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
Uut
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
Uup
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
Uus
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
Uuo
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).
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The simplest atom, with Z = 1, is hydrogen (chemical symbol H) – the only atom for that the theory discussed above is quantitatively correct.86 According to Eq. (191), the 1 s 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) is filled with two electrons. As will be discussed in detail in Chapter 8, electrons of the same atom are actually indistinguishable, so that 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 counter-intuitively, for the atom classification purposes, they are not crucial, and we may pretend that two electrons of a helium atom just have “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 4-fold increase of the constant E 0 given by Eq. (192), the binding energy of each electron is crudely 4
times higher than that of the hydrogen atom – though the electron interaction decreases it by about 25%
– see Sec. 8.2. This is why taking one electron away (i.e. the positive ionization of a helium atom) requires relatively high energy, ~23.4 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, the 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.
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 two-thirds of its electric charge +3 e. As a result, the 2 s-state electron is approximately, but reasonably well described by Eq. (201) with Z = 1 and n = 2, giving binding energy close to just 3.4 eV (experimentally, ~5.39 eV), so that 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, it is a typical alkali metal. The similarity of chemical properties of lithium and hydrogen, with the chemical valence of one,87 places Li as the starting element of the second period (row), with the first period limited to only H and He – see Fig. 23.
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 that 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.
86 Besides very small fine-structure and hyperfine-splitting corrections – to be discussed, respectively, in Chapters 6 and 8.
87 The chemical valence (or “valency”) is a not very precise term describing the number of atom’s electrons involved in chemical reactions. For the same atom, especially with a large number of electrons in its outer, unfilled 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.)
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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, 1,88 with a gradually growing ionization potential (up to ~21.6 eV in Ne with Z =
10), i.e. a growing reluctance to 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 an at least 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 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 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 the ionization energy of ~15.7 eV due to the fully filled 3 s and 3 p shells, 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 inter-electron interactions, and gradually accumulates with the growth of Z. It may be semi-quantitatively understood from the results of 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 much stronger than that of l, so that 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, are the so-called transition metals whose (comparable) ionization energies and chemical properties are determined by the 4 s electrons.
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 88 The specific order of filling of the states within each shell follows the so-called Hund rules – see Sec. 8.3.
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each time and only when the first electron with the next principal quantum number ( n) appears.89 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 states) and actinides ( Z from 89 to 103, of the 7th row, with a gradual filling of the 5 f and 6 d states), are usually represented as outlet rows – see Fig. 23. This is quite acceptable for basic chemistry, because the chemical properties of the elements within each such group are rather close.
To summarize my very short review of this extremely important topic,90 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 set 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 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 . (The quantification of this condition is left for the reader’s exercise.) 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 that its expansion in the series over the spherical harmonics includes only the terms with m = 0. Hence, the solution (64) of the stationary Schrödinger equation (63) in this case may be represented as91
ikz
a e
R r P
,
(3.213)
i
s
i
l lcos
l0
89 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).
90 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.
91 The particular terms in this sum are frequently called partial waves.
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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, and our former argument for dropping the functions yl( kr) is no more valid. 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 that 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,92
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
A
B
R r
h
h
h
h
l
l
1
l 2 l l 1
l 2 l
2
2 i
(3.218)
A iB
A iB
l
l
1
h
h
l
l
l
2
l
,
2
2
this means that the combination ( Al + iBl) has to be equal zero, so that Bl = iAl. Hence we have just one unknown coefficient (say, Al) for each l,93 and may rewrite Eq. (218) in an even simpler form: Spherically-
1
R
,
at
,
(3.219) symmetric
l r
Al jl kr iyl kr
A h
l l
kr
r
R
scatterer: Al
92 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.
93 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.
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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 ,
at
, ,
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 that 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):94
2
Spherically-
4 A
symmetric
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 Rayleigh formula for the expansion of the incident plane wave’s exponent into a series over the Legendre polynomials,95
eikz ikr cos
e
il2 l 1 j kr P .
(3.225)
l
l cos
l0
As the simplest example, let us calculate scattering by a completely opaque 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.
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
94 Physically, this reduction of the double sum to a single one means that due to the orthogonality of the sherical harmonics, the total scattering probability flows due to each partial wave just add up.
95 It may be proved using the Rodrigues formula (165) and integration by parts – the task left for the reader’s exercise.
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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 that 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 that 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 describes 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 opaque, 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.
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 always finite, because the theory describes effective bending of the de Broglie waves along the back side of the sphere, which smears the interference pattern.
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The same bending is also responsible for a rather counter-intuitive fact, 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 g as in the purely-classical scattering theory. First discovered for optical wave scattering, this fact (common for all large non-absorbing scatterers) is sometimes interpreted as a manifestation of the so-called wave extinction paradox.
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 longer than the sphere’s radius R, so that the wave’s propagation is affected by the whole sphere.
The above analysis may be readily generalized to the case 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), plugging Eq. (225) into Eq. (213) and the result into Eq. (66), requiring the coefficients before each angular function Pl(cos) to be balanced, and canceling the common coefficient a 0, 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). Unfortunately, due to the lack of time, I have to refer the reader interested in such cases to special literature.96
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 ; sketch and discuss this function for various magnitudes and signs of U 0.
3.2.* Analyze how are the Landau levels
(50) modified by an additional uniform electric
V 0
B
V 0
g
gate
g
gate
field E directed along the plane of the particle’s
w
motion. Contemplate the physical meaning of 2D electron
your result and its implications for the quantum gas plane
semiconductor
Hall effect in a gate-defined Hall bar. (The area
96 See, e.g., J. Taylor, Scattering Theory, Dover, 2006.
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l w of such a bar [see Fig. 6] is defined by metallic gate electrodes parallel to the 2D electron gas plane
– see the figure on the right. The negative voltage V g, applied to the gates, squeezes the 2D gas from the area under them into the complementary, Hall-bar part of the plane.)
3.3. 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.
3.4. 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.
3.5. 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.
3.6. 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.
3.7. 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 results in the limits kR << 1 and kR >> 1, and compare them with those for a uniform sphere considered in Sec. 3.
3.8. 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 potential
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”) formula97 for the unscreened Coulomb potential ( 0), and formulate the condition of Born approximation’s validity in this limit.
3.9. A quantum particle with electric charge Q is scattered by a localized distributed charge with a spherically-symmetric density ( r), and zero total charge. Use the Born approximation to calculate the 97 See, e.g., CM Sec. 3.5, in particular Eq. (3.73).
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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.
3.10. 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 does their relationship change in the eikonal approximation.
3.11. In the tight-binding approximation, find the lowest stationary states of a particle placed into a system of three similar, weakly coupled potential wells located in the vertices of an equilateral triangle.
3.12. 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 axis x, at that 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
x
peak when the local potentials, represented by the different points, become
similar.
Hint: The order of diffraction on a multidimensional Bravais lattice is a somewhat ambiguous notion, usually associated with the sum of magnitudes of all integers lj in Eq. (109), for the vector Q that is equal to q k – ki.
3.13. 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 energy of the axially-symmetric states quasi-localized at the potential minima;
(iii) analyze and sketch (or plot) the resulting dispersion relation E(q) inside the 1st Brillouin zone.
3.14. Complete the tight-binding-approximation calculation of the band structure of the honeycomb lattice, 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 axially-symmetric, but are proportional to exp{ im} – as they are, with m = 1, for the 2 pz electrons of carbon atoms in graphene, that are responsible for its transport properties.
3.15. Examine basic properties of the so-called Wannier functions defined as
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(r) const (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, and the integration over the quasimomentum q is extended over any (e.g., the first) Brillouin zone.
3.16. 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: 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. 98
3.17. 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%.
3.18. 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
3.19. Estimate the energy E of the localized ground state of a particle of mass m, in an axially-symmetric 2D potential well of a finite radius R, with an arbitrary but very small potential U().
(Quantify this condition.)
3.20. Spell out the explicit form of the spherical harmonics 0
Y ( ,) and 4
Y ( ,) .
4
4
3.21. Calculate x and x 2 in the ground states of the planar and spherical rotators of radius R.
What can you say about p
2
x and px ?
3.22. A spherical rotator, with r ( x 2 + y 2+ z 2)1/2 = R = const, of mass m is in a state with the following wavefunction: = const(⅓ + sin2). Calculate its energy.
3.23. According to the discussion at the beginning of Sec. 5, stationary wavefunctions of a 3D
harmonic oscillator may be calculated as products of three 1D “Cartesian oscillators” – see, in particular Eq. (125), with d = 3. However, according to 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:
98 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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(i) the ground state of the oscillator, and
(ii) each of its lowest excited states,
as linear combinations of products of 1D oscillator’s stationary wavefunctions. Also, calculate the degeneracy of the n th energy level of the oscillator.
3.24. Calculate the smallest depth U 0 of a spherical, flat-bottom potential well U r U , for r R
,
0
,
0
for R r,
at that it has a bound (localized) stationary state. Does such a state exist for a very narrow and deep well U(r) = – W(r), with a positive and finite W?
3.25. 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
In the light of the found relation, discuss the origin of the difference between the solutions of the previous problem and Problem 2.17.
3.26. 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.
Hint: You may like to introduce the following new variables: f rR and Ce– r/2 R, with an appropriate choice of the constant C.
3.27. A particle 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.
3.28. Use the variational method to estimate the ground-state energy of a particle of mass m, moving in the following spherically-symmetric potential:
U r
4
ar .
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3.29. Use the variational method, with the trial wavefunction trial = A/( r + a) b, where both a > 0
and b > 1 are fitting parameters, to estimate the ground-state energy of the hydrogen-like atom/ion with the nuclear charge + Ze. Compare the solution with the exact result.
3.30. Calculate the energy spectrum of a particle moving in a monotonic, but otherwise arbitrary attractive spherically-symmetric 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, 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.
3.31. 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. 99 Calculate the probabilities for the electron of the changed system to be:
(i) in the ground state, and
(ii) in the lowest excited state.
3.32. 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%.
3.33. 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?
3.34. Use the Hellmann-Feynman theorem (see Problem 1.5) 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).
3.35. For the ground state of a hydrogen atom, calculate the expectation values of E and E2, where E is the electric field created by the atom, at distances r >> r 0 from its nucleus. Interpret the resulting relation between E2 and E 2, at the same observation point.
3.36. Calculate the condition at that a particle of mass m, moving in the field of a very thin spherically-symmetric shell, with
U r W r R,
and W < 0, has at least one localized (“bound”) stationary state.
99 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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3.37. 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.)
3.38. 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 for this geometry.
(ii) Spell out the contribution 0 to the total cross-section , due to 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.
3.39. 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 U 0. Analyze the result in detail, and give an interpretation of its most remarkable features.
3.40. 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 side a, and
(ii) a cube with side a,
with hard walls. For the square, repeat the calculations, using a 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 1.15, or start from reading about the finite difference method.100 Also: try to exploit the symmetry of the systems.
100 See, e.g., CM Sec. 8.5 or EM Sec. 2.11.
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intentionally left
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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 allows 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 such 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 simple 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 quark flavors or lepton numbers. 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 electrons 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 that both B z and d B z/ dz are not equal to zero. The experiment shows that 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 electron has an intrinsic, permanent magnetic dipole moment m. Indeed, classical electrodynamics2 tells us that the potential energy U of a magnetic dipole in an external magnetic field B is equal to (–m ꞏ B), so that the force acting on the electron, F U m B ,
(4.3)
has a non-zero vertical component
F
B
B .
(4.4)
z
mz z m
z
z
z
z
Hence if we further assume that electron’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 original 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 3
e
23 J
9274
.
0
10
.
(4.5)
Bohr
B
2 m
T
magneton
e
1 To the best of my knowledge, the concept of spin as a measure of the internal rotation of a particle was first suggested by Ralph Kronig, then a 20-year-old student, in January 1925, a few months before two other students, G. Uhlenbeck and S. Goudsmit – to whom the idea is usually attributed. The concept was then accepted (rather reluctantly) and developed quantitatively by Wolfgang Pauli.
2 See, e.g., EM Sec. 5.4, in particular Eq. (5.100).
3 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 electron, 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 multi-stage Stern-Gerlach experiments shown in Fig. 2. In the first of the experiments, the electron 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 electrons 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 multi-stage
Stern-Gerlach experiments.
The boxes SG (…) denote
100% magnets similar to 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 electrons 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, electrons somehow interact with the magnetic field so that the x-
polarized (non-absorbed) 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 a 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.4 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, a quantum state as such is not a mathematical object,5 and can participate in mathematical formulas only as a “label” – e.g., the index of the wavefunction . On the other hand, such 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 bra-vector ,6
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 its possible bra- and ket-vectors of a given system.7 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 rule corollaries. While there is a general consensus among physicists what the corollaries are, there are many possible ways to carve from them the basic postulate sets. Just as in Sec. 1.2, I will not try too hard to beat the number of the postulates to the smallest possible 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
4 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.
5 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”.
6 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.
7 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.8
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 evident from Eq. (6), the complex 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 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 ,9 very similar (though not identical) to that between a wavefunction and its complex conjugate *. 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 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 8 One may express the same statement by saying that the vector belongs to the same Hilbert space as all j.
9 Mathematicians like to say that the ket- and bra-vectors of the same quantum system are defined in two isomorphic Hilbert spaces.
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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 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” of one state from another: if is a possible ket of the system, and A îs a legitimate10
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 clarification of the notion of the Hilbert space: for the given set of linear operators of a system, its Hilbert state 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 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 10 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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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 that particle.
As the adjective “linear” in the operator definition implies, the main rules 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.) 11
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, not including 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
This last 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: 11 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 that only real c-numbers may be considered as a particular type of the Hermitian operators (22).
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ˆ
A ˆ A
†
ˆ
*
A ˆ
A *
†
,
(4.24)
which is most frequently rewritten as
Long
bracket:
ˆ A * †
ˆ A .
(4.25) 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 to 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 the rule (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
A ˆ B ˆ
, A ˆ B ˆ B ˆ A ˆ ,
(4.33) Commutator
Chapter 4
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is a non-trivial and very useful notion. Another similar notion is the anticommutator:12
Anti-
A ˆ B ˆ
, A ˆ B ˆ B ˆ A ˆ .
(4.34)
commutator
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, 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 first of these terms means that any possible state vector of the system (i.e. 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 that, in particular, if is one of the basis states, say uj’, then j = jj’. The second 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 eigenfunctions 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.
12 Another popular notation for the anticommutator (34) is A ˆ B ˆ
, ; it will not be used in these notes.
Chapter 4
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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 used on numerous occasions 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
vector of the type (37), singles out just one of its components, for example,
ˆ u u u ,
(4.43)
j
j
j
j
j
i.e. “kills” all components of the linear superposition but one. In the geometric analogy, such operator
“projects” the state vector on the j th “direction”, hence its name – the projection operator. Probably, the most important property of the projection operators, called the closure (or “completeness”) relation, immediately follows from Eq. (41): their sum over the full basis is equivalent to the identity operator
u u
I ˆ
.
(4.44) Closure
j
j
relation
j
This means in particular that we may insert the left-hand side of Eq. (44), for any basis, into any bra-ket relation, at any place – the trick that we will use again and again.
Now let us see how the expansions (37) transform the key notions introduced in the last section, starting from the short bracket (11), i.e. the inner product of two state vectors:
*
*
*
u u .
(4.45)
j
j
j'
j'
j
j'
jj'
j
j
j, j'
j, j'
j
Besides the complex conjugation, this expression is similar to the scalar product of the usual, geometric vectors. Now, let us explore the long bracket (23):
A ˆ * u A ˆ u * A .
(4.46)
j
j
j'
j'
j
jj'
j'
j, j'
j, j'
Here, the last step uses the very important notion of matrix elements of the operator, defined as Chapter 4
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Operator’s
matrix
A
u A ˆ
u .
(4.47)
jj'
j
j'
elements
As evident from Eq. (46), the full set of the matrix elements completely characterizes the operator, just as the full set of the expansion coefficients (40) fully characterizes a quantum state. The term “matrix”
means, first of all, that it is convenient to represent the full set of Ajj’ as a square table ( matrix), with the linear dimension equal to the number of basis states uj of the system under the consideration. By the way, this number (which may be infinite) is called the dimensionality of its Hilbert space.
As two simplest examples, all matrix elements of the null-operator, defined by Eqs. (35), are evidently equal to zero (in any basis), and hence it may be represented as a matrix of zeros (called the null-matrix):
0 0 ...
Null
0 0 0 ... ,
(4.48)
matrix
... ... ...
while for the identity operator I ˆ , defined by Eqs. (36), we readily get
ˆ
I u I u
u u
,
(4.49)
jj'
j
j'
j
j'
jj'
i.e. its matrix (naturally called the identity matrix) is diagonal – also in any basis:
1 0 ...
Identity
I 0 1 ... .
(4.50)
matrix
... ... ...
The convenience of the matrix language extends well beyond the representation of particular operators. For example, let us use the definition (47) to calculate matrix elements of a product of two operators:
ˆ ˆ
( AB)
u
B
A u
.
(4.51)
jj"
j
j"
Here we may use Eq. (44) for the first (but not the last!) time, inserting the identity operator between the two operators, and then expressing it via a sum of projection operators:
Matrix
element
of an
( AB)
ˆ ˆ
u AB u
ˆ ˆˆ
