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

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138

L. D. Faddeev and 0. A. Yakubovski

Further. setting

g =

r1+1 A,

tr1

A(l+1)

+A'

1+1

l(1 + 1) A 2(1

+ 1) A'

+ A

r2 r

we get that

n + 2(1+1)

_ 2x I A' +

r2Z

2x(l + 1)1

Al = 0.

r / `r r

Let us look for a solution of this equation in the form of a series

(3).

aa[i(i- l )ri-2+2(l+l)ira-2 -2ixr'-1 0.

i=O

We make the change of summation index i --> i + I in the first two

terms in the square brackets, and then

E ri {ai+1 [(i + 1)i + 2(i + 1) (1 + 1)] - ai [2x(i + l + 1) -- 2Z] } = 0.

z=O

Equating the coefficients of the powers of r, we get that

(4)

1 ) ( i 2 1 +

= 2

From d'Alembert's criterion it

is clear that the series converges

for all r. We estimate the behavior of the series with the coefficients

defined by (4) for large r.

Of course, the asymptotic behavior as

r -p oc is determined by the coefficients of the higher powers, but

then

ai+1

1 ai;

that is,

and

(2x)z

ti C

§ 32. The hydrogen atom. The alkali metal atoms

139

Thus. for the solution f we get that

Cri+lexr

as to _+ 30 (This argurzient could be made more precise, of course.)

We see that a solution of the radial equation with the correct

behavior as r - 0 grows exponentially as r -+ oo. However, it is clear

from. (4) that there are values of x such that the series breaks off at,

some term. In this case the function Al turns out to be a polynomial,

and the solution fl (r) is square integrable. Let k denote the index

of the highest nonzero coefficient. that is, ak 34 0 and ak+1 = 0 for

k = 1, 2..... It is obvious from (I) that t his is possible if

x= xk t =

k+7_471-

From the formula -2E = x2 . we get that

(5)

Ekl _

-2(k + l +1 ) 2

The parameter k is the radial quantumn number introduced ear-

liei We see that the eigenvalues Eki depend only on n = k + 1 + 1.

This number is called the principal quantum number. Recalling that

k = 0, 1, 2.... and l = 0, 1, 2,

... we get

that, n -- 1, 2, 3, .... Fur-

rherruore, for a given n the quantumn number l can take the values

0, 1.2. ..,n - 1

Accordingly, we have obtained the, following results The eigen-

values E are given by the formula

(6)

Eit

2n2

and the eigenfunctions have the form

(7)

Ortlrrt =

rte-x"''A.t(r) Ym(O.

where Ar1 1

is a polynomial of degree n. -- I - 1 whose coefficients are

found from the formula (4), with ae found from the normalization con-

dition. We see that the number of eigenvalues is infinite and has accu-

mulation point E = 0. It is not hard to determine the multiplicity of

the eigenvalire E,, To each E,, there correspond eigenfunctions

140

L. D. Faddeev and 0. A. Yakubovskii

differing by the quantum numbers 1 and m, with 1 = 0. 11 2..... n - 1

and m = -1,

-t -- 1,. .. ,1. For the multiplicity q we have

n-1

q=j:(21+1)=n

t=v

The multiplicity of the eigenvalues for the Coulomb field turns out

to be greater than in the general case of a central field: there is an

additional degeneracy with respect to 1. We have already mentioned

that this "accidental" degeneracy is explained by the presence of a

symmetry group richer than SO(3) for the Schrodinger operator for

the hydrogen atom.

Let us now consider what physical information is given to us by

the solution of the Schrodinger equation for the physical atom. First

of all, we have found the admissible values of the energy, which it is

reasonable to give in the usual units. For this it suffices to multiply

the expression (6) for F,t by the atomic unit of energy equal to

= 4.36. 10erg = 27.21 eV.

We assume that Z = 1, that is, we consider the hydrogen atom, and

then

(s)

le4

Ert = -2n2h2 .

For the energy of the ground state of the hydrogen atom (n = 1), we

have

El = 2h2

_ -13.6 eV

The absolute value of I his energy is called the ionization potential or

the binding energy of the electron in the atom and is equal to the

work that must he done to strip the electron from the atom.

The formula (8) enables us to compute the frequencies of t he spec-

tral lines for the hydrogen atom. Quantum electrodynamics confirms

Bohr's hypothesis that, the frequency of a spectral line is determined

by the formula

hwrrt n = Err - Ern

Fn > Err, 1

§ 32. The hydrogen atom. The alkali metal atoms

141

and moreover, there is absorption of a quantum of light if the atom

passes from a state with less energy to a state with greater energy,

and emission of a quantum for the reverse transition. 28

For the frequencies of the spectral lines we have the formula

jie4

1 1

(9)

Wnm =

2h3 rte

n < m.

This is called Balmer's formula and was discovered by him empirically

long before the creation of quantum mechanics.

We direct attention to the dependence of the frequencies

on the reduced mass p. In nature there are two forms of hydrogen:

the ordinary hydrogen H whose nucleus if a proton with mass M =

1836 m (m is the mass of the electron), along with a small quantity

of heavy hydrogen or deuterium D whose nucleus is twice as heavy

as the proton. Using the formula it = rnM/(m + M), we easily

compute that /2D11LH = 1.000272, that is, the reduced masses are

very close. Nevertheless, the precision of spectroscopic measurements

(wavelengths can be measured with an accuracy of 7-8 significant

figures) make it possible to accurately measure the ratio W D /W H for

the corresponding lines. This ratio is also equal to 1.000272 (for

some lines there can be a deviation in the last significant figure). In

general the values computed theoretically according to the formula

(9) coincide with the experimental values of the frequencies to five

significant figures. The existing deviations, however, can be removed

by taking into account relativistic corrections.

Along with the transitions between stationary states of the dis-

crete spectrum there can be transitions from the discrete spectrum

to the continuous spectrum and the reverse transitions; physically,

they correspond to ionization and recombination processes (capture

of an electron by the nucleus). In these cases a continuous spectrum

of absorption or emission is observed.29

28An absorption spectrum (dark lines on a bright background) arises if a light

beam with continuous spectrum passes through a medium containing atomic hydrogen

Absorption lines are observed in the spectra of stars A line spectrum of emission

will be observed, for example, if an cbs trical discharge takes place in a medium with

atomic Hydrogen Under the action of impacts with electrons, the hydrogen atoms will

then pass into excited states The transitions to levels with less energy lead to the

appearance of bright lines

29Tlie word `spectrum" is used here in two senses the spec trump of an operator

and the admissible values of the frequency of electromagnetic radiation

142 L. D. Faddeev and 0. A. Yakubovsk j

The spectral lines of hydrogen on spectrograms are grouped into

series corresponding to a definite value of n in the formula (9) and

m = n + 1, n + 2,.

.The first few series have been given names: the

Lymian series (n = 1), the Balmer series (n = 2), and the Paschen

series (n = 3). The lines of the Lyman series lie in the ultraviolet

part of the spect ruin, the first four lines of the Balmier series lie in the

visible part of the spectrum, and the lines of the Paschen series and

subsequent series lie in the infrared part of the spectruni. Toward

the end of each series the lines converge to the so-called limit of the

series, after which the continuous spectrum begins.

In Figure 8 the horizontal lines represent the energy levels of

a hydrogen atom, and the vertical segments represent the possible

transitions between there. The region of the continuous spectrum is

shaded.

_2 `-V--

Lyman series

Paschen s

Balmer series

eries

n--3

n=2

n=1

Figure 8

§ 32. The hydrogen atom. The alkali metal atoms 143

Figure 9

In Figure 9 we represent schematically the form of a spectral

series, with the limit of the series represented by the dashed line.

The probabilities of transitions between states are important char-

acteristics of atoms. The intensities of the spectral lines depend on the

transition probabilities. The transitions that take place are sponta-

neous from an upper level to a lower level with emission of a quantum,

forced (under the action of a light beam), or, finally, due to collisions

with charged particles. Formulas for computing the probabilities of

spontaneous and forced transitions are given by quantum electrody-

namics, and transitions due to collisions are studied in the quantum

theory of scattering. To compute all these characteristics, one must

know the wave functions. Moreover, knowledge of the wave functions

makes it possible to judge the size of an atom, the distribution of

charge in the atom, and even the shape of the atom. We recall that

I is the density of the coordinate distribution function. By

the size of the atom we understand the size of the region in which

k'(x)I2 is not negligibly small. It is clear that the size of an atom is

a conditional concept.

As an example let us consider the ground state of a hydrogen

atom (n = 1,1 = 0, m = 0). Since Yoo(n) = const and x, = 1, we get

by the formula (7) that

P]OUX = CC-r.

We find the constant C from the normalization condition

,012 dx

= IC12 47r

e-2rr2

dr = IC12 7r =1,

JR3

from which C = 1 / f and

V)lUU(X) =

8-r.

144

L. D. Faddeev and 0. A. Yakubovskil

Figure 10

It is easy to see that

p(r) =

4irIb100(r)12r2

= 4e'

2rr2

is the density of the distribution function of the coordinate r. The

graph if this function is pictured in Figure 10. The maximum of

p(r) is attained for ro = 1; that is, r0 = 1 is the most probable

distance of the electron from the nucleus. In the usual units, ro =

h2lpe2 = 0.529. 10-8 cm. It is interesting to note that this number

coincides with the radius of the first Bohr orbit. We see that the size

of a hydrogen atom has order 10-8 cm.

By the density of the charge in an atom we understand the quan-

tity -- e b (x) 2, that is. we consider that because of its rapid motion

about the nucleus, the electron is as if smeared over the volume of

the atone, forming an electron cloud.

Finally, the form of the function (7) shows that for 1 34 0 the

density of the coordinate distribution is not spherically symmetric.

The dependence of this function on the angles lets us say something

about the shape of the atom in various states.

In this same section we consider a simple model of alkali metal

atoms, based on the assumption t hat their optical properties are ex-

plained by the motion of the valence electron in a certain central field

V (r) . This potential V (r) can be written as a suns of two terms,

V(r)

=-Z +V r ,

§ 32. The hydrogen atom. The alkali metal atoms

145

where the first term describes the interaction of the electron with the

nucleus, and Vl (r} can be interpreted as the potential of the inter-

action of the electron with the negative charge of the other electrons

distributed over the volume of the atom. The reasonableness of such

a model for atoms of the alkali metals becomes clear only after we

become acquainted with the properties of complex atoms and the

_l lend eleev periodic table.

We know very little about the potential V (r), but nevertheless

we can assert that

V(r) -

1 forr --+oc

and

V (r)

- -- for r -* 0.

The first condition follows from the obvious fact that when the valence

elect ron is removed to infinity, it finds itself in the field of an ion

with a single positive charge. The second condition follows from the

continuity of the potential Vl (r) of the volume distribution of charges.

As a model potential we choose

(10) V(r)

a>0.

Despite the fact that this potential has the correct behavior at infin-

ity, its behavior at zero is different from that of the "true" potential.

At the same time, the model potential correctly reflects the fact that

upon approach to the nucleus the field becomes stronger than the

Coulomb field -1 /r. We assume that the parameter a is small (in

what sense is indicated below). The numerical values of this parame-

ter for the different alkali metal atoms is most reasonably chosen from.

a comparison of the results of computations of the energy levels with

those found experimentally.

The radial equation for such a potential is very simple to solve.

Indeed, it has the form

2 2a

_l(1+1}

fr +

.fi

.f i - 0.

We introduce the number 1' satisfying the equation

l'(l'+1)+2a-l(1+1) =0

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

and the condition lima--.o 1'= 1, so that 1' = --1/2 + /fl1/2)2 --- 2a.

The equation (11) can be rewritten in the form

, 2 _ 11(11 +1) _x2 =0

fit

,ft

.ft

that is, it coincides formally with the equation for the Coulomb field.

All this can make sense only under the condition that 1(1 + 1) + 1 /4 --

2a > 0 Otherwise we get complex values for 11.30

If a < 1/8, then the condition 1(1 + 1) + 1 /4 -- 2a > 0 holds for

all 1. One usually writes 1' up to terms of order a2; that is,

1+1/2

Using the formula (5) with Z = 1, we then get that

Ekl _

2(k + 1 -

a.+1)2 1

or, introducing the principal quantum number n = k + 1 + 1,

F_. - -

2(n--aj)It

is clear from (12) that for the potential (9) the Coulomb degen-

eracy with respect to 1 is removed. The energy levels Ent lie deeper

than the levels E7 of the hydrogen atom, and the levels Ent and En

become close as n increases. The formula (12) describes the energy

levels of the alkali metal atoms fairly well for an appropriate value

of a. This formula was first obtained by Rydberg by analyzing ex-

perimental data. We remark that for the alkali metal atoms, as for

hydrogen, the principal quantum number takes integer values, but

the minimal value of n is not 1 but 2 f o r Li, 3 f o r Na, ... , since the

states with smaller principal quantum number are occupied by the

electrons of the inner shells of the atom (this assertion will become

clear after we become familiar with the structure of complex atoms).

In conclusion we note that this model illustrates a semi-empirical

approach to the solution of complex quantum mechanics problems.

'ieit can be shown that for 2c, - 1(1 + 1) > 1/4 the radial Sc hrodinger operator

1 d2 1(t+1)-2Q

2 dr2 + 2r2

becomes unbounded below

§ 33. Perturbation theory

147

Such an approach consists in the following: instead of solving the

problem in the exact formulation, one uses physical considerations to

construct a simplified model of the system. The Schri dinger operator

for the model problem usually depends on parameters which are just

as difficult to find theoretically as it is to solve the problem in full

scope. Therefore, the parameters are found by comparing the results

of computations of the model problem with experimental data.

§ 33. Perturbation theory

In quantum mechanics there are relatively few interesting problems

that admit the construction of exact solutions. Therefore, approxima-

tion methods play an important role. Approximation theories often

turn out to be more valuable for understanding physical phenomena

than exact numerical solutions of the corresponding equations. The

main approximation methods in quantum mechanics are based on

perturbation theory and the variational principle.

We describe the formulation of a problem in perturbation theory.

Suppose that A is a given self-adjoint operator whose spectrum is

known. It is required to find the spectrum of the operator B = A+ C

under the condition that the self-adjoint operator C is small in some

sense. We do not specify what is meant by smallness of C, since we

are considering only a formal scheme in perturbation theory. Giving a

rigorous basis for such a scheme requires the solution of some complex

mathematical problems.

We shall analyze the case when A has a pure point spectrum, and

we begin with the problem of perturbation of a simple eigenvalue. Let

us consider the one-parameter family of operators

(1)

Ae = A+eC.

It is clear that A o = A and A 1 = B. We know the cigenvectors V)n

and eigenvalues A,, of A, which satisfy the equation

(2)

Ann =ArY)n

I t is assumed that the spectrum of A is simple; that is, to each An

there corresponds one eigenvector V)n.