F
U
2
2
() E .
(3.143)
2m R d d F d 2
The fraction ( d 2 F/ d2)/ F should be a constant (because all other terms of the equation may be functions only of ρ), so that for the function F() we get an ordinary differential equation, 2
d
F
2
F 0,
(3.144)
2
d
where 2 is the variable separation constant. The fundamental solutions of Eq. (144) are evidently F
exp{ i}. Now requiring, as we did for the planar rotator, the 2 periodicity of any observable, i.e.
2 im
F
(
2 ) F ( ) e
,
(3.145)
where m is an integer, we see that the constant has to be equal to m, and get, for the angular factor, the same result as for the full wavefunction of the planar rotator – cf. Eq. (129):
C eim
F
,
with m ,
0 ,
1 ,...
2
(3.146)
m
m
Plugging the resulting relation ( d 2 F/ d2)/ F = – m 2 back into Eq. (143), we may rewrite it as 2
1 d dR m 2
.
(3.147)
2 U (
) E
2m R d d
The physical interpretation of this equation is that the full energy is a sum,
E E E ,
(3.148)
of the radial-motion part
2
1 d dR
E
()
U
.
(3.149)
2m
d
d
and the angular-motion part
62 See, e.g., MA Eq. (10.3) with / z = 0.
63 At this stage, I do not want to mark the particular solution (eigenfunction) and corresponding eigenenergy E
with any single index, because based on our experience in Sec. 1.7, we already may expect that in a 2D problem the role of this index will be played by two integers – two quantum numbers.
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2
2
m
E
.
(3.150)
2
2m
Now let us recall that a similar separation exists in classical mechanics,64 because the total energy of a particle moving in a central field may be represented as
m 2
E
v U () m 2
2
2
U() E E ,
(3.151)
2
2
2
2
2
p
p
L
m 2 2
with E
U (),
and E
z
.
(3.152)
2
2m
2
2m
2
m
The comparison of the latter relation with Eqs. (139) and (150) gives us grounds to expect that the quantization rule Lz = m may be valid not only for this 2D problem but in 3D cases as well. In Sec. 5.6, we will see that this is indeed the case.
Returning to Eq. (147), with our 1D wave mechanics experience we may expect that any fixed m this ordinary, linear, second-order differential equation should have (for a motion confined to a certain final region of its argument ρ) a discrete energy spectrum described by another integer quantum number
– say, n. This means that the eigenfunctions (142) and corresponding eigenenergies (148) and R( ρ) should be indexed by two quantum numbers, m and n. So, the variable separation is not so “clean” as it was for the rectangular potential well. Normalizing the angular function F to the full circle, = 2, we may rewrite Eq. (142) as
1
im
R () F
( )
R ( )
.
(3.153)
m, n
m,n
m
2
e
1/2 m,n
A good (and important) example of an analytically solvable problem of this type is a 2D particle whose motion is rigidly confined to a disk of radius R, but otherwise free:
,
0
0
for R,
U ()
(3.154)
, for R .
In this case, the solutions Rm,n() of Eq. (147) are proportional to the first-order Bessel functions Jm( knρ),65 with the spectrum of possible values kn following from the boundary condition Rm,n( R) = 0.
Let me leave a detailed analysis of this problem for the reader’s exercise.
3.6. Spherically-symmetric systems: Brute force approach
Now let us proceed to the mathematically more involved, but practically even more important case of the 3D motion, in a spherically-symmetric potential
U (r) U ( r).
(3.155)
64 See, e.g., CM Sec. 3.5.
65 A short summary of properties of these functions, including the most important plots and a useful table of values, may be found in EM Sec. 2.7.
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Let us start, again, with solving the eigenproblem for a rigid rotator – now a spherical rotator, i.e. a particle confined to move on the spherical surface of radius R. The rotator has two degrees of freedom because its position on the surface is completely described by two coordinates – say, the polar angle and the azimuthal angle . In this case, the kinetic energy we need to consider is limited to its angular part, so that in the Laplace operator in spherical coordinates66 we may keep only those parts, with fixed r = R. Because of this, the stationary Schrödinger equation becomes 2
1
1
2
sin
E .
(3.156)
2
2
2
2 R
m
sin
sin
(Again, we will attach indices to and E in a minute.) With the natural variable separation,
( ) F () ,
(3.157)
Eq. (156), with all terms multiplied by sin2/ F, yields
2
sin
2
d
d 1 d F
sin
2
E sin .
(3.158)
2
2 R
m
d
d
2
F d
Just as in Eq. (143), the fraction ( d 2 F/ dx 2)/ F may be a function of only, and hence has to be constant, giving Eq. (144) for it. So, with the same periodicity condition (145), the azimuthal functions are expressed by (146) again; in the normalized form,
1
F
im
( )
e
.
(3.159)
m
2 1/ 2
With that, the fraction ( d 2 F/ d2)/ F in Eq. (158) equals (- m 2), and after the multiplication of all terms of that equation by /sin2, it is reduced to the following ordinary linear differential equation for the polar eigenfunctions ():
2
2
1
d
d
m
sin
,
with
E
.
(3.160)
2
2
sin d
d sin
2 R
m
It is common to recast it into an equation for a new function P() (), with cos : 2
d
dP
m
1
2
l l
1
P 0
,
(3.161)
d
d
1
2
where a new notation for the normalized energy is introduced: l( l+1) . The motivation for such notation is that, according to the mathematical analysis of Eq. (161) with integer m,67 it has solutions only if the parameter l is an integer: l = 0, 1, 2,…, and only if that integer is not smaller than m, i.e. if
l m l .
(3.162)
This fact immediately gives the following spectrum of the spherical rotator’s energy E – and, as we will see later, the angular part of the energy of any spherically-symmetric system:
66 See, e.g., MA Eq. (10.9).
67 This analysis was first carried out by A.-M. Legendre (1752-1833). Just as a historic note: besides many original mathematical achievements, Dr. Legendre had authored a famous textbook, Éléments de Géométrie, which dominated teaching geometry through the 19th century.
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2 l l
1
Angular
E
,
(3.163) energy
l
2
2
R
m
spectrum
so that the only effect of the magnetic quantum number m here is imposing the restriction (162) on the non-negative integer l – the so-called orbital quantum number. This means, in particular, that each energy (163) corresponds to (2 l + 1) different values of m, i.e. is (2 l + 1)–degenerate.
To understand the nature of this degeneracy, we need to explore the corresponding eigenfunctions of Eq. (161). They are naturally numbered by two integers, m and l, and are called the a ssociated Legendre functions P m
l . (Note that here m is an upper index, not a power!) For the particular, simplest case m = 0, these functions are the so-called Legendre polynomials P
0
l() P l (), which may
be defined as the solutions of the following Legendre equation, resulting from Eq. (161) at m = 0: d
d
Legendre
1
2
P l l
1 P 0
,
(3.164) equation
d
d
but also may be calculated explicitly from the following Rodrigues formula:68
1 d l
P ( )
( 2
)
1 l , l ,
0 ,
1 ,...
2 .
(3.165) Legendre
l
2 l !
l d l
polynomials
Using this formula, it easy to spell out a few lowest Legendre polynomials:
1
2
1
P ( ) ,
1
P ( ) , P ( )
P
,
(3.166)
0
1
2
3
1, ( )
3
5 3 3 ,...
2
2
though such explicit expressions become bulkier and bulkier as l is increased. As these expressions (and Fig. 19) show, as the argument is increased, all these functions end up at the same point, Pl(+1) = + 1, while starting at either at the same point or at the opposite point: Pl(-1) = (-1) l. On the way between these two end points, the l th polynomial crosses the horizontal axis exactly l times, i.e. Eq. (164) has l roots.69
1
.
0.5
P ( ) 0
l
4
3
0.5
2
l 1
Fig. 3.19. A few lowest Legendre polynomials.
1
1
0.5
0
0.5
1
cos
68 This wonderful formula may be readily proved by plugging it into Eq. (164), but was not so easy to discover!
This was done (independently) by B. O. Rodrigues in 1816, J. Ivory in 1824, and C. Jacobi in 1827.
69 In this behavior, we may readily recognize the “standing wave” pattern typical for all 1D eigenproblems – cf.
Figs. 1.8 and 2.35, as well as the discussion of the Sturm oscillation theorem at the end of Sec. 2.9.
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It is also easy to use the Rodrigues formula (165) and the integration by parts to show that on the segment –1 +1, the Lagrange polynomials form a full orthogonal set of functions, with the following normalization rule:
1
2
P ( ) P ( ) d
.
(3.167)
l
l '
ll '
2 l 1
1
For m > 0, the associated Legendre functions (now not necessarily polynomials!), may be expressed via the Legendre polynomials (165) using the following formula:70
m
Associated
d
Legendre
m
P ( ) ( )
1 m 1
(
2
) m/2
P ( ) ,
(3.168)
l
functions
m l
d
while the functions with a negative magnetic quantum number may be found as
m
m ( l m)!
P ( ) ( )
1
Pm ( ),
for m 0 .
(3.169)
l
( l m)! l
On the segment –1 +1, the associated Legendre functions with a fixed index m form a full orthogonal set, with the normalization relation,
1
l m
m
m
2 (
)!
P ( ) P ( ) d
,
(3.170)
l
l '
ll '
2 l 1 ( l m)!
1
which is evidently a generalization of Eq. (167) for arbitrary m.
Since the difference between the angles and is to large extent artificial (due to an arbitrary direction of the polar axis), physicists prefer to use not the functions () m
P (cos) and
l
Fm() eim
separately, but normalized products of the type (157), which are called the s pherical harmonics: 1/ 2
Spherical
m
l 2 1 l( m)!
harmonics
m
im
Y
( ,)
P (cos ) e
.
(3.171)
l
4
l
( m
l
)!
The specific front factor in Eq. (171) is chosen in a way to simplify the following two expressions: the relation of the spherical harmonics with opposite signs of the magnetic quantum number,
m
Y
( ,) ( )
1 m Y
,
(3.172)
l
m(,)
l
and the following normalization relation:
m
Y ( , Y
d
,
(3.173)
l
m'
)
( , )
l '
ll ' mm'
4
with the integration over the whole solid angle. The last formula shows that on a spherical surface, the spherical harmonics form an orthonormal set of functions. This set is also full, so that any function defined on the surface, may be uniquely represented as a linear combination of Y m
l .
Despite a somewhat intimidating character of the formulas given above, they yield quite simple expressions for the lowest spherical harmonics, which are most important for applications: 70 Note that some texts use different choices for the front factor (called the Condon-Shortley phase) in the functions P m
m
l , which do not affect the final results for the spherical harmonics Yl .
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0
l 0 : Y 1/ 4
,
(3.174)
0
1/2
1
Y
i
e
1
3/8 1/2
sin
,
l 1:
0
Y
(3.175)
1
3/ 4 1/2 cos
,
1
1/ 2
i
Y
e
1
3/8
sin
,
2
Y
i
e
2
15/32 1/2 sin2
2
,
1
Y
i
e
2
15/8 1/2 sin cos
,
l 2
:
0
Y
(3.176)
2
3/16 1/2 3
( cos2
),
1
.
etc
1
i
Y
e
2
15/8 1/2 sin cos
,
2
i
Y
e
2
15/32 1/2 sin2 2
,
It is important to understand the general structure and symmetry of these functions. Since the sherical harmonics with m 0 are complex, the most popular way of their graphical representation is to normalize their real and imaginary parts as71
m
Im Y m
l
sin m , for m ,0
Y 2
(3.177)
lm
1 Re Y m
l
cos m , for m ,0
(for m = 0, Y
0
l 0 Yl ), and then plot the magnitude of these real functions in the spherical coordinates as the distance from the origin, while using two colors to show their sign – see Fig. 20.
Let us start from the simplest case l = 0. According to Eq. (162), for this lowest orbital quantum number, there may be only one magnetic quantum number, m = 0. According to Eq. (174), the spherical harmonic corresponding to that state is just a constant, so that the wavefunction of this so-called s state 72
is uniformly distributed over the sphere. Since this function has no gradient in any angular direction, it is only natural that the angular kinetic energy (163) of the particle equals zero.
According to the same Eq. (162), for l = 1, there are 3 different p states, with m = –1, m = 0, and m = +1 – see Eq. (175). As the second row of Fig. 20 shows, these states are essentially identical in structure and are just differently oriented in space, thus readily explaining the 3-fold degeneracy of the kinetic energy (163). Such a simple explanation, however, is not valid for the 5 different d states ( l = 2), shown in the third row of Fig. 20, as well as the states with higher l: despite their equal energies, they differ not only by their spatial orientation but their structure as well. All states with m = 0 have a nonzero gradient only in the direction. On the contrary, the states with the ultimate values of m ( l), change only monotonically (as sin l) in the polar direction, while oscillating in the azimuthal direction.
The states with intermediate values of m provide a gradual transition between these two extremes, oscillating in both directions, stronger and stronger in the azimuthal direction as m is increased. Still, the magnetic quantum number, surprisingly, does not affect the angular energy for any l.
71 Such real functions Ylm, which also form a full orthonormal set, and are frequently called the real (or “tesseral”) spherical harmonics, are more convenient than the complex harmonics Y m l for several applications, especially
when the variables of interest are real by definition.
72 The letter names for the states with various values of l stem from the history of optical spectroscopy – for example, the letter “s” used for states with l = 0, originally denoted the “sharp” optical line series, etc. The sequence of the letters is as follows: s, p, d, f, g, and then continuing in alphabetical order.
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l = 0
( s state)
l = 1
( p states)
l = 2
( d states)
l = 3
( f states)
m = –3
–2
–1
0
+1 +2
+3
Fig. 3.20. Radial plots of several lowest real spherical harmonics Ylm. (Adapted from https://en.wikipedia.org/wiki/Spherical_harmonics under the CC BY-SA 3.0 license.)
Another counter-intuitive feature of the spherical harmonics follows from the comparison of Eq.
(163) with the second of the classical relations (152). These expressions coincide if we interpret the constant
2
2
L l( l )
1 ,
(3.178)
as the value of the full angular momentum squared, L 2 = L2 (including its both and components) in the eigenstate with eigenfunction Y m
l . On the other hand, the structure (159) of the azimuthal component
F() of the wavefunction is exactly the same as in 2D axially-symmetric problems, implying that Eq.
(139) still gives correct values Lz = m for the z-component of the angular momentum. This fact invites a question: why for any state with l > 0, ( Lz)2 = m22 l 22 is always less than L 2 = l( l + 1)2? In other words, what prevents the angular momentum vector to be fully aligned with the axis z?
Besides the difficulty of answering this question using the above formulas, this analysis (though mathematically complete), is as intellectually unsatisfactory as the harmonic oscillator analysis in Sec.
2.9. In particular, it does not explain the meaning of the extremely simple relations for the eigenvalues of the energy and the angular momentum, coexisting with rather complicated eigenfunctions.
We will obtain natural answers to all these questions and concerns in Sec. 5.6 below, and now proceed to the extension of our wave-mechanical analysis to the 3D motion in an arbitrary spherically-symmetric potential (155). In this case, we have to use the full form of the Laplace operator in spherical Chapter 3
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coordinates.73 The variable separation procedure is an evident generalization of what we have done before, with the particular solutions of the type
R ()Θ( ) F (),
(3.179)
whose substitution into the stationary Schrödinger equation yields
2
1 d 2 dR 1 1 d
d Θ
1
1 d 2 F
r
sin
U ( r) E .
(3.180)
2
2
2
2 r
m
R dr
dr Θ sin d
d
sin F d
It is evident that the angular part of the left-hand side (the two last terms in the square brackets) separates from the radial part, and that for the former part we get Eq. (156) again, with the only change, R r. This change does not affect the fact that the eigenfunctions of that equation are still the spherical harmonics (171), which obey Eq. (164). As a result, Eq. (180) gives the following equation for the radial function R( r):
2
1 d 2 dR
r
l l( )
1 U ( r) E .
(3.181)
2 r 2
m
R dr
dr
Note that no information about the magnetic quantum number m has crept into this radial equation (besides setting the limitation (162) for the possible values of l) so that it includes only the orbital quantum number l.
Let us explore the radial equation for the simplest case when U( r) = 0 – for example, to solve the eigenproblem for a 3D particle free to move only inside the sphere of radius R – say, confined there by the potential74
,
0
0
for
r R,
U
(3.182)
,
for
R r
.
In this case, Eq. (181) is reduced to
2
1 d dR
2
r
l( l )
1 E.
(3.183)
2
2
r
m
R dr
dr
Multiplying both parts of this equality by r 2 R, and introducing the dimensionless argument kr, where k 2 is defined by the usual relation 2 k 2/2m = E, we obtain the canonical form of this equation, 2
d
d
2
R
2
R
l l R
(3.184)
2
2 1
,
0
d
d
Satisfied by so-called spherical Bessel functions of the first and second kind, jl() and yl().75 These functions are directly related to the Bessel functions of semi-integer order,76
73 Again, see MA Eq. (10.9).
74 This problem, besides giving a simple example of the quantization in spherically-symmetric systems, is also an important precursor for the discussion of scattering by spherically-symmetric potentials in Sec. 8.
75 Alternatively, yl() are called “spherical Weber functions” or “spherical Neumann functions”.
76 Note that the Bessel functions J() and Y() of any order obey the universal recurrence relations and asymptotic formulas (discussed, e.g., in EM Sec. 2.7), so that many properties of the functions jl() and yl() may be readily derived from these relations and Eqs. (185).
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1/ 2
1/ 2
j ( )
J
( ),
y ( )
Y ( ),
(3.185)
l
2
1
l
l
2
1
l
2
2
but are actually much simpler than even the “usual” Bessel functions, such as Jn() and Yn() of an integer order n, because the former ones may be directly expressed via elementary functions: sin
sin
cos
3
1
3
j ( )
,
j ( )
,
j ( )
sin
cos ,...,
0
1
2
2
3
2
(3.186)
cos
cos
sin
3
1
3
y ( )
,
y ( )
,
y ( )
cos
sin ,...,
0
1
2
2
3
2
A few lowest-order spherical Bessel functions are plotted in Fig. 21.
1
0.5
l 0
l 0
1
2
3
0.5
0
1 2
j
3
y
l
l
0
0.5
0.5
1
0
5
10
15
0
5
10
15
Fig. 3.21. Several lowest-order spherical Bessel functions.
As these formulas and plots show, the functions yl() are diverging at 0, and thus cannot be used in the solution of our current problem (182), so that we have to take
R
const
.
(3.187)
l r
jl kr
Still, even for these functions, with the sole exception of the simplest function j 0(), the characteristic equation jl( kR) = 0, resulting from the boundary condition R( R)= 0, can be solved only numerically.
However, the roots l,n of the equation jl() = 0, where the integer n (= 1, 2, 3,…) is the root’s number, are tabulated in virtually any math handbook, and we may express the eigenvalues we are interested in, 2
2
2
2
k
l, n
l, n
l,
k
,
E
n
,
(3.188)
l, n
l, n
2
R
2m
2 R
m
via these tabulated numbers. The table below lists several smallest roots, and the corresponding eigenenergies (normalized to their natural unit E 0 2/2m R 2), in the order of their growth. It shows a Chapter 3
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very interesting effect: going up the energy spectrum, first the eigenenergies grow because of increases of the orbital quantum number l, at the same (lowest) radial quantum number n = 1, due to the growth of the first roots of functions jl(), but then suddenly the second root of j 0() cuts into this orderly sequence, just to be followed by the first root of j 3(). With the further growth of energy, the sequences of l and n become even more entangled.
l n
l,n
El,n/ E 0 = ( l,n)2
0
1
3.1415
2 9.87
1
1
4.493
20.19
2
1
5.763
33.21
0
2
2 6.283
42 39.48
3
1
6.988
48.83
To complete the discussion of our current problem (182), note again that the energy levels, listed in the table above, are (2 l +1)-degenerate because each of them corresponds to (2 l + 1) different eigenfunctions, each with a specific value of the magnetic quantum number m:
r
,
C j
l n
Y m , ,
with
.
(3.189)
n, l, m
l, n l
l
l m
l
R
3.7 Atoms
Now we are ready to discuss atoms, starting from the simplest, exactly solvable Bohr atom problem, i.e. that of a single particle’s motion in the so-called attractive Coulomb potential 77
C
Attractive
U ( r) , with C .
0
(3.190) Coulomb
r
potential
The natural scales of E and r in this problem are commonly defined by the requirement of equality of the kinetic and potential energy magnitude scales (dropping all numerical coefficients):
2
C
E
,
(3.191)
0
2
r
m
r
0
0
similar to its particular case (1.13b). Solving this system of two equations, we get78
2
2
2
C
E
m ,
and r
.
(3.192)
0
2
0
r
m
C
0
m
77 Historically, the solution of this problem in 1928, that reproduced the main result (1.12)-(1.13) of the “old”
quantum theory developed by N. Bohr in 1912, without its phenomenological assumptions, was the decisive step toward the general acceptance of Schrödinger’s wave mechanics.
78 For the most important case of the hydrogen atom, with C = e 2/40, these scales are reduced, respectively, to the Bohr radius r B (1.10) and the Hartree energy E H (1.13a). Note also that according to Eq. (192), for the so-called hydrogen-like atom (actually, a positive ion) with C = Z( e 2/40), these two key parameters are rescaled as r 0 = r B/ Z and E 0 = Z2 E H.
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In the normalized units E/ E 0 and r/ r 0, equation (181) for our current case (190), looks relatively simple,
2
d R
2 R
d
1
l l
1 R 2 R ,
0
(3.193)
2
d
d
but unfortunately, its eigenfunctions may be called elementary only in the most generous meaning of the word. With the adequate normalization,
2
R R r dr ,
(3.194)
n, l
n ,' l
nn'
0
these (mutually orthogonal) functions may be represented as
1/ 2
Bohr
3
atom:
l
2 ( n l )!
1
r 2 r
l
r
2 1
2
radial
R ( r)
(3.195)
n l
L
,
n l
functions
nr n n l
nr
nr
nr
0
2 (
exp
.
)! 3
1
0
0
0
Here q
L ( ) are the so-called associated Laguerre polynomials, which may be calculated as p
q
Associated
d
Laguerre
q
L ( ) ( )
1 q
L
( ) .
(3.196)
p
polynomials
q p q
d
from the simple Laguerre polynomials L
0
p() L
.79 In turn, the easiest way to obtain L
p
p() is to use
the following Rodrigues formula:80
Rodrigues
formula for
d p
L
( )
p
e
e
.
(3.197)
p
Laguerre
d p
polynomials
Note that in contrast with the associated Legendre functions P m
l , participating in the spherical
harmonics, all L q
p are just polynomials, and those with small indices p and q are indeed quite simple: 0
L
L
L
0
,
1
0
1
,
1
0
2
2
4
,
2
1
L
L
L
(3.198)
0
,
1
1
1
2
,
4
1
2
3 2 18
,
18
2
L
L
L
0
,
2
2
1
6
,
18
2
2
12 2 96
,...
144
Returning to Eq. (195), we see that the natural quantization of the radial equation (193) has brought us a new integer quantum number n. To understand its range, we should notice that according to Eq. (197), the highest power of terms in the polynomial Lp+ q is ( p + q), and hence, according to Eq.
(196), that of q
L is p, so that the highest power in the polynomial participating in Eq. (195) is ( n – l –
p
1). Since the power cannot be negative to avoid the unphysical divergence of wavefunctions at r 0, the radial quantum number n has to obey the restriction n l + 1. Since l, as we already know, may take values l = 0, 1, 2,…, we may conclude n may only take the following values: 79 In Eqs. (196)-(197), p and q are non-negative integers, with no relation whatsoever to the particle’s momentum or electric charge. Sorry for this notation, but it is absolutely common, and can hardly result in any confusion.
80 Named after the same B. O. Rodrigues, and belonging to the same class as his other famous result, Eq. (165) for the Legendre polynomials.
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n ,
1 ,
2 ,...
3
(3.199)
What makes this relation very important is the following, most surprising result: the eigenenergies corresponding to the wavefunctions (179), which are indexed with three quantum numbers:
R ( r) m
Y ( ,) ,
(3.200)
n, l. m
n, l
l
depend only on one of them, n:
2
1
E
1
C
0
,
i.e. E
.
(3.201)
n
m
2
2
2
2 n
n
2 n
2 n
i.e. agree with Bohr’s formula (1.12). Because of this reason, n is usually called the principal quantum number, and the above relation between it and the “more subordinate” orbital quantum number l is rewritten as
l n 1.
(3.202)
Together with the inequality (162), this gives us the following, very important hierarchy of the three quantum numbers involved in the Bohr atom problem:
Bohr
1 n
0 l n 1
l m l .
(3.203) atom:
quantum
numbers
Taking into account the (2 l +1)-degeneracy related to the magnetic number m, and using the well-known formula for the arithmetic progression,81 we see that the n th energy level (201) has the following orbital degeneracy:
n 1
n 1
n 1
n n
1
g
(2 l )
1 2 l 1 2
2
n n .
(3.204)
l 0
l 0
l 0
2
Due to its importance for atoms, let us spell out the hierarchy (203) of a few lowest-energy states, using the traditional state notation, in which the value of n is followed by the letter that denotes the value of l: n 1: l 0
(
1
one s state
) m 0 .
(3.205)
n 2 : l 0
(
2
one s state
)
m ,
0
(3.206)
l 1 (
2
three p states
) m ,
0 .
1
n 3 : l 0
(
3
one s state
)
m ,
0
l 1 (
3
three p states
)
m ,
0 ,
1
(3.207)
l 2
(
3
five d states
)
m ,
0 ,
1 2.
Figure 22 shows plots of the radial functions (195) of the listed states. The most important of them is of course the ground (1 s) state with n = 1 and hence E = – E 0/2. According to Eqs. (195) and (198), its radial function is just a simple decaying exponent
Ground
2
state:
r / r 0
R ( r)
e
,
(3.208) radial
,
1 0
3 / 2
r
function
0
81 See, e.g., MA Eq. (2.5a).
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while its angular distribution is uniform – see Eq. (174). The gap between the ground energy and the energy E = – E 0/8 of the lowest excited states (with n = 2) in a hydrogen atom (in which E 0 = E H 27.2
eV) is as large as ~ 10 eV, so that their thermal excitation requires temperatures as high as ~105 K, and the overwhelming part of all hydrogen atoms in the visible Universe are in their ground state. Since the atomic hydrogen makes up about 75% of the “normal” matter,82 we are very fortunate that such simple formulas as Eqs. (174) and (208) describe the atomic states prevalent in Mother Nature!
0.25
0.25
n 1
2 p ( l )
1
n 2
1 s ( l 0)
3 / 2
3 / 2
R r
R r
2, l 0
,
1 l 0
0
0
2 s ( l 0)
0.25
0.25
0
2
4
6
8
10
0
2
4
6
8
10
r/r
r/r
0
0
0.25
n 3
3 / 2
R r
3 p ( l )
1
3, l 0
3 d ( l 2)
0
3 s ( l 0)
Fig. 3.22. The lowest radial functions
of the Bohr atom.
0.25
0
2
4
6
8
10
r/r 0
According to Eqs. (195) and (198), the radial functions of the lowest excited states, 2 s (with n =
2 and l = 0), and 2 p (with n = 2 and l = 1) are also not too complicated: 1
r
1
r
r / 2 r
R ( r)
r / 2 r 0
e
0
R ( r)
e
,
(3.209)
2,0
2 r
r
2 1
,
2 r
3 r
0 3 / 2
1/ 2
0
2
,
3 / 2
0
0
with the former of these states (2 s) having a uniform angular distribution, and the three latter (2 p) states, with different m = 0, 1, having simple angular distributions, which differ only by their spatial orientation – see Eq. (175) and the second row of Fig. 20. The most important trend here, clearly visible from the comparison of the two top panels of Fig. 22 as well, is a larger radius of the decay exponent in 82 Excluding the so-far hypothetical dark matter and dark energy.
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the radial functions (2 r 0 for n = 2 instead of r 0 for n = 1), and hence a larger radial extension of the states. This trend is confirmed by the following general formula:83
r
0
r
3 2 n l( l )1.
(3.210)
n, l
2
The second important trend is that at a fixed n, the orbital quantum number l determines how fast does the wavefunction change with r near the origin, and how much it oscillates in the radial direction at larger values of r. For example, the 2 s eigenfunction R 2,0( r) is different from zero at r = 0, and “makes one wiggle” (has one root) in the radial direction, while the eigenfunctions 2 p equal zero at r = 0 but do not cross the horizontal axis after that. Instead, those wavefunctions oscillate as the functions of an angle – see the second row of Fig. 20. The same trend is clearly visible for n = 3 (see the bottom panel of Fig. 22), and continues for the higher values of n.
The states with l = l max n – 1 may be viewed as crude analogs of the circular motion of a particle in a plane whose orientation defines the quantum number m. On the other hand, the best classical image of the s-state ( l = 0) is a purely radial, spherically-symmetric motion of the particle to and from the attracting center. (The latter image is especially imperfect because the motion needs to happen simultaneously in all radial directions.) The classical language becomes reasonable only for the highly degenerate Rydberg states, with n >> 1, whose linear superpositions may be used to compose wave packets closely following the classical (circular or elliptic) trajectories of the particle – just as was discussed in Sec. 2.2 for the free 1D motion.
Besides Eq. (210), mathematics gives us several other simple relations for the radial functions Rn,l (and, since the spherical harmonics are normalized to 1, for the eigenfunctions as the whole), including those that we will use later in the course:84
1
1
1
1
1
1
,
,
.
(3.211)
2
2
3
r
n r
r
n l ½ r
r
n l l ½ ( l )
1 r
n, l
0
n, l
2
3
3
0
n, l
3
0
In particular, the first of these formulas means that for any eigenfunction n,l,m, with all its complicated radial and angular dependencies, there is a simple relation between the potential and full energies: 1
C
E
0
U
C
2 E ,
(3.212)
n, l
2
2
n
r
n r
n
n, l
0
so that the average kinetic energy of the particle, T n,l = En – U n,l, is equal to En – 2 En = En > 0.
As in the several previous cases we have met, simple results (201), (210)-(212) are in sharp contrast with the rather complicated expressions for the corresponding eigenfunctions. Historically this contrast gave an additional motivation for the development of more general approaches to quantum mechanics, that would replace, or at least complement our brute-force (wave-mechanics) analysis. A discussion of such an approach will be the main topic of the next chapter.
83 Note that even at the largest value of l, equal to ( n –1), the second term l( l + 1) in Eq. (210) is equal to ( n 2 – n), and hence cannot over-compensate the first term 3 n 2.
84 The first of these relations may be readily proved using the Heller-Feynman theorem (see Chapter 1); this proof will be offered for the reader’s exercise after a more general form of this theorem has been proved in Chapter 6.
Chapter 3
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Rather strikingly, the above classification of the quantum numbers, with minor steals from the later chapters of this course, allows a semi-quantitative explanation of the whole system of chemical elements. The “only” two additions we need are the following facts:
(i) due to their unavoidable interaction with relatively low-temperature environments, atoms tend to relax into their lowest-energy state, and
(ii) due to the Pauli principle (valid for electrons as the Fermi particles), each orbital eigenstate discussed above may be occupied by two electrons with opposite spins.
Of course, atomic electrons do interact, so that their quantitative description requires quantum mechanics of multiparticle systems, which is rather complex. (Its main concepts will be discussed in Chapter 8.) However, the lion’s share of this interaction is reduced to simple electrostatic screening, i.e.
a partial compensation of the electric charge of the atomic nucleus, as felt by a particular electron, by other electrons of the atom. This screening changes quantitative results (such as the energy scale E 0) dramatically; however, the quantum number hierarchy, and hence their classification, is not affected.
The hierarchy of atoms is most often represented as the famous periodic table of chemical elements,85 whose simple version is shown in Fig. 23. (The table in Fig. 24 presents a sequential list of the elements and their electron configurations, following the convention already used in Eqs. (205)-
(207), with the additional upper index showing the number of electrons with the indicated values of quantum numbers n and l.) The number in each table’s cell, and in the first column of the list, is the so-called atomic number Z, which physically is the number of protons in the particular atomic nucleus, and hence the number of electrons in an electrically-neutral atom.
1
2
Property legend:
H
He
3
4
alkali metals
transition metals
metalloids
5
6
7
8
9
10
Li
Be
B
C
N
O
F
Ne
alkali-earth metals
nonmetals
halogens
11
12
13
14
15
16
17
18
Na
Mg rare-earth metals
other metals
noble gases
Al
Si
P
S
Cl
Ar
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
K
Ca
Sc
Ti
V
Cr
Mn
Fe
Co
Ni
Cu
Zn
Ga
Ge
As
Se
Br
Kr
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
Rb
Sr
Y
Zr
Nb Mo
Tc
Ru
Rh
Pd
Ag
Cd
In
Sn
Sb
Te
I
Xe
55
56
57-
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
Cs
Ba
71
Hf
Ta
W
Re
Os
Ir
Pt
Au
Hg
Tl
Pb
Bi
Po
At
Rn
87
88
89- 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118
Fr
Ra 102
Rf
Db
Sg
Bh
Hs
Mt
Ds
Rg
Cn Uut
Fl
Uup Lv Uus
Uuo
Lanthanides: 57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
La
Ce
Pr
Nd
Pm Sm
Eu
Gd
Tb
Dy
Ho
Er
Tm Yb
Lu
Actinides: 89
89
90
91
92
93
94
95
96
97
98
99
100 101 102
Ac
Ac
Th
Pa
U
Np
Pu
Am Cm Bk
Cf
Es
Fm Md
Lr
Fig. 3. 23. The periodic table of elements, showing their atomic numbers and chemical symbols, as well as their color-coded basic physical/chemical properties at the so-called ambient (meaning usual laboratory) conditions.
85 Also called the Mendeleev table, after Dmitri Ivanovich Mendeleev who put forward the concept of the quasi-periodicity of chemical element properties as functions of Z phenomenologically in 1869. (The explanation of this periodicity had to wait for 60 more years until the advent of quantum mechanics in the late 1920s.) Chapter 3
Page 49 of 64
QM: Quantum Mechanics
Atomic Atomic Electron
Atomic Atomic Electron
Atomic Atomic
Electron
number symbol
states
number symbol
states
number symbol
states
77
Ir
4 f 145 d 76 s 2
Period 1
Period 5
[Kr] shell,
plus:
78
Pt
4 f 145 d 96 s 1
1
H
1 s 1
37
Rb
5 s 1
79
Au
4 f 145 d 106 s 1
2
He
1 s 2
38
Sr
5 s 2
80
Hg
4 f 145 d 106 s 2
39
Y
4 d 15 s 2
81
Tl
4 f 145 d 106 s 26 p 1
Period 2
[He] shell,
plus:
40
Zr
4 d 25 s 2
82
Pb
4 f 145 d 106 s 26 p 2
3
Li
2 s 1
41
Nb
4 d 45 s 1
83
Bi
4 f 145 d 106 s 26 p 3
4
Be
2 s 2
42
Mo
4 d 55 s 1
84
Po
4 f 145 d 106 s 26 p 4
5
B
2 s 22 p 1
43
Tc
4 d 65 s 1
85
At
4 f 145 d 106 s 26 p 5
6
C
2 s 22 p 2
44
Ru
4 d 75 s 1
86
Rn
4 f 145 d 106 s 26 p 6
7
N
2 s 22 p 3
45
Rh
4 d 85 s 1
Period 7
[Rn] shell,
8
O
2 s 22 p 4
46
Pd
4 d 10
plus:
9
F
2 s 22 p 5
47
Ag
4 d 105 s 1
87
Fr
7 s 1
10
Ne
2 s 22 p 6
48
Cd
4 d 105 s 2
88
Ra
7 s 2
49
In
4 d 105 s 25 p 1
89
Ac
6 d 17 s 2
Period 3
[Ne] shell,
plus:
50
Sn
4 d 105 s 25 p 2
90
Th
6 d 27 s 2
11
Na
3 s 1
51
Sb
4 d 105 s 25 p 3
91
Pa
5 f 26 d 17 s 2
12
Mg
3 s 2
52
Te
4 d 105 s 25 p 4
92
U
5 f 36 d 17 s 2
13
Al
3 s 23 p 1
53
I
4 d 105 s 25 p 5
93
Np
5 f 46 d 17 s 2
14
Si
3 s 23 p 2
54
Xe
4 d 105 s 25 p 6
94
Pu
5 f 67 s 2
15
P
3 s 23 p 3
95
Am
5 f 77 s 2
Period 6
[Xe] shell,
16
S
3 s 23 p 4
plus:
96
Cm
5 f 76 d 17 s 2
17
Cl
3 s 23 p 5
55
Cs
6 s 1
97
Bk
5 f 97 s 2
18
Ar
3 s 23 p 6
56
