Faddeev L.D., Yakubovskii O.A. Lectures on Quantum Mechanics for Mathematics Students

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128

L. D. Faddeev and 0. A. YakubovskiT

Somewhat cumbersome computations lead to the following form

for the angular momentum operators in spherical coordinates:

got e Cos

L2 = -i

cos O To

cot ©sixl

}

In these variables the operators L± = L1 ± iL2 have the form

L = ei-P

i cot B

L_ = e--

i Cot o

The common eigenfunctions of the operators L2 and L3 will he de-

noted by l

(n) (the number j for L is usually denoted by 1). In

spherical coordinates the equations L, Y11 = 0 and 13 Y11= l Y11 have

the form

OY/I

-i-

=lY

tt,

aYl

iCoto

y =0.

From the first equation in (1) we see that

Yt (0, P) =

Fat (0),

where 1 can take only the integer values l = 0, 1, 2, .... The second

equation in (1) enables us to get an equation for F411(0):

(2)

OF,, (e)

- i cot e F11 (e) - v.

Solving this equation, we get that

F11(0) = C since 0.

We note that for each 1 there exists a single solution of the equation

(2). Thus, we have found that

Y1(0, co) = C

sin1 0

ei10

§ 30. Representations of the rotation group on L' (S2) 129

where the constant C can be found from the normalization condition.

The remaining functions Ym (n) can be computed by the formula

Y -- -

rl-i42

(-570-+

icote

V(1 + rn)(l-m+1)

We shall riot go through the corresponding comput ations. The funs--

t ions Yrra (n) are called the normalized spherical functions of order 1.

They can be expressed as

irnSp

Yrrm (e1

) _ -

1u1rn(cos O)

2ir

where the functions

P1, (A) -

(l_m)!{2

]

1 z 1)1

11! 1-

)

dot -"d

are called the normalized associated Legendre polynomials.

Accordingly, a basis of an irreducible representation in the space

L2(S2) consists of the spherical functions Ym (n) for a fixed 1 and

m = -1, -1 + 1, ... -, 1The space L2 (S2) contains a subspace for an

irreducible representation for each odd dimension 21 + 1 (1 an integer).

The theorem on decomposition of a representation of the rotation

group into irreducible representations is equivalent in this case to the

assertion that the spherical functions are complete in L2 (S2)

Any

function 0(n) E L2(S2) can be expanded in a convergent series

(3)

(n)= E 1: CimYm (n).

1=0 m=-l

We recall that irreducible representations of both even and odd

dimensions acted in D2. The space D2 can be represented as a direct

sum D2 eV2 , where D2 and D2 are orthogonal subspaces of even and

odd functions, respectively. In E)2+, as in L2 (S2), only representations

of odd dimensions act. Any element f (z) E D2 can be represented in

the form

l-mil+m

(4)

f (Z) = E E C1M-

1=0 M=-1

V(1 - M)! (I + M)!

130

L. D. Faddeev and 0. A. Yakubovskjr

It is clear that the one-to-one correspondence

f(z)

V) (n)

establishes an isomorphism between the spaces Di and L2 (S2) for

which

l l

rrt n)

(l -- rrt)!(l + m)}

MkHLk, k=1,2,3.

In the conclusion of this section we consider a representation of

the rotation group on the state space L2 (R3) = L2 (S2) 0 L2 (R+).

Lot {f,L(r)} be an arbitrary basis in L2 (R+). Then {fit(r)Yirn(n)} is

a basis in the space L2 (R,"), and any function b (x) e L2 (R3) can be

expanded in a series:

00 1

V(x) = 1: E 1: Cnlmfrt(r) Yim(0, o)

n l=O m=-l

From this formula it is clear that 12 (R3) can also be decomposed

(in many ways) into subspaces in which the irreducible representations

of the rotation group of order (21+1) act, and each representation DI

is encountered infinitely many times. Any of the invariant subspaces

in which the irreducible representation Dl acts is a set of functions of

the form f (r) Elrt=-1 C Yl., (0, o), where f (r) E L2(R+).

§ 31. The radial Schrodinger equation

Let us return to the problem of the motion of a particle in a central

field We shall look for solutions of the equation

I D

+ V(r) 0 = E01

or, using the formula (23.5), the equation

a 2 8 L2

(1)

2 r2

r ar

2 r2

V (r) 0= F.

We have seen that in the absence of accidental degeneracies the eigen-

spaces of the Schrodinger operator H must coincide with the sub-

spaces of the irreducible representations D1, while in the presence

accidental degeneracies they are direct sums of such subspaces. It is

§ 31. The radial Schrodinger equation

131

clear that all the independent eigenfunctions of H can be constructed

if we look for them in the form

(2)

/'(r, n) = Ri (r) Ym (n).

These functions are already eigenfunctions of the operators L2

and L3:

L2TP =l(l+l)b, L3ip="IV),

and therefore they describe states of the particle with definite values

of the square of the angular momentum and of its third projection.

Substitution of (2) in (1) gives us an equation for Ri (r):

d (r2)

dR1(r)

1(1 + 1 )

R r

V r R r= ER r.

2 r2 dr

dr + 2 r2

, (

) + ()

t ()

I (

}

We introduce a new unknown function by

R1 r

=fa(r)

and the equation for fa (r) takes the form

_ 1 d2.fi 1(1+1)

(3)

dr2 + 2 r2

+ V (r)

fl = Et fl.

This equation is called the radial Schrodinger equation. We point out

some of its features. First of all, the parameter m does not appear in

the equation, which physically means that the energy of the particle

does not depend on the projection of the angular momentum on the

x3-axis. For each 1 its own radial equation is obtained. The spectrum

of the radial equation is always simple (this can be proved), and

therefore accidental degeneracies are possible if the equations (3) with

different 1 have the same eigenvalues.

The radial equation coincides in form with the Schrodinger equa-

tion

1 d`2 /

,i dx2

+VVY = F,

for a one-dimensional particle if we introduce the so-called effective

potential

Veff Y V M +

132 L. D. Faddeev and 0. A. Yakubovskij

Figure 7

However, there is one essential difference. The function p(x) is defined

on R while f (r) is defined on R+, and hence the radial equation is

equivalent to the one-dimensional Schrodinger equation for the prob-

lem with the potential V (x) under the condition that V (x) = oo for

x < 0.

Figure 7 shows the graphs of the functions V (r), 1(1 + 1) / (2/ tr2 ),

and V0 ff (r), with V (r) taken to be the Coulomb potential -a/r of

attraction (a > 0).

The expression 1(1 + 1) / (2i,ir2) can be interpreted as the poten-

tial of repulsion arising due to the centrifugal force. Therefore, this

expression is usually called the centrifugal potential.

In quantum mechanics one has to solve problems with very diverse

potentials V (x). The most important of them seem to be the Coulomb

potential V (r) = a/r describing the interaction of charged particles

and the Yukawa potential V (r) = g e r

often used in nuclear physics.

§ 31. The radial Schrodinger equation

133

One usually considers potentials that are less singuar than 1/r 2_e

as r - 0 (e > 0). In dependence on their behavior as r --* oc, decay-

ing potentials (V(r) --+ 0) are divided into the short-range potentials,

which satisfy V (r) = o( 1l r24c ) for some e > 0, arid the long range

potentials, which do not satisfy t his condition The Yukawa potential

is a short,-range potential, while the Coulomb potential is a long-range

potential. The spectrum of the radial Schrodinger operator

t(l + 1)

dr2 +

V ( r )

(p=)

is well known for a very broad class of potentials.

In the case of

an increasing potential with V (r) - oo as r --+ oc, the spectrum

is a pure point, spectrum arid is simple

In the case of a decaying

potential the interval 0 < E < oc is filled by the continuous spectrum,

and the negative spectrum is discrete. For a short-range potential

the positive spectrum is simple and continuous, while the negative

spectrum consists of a finite number of eigenvalues.

We shall give some simple arguments enabling us to understand

the basic features of the spectrum of H1. To this end we consider how

the solutions of the radial equation behave as r - 0 and r --p oo. If as

r -+ oo we ignore the term Veff in the radial equation, then it reduces

+F;fi=0.

For E > 0 this equation has the two linearly independent solutions

e -ikr said eik'", where

= 2 jiE > 0 and k > 0.

For E < 0 the

linearly independent solutions have the form e-"' and e". where

x2=--2pE>0and x>0.

In the case r --- 0, we can hope to get the correct behavior of the

solutions of the radial equation if in this equation we leave the most

singular

r {2i ,'

f j of the terms linear in fl, so that

fill

aft + i)

fi = o.

This equation has the two linearly independent solutions r`l and

rl--i

134 L. D. Faddeev and 0. A. Yakubovskil

Let us now consider what conditions should reasonably he im-

posed on the solutions of the radial equation. We are interested in

the solutions V) (x) =

} Y

(n) of the Schrtidiiiger equation. The

functions b(x) must be continuous in R3 and either square-integrable

or bounded on the whole space. In the first case they are eigenfunc-

tions in the usual sense of the word, and in the second case they can

be used to describe the continuous spectrum.

The continuity of bb(x) implies the condition fl (0) = 0. Therefore,

only solutions of the radial equation that behave like Crl+1 as r - 0

are of interest.

This condition determines fl(r) up to a numerical

factor. Next, for E < 0 we must, find a solution f (r) that behaves

like Ce x" as r - oo (otherwise, the solution will be unbounded).

For arbitrary negative E these conditions turn out to be incompatible.

Those values of E for which we can construct a solution having the

correct behavior at zero and at infinity are precisely the eigenvalues.

For any E > 0 the solution fl (r) is bounded, and thus it suffices

that it have the correct behavior at zero. The spectrum is continuous

for E > 0.

The eigenfunctions of the discrete spectrum describe a particle

localized in a neighborhood of the center of force; that is, they cor-

respond to a finite motion. The eigenfunctions of the continuous

spectrum can be used to describe the states in which the motion of

the particle is infinite.

The eigenfunctions of the discrete spectrum of the radial equation

will be denoted by fkl (r), where k indexes the eigenvalues Ekl of the

equation for a given 1

Hlfkl = Eklfkl

The eigenfunctions of the continuous spectrum corresponding to

the energy E will he denoted by f hl :

Hl f F,l = Ef EI.

For a broad class of potentials it has been proved that the system

{ f kl

, f r, t

} of functions is complete for each 1 = 0, 1, 2,.

. This means

§ 31. The radial Schrodinger equation

1 35

that an arbitrary function27 f (r) E L2 (0, c) has a representation

f (r) =

Ck f kl (r) +

C(E) IE1(r) dE,

k c

where

Ck = fx

f (r) .fki (r) dr,

C(E) = f .f (r) .fFt (r) dr.

Let us return to the three-dimensional problem. The functions of

the discrete spectrum have the form

x `

f kt (r)

ktrra ( }

lira }

and the multiplicity of the eigenvalue Eki is equal to 21 + I (in the

absence of accidental degeneracies).

For the eigenfunctions of the

continuous spectrum we have

fEt(r)()

Et7n

The multiplicity of the continuous spectrum is infinite, since for any

E > 0 there are solutions of the radial equations for all l and, more-

over, m = -l,--1+

1,...,1.

The parameters k, 1, and m that determine the eigenfunctions

of the point spectrum are called the radial, orbital, and magnetic

quantum numbers, respectively.

These names go back to the old

Bohr-Soxnnmerfeld quantum theory, in which a definite classical orbit

(or several such orbits) corresponded to each admissible value of the

energy. The numbers k and 1 determined the size and shape of the

orbit, and the number m determined the orientation of the plane of

the orbit in space. The number m plays an essential role in magnetic

phenomena, which explains its name.

The completeness of the system {fkz

, f El

} of functions in L2 (0, oc)

implies the completeness of the system {'kjm(X), LE1m (x) } in L2 (R3) .

For brevity of notation we consider the case when the point spec-

trum of the operator H is absent. In this case an arbitrary function

27L2(0, oc) denotes the space of square-integrable functions (without the weight

7 2) on R+ If f E

L2(0,00), then H = f /r E L2 (R-"')

136 L. D. Faddeev and 0. A. YakubovskiI

Z;' (x) E L2 (R3) can be represented in the form

x I

VI(x) _ 1:

Cim(fE) ,O Fim(x) dE.

1-0 m--l U

I t is clear that p is determined by the sequence {C'im(E)} of

functions, and therefore we get the representation

H {Clrn(E)}

in the space of sequences of functions. In this space the scalar product

is given by

cc t

(1,i/'2)= E

Ctrn C(" dE.

1=0 m=-l

1(0)

17n

From the fact that JBam (x) is an eigenfunctiori of the operators H,

L2, and L3, it follows easily that these operators act as follows in the

representation constructed:

IICim(E) = ECjm(E),

(5)

L2CIm(E) =1(l + 1) Cim(E),

L3Cim(E) = mCim(E).

Therefore, this representation is an eigenrepresentation for the three

commuting operators H, L2, and L3. (The equation (5) should not

be confused with the equations for the eigenvectors.)

§ 32. The hydrogen atom. The alkali metal

atoms

The hydrogen atom is a bound state of a positively charged nucleus

with charge e and an electron with charge -e (e > 0 is the absolute

value of the charge of an electron). Therefore, the potential V (,r) has

the form

V(r) = --.

We consider the problem of motion in the field

V(r) Zee

§ 32. The hydrogen atom. The alkali metal atoms

137

Such a potential corresponds to a hydrogen atom with Z = 1 and

to the hydrogen-like ions He+, Li+-, ... with Z = 2,3,. ..

In the

coordinate representation the Schrodinger operator has the form

Zee

2/4

where It = m M/ (m + M) is the reduced mass, and m and M are

the masses of the electron and nucleus, respectively. We shall solve

the problem in the so-called atomic system of units in which h = 1,

A = 1, and

e2 = 1. Then the radial Schrodinger equation takes the

form

1(1+1)

ft () + 2

f1--f1=Ef1.

We are interested in the discrete spectrum, so we consider the

case E < 0. It is convenient to use the notation -2E = x2. Then

2Z -ly+1)

-2 =0.

()

fit

f t

The arguments in the previous section about the behavior of a

solution as r -+ 0 and r -* oo suggest that it is convenient to look for

a solution in the form

(2)

fft (r) = rt+1e-x' At (r).

If we can find Al (r) representable by a convergent power series,

(3)

A,(r) = >ajr'

i=u

with a0

0 and such that f j (r) satisfies (1), then the correct behavior

of ft (r) as r

0 will also be ensured Of course, the behavior of ft (r)

as r -- oo depends on the asyxnptotics of the function At (r) as r --+ oo.

It is convenient to make the substitution of (2) into the equation

(1) in two steps. Introducing the function g by

f=e-xrg,

f , =

e-X' (x2q

- 2xg' + y"),

we have

1(1 +1)

=0.