Quantum Mechanics by Konstantin K. Likharev - HTML preview
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(3.188)
l, n
l, n
2
R
2m
2 R
m
via these tabulated numbers. The table on the right lists
several smallest roots, and the corresponding eigenenergies
l n
l,n
El,n/ E 0 = ( l,n)2
(normalized to their natural unit E 0 2/2m R 2), in the order 0
1
3.1415
2 9.87
of their growth. It shows a very interesting effect: going up
1
1
4.493
20.19
the energy spectrum, the first energies grow due to unit
increments of the orbital quantum number l and the
2
1
5.763
33.21
corresponding increases of the first roots of the functions
0
2
2 6.283
42 39.48
jl(), at the same (lowest) radial quantum number n = 1.
3
1
6.988
48.83
Then, suddenly, the second root of j 0(), accompanied by a
jump to n = 2, cuts into this orderly sequence, just to be followed by the first root of j 3(), returning to the initial sequence with n = 1. With the further growth of energy, the sequences of l and n become even more entangled.
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 78
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): 78 Historically, the solution of this problem in 1928, which reproduced the main results (1.12)-(1.13) of the “old”
quantum theory developed by N. Bohr in 1912, but without its phenomenological assumptions, was the decisive step toward the general acceptance of Schrödinger’s wave mechanics.
Chapter 3
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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 get79
2
2
2
C
E
m ,
and r
.
(3.192)
0
2
0
r
m
C
0
m
In the normalized units E/ E 0 and r/ r 0, Eq. (181) for our 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 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
.80 In turn, the easiest way to obtain L
p
p() is to use
the following Rodrigues formula:81
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: 79 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.
80 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.
81 Named after the same B. O. Rodrigues, and belonging to the same class as his other famous result, Eq. (165) for the Legendre polynomials.
Chapter 3
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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
etc.
,
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 the highest power in the polynomial participating in Eq. (195) is ( n – l – 1).
p
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 the values l = 0, 1, 2,…, we may conclude n may only take the following values: 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
atom:
1 n
0 l n 1
l m l .
(3.203) quantum
numbers
Taking into account the (2 l +1)-degeneracy related to m, and using the well-known formula for the arithmetic progression,82 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 that 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
82 See, e.g., MA Eq. (2.5a).
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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
state:
2
radial
r / r 0
R ( r)
e
,
(3.208)
,
1 0
3 / 2
function
r 0
while its angular distribution is uniform – see Eq. (174). The gap between the ground state energy E g = –
E 0/2 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 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 atomic hydrogen makes up about 75% of the “normal” matter,83 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: 83 Excluding the so-far hypothetical dark matter and dark energy.
Chapter 3
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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 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:84
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 the wavefunction changes 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:85
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 the average kinetic energy of the particle, T n,l = En – U n,l, is equal to En – 2 En = En > 0.
84 Note that even at the largest value of l, equal to ( n –1), the second term l( l + 1) in the square brackets of Eq.
(210) is equal to ( n 2 – n), and hence cannot over-compensate the first term 3 n 2.
85 The first of these relations may be proved using the Hellmann-Feynman theorem (see Sec. 1.8); 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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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 the brute-force (wave-mechanics) analysis. A discussion of such an approach will be the main topic of the next chapter.
Rather strikingly, the above classification of the quantum numbers, with a few 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 house two electrons with opposite spins.
Of course, atomic electrons do interact, so 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) much; however, the quantum number hierarchy and hence the state classification, are not affected.
The system of atoms is most often represented as the famous periodic table of chemical elements,86 whose simple version is shown in Fig. 23. (The table in Fig. 24 presents a sequential list of the elements with 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.
The simplest atom, with Z = 1, is hydrogen (chemical symbol H) – the only atom for which the theory discussed above is quantitatively correct.87 According to Eq. (191), the ground state of its only electron corresponds to the quantum number values n = 1, l = 0, and m = 0 – see Eq. (205). In most versions of the periodic table, the cell of H is placed in the top left corner.
In the next atom, helium (symbol He, Z = 2), the same orbital quantum state (1 s) houses two electrons. As will be discussed in detail in Chapter 8, electrons of the same atom are actually indistinguishable, so their quantum states are not independent and may be entangled. These factors are important for several properties of helium atoms (and heavier elements as well); however, a bit counterintuitively, for atom classification purposes, they are not crucial, and we may think about the two electrons of a helium atom just having “opposite spins”. Due to the twice higher electric charge of the nucleus of the helium atom, i.e. the twice higher value of the constant C in Eq. (190), resulting in a four-fold increase of the constant E 0 given by Eq. (192), the binding energy of each electron is crudely four times higher than that of the hydrogen atom – though the electron interaction decreases it by about 25%
86 Also called the Mendeleev table, after D. I. 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.) 87 Besides very small fine-structure and hyperfine-splitting corrections – to be discussed, respectively, in Chapters 6 and 8.
Chapter 3
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– see Sec. 8.2 below. This is why taking one electron away (i.e. the positive ionization of a helium atom) requires relatively high energy, ~24.6 eV, which is not available in the usual chemical reactions. On the other hand, a neutral helium atom cannot bind one more electron (i.e. form a negative ion) either. As a result, helium, and all other elements with fully completed electron shells (the term meaning the sets of states with eigenenergies well separated from higher energy levels) is a chemically inert noble gas, thus starting the whole right-most column of the periodic table, allocated for such elements.
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
Nh
Fl
Mc
Lv
Ts
Og
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 the color-coded basic physical/chemical properties at the so-called ambient (meaning usual laboratory) conditions.
The situation changes rather dramatically as we move to the next element, lithium (Li), with Z =
3 electrons. Two of them are still accommodated by the inner shell with n = 1 (listed in Fig. 24 as the helium shell [He]), but the third one has to reside in the next shell with n = 2, l = 0, and m = 0, i.e. in the 2 s state. According to Eq. (201), the binding energy of this electron is much lower, especially if we take into account that according to Eqs. (210)-(211), the 1 s electrons of the [He] shell are much closer to the nucleus and almost completely compensate for two-thirds of its electric charge +3 e. As a result, the 2 s-
state electron is approximately but reasonably described by Eq. (201) with Z = 1 and n = 2, giving its binding energy close to 3.4 eV (actually, ~5.39 eV), so a lithium atom can give out that electron rather easily – to either an atom/ion of another element to form a chemical compound or to the common conduction band of the solid-state lithium; as a result, at the ambient conditions, this is a typical alkali metal. The similarity of chemical properties of lithium and hydrogen, with the chemical valence of one,88 places Li as the starting element of the second period (row), with the first period limited to only H
and He – see Fig. 23.
88 Chemical valence (or “valency”) is a not very precise term describing the number of the atom’s electrons involved in chemical bonding. For the same atom, especially with a large number of electrons in its outer shell, this number may depend on the chemical compound formed. (For example, the valence of iron is two in the ferrous oxide, FeO, and three in the ferric oxide, Fe2O3.)
Chapter 3
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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
Ba
6 s 2
98
Cf
5 f 107 s 2
57
La
5 d 16 s 2
99
Es
5 f 117 s 2
Period 4
[Ar] shell,
plus:
58
Ce
4 f 15 d 16 s 2
100
Fm
5 f 127 s 2
19
K
4 s 1
59
Pr
4 f 36 s 2
101
Md
5 f 137 s 2
20
Ca
4 s 2
60
Nd
4 f 46 s 2
102
No
5 f 147 s 2
21
Sc
3 d 14 s 2
61
Pm
4 f 56 s 2
103
Lr
5 f 146 d 17 s 2
22
Ti
3 d 24 s 2
62
Sm
4 f 66 s 2
104
Rf
5 f 146 d 27 s 2
23
V
3 d 34 s 2
63
Eu
4 f 76 s 2
105
Db
5 f 146 d 37 s 2
24
Cr
3 d 44 s 2
64
Gd
4 f 75 d 16 s 2
106
Sg
5 f 146 d 47 s 2
25
Mn
3 d 54 s 2
65
Tb
4 f 96 s 2
107
Bh
5 f 146 d 57 s 2
26
Fe
3 d 64 s 2
66
Dy
4 f 106 s 2
108
Hs
5 f 146 d 67 s 2
27
Co
3 d 74 s 2
67
Ho
4 f 116 s 2
109
Mt
5 f 146 d 77 s 2
28
Ni
3 d 84 s 2
68
Er
4 f 126 s 2
110
Ds
5 f 146 d 87 s 2
29
Cu
3 d 94 s 1
69
Tm
4 f 136 s 2
111
Rg
5 f 146 d 97 s 2
30
Zn
3 d 104 s 2
70
Yb
4 f 146 s 2
112
Cn
5 f 146 d 107 s 2
31
Ga
3 d 104 s 24 p 1
71
Lu
4 f 145 d 16 s 2
113
Nh
5 f 146 d 107 s 27 p 1
32
Ge
3 d 104 s 24 p 2
72
Hf
4 f 145 d 26 s 2
114
Fl
5 f 146 d 107 s 27 p 2
33
As
3 d 104 s 24 p 3
73
Ta
4 f 145 d 36 s 2
115
Mc
5 f 146 d 107 s 27 p 3
34
Se
3 d 104 s 24 p 4
74
W
4 f 145 d 46 s 2
116
Lv
5 f 146 d 107 s 27 p 4
35
Br
3 d 104 s 24 p 5
75
Re
4 f 145 d 56 s 2
117
Ts
5 f 146 d 107 s 27 p 5
36
Kr
3 d 104 s 24 p 6
76
Os
4 f 145 d 66 s 2
118
Og
5 f 146 d 107 s 27 p 6
Fig. 3.24. Atomic electron configurations. The upper index shows the number of electrons in the states with the indicated quantum numbers n (the first digit) and l (letter-coded as was discussed above).
Chapter 3
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In the next element, beryllium (symbol Be, Z = 4), the 2 s state ( n = 2, l = 0, m = 0) houses one more electron, with the “opposite spin”. Due to the higher electric charge of the nucleus, Q = +4 e, with only half of it compensated by 1 s electrons of the [He] shell, the binding energy of the 2 s electrons is somewhat higher than that in lithium, so the ionization energy increases to ~9.32 eV. As a result, beryllium is also chemically active with the valence of two, but not as active as lithium, and is also is metallic in its solid-state phase, but with a lower electric conductivity than lithium.
Moving in this way along the second row of the periodic table (from Z = 3 to Z = 10), we see a gradual filling of the rest of the total 2 n 2 = 222 = 8 different electron states of the n = 2 shell (see Eq.
(204), with the additional spin degeneracy factor of 2), including two 2 s states with m = 0, and six 2 p states with m = 0, 1, with a gradually growing ionization potential (up to ~21.6 eV in Ne with Z = 10), i.e. a growing reluctance to either conduct electricity or form positive ions. However, the last elements of the row, such as oxygen (O, with Z = 8) and especially fluorine (F, with Z = 9) can readily pick up extra electrons to fill up their 2 p states, i.e. form negative ions. As a result, these elements are chemically active, with a double valence for oxygen and a single valence for fluorine. However, the final element of this row, neon, has its n = 2 shell completely full, and cannot form a stable negative ion.
This is why it is a noble gas, like helium. Traditionally, in the periodic table, such elements are placed right under helium (Fig. 23), to emphasize the similarity of their chemical properties. But this necessitates making at least a 6-cell gap in the 1st row. (Actually, the gap is often made larger, to accommodate the next rows – keep reading.)
Period 3, i.e. the 3rd row of the table, starts exactly like period 2, with sodium (Na, with Z = 11), also a chemically active alkali metal whose atom features 2+8 = 10 electrons filling the shells with n = 1
and n = 2 (in Fig. 24, collectively called the neon shell [Ne]), plus one electron in the 3 s state ( n = 3, l =
0, m = 0), which may be again reasonably well described by the hydrogen atom theory – see, e.g., the red curve on the last panel of Fig. 22. Continuing along this row, we could naively expect that, according to Eq. (204), and with the account of double spin degeneracy, this period of the table should have 2 n 2 = 232 = 18 elements, with a gradual, sequential filling of first, two 3 s states, then six 3 p states, and then ten 3 d states. However, here we run into a big surprise: after argon (Ar, with Z = 18), a relatively inert element with an ionization energy of ~15.7 eV due to the fully filled 3 s and 3 p subshells, the next element, potassium (K, with Z = 19) is an alkali metal again!
The reason for that is the difference of the actual electron energies from those of the hydrogen atom, which is due mostly to electron-electron interactions, and gradually accumulates with the growth of Z. It may be semi-quantitatively understood from the results described in Sec. 6. In hydrogen-like atoms/ions, the electron state energies do not depend on the quantum number l (as well as m) – see Eq.
(201). However, the orbital quantum number does affect the wavefunction of an electron. As Fig. 22
shows, the larger l the less the probability for an electron to be close to the nucleus, where its positive charge is less compensated by other electrons. As a result of this effect (and also the relativistic corrections to be discussed in Sec. 6.3), the electron’s energy grows with l. Actually, this effect is visible already in period 2 of the table: it manifests itself in the filling order – the p states after the s states. However, for potassium (K, with Z = 19) and calcium (Ca, with Z = 20), the energies of the 3 d states become so high that the energies of the two 4 s states are lower, and the latter states are filled first.
As described by Eq. (210), and also by the first of Eqs. (211), the effect of the principal number n on the distance from the nucleus is stronger than that of l, so the 4 s wavefunctions of K and Ca are relatively far from the nucleus, and determine the chemical valence (equal to 1 and 2, correspondingly) of these elements. The next atoms, from Sc ( Z = 21) to Zn ( Z = 30), with the gradually filled “internal” 3 d states, Chapter 3
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are the so-called transition metals whose (comparable) ionization energies and chemical properties are determined by the 4 s electrons.89
This fact is the origin of the difference between various forms of the “periodic” table. In its most popular option, shown in Fig. 23, K is used to start the next period 4, and then a new period is started each time and only when the first electron with the next principal quantum number ( n) appears.90 This topology of the table provides a very clear match of the chemical properties of the first element of each period (an alkali metal), as well as its last element (a noble gas). It also automatically means making gaps in all previous rows. Usually, this gap is made between the atoms with completely filled s states and with those with the first electron in a p state, because here the properties of the elements make a somewhat larger step. (For example, the step from Be to B makes the material an insulator, but the step from Mg to Al makes a smaller difference.) As a result, the elements of the same column have only approximately similar chemical valences and physical properties.
In order to accommodate the lower, longer rows, such representation is inconvenient, because the whole table would be too broad. This is why the so-called rare earth elements, including lanthanides (with Z from 57 to 70, of the 6th row, with a gradual filling of the 4 f and 5 d subshells) and actinides ( Z
from 89 to 103, of the 7th row, with a gradual filling of the 5 f and 6 d subshells), are usually represented as outlet rows – see Fig. 23. This is quite acceptable for basic chemistry because chemical properties of the elements within each such group are rather close.
To summarize my very short review of this extremely important topic,91 the “periodic table of elements” is not periodic in the strict sense of the word. Nevertheless, it has had an enormous historic significance for chemistry, as well as atomic and solid-state physics, and is still very convenient for many purposes. For our course, the most important aspect of its discussion is the surprising possibility to describe, at least for classification purposes, such a complex multi-electron system as an atom as a system of quasi-independent electrons in certain quantum states indexed with the same quantum numbers n, l, and m as those of the hydrogen atom. This fact enables the use of various perturbation theories, which give a more quantitative description of atomic properties. Some of these techniques will be reviewed in Chapters 6 and 8.
3.8. Spherically symmetric scatterers
The machinery of the Legendre polynomials and the spherical Bessel functions, discussed in Sec.
6, may also be used for the analysis of particle scattering by spherically symmetric potentials (155) beyond the Born approximation (Sec. 3), provided that such a potential U( r) is also localized, i.e.
reduces sufficiently fast at r .92 Indeed, directing the z-axis along the propagation of the incident plane de Broglie wave i, and taking its origin in the center of the scatterer, we may expect the scattered wave s to be axially symmetric, so its expansion in the series over the spherical harmonics includes 89 The sequence of shell and subshell formation with the atomic number’s growth approximately follows the so-called Madelung rule (saying that the orbitals with the lowest sum ( n + l) are filled first), while the order of state filling inside each subshell closely follows the Hund rules, to be discussed in Sec. 8.3.
90 Another popular option is to return to the first column as soon an atom has one electron in the s state (like it is in Cu, Ag, and Au, in addition to the alkali metals).
91 For a bit more detailed (but still succinct) discussion of the valence and other chemical aspects of atomic structure, I can recommend Chapter 5 of the very clear text by L. Pauling, General Chemistry, Dover, 1988.
92 The quantification of this condition is left for the reader’s exercise.
Chapter 3
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only the terms with m = 0. Hence, the solution (64) of the stationary Schrödinger equation (63) in this case may be represented as93
ikz
a e
R r P
,
(3.213)
i
s
i
l lcos
l0
where k (2m E)1/2/ is defined by the energy E of the incident particle, while the radial functions Rl( r) have to satisfy Eq. (181), and be finite at r 0. At large distances r >> R, where R is the effective radius of the scatterer, the potential U( r) is negligible, and Eq. (181) is reduced to Eq. (183). In contrast to its analysis in Sec. 6, we should look for its solution using a linear superposition of the spherical Bessel functions of both kinds:
R
,
at
,
(3.214)
l r
A j
l l kr
B y
l
l kr
r
R
because Eq. (183) is now invalid at r 0, so our former argument for dropping the functions yl( kr) is no more valid as well. In Eq. (214), Al and Bl are some complex coefficients, determined by the scattering potential U( r), i.e. by the solution of Eq. (181) at r ~ R.
As the explicit expressions (186) show, the spherical Bessel functions jl() and yl() represent standing de Broglie waves, with equal real amplitudes, so their simple linear combinations (called the spherical Hankel functions of the first and second kind),
1
h
j
iy
,
h 2
and
j
iy
,
(3.215)
l
l l
l
l l
represent traveling spherical waves propagating, respectively, from the origin (i.e. from the center of the scatterer), and toward the origin. In particular, at >> 1, l, i.e. at large distances r >> 1/ k, l/ k,94
i l1
1
2
il1
h
,
.
(3.216)
l
kr ikr
e
hl kr
ikr
e
kr
kr
But using the same physical argument as at the beginning of Sec. 1, we may argue that in the case of a localized scatterer, there should be no latter waves at r >> R; hence, we have to require the amplitude of the term proportional to h (2)
l
to be zero. With the relations reciprocal to Eqs. (215),
1
1
2
1
j
h
h
y
h
h
(3.217)
l
l l,
l
1
l 2 l,
2
2 i
which enable us to rewrite Eq. (214) as
R
l r
Al
B
1
hl kr 2
hl kr
l
1
hl kr 2
hl kr
2
2 i
(3.218)
A iB
l
l
A
iB
1
h
l
kr
l
l
2
hl kr,
2
2
this means that the combination ( Al + iBl) has to be equal zero, i.e. Bl = iAl. Hence we have just one unknown coefficient (say, Al) for each l,95 and may rewrite Eq. (218) in an even simpler form: 93 The particular terms in this sum are frequently called partial waves.
94 For arbitrary l, this result may be confirmed using Eqs. (185) and the asymptotic formulas for the “usual”
Bessel functions – see, e.g., EM Eqs. (2.135) and (2.152), valid for an arbitrary (not necessarily integer) index n.
Chapter 3
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1
R
,
at
,
(3.219)
l r
Al jl kr iyl kr
A h
l l
kr
r
R
and use Eqs. (213) and (216) to write the following expression for the scattered wave at large distances: a i ikr
l 1
1 l
e
.
(3.220)
s
i A P cos ,
for
, ,
l l
r
R
kr
l0
k k
Comparing this expression with the general Eq. (81), we see that for a spherically symmetric, localized scatterer,
1
f
i l 1
A P
,
(3.221)
l l cos
k l0
so the differential cross-section (84) is
d
1
2
l 1
1
i A P
il' l
A A P
P
.
(3.222)
l l cos
*
2
2
l
l '
l cos l ' cos
d
k l0
k l, l'0
The last expression is more convenient for the calculation of the total cross-section (59):
1
1
d
d
2
d 2
dcos
l' l
*
i
A A
P P
d ,
(3.223)
2
l
l '
l l '
d
d
k
4
1
l, l '0
1
where cos , because this result may be much simplified by using Eq. (167):96
2
Spherically
symmetric
4 A
scatterer:
, with
l
.
(3.224)
l
l
2
l
k
l
0
2
1
Hence the solution of the scattering problem is reduced to the calculation of the partial wave amplitudes Al defined by Eq. (219) – and for the total cross-section, merely of their magnitudes. This task is much facilitated by using the following formula97 for the expansion of the incident plane wave into a series over the Legendre polynomials,
eikz ikr cos
e
il2 l 1 j kr P .
(3.225)
l
l cos
l0
As the simplest example, let us consider scattering by a completely impenetrable and “hard”
(meaning sharp-boundary) sphere, which may be described by the following potential:
, for r R,
U r
(3.226)
,
0
for
R r.
95 Moreover, using the conservation of the orbital momentum, to be discussed in Sec. 5.6, it is possible to show that this complex coefficient may be further reduced to just one real parameter, usually recast as the partial phase shift l between the l th spherical harmonics of the incident and scattered waves. However, I will not use this notion, because practical calculations are more physically transparent (and not more complex) without it.
96 Physically, this reduction of the double sum to a single one means that due to the orthogonality of the spherical harmonics, the total scattering probability flows due to each partial wave just add up.
97 It may be proved by using the Rodrigues formula (165) and integration by parts – the task left for the reader’s exercise.
Chapter 3
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In this case, the total wavefunction has to vanish at r R, and hence for the external problem ( r R), the sphere enforces the boundary condition 0 + s = 0 for all values of , at r = R. With Eqs. (213), (220), and (225), this condition becomes
a
R R il l j kR P
.
(3.227)
i
l 2 1 l lcos 0
l0
Due to the orthogonality of the Legendre polynomials, this condition may be satisfied for all angles only if all the coefficients before all Pl(cos) vanish, i.e. if l
R
2 1
.
(3.228)
l R
i l
jl kR
On the other hand, for r > R, U( r) = 0, so Eq. (183) is valid, and its outward-wave solution (219) has to be valid even at r R, giving
R
.
(3.229)
l R
Al jl kR iyl kR
Requiring the two last expressions to give the same result, we get
j kR
A il
2 1
,
(3.230)
l
l
l
j
l kR
iyl kR
so Eqs. (222) and (224) yield:
2
d
1
j kR
2
4 2 1
l
l
j kR
l 2
l
1
cos
,
.
(3.231)
d
k 2
2
2
2
l 0
jl kR iyl kR Pl
l
k
jl kR yl kR
As Fig. 25a shows, the first of these results gives an angular structure of the scattered de Broglie wave, which is qualitatively similar to that given by the Born approximation – cf. Eq. (98) and Fig. 10.
(a)
(b)
100
4
kR 30
10
3
l
l 0
10
0
1 d
1
d
1.0
2
g
0.1
g
1
0.1
1
2
0.01
0
0
0.2
0.4
0.6
0.8
0
2
4
6
8
10
/
kR
Fig. 3.25. Particle scattering by an impenetrable hard sphere: (a) the differential cross-section normalized to the geometric cross-section g R 2 of the sphere, as a function of the scattering angle , and (b) the (similarly normalized) total cross-section and its lowest spherical components, as functions of the dimensionless product kR E 1/2.
Chapter 3
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Namely, at low particle’s energies ( kR << 1), the scattering is essentially isotropic, while in the opposite, high-energy limit kR >> 1, it is mostly confined to small angles ~ / kR << 1, and exhibits numerous local destructive-interference minima at angles n ~ n/ kR. However, in our current (exact!) theory, these minima are different from zero because the theory describes an effective bending of the de Broglie waves along the back side of the sphere, which smears the interference pattern.
This bending is also responsible for a rather counter-intuitive fact (sometimes called wave extinction paradox), described by the second of Eqs. (231) and clearly visible in Fig. 25b: even at kR
, the total cross-section of scattering tends to 2g 2 R 2, rather than to the geometric cross-section
g as in the purely-classical scattering theory. First discovered for optical waves, this effect is common for all large non-absorbing scatterers. The fact that at kR << 1, the cross-section is also larger than g, approaching 4g at kR 0, is much less surprising, because in this limit the de Broglie wavelength =
2/ k is much larger than the sphere’s radius R, so the sphere effect on the incident wave extends to the area of the order of 2 >> g R 2.
The above analysis may be readily generalized to the case of a step-like ( sharp but finite) potential (97) – the problem left for the reader’s exercise. On the other hand, for a finite and smooth scattering potential U( r), by plugging Eq. (225) into Eq. (213) and the result into Eq. (66), and requiring the coefficients before each angular function Pl(cos) to be balanced, we get the following inhomogeneous generalization of Eq. (181) for the radial functions defined by Eq. (213): 2
d
2 d
E U r R
r
l l( )
1 R ( r) U
l 2 1
.
(3.232)
2
l
r i l jl kr
l
2 r
m
dr
dr
This differential equation has to be solved in the whole scatterer volume (i.e. for all r ~ R) with the boundary conditions for the functions Rl( r) to be finite at r 0, and to tend to the asymptotic form (219) at r >> R. The last requirement enables the evaluation of the coefficients Al that are needed for spelling out Eqs. (222) and (224), for any particular potential U( r). Unfortunately, due to the lack of time/space, for particular examples, I have to refer the interested reader to special literature.98
3.9. Exercise Problems
3.1. A particle of energy E is incident (in the figure on the right, within y
the plane of the drawing) on a sharp potential step:
,
0
for x ,
0
U (r)
x
0
U ,
0
for x
.
0
k
Calculate the particle reflection probability R as a function of the incidence angle , and discuss this function for various magnitudes and signs of U 0.
3.2. For a charged particle moving in a magnetic field B, calculate the commutation relations between Cartesian components of the kinetic (“mv-”) momentum operator defined by Eq. (20). Can the result be represented in a vector form?
98 See, e.g., J. Taylor, Scattering Theory, Dover, 2006.
Chapter 3
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3.3. In the classical mechanics version of the Landau-level problem discussed in Sec. 3.2 of the lecture notes, the geometric center of the particle’s orbit is an integral of motion, determined by initial conditions. Calculate the commutation relation between the quantum-mechanical operators corresponding to the Cartesian coordinates of the center.
3.4.* Analyze how are the Landau levels (50) modified by an additional uniform electric field E
directed along the plane of the particle’s motion. Contemplate the physical meaning of your result and its implications for the quantum Hall effect in a
gate-defined Hall bar. (The area l w of such a
V 0
B
V 0
g
gate
g
gate
bar is defined by metallic “gate” electrodes
parallel to the 2D electron gas plane – see the
w
2D electron
figure on the right. The negative voltage V g gas plane
applied to the gates squeezes the 2D gas from the
semiconductor
area under them into the complementary, Hall-
bar part of the plane.)
3.5. Analyze how are the Landau levels (50) modified if a 2D particle is confined in an additional 1D potential well U( x) = m 2
0 x 2/2.
3.6. Find the stationary states of a spinless, charged 3D particle moving in “crossed” (mutually perpendicular) uniform electric and magnetic fields, with E << c B. For such states, calculate the expectation values of the particle’s velocity in the direction perpendicular to both fields and compare the result with the solution of the corresponding classical problem.
Hint: You may like to generalize Landau’s solution for 2D particles, discussed in Sec. 2, to the 3D case.
3.7. Use the Born approximation to calculate the angular dependence and the total cross-section of scattering of an incident plane wave propagating along the x-axis, by the following pair of similar point inhomogeneities:
a
a
U (r) W r n
r n
.
z
z
2
2
Analyze the results in detail. Derive the condition of the Born approximation’s validity for such delta-functional scatterers.
3.8. Use the Born approximation to analyze the scattering of particles of energy E by a very thin, straight, uniform rod of length l, oriented normally to the incident particle’s velocity. In particular, calculate the differential and total cross-sections of scattering and analyze the results in the low-energy and high-energy limits.
3.9. Complete the analysis of the Born scattering by a uniform spherical potential (97), started in Sec. 3, by calculation of its total cross-section. Analyze the result in the limits kR << 1 and kR >>1.
3.10. Use the Born approximation to calculate the differential cross-section of particle scattering by a very thin spherical shell, whose potential may be approximated as U(r) = W( r – R). Analyze the Chapter 3
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results in the limits kR << 1 and kR >> 1, and compare them with those for a uniform sphere considered in Sec. 3.
3.11. Use the Born approximation to calculate the differential and total cross-sections of electron scattering by a screened Coulomb field of a point charge Ze, with the electrostatic potential99
r
Ze
r
e
,
4
r
0
neglecting spin interaction effects, and analyze the result’s dependence on the screening parameter .
Compare the results with those given by the classical (“Rutherford”) formula100 for the unscreened Coulomb potential ( 0), and formulate the condition of Born approximation’s validity in this limit.
3.12. A quantum particle with electric charge Q is scattered by the field of a localized distributed charge with a spherically symmetric density ( r) and zero total charge. Use the Born approximation to calculate the differential cross-section of the forward scattering (with the scattering angle = 0), and evaluate it for the scattering of electrons by a hydrogen atom in its ground state.
3.13. Prove the optical theorem (99).
Hint: For the general solution (64) of the scattering problem, with i given by Eq. (6) and s in the form (81), calculate the full probability current I through a spherical surface of radius r >> k–1, and then require that in accordance with the continuity relation (1.48), in this stationary situation, I = 0.
3.14. Reformulate the Born approximation for the 1D case. Use the result to find the scattering and transfer matrices of a “rectangular” (flat-top) scatterer
U , for x d / ,
2
U ( x) 0
,
0
otherwise.
Compare the results with those of the exact calculations carried out earlier in Chapter 2 and analyze how their relationship changes in the eikonal approximation.
3.15. In the tight-binding approximation, find the lowest stationary states of a particle placed into a system of three similar, isotropic, weakly coupled potential wells located in the vertices of an equilateral triangle.
3.16. The figure on the right shows a fragment of a periodic 2D lattice, y
a
with the red and blue points showing the positions of different local potentials.
(i) Find the reciprocal lattice and the 1st Brillouin zone of the system.
(ii) Calculate the wave number k of the monochromatic de Broglie wave
a
incident along the x-axis, at which the lattice creates the lowest-order diffraction peak within the [ x, y] plane, and the direction toward this peak.
(iii) Semi-quantitatively, describe the evolution of the intensity of the peak when all local potentials become similar.
x
99 This Yukawa potential was first suggested in 1935 by H. Yukawa as a model for strong interactions.
100 See, e.g., CM Sec. 3.5, in particular Eq. (3.73).
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Hint: The order of diffraction on a multidimensional Bravais lattice is a somewhat ambiguous notion dependent on the lattice type, but the lowest-order peak is always that corresponding to the smallest non-zero magnitude of the vector Q.
3.17. For the 2D hexagonal lattice (Fig. 12b):
(i) find the reciprocal lattice Q and the 1st Brillouin zone;
(ii) use the tight-binding approximation to calculate the dispersion relation E(q) for a 2D particle moving through a potential profile with such periodicity, with an energy close to the eigenenergy of similar isotropic states quasi-localized at the lattice points;
(iii) analyze and sketch/plot the resulting dispersion relation E(q) inside the 1st Brillouin zone.
3.18. Complete the tight-binding-approximation calculation of the band structure of the honeycomb lattice, that was started at the end of Sec. 4. Analyze the results; in particular, prove that the Dirac points qD are located in the corners of the 1st Brillouin zone, and express the velocity vn participating in Eq. (122), in terms of the coupling energy n. Show that the final results do not change if the quasi-localized wavefunctions are not isotropic but are proportional to exp{ im} – as they are, with m = 1, for the 2 pz electrons of carbon atoms in graphene, which are responsible for its transport properties.
3.19. Examine the basic properties of the so-called Wannier functions 101 defined as
(r) C (r) e iq R d 3 q , R
q
BZ
where q(r) is the Bloch wavefunction (108), R is any vector of the Bravais lattice, C is a normalization constant, and the integration over the quasimomentum q is extended over any (e.g., the first) Brillouin zone.
3.20. Evaluate the long-range interaction (the so-called London dispersion force) between two similar, electrically neutral atoms or molecules, modeling each of them as an isotropic 3D harmonic oscillator with the electric dipole moment d = qs, where s is the oscillator’s displacement from its equilibrium position.
Hint: You may like to represent the total Hamiltonian of the system as a sum of Hamiltonians of independent 1D harmonic oscillators, and calculate their total ground-state energy as a function of the distance between the dipoles. 102
3.21. Derive expressions for the stationary wavefunctions and the corresponding energies of a 2D particle of mass m, free to move inside a round disk of radius R. What is the degeneracy of each energy level? Calculate the five lowest energy levels with an accuracy better than 1%.
101 Named after G. Wannier who introduced these functions in 1939.
102 This explanation of the interaction between electrically-neutral atoms was put forward in 1930 by F. London, on the background of a prior (1928) work by C. Wang. Note that in some texts this interaction is (rather inappropriately) referred to as the “van der Waals force”, though it is only one (long-range) component of the van der Waals model – see, e.g., SM Sec. 4.1.
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3.22. Calculate the ground-state energy of a 2D particle of mass m, localized in a very shallow flat-bottom potential well
U
2
U ,
for
R
,
0
0
with U
.
0
2
,
0
for R,
R
m
3.23. Estimate the energy E of the localized ground state of a 2D particle of mass m, in an axially symmetric potential well of a finite radius R, with an arbitrary but very small potential U(). (Quantify this condition.)
3.24. Spell out the spherical harmonics 0
Y ( ,) and 4
Y ( ,) .
4
4
3.25. Calculate x and x 2 in the ground states of the planar and spherical rotors of radius R.
What can you say about p
2
x and px ?
3.26. A spherical rotor with r = R = const and mass m is in a state with the following wavefunction: = C(⅓ + sin2), where C is a constant. Calculate the energy of its angular motion.
3.27. According to the discussion at the beginning of Sec. 5, stationary wavefunctions of a 3D
isotropic harmonic oscillator may be calculated as products of three similar 1D “Cartesian oscillators” –
see, in particular Eq. (125), with d = 3. However, per the discussion in Sec. 6, the wavefunctions of the type (200), proportional to the spherical harmonics Y m
l , also describe stationary states of this spherically
symmetric system. Represent the wavefunctions (200) of:
(i) the ground state of the oscillator, and
(ii) each of its lowest excited states,
as linear combinations of products of the 1D oscillator’s stationary wavefunctions. Also, calculate the degeneracy of the n th energy level of the oscillator.
3.28. A particle of mass m is placed into a spherical, flat-bottom potential well U r U , for r R
,
0
with U 0.
,
0
for R r,
0
(i) Calculate the smallest U 0 at which the particle has a bound (localized) stationary state.
(ii) Calculate the energy of this state if U 0 is barely larger than that minimum value.
(iii) Does such a localized state exist in a very narrow and deep well that may be described as U(r) = – W(r) with a positive and finite W?
3.29. A 3D particle of mass m is placed into a spherically symmetric potential well with – < U( r) U() = 0. Relate its ground-state energy to that of a 1D particle of the same mass, moving in the following potential well:
U' x U x,
for x ,
0
,
for
x .
0
Use the found relation to:
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(i) discuss the origin of the difference between the solutions of Task (i) of the previous problem and of Problem 2.21, and
(ii) calculate the energy spectrum of an electron moving over an impenetrable plane surface of a perfect conductor.
3.30. Calculate the smallest value of the parameter U 0, for that the following spherically symmetric potential well:
U r
r /
U e
R ,
with U , R 0 ,
0
0
has a bound (localized) eigenstate for a particle of mass m.
Hint: You may like to introduce the following new variables: f rR and Ce– r/2 R, with a proper constant C.
3.31.* A particle of mass m moves in the field of an attractive spherically symmetric potential U( r) U() 0. Find a condition necessary for it to have at least one bound state. Compare the result with those of Problems 28 and 30.
3.32. A particle of mass m, moving in a certain central potential U( r), has a stationary state with the following wavefunction:
r
Cr e
cos ,
where C, , and > 0 are constants. Calculate:
(i) the probabilities of all possible values of the quantum numbers m and l, and (ii) the confining potential and the state’s energy.
3.33. For an isotropic 3D harmonic oscillator, calculate:
(i) the energy spectrum resulting from the Bohr quantization of circular classical orbits, and (ii) the energy spectrum of the s-states in the WKB approximation.
Compare the results with the exact energy spectrum of the oscillator, and comment.
3.34. For a particle of mass m, moving in the spherically symmetric potential U(r) = ar 4: (i) use the variational method to estimate the ground-state energy,
(ii) calculate the energy spectrum resulting from the Bohr quantization of circular orbits, and (iii) calculate the energy spectrum of the s-states in the WKB approximation.
Compare the results and comment.
3.35. For a particle of mass m, moving in the attracting Coulomb potential U( r) = – C/ r (e.g., the electron in a hydrogen atom):
(i) estimate the ground state energy by using the trial wavefunction trial = A/( r + a) b, where both a > 0 and b > 1 are fitting parameters, and
(ii) calculate the energy spectrum of the s-states in the WKB approximation.
Compare the results with the exact energy spectrum of the atom.
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3.36. Calculate the energy spectrum of a particle moving in a monotonic but otherwise arbitrary spherically symmetric attractive potential U( r) < 0, in the approximation of very large orbital quantum numbers l. Formulate the quantitative condition(s) of validity of your theory. Check that for the Coulomb potential U( r) = – C/ r, your result agrees with Eq. (201).
Hint: Try to solve Eq. (181) approximately by introducing the same new function f( r) rR( r) that was already used in Sec. 1 and in the solutions of a few earlier problems.
3.37. Prove Eq. (210) and the first two of Eqs. (211) for the ground state of a hydrogen-like atom/ion.
3.38. For the ground state of a particle in the Coulomb potential (190), calculate the probability to find it farther from the attracting center than the radius the same particle with the same energy would have on a classical circular orbit.
3.39. For the ground state and the lowest excited states of the hydrogen atom:
(i) calculate the spatial distribution of the electric current flowing around the nucleus, (ii) evaluate its highest density, and
(iii) calculate and evaluate its magnetic field at the position of the nucleus.
3.40. An electron had been in the ground state of a hydrogen-like atom/ion with nuclear charge Ze when the charge suddenly changed to ( Z + 1) e.103 Calculate the probabilities for the electron of the changed system to be:
(i) in its ground state, and
(ii) in one of the lowest excited states.
3.41. Due to a very short pulse of an external force, the nucleus of a hydrogen-like atom/ion, initially at rest in its ground state, starts moving with velocity v. Calculate the probability W g that the atom remains in its ground state. Evaluate the energy to be given, by the pulse, to a hydrogen atom in order to reduce W g to 50%.
3.42. Calculate x 2 and p 2
x in the ground state of a hydrogen-like atom/ion. Compare the results with Heisenberg’s uncertainty relation. What do these results tell about the electron’s velocity in the system?
3.43. Use the Hellmann-Feynman theorem (see Problem 1.7) to prove:
(i) the first of Eqs. (211), and
(ii) the fact that for a spinless particle in an arbitrary spherically symmetric attractive potential U( r), the ground state is always an s-state (with the orbital quantum number l = 0).
3.44. For the ground state of a hydrogen atom, calculate:
103 Such a fast change happens, for example, at the beta-decay, when one of the nucleus’ neurons spontaneously turns into a proton, emitting a high-energy electron and a neutrino, which leave the system very fast (instantly on the atomic time scale), and do not affect directly the atom transition’s dynamics.
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(i) the expectation value of E, where E is the electric field created by the atom as a whole, and (ii) the expectation value of E2 at distances r >> r 0 from the nucleus.
Interpret the obtained relation between E2 and E 2 at distant observation points.
3.45. Find the condition at which a particle of mass m, moving in the field of a very thin spherical shell with U(r) = W( r – R) and W < 0, has at least one localized (“bound”) stationary state.
3.46. Calculate the lifetime of the lowest metastable state of a particle in the same spherical shell potential as in the previous problem, but now with W > 0, for sufficiently large W. (Quantify this condition.)
3.47. A particle of mass m and energy E is incident on a very thin spherical shell of radius R, whose localized states were the subject of two previous problems, with an arbitrary “weight” W.
(i) Derive general expressions for the differential and total cross-sections of scattering.
(ii) Spell out the contribution 0 to the total cross-section , given by the spherically symmetric component of the scattered de Broglie wave.
(iii) Analyze the result for 0 in the limits of very small and very large magnitudes of W, for both signs of this parameter. In particular, in the limit W +, relate the result to the metastable state’s lifetime calculated in the previous problem.
3.48. Calculate the spherically symmetric contribution 0 to the total cross-section of particle scattering by a uniform sphere of radius R, described by the following potential: U r U , for r R
,
0
,
0
otherwise,
with an arbitrary constant U 0. Analyze the result in detail, and give an interpretation of its most remarkable features.
3.49. Use the finite difference method with the step h = a/2 to calculate as many energy levels as possible, for a particle confined to the interior of:
(i) a square with sides a, and
(ii) a cube with sides a,
with hard walls. For the square, repeat the calculations by using the finer step h = a/3. Compare the results for different values of h with each other and with the exact formulas.
Hint: It is advisable to either first solve (or review the solution of) the similar 1D Problem 1.18, or start from reading about the finite difference method.104 Also: try to exploit the symmetry of the systems.
104 See, e.g., CM Sec. 8.5 or EM Sec. 2.11.
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Chapter 4. Bra-ket Formalism
The objective of this chapter is to describe Dirac’s “bra-ket” formalism of quantum mechanics, which not only overcomes some inconveniences of wave mechanics but also enables a natural description of such intrinsic properties of particles as their spin. In the course of the formalism’s discussion, I will give only a few simple examples of its application, leaving more involved cases for the following chapters.
4.1. Motivation
As the reader could see from the previous chapters of these notes, wave mechanics gives many results of primary importance. Moreover, it is mostly sufficient for many applications, for example, solid-state electronics and device physics. However, in the course of our survey, we have filed several grievances about this approach. Let me briefly summarize these complaints:
(i) Attempts to analyze the temporal evolution of quantum systems, beyond the trivial time behavior of the stationary states, described by Eq. (1.62), run into technical difficulties. For example, we could derive Eq. (2.151) describing the metastable state’s decay and Eq. (2.181) describing the quantum oscillations in coupled wells, only for the simplest potential profiles, though it is intuitively clear that these simple results should be common for all problems of this kind. Solving such problems for more complex potential profiles would entangle the time evolution analysis with the calculation of the spatial distribution of the evolving wavefunctions – which (as we could see in Secs. 2.9 and 3.6) may be rather complex even for time-independent potentials. Some separation of the spatial and temporal dependencies is possible using perturbation approaches (to be discussed in Chapter 6) but even those would lead, in the wavefunction language, to very cumbersome formulas.
(ii) The last statement can also be made concerning other issues that are conceptually addressable within the wave mechanics, e.g., the Feynman path integral approach, coupling to the environment, etc. Pursuing them in the wave mechanics language would lead to formulas so bulky that I had postponed their discussion until we would have a more compact formalism on hand.
(iii) In the discussion of several key problems (for example the harmonic oscillator and spherically-symmetric potentials), we have run into rather complicated eigenfunctions coexisting with very simple energy spectra – that infer some simple background physics. It is very important to get this physics revealed.
(iv) In the wave-mechanics postulates formulated in Sec. 1.2, the quantum mechanical operators of the coordinate and momentum are treated rather unequally – see Eqs. (1.26b). However, some key expressions, e.g., for the fundamental eigenfunction of a free particle,
p r
exp i
,
(4.1)
or the harmonic oscillator’s Hamiltonian,
2
1
m
2
0
2
ˆ
H
ˆ p
ˆ r ,
(4.2)
2 m
2
just beg for a similar treatment of coordinates and momenta.
© K. Likharev
Essential Graduate Physics
QM: Quantum Mechanics
However, the strongest motivation for a more general formalism comes from wave mechanics’
conceptual inability to describe elementary particles’ spins 1 and other internal quantum degrees of freedom, such as quarks’ flavors. In this context, let us review the basic facts on spin (which is very representative and experimentally the most accessible of all internal quantum numbers), to understand what a more general formalism has to explain – as a minimum.
Figure 1 shows the conceptual scheme of the simplest spin-revealing experiment, first conceived by Otto Stern in 1921 and implemented by Walther Gerlach in 1922. A collimated beam of particles2
from a natural source, such as a heated cathode, is passed through a gap between the poles of a strong magnet, whose magnetic field B, (in Fig. 1, directed along the z-axis) is nonuniform, so both B z and d B z/ dz are not equal to zero. The experiment shows that even if all particles are in the ground orbital state, the beam splits into two beams of equal intensity.
collimator z
magnet
N
y
W = 50%
B
electron
B ,
z 0 S
W = 50%
Fig. 4.1. The simplest Stern-
z
z
Gerlach experiment.
source
particle detectors
This result may be semi-quantitatively explained on classical (if somewhat phenomenological) grounds by assuming that each particle has an intrinsic, permanent magnetic dipole moment m. Indeed, classical electrodynamics tells us3 that the potential energy U of a magnetic dipole in an external magnetic field B is equal to (–m ꞏ B), so the force acting on the particle, F U m B ,
(4.3)
has a non-zero z-component
F
B
B .
(4.4)
z
mz z m
z
z
z
z
Hence if we further assume that the particle’s magnetic moment may take only two equally probable discrete values of mz = (though such discreteness does not follow from any classical model of the particle), this may explain the basic Stern-Gerlach effect qualitatively. The quantitative explanation of the beam splitting angle requires the magnitude of to be equal (or very close) to the so-called Bohr magneton 4
e
23 J
9274
.
0
10
.
(4.5)
Bohr
B
2 m
T
magneton
e
1 Reportedly, the concept of spin as a measure of the internal rotation of a particle was first suggested (though later rejected) by Ralph Kronig, then a 20-year-old student, in January 1925, a few months before two other students, George Uhlenbeck and Samuel Goudsmit, came to this idea independently. The concept was then accepted (first, rather reluctantly) and developed quantitatively by Wolfgang Pauli.
2 The initial Stern-Gerlach experiments used silver atoms because their larger mass helps to decrease the spit beam widths. However, the discussion below is valid for any spin-½ particles including electrons.
3 See, e.g., EM Sec. 5.4, in particular Eq. (5.100).
4 A good mnemonic rule is that it is close to 1 K/T. In the Gaussian units, B e/2 m e c 0.927410-20 erg/G.
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However, as we will see below, this value cannot be explained by any internal motion of the particle, say its rotation about the z-axis. More importantly, this semi-classical phenomenology cannot explain, even qualitatively, other experimental results, for example those of the set of multistage Stern-Gerlach experiments shown in Fig. 2. In the first of the experiments, the particle beam is first passed through a magnetic field (and its gradient) oriented along the z-axis, just as in Fig. 1. Then one of the two resulting beams is absorbed (or removed from the setup in some other way), while the other one is passed through a similar but x-oriented field. The experiment shows that this beam is split again into two components of equal intensity. A classical explanation of this experiment would require an even more unnatural additional assumption that the initial particles had random but discrete components of the magnetic moment simultaneously in two directions, z and x.
50%
100 %
SG
SG
( x)
( z)
50%
absorber
100 %
SG
50%
( z)
SG
SG
50%
( x)
( z)
Fig. 4.2. Three multistage
Stern-Gerlach experiments.
The boxes SG (…) denote
100% magnets similar to the one
100 %
SG
SG
( z)
shown in Fig. 1, with the
( z)
0%
field oriented in the
indicated direction.
However, even this assumption cannot explain the results of the three-stage Stern-Gerlach experiment shown on the middle panel of Fig. 2. Here, the previous two-state setup is complemented with one more absorber and one more magnet, now with the z-orientation again. Completely counterintuitively, it again gives two beams of equal intensity, as if we have not yet filtered out the particles with mz corresponding to the lower beam, at the first z-stage. The only way to save the classical explanation here is to say that maybe, particles somehow interact with the magnetic field, so the x-
polarized beam becomes spontaneously depolarized again somewhere between the two last stages. But any hope for such an explanation is ruined by the control experiment shown on the bottom panel of Fig.
2, whose results indicate that no such depolarization happens.
We will see below that all these (and many more) results find a natural explanation in the so-called matrix mechanics pioneered by Werner Heisenberg, Max Born, and Pascual Jordan in 1925.
However, the matrix formalism is rather inconvenient for the solution of most problems discussed in Chapters 1-3, and for a short time, it was eclipsed by E. Schrödinger’s wave mechanics, which had been put forward just a few months later. However, very soon Paul Adrien Maurice Dirac introduced a more general bra-ket formalism of quantum mechanics, which provides a generalization of both approaches and proves their equivalence. Let me describe it, begging for the reader’s patience because (in contrast with my usual style), I will not be able to give particular examples of its application for a while – until all the basic notions of the formalism have been introduced.
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4.2. States, state vectors, and linear operators
The basic notion of the general formulation of quantum mechanics is the quantum state of a system.5 To get some gut feeling of this notion, if a quantum state of a particle may be adequately described by wave mechanics, this description is given by the corresponding wavefunction (r, t).
Note, however, that a quantum state as such is not a mathematical object,6 and can participate in mathematical formulas only as a “label” – e.g., the index of the wavefunction . On the other hand, such a wavefunction is not a state, but a mathematical object (a complex function of space and time) giving a quantitative description of the state – just as the classical radius vector r and velocity v as real functions of time are mathematical objects describing the motion of the particle in its classical description – see Fig. 3. Similarly, in the Dirac formalism, a certain quantum state is described by either of two mathematical objects, called the state vectors: the ket-vector and the bra-vector ,7
whose relationship is close to that between the wavefunction
*
and its complex conjugate .
mechanics
classical
:
r t, v t, etc.
in
system
mathematical
*
mechanics
wave
either
:
(r, t
Ψ
or
)
(r, t)
state
descriptions:
α
bra -
formalism
ket
either
:
or
Fig. 4.3. Physical state of a system and its descriptions.
One should be cautious with the term “vector” here. The usual geometric vectors, such as r and v, are defined in the usual geometric (say, Euclidean) space. In contrast, the bra- and ket-vectors are defined in a more abstract Hilbert space – the full set of all possible state vectors of a given system.8 So, despite certain similarities with the geometric vectors, the bra- and ket-vectors are different mathematical objects, and we need to define the rules of their handling. The primary rules are essentially postulates and are justified only by the correct description of all experimental observations of the rules’
corollaries. While there is a general consensus among physicists about what the corollaries are, there are many possible ways to carve from them the different sets of basic postulates. Just as in Sec. 1.2, I will not try too hard to beat the number of the postulates down to the minimum, trying instead to keep their physical meaning transparent.
(i) Ket-vectors. Let us start with ket-vectors – sometimes called just kets for short. Their most important property is the linear superposition. Namely, if several ket-vectors j describe possible states of a quantum system, numbered by the index j, then any linear combination ( superposition) Linear
c ,
(4.6)
superposition
j
j
of ket-vectors
j
5 An attentive reader could notice my smuggling the term “system” instead of “particle”, which was used in the previous chapters. Indeed, the bra-ket formalism allows the description of quantum systems much more complex than a single spinless particle that is a typical (though not the only possible) subject of wave mechanics.
6 As was expressed nicely by Asher Peres, one of the pioneers of the quantum information theory, “quantum phenomena do not occur in the Hilbert space, they occur in a laboratory”.
7 The terms bra and ket were suggested to reflect the fact that the pair and may be considered as the parts of the combinations like (see below), which remind expressions in the usual angle brackets.
8 I have to confess that this is a bit loose definition; it will be refined soon.
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where cj are any (possibly complex) c-numbers, also describes a possible state of the same system.9
Actually, since ket-vectors are new mathematical objects, the exact meaning of the right-hand side of Eq. (6) becomes clear only after we have postulated the following rules of summation of these vectors,
,
(4.7)
j
j'
j'
j
and their multiplication by an arbitrary c-number:
c
c .
(4.8)
j
j
Note that in the set of wave-mechanics postulates, the statements parallel to Eqs. (7) and (8) were unnecessary because the wavefunctions are the usual (albeit complex) functions of space and time, and we know from the usual algebra that such relations are indeed valid.
As Eq. (6) shows, the coefficient cj may be interpreted as the “weight” of the state j in the linear superposition . One important particular case is cj = 0, showing that the state j does not participate in the superposition . The corresponding term of the sum (6), i.e. the product Null-state
0 ,
(4.9)
vector
j
has a special name: the null-state vector. (It is important to avoid confusion between the null state corresponding to vector (9), and the ground state of the system, which is frequently denoted by the ket-vector 0. In some sense, the null state does not exist at all, while the ground state not only does exist but frequently is the most important quantum state of the system.)
(ii) Bra-vectors and inner products. Bra-vectors , which obey the rules similar to Eqs. (7) and (8), are not new, independent objects: a ket-vector and the corresponding bra-vector describe the same state. In other words, there is a unique dual correspondence between and ,10 very similar (though not identical) to that between a wavefunction and its complex conjugate *.11 The correspondence between these vectors is described by the following rule: if a ket-vector of a linear superposition is described by Eq. (6), then the corresponding bra-vector is
Linear
superposition
c*
c* .
(4.10)
j
j
j j
of bra-vectors
j
j
The mathematical convenience of using two types of vectors rather than just one becomes clear from the notion of their inner product (due to its second, shorthand form, also called the short bracket): Short
bracket
(inner
,
(4.11)
product)
which is a scalar c-number, in a certain but limited analogy with the scalar product of the usual geometric vectors. (For one difference, the product (11) may be a complex number.) The main property 9 One may express the same statement by saying that the vector belongs to the same Hilbert space as all j.
10 Mathematicians like to say that the ket- and bra-vectors of the same quantum system are defined in two isomorphic Hilbert spaces.
11 This analogy is not occasional: we will see very soon that the wavefunction of a quantum state is just a special (“coordinate”) representation of its state vector.
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of the inner product is its linearity with respect to any of its component vectors. For example, if a linear superposition is described by the ket-vector (6), then
c ,
(4.12)
j
j
j
while if Eq. (10) is true, then
c* .
(4.13)
j
j
j
In plain English, c-number factors may be moved either into or out of the inner products.
The second key property of the inner product is
Inner
*
.
(4.14) product::
complex
conjugate
It is compatible with Eq. (10); indeed, the complex conjugation of both parts of Eq. (12) gives:
*
* c*
c*
.
(4.15)
j
j
j
j
j
j
Finally, one more rule: the inner product of the bra- and ket-vectors describing the same state (called the norm squared) is real and non-negative,
State’s
2
.
0
(4.16) norm
squared
In order to give the reader some feeling about the meaning of this rule: we will see below that if some state may be described by the corresponding wavefunction (r, t), then
*
3
d 0
r
.
(4.17)
Hence the role of the bra- and ket-vectors of the same state is very similar to that of complex-conjugate pairs of its wavefunctions.
(iii) Operators. One more key notion of the Dirac formalism is quantum-mechanical linear operators. Just as for the operators discussed in wave mechanics, the function of an operator is to
“generate” one state from another: if is a possible ket of the system, and A îs a legitimate12 operator, then the following combination,
A ˆ ,
(4.18)
is also a ket-vector describing a possible state of the system, i.e. a ket-vector in the same Hilbert space as the initial vector . An alternative formulation of the same rule is the following refinement of the notion of the Hilbert space: for a given set of linear operators of a system, its Hilbert space includes all vectors that may be obtained from each other using the operations of the type (18). In this context, let me note that the operator set, and hence the Hilbert space of a system, usually (if not always) implies its 12 Here the term “legitimate” means “having a clear sense in the bra-ket formalism”. Some examples of
“illegitimate” expressions are: A ˆ , A ˆ , , and . Note, however, that the last two expressions may be legitimate if and are states of different systems, i.e. if their state vectors belong to different Hilbert spaces.
We will run into such direct products of the bra- and ket-vectors (sometimes denoted, respectively, as and
) in Chapters 6-10.
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certain approximate model. For example, if the coupling of orbital degrees of freedom of a particle to its spin may be ignored (as it may be for a non-relativistic particle in the absence of an external magnetic field), we may describe the dynamics of the particle using spin operators only. In this case, the set of all possible spin vectors of the particle forms a Hilbert space separate from that of the orbital-state vectors of the same particle.
As the adjective “linear” in the operator definition implies, the main rule governing the operators is their linearity with respect to both any superposition of vectors:
A ˆ
ˆ
c
c A ,
(4.19)
j
j
j
j
j
j
and any superposition of operators:
ˆ
ˆ
c A
c A .
(4.20)
j
j
j j
j
j
These rules are evidently similar to Eqs. (1.53)-(1.54) of wave mechanics.
The above rules imply that an operator “acts” on the ket-vector on its right; however, a combination of the type
A ˆ
is also legitimate and represents a new bra-vector. It is important that,
generally, this vector does not represent the same state as the ket-vector (18); instead, the bra-vector isomorphic to the ket-vector (18) is
Conjugate
†
ˆ
A .
(4.21)
operator
This statement serves as the definition of the Hermitian conjugate (also called “Hermitian adjoint”) †
ˆ A of the initial operator A ˆ . For an important class of operators, called the Hermitian operators, the conjugation is inconsequential, i.e. for them
Hermitian
operator
A ˆ† A ˆ .
(4.22)
(This equality, as well as any other operator equation below, means that these operators act similarly on any bra- or ket-vector of the given Hilbert space.) 13
To proceed further, we need one more additional postulate, sometimes called the associative axiom of multiplication: just as an ordinary product of scalars, any legitimate bra-ket expression that does not include explicit summations, does not change from an insertion or removal of a pair of parentheses – meaning as usual that the operation inside them has to be performed first. The first two examples of this postulate are given by Eqs. (19) and (20), but the associative axiom is more general and means, for example, that
Long
bracket:
A ˆ A ˆ A ˆ ,
(4.23)
definition
13 If we consider c-numbers as a particular type of operators (which is legitimate for any Hilbert space), then according to Eqs. (11) and (21), for them the Hermitian conjugation is equivalent to the simple complex conjugation, so only real c-numbers may be considered as a particular type of Hermitian operators (22).
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This equality serves as the definition of the last form, called the long bracket (evidently, also a scalar), with an operator sandwiched between a bra-vector and a ket-vector. This definition, when combined with the definition of the Hermitian conjugate and Eq. (14), yields an important corollary: ˆ
A ˆ A
†
ˆ
*
A ˆ
A *
†
,
(4.24)
which is most frequently rewritten as
Long
ˆ A * †
ˆ A .
(4.25) bracket:
complex
conjugate
The associative axiom also enables us to comprehend the following definition of one more, outer product of bra- and ket-vectors:
Outer
.
(4.26) bra-ket
product
In contrast to the inner product (11), which is a scalar, this mathematical construct is an operator.
Indeed, the associative axiom allows us to remove parentheses in the following expression:
.
(4.27)
But the last short bracket is just a scalar; hence the mathematical object (26) acting on a ket-vector (in this case, ) gives a new ket-vector, which is the essence of the operator’s action. Very similarly,
(4.28)
– again a typical operator’s action on a bra-vector. So, Eq. (26) defines an operator.
Now let us perform the following calculation. We may use the parentheses’ insertion into the bra-ket equality following from Eq. (14),
*
,
(4.29)
to transform it into the following form:
*
.
(4.30)
Since this equality should be valid for any state vectors and , its comparison with Eq. (25) gives the following operator equality
Outer
† .
(4.31) product:
Hermitian
conjugate
This is the conjugate rule for outer products; it reminds Eq. (14) for inner products but involves the Hermitian (rather than the usual complex) conjugation.
The associative axiom is also valid for the operator multiplication:
ˆ B
A ˆ
ˆ
A B ˆ
ˆ
,
B
A ˆ A ˆ B ˆ ,
(4.32)
showing that the action of an operator product on a state vector is nothing more than the sequential action of its operands. However, we have to be careful with the operator products; generally, they do not commute: ˆ B
A ˆ
ˆ
A
B ˆ . This is why the commutator – the operator defined as
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Commutator
A ˆ B ˆ
, A ˆ B ˆ B ˆ A ˆ ,
(4.33)
is a non-trivial and very useful notion. Another similar notion is the anticommutator:14
Anti-
commutator
A ˆ B ˆ
, A ˆ B ˆ B ˆ A ˆ .
(4.34)
Finally, the bra-ket formalism broadly uses two special operators. The null operator 0îs defined by the following relations:
Null
0ˆ 0
0ˆ
,
0 ,
(4.35)
operator
where is an arbitrary state; we may say that the null operator “kills” any state by turning it into the null state. Another useful notion is the identity operator, which is defined by the following action (or rather “inaction” :-) on an arbitrary state vector:
Identity
I ˆ
I ˆ
,
.
(4.36)
operator
These definitions show that the null operator and the identity operator are Hermitian.
4.3. State basis and matrix representation
While some operations in quantum mechanics may be carried out in the general bra-ket formalism outlined above, many calculations are performed for quantum systems that feature a full and orthonormal set { u} { u 1, u 2, …, u j, …} of its states uj, frequently called a basis. The former of these terms means that any possible state vector of the system (i.e. any vector of its Hilbert space) may be represented as a unique sum of the type (6) or (10) over its basis vectors:
Expansion
over
u
*
,
u ,
(4.37)
j
j
j j
state basis
j
j
so, in particular, if is one of the basis states, say uj’, then j = jj’. The latter term means that Basis
vectors:
u u
.
(4.38)
ortho-
j
j'
jj'
normality
For the systems that may be described by wave mechanics, examples of the full orthonormal bases are represented by any full and orthonormal set of stationary functions calculated in the previous three chapters of this course – for the simplest example, see Eq. (1.87).
Due to the uniqueness of the expansion (37), the full set of the coefficients j involved in the expansion of a state in certain basis { u} gives its complete description – just as the Cartesian components Ax, Ay, and Az of a usual geometric 3D vector A in certain reference frame give its complete description. Still, let me emphasize some differences between such representations of the quantum-mechanical state vectors and 3D geometric vectors:
(i) a quantum state basis may have a large or even infinite number of states uj, and (ii) the expansion coefficients j may be complex.
14 Another popular notation for the anticommutator (34) is A ˆ B ˆ
, ; it will not be used in these notes.
Chapter 4
Page 9 of 52
QM: Quantum Mechanics
With these reservations in mind, the analogy with geometric vectors may be pushed further on.
Let us inner-multiply both parts of the first of Eqs. (37) by a bra-vector uj’ and then transform the resulting relation using the linearity rules discussed in the previous section, and Eq. (38): u u
u
u u
(4.39)
j'
j'
j
j
.
j
j'
j
j'
j
j
Together with Eq. (14), this means that any of the expansion coefficients in Eq. (37) may be represented as an inner product:
Expansion
u
*
,
u ;
(4.40) coefficients
j
j
j
j
as inner
products
these important equalities relations are analogs of equalities Aj = n jA of the usual vector algebra and will be repeatedly used in this course. With them, the expansions (37) may be rewritten as
u u
ˆ
u u
ˆ
,
,
(4.41)
j
j
j
j
j
j
j
j
j
j
where
ˆ u u .
(4.42) Projection
j
j
j
operator
Eqs. (41) show that ˆ so defined is a legitimate linear operator. This operator, acting on any state j