(k)_n +

k−1

i=1

akiu⁽ⁱ⁾_n , aki = −u_n⁽ⁱ⁾|v^(k)_n

u⁽ⁱ⁾n|u⁽ⁱ⁾n . (8.30) The denominators in (8.30) are different from zero because they are the squared norms of the previously-defined combinations.

8.4.3 Completeness

As discussed in Sect.8.2.1, the eigenfunctions of the Schrödinger equation for a free particle, for a givenk = √

2m E/h¯ and apart from a multiplicative constant, arew_+k = exp (ik x) andw_−k =exp (−ik x). They may be written equivalently asw(x,k)=exp (ik x), withk = ±√

2m E/h. Taking the multiplicative constant¯ equal to 1/√

2π, and considering a functionf that fulfills the condition (C.19) for the Fourier representation, one applies (C.16) and (C.17) to find

f(x)= _+∞

−∞

exp (ik x)

√2π c(k) dk, c(k)= _+∞

−∞

exp (−ik x)

√2π f(x) dx. (8.31) Using the definition (8.13) of scalar product one recasts (8.31) as

f(x)= _+∞

−∞

c(k)w(x,k) dk, c(k)= w|f. (8.32) In general the shorter notationwk(x),ck is used instead ofw(x,k), c(k). A set of functions likew_k(x) that allows for the representation offgiven by the first relation in (8.32) is said to becomplete. Each member of the set is identified by the value of the continuous parameterkranging from−∞to+∞. To eachkit corresponds a coefficientof the expansion, whose value is given by the second relation in (8.32).

Expressions (8.31) and (8.32) hold true because they provide the Fourier transform or antitransform of a function that fulfills (C.19). On the other hand,wk(x) is also the set of eigenfunctions of the free particle. In conclusion, the eigenfunctions of the Schrödinger equation for a free particle form a complete set.

The same conclusion is readily found for the eigenfunctions of the Schrödinger equation for a particle in a box. To show this, one considers a functionf(x) defined in an interval [−α/2,+α/2] and fulfilling_+α/2

−α/2 |f(x)|dx <∞. In this case the expansion into a Fourier series holds:

f(x)=1 2a₀+

∞

n=1

[an cos (2π n x/α)+bnsin (2π n x/α)] , (8.33)

witha₀/2= ¯f =(1/α)_+α/2

−α/2 f(x) dxthe average off over the interval, and

$an

= 2 α

₊α/2

−α/2

(cos sin

) 2π n x α

f(x) dx, n=1, 2,. . . (8.34) Equality (8.33) indicates convergence in the mean, namely, usingg=f− ¯f for the sake of simplicity, (8.33) is equivalent to

Nlim→∞

₊α/2

−α/2

* g−

n=1

[ancos (2π n x/α)+b_n sin (2π n x/α)]

dx=0.

(8.35)

166 8 Time-Independent Schrödinger Equation

Defining the auxiliary functions χ_n=

2/αcos (2π n x/α), σ_n=

2/αsin (2π n x/α), (8.36) a more compact notation is obtained, namely,f = ¯f +_∞

n=1(χ_n|fχ_n+ σ_n| fσ_n)or, observing thatσ_n|const = χ_n|const =0,

g= ∞

n=1

(χ_n|gχ_n+ σ_n|gσ_n) . (8.37) The norm of the auxiliary functions (8.36) is unity,χ_n|χ_n = σ_n|σ_n =1 forn= 1, 2,. . ., and all auxiliary functions are mutually orthogonal:χ_m|χ_n = σ_m|σ_n =0 for n,m = 0, 1, 2,. . ., m = n, and σm|χn = 0 for n,m = 0, 1, 2,. . . A set whose functions have a norm equal to unity and are mutually orthogonal is called orthonormal. Next, (8.37) shows that the setχn,σn,n=0, 1, 2,. . . is complete in [−α/2,+α/2] with respect to anygfor which the expansion is allowed. Letting c_2n₋₁ = χn|g,c_2n = σn|g,w_2n₋₁ =χn,w_2n=σn, (8.37) takes the even more compact form

g= ∞

m=1

cmwm, cm= wm|g. (8.38) From the properties of the Fourier series it follows that the set of theσnfunctions alone is complete with respect to any function that is odd in [−α/2,+α/2], hence it is complete with respect to any function over the half interval [0,+α/2]. On the other hand, lettinga = α/2 and comparing with (8.8) shows that σn(apart from the normalization coefficient) is the eigenfunction of the Schrödinger equation for a particle in a box. In conclusion, the set of eigenfunctions of this equation is complete within [0,a].

One notes the striking resemblance of the first relation in (8.38) with the vector- algebra expression of a vector in terms of its components c_m. The similarity is completed by the second relation in (8.38), that provides each component as the projection ofgoverwm. The latter plays the same role as the unit vector in algebra, the difference being that the unit vectors here are functions and that their number is infinite. A further generalization of the same concept is given by (8.32), where the summation indexkis continuous.

Expansions like (8.32) or (8.38) hold becausewk(x) andwm(x) are complete sets, whose completeness is demonstrated in the theory of Fourier’s integral or series;

such a theory is readily extended to the three-dimensional case, showing that also the three-dimensional counterparts ofwk(x) orwm(x) form complete sets (in this case the indiceskormare actually groups of indices, see, e.g., (9.5)). One may wonder whether other complete sets of functions exist, different from those considered in this section; the answer is positive: in fact, completeness is possessed by many

other sets of functions,⁷and those of interest in Quantum Mechanics are made of the eigenfunctions of equations like (8.23). A number of examples will be discussed later.

8.4.4 Parseval Theorem

Consider the expansion of a complex function f with respect to a complete and orthonormal set of functionswn,

f =

cnwn, cn= wn|f, wn|wm =δnm, (8.39) where the last relation on the right expresses the set’s orthonormality. As before,m indicates a single index or a group of indices. The squared norm off reads

||f||²=

|f|²d= ,

cnwn|

cmwm

. (8.40)

Applying (8.17,8.18) yields

||f||²=

c_n^∗

c_mw_n|w_m =

c_n^∗

c_mδ_nm=

|c_n|², (8.41) namely, the norm of the function equals the norm of the vector whose components are the expansion’s coefficients (Parseval theorem). The result applies irrespective of the set that has been chosen for expandingf. The procedure leading to (8.41) must be repeated for the continuous spectrum, where the expansion reads

f =

cαwαdα, cα = wα|f. (8.42) Here a difficulty seems to arise, related to expressing the counterpart of the third relation in (8.39). Considering for the sake of simplicity the case where a single index is present, the scalar productwα|wβmust differ from zero only forβ =α, while it must vanish forβ = αno matter how small the differenceα−β is. In other terms, for a given value ofαsuch a scalar product vanishes for anyβ apart from a null set. At the same time, it must provide a finite value when used as a factor within an integral. An example taken from the case of a free particle shows that the requirements listed above are mutually compatible. In fact, remembering the analysis of Sect.8.4.3, the scalar product corresponding to the indicesαandβreads

wα|wβ = 1 2π

_+∞

−∞

exp [i (β−α)x] dx =δ(α−β), (8.43)

7The completeness of a set of eigenfunctions must be proven on a case-by-case basis.

168 8 Time-Independent Schrödinger Equation where the last equality is taken from (C.43). As mentioned in Sect. C.4, such an equality can be used only within an integral. In conclusion,⁸

|f|²d= f|f = _+∞

−∞

c_α^∗dα _+∞

−∞

c_βδ(α−β) dβ = _+∞

−∞ |c_α|²dα.

(8.44) One notes that (8.44) generalizes a theorem of Fourier’s analysis that states that the norm of a function equals that of its transform.

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