1
(
(0) .
n
n
n" 1
For each n’ = 1, 2, … N, this is a system of N linear, homogenous equations (with N terms each) for N
unknown coefficients n” (0) n’ . In this problem, we may readily recognize the problem of diagonalization of the perturbation matrix H(1) – cf. Sec. 4.4 and in particular Eq. (4.101). As in the general case, the condition of self-consistency of the system is:
)
1
(
1
)
1
(
H
E
H
...
11
n
12
Initially
degenerate
)
1
(
)
1
(
1
H
H
E
... 0 ,
(6.26)
21
22
n
system:
energy levels
...
...
...
where now the index n numbers the N roots of this equation, in an arbitrary order. According to the definition (25) of E (1)
(1)
n
, the resulting N energy levels En may be found as E(0) + En . If the perturbation matrix is diagonal in the chosen basis n(0), the result is extremely simple,
(0)
)
1
(
)
1
(
E E
E H ,
(6.27)
n
n
nn
4 This is exactly the reason why such simple perturbation approach runs into serious problems for systems with a continuous spectrum, and other techniques (such as the WKB approximation) are often necessary.
5 Note that here the choice of the basis is to some extent arbitrary, because due to the linearity of equations of quantum mechanics, any linear combination of the states n” (0) is also an eigenstate of the unperturbed Hamiltonian. However, for using Eq. (25), these combinations have to be orthonormal, as was supposed at the derivation of Eq. (7).
Chapter 6
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and formally coincides with Eq. (14) for the non-degenerate case, but now it may give a different result for each of N previously degenerate states n.
Now let us see what this general theory gives for several important examples. First of all, let us consider a system with just two degenerate states with energy sufficiently far from all other levels. Then, in the basis of these two degenerate states, the most general perturbation matrix is
H
H
)
1
(
H
11
12
(6.28)
H
H
21
22
This matrix coincides with the general matrix (5.2) of a two-level system. Hence, we come to the very important conclusion: for a weak perturbation, all properties of any double-degenerate system are identical to those of the genuine two-level systems, which were the subject of numerous discussions in Chapter 4 and again in Sec. 5.1. In particular, its eigenenergies are given by Eq. (5.6), and may be described by the level-anticrossing diagram shown in Fig. 5.1.
6.2. The linear Stark effect
As a more involved example of the level degeneracy lifting by a perturbation, let us discuss the Stark effect 6 – the atomic level splitting by an external electric field. Let us study this effect, in the linear approximation, for a hydrogen-like atom/ion. Taking the direction of the external electric field E (which is practically always uniform on the atomic scale) for the z-axis, the perturbation may be represented by the following Hamiltonian:
ˆ )1
(
H
ˆ z
F q Eˆ z q r
E cos .
(6.29)
(In the last form, the operator sign is dropped, because we will work in the coordinate representation.) As you (should :-) remember, energy levels of a hydrogen-like atom/ion depend only on the principal quantum number n – see Eq. (3.201); hence all the states, besides the ground 1 s state in which n = 1 and l = m = 0, have some orbital degeneracy, which grows rapidly with n. Let us consider the lowest degenerate level with n = 2. Since, according to Eq. (3.203), 0 l n –1, at this level the orbital quantum number l may equal either 0 (one 2 s state, with m = 0) or 1 (three 2 p states, with m = 0, 1).
Due to this 4-fold degeneracy, H(1) is a 44 matrix with 16 elements:
l 0
l
1
m
0 m
0 m
1 m 1
H
H
H
H
m ,
0
l ,
0
11
12
13
14
H
H
H
H
m ,
0
(6.30)
H )1
(
21
22
23
24
H
H
H
H
m ,
1 l .
1
31
32
33
34
H
H
H
H
m ,
1
41
42
43
44
6 This effect was discovered experimentally in 1913 by Johannes Stark and independently by Antonio Lo Surdo, so it is sometimes (and more fairly) called the “Stark – Lo Surdo effect”. Sometimes this name is used with the qualifier “dc” to distinguish it from the ac Stark effect – the energy level shift under the effect of an ac field – see Sec. 5 below.
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However, there is no need to be scared. First, due to the Hermitian nature of the operator, only ten of these matrix elements (four diagonal and six off-diagonal ones) may be substantially different from each other. Moreover, due to a high symmetry of the problem, there are a lot of zeros even among these elements. Indeed, let us have a look at the angular components Y m l of the corresponding
wavefunctions, with l = 0 and l = 1, described by Eqs. (3.174)-(3.175). For the states with m = 1, the azimuthal parts of wavefunctions are proportional to exp{ i}; hence the off-diagonal elements H 34 and H 43 of the matrix (30), relating these functions, are proportional to
2
*
* ˆ
Ω
)
1
(
i
d Y H Y
d e
i
e
.
0
(6.31)
1
1
0
The azimuthal-angle symmetry also kills the off-diagonal elements H 13, H 14, H 23, H 24 (and hence their complex conjugates H 31, H 41, H 32, and H 42), because they relate states with m = 0 and m = 1, and hence are proportional to
2
0* ˆ
Ω
)
1
(
1
d Y H Y
i
d e
.
0
(6.32)
1
1
0
For the diagonal matrix elements H 33 and H 44, corresponding to l = 1 and m = 1, the azimuthal-angle integrals do not vanish, but since the corresponding spherical harmonics depend on the polar angle as sin, these elements are proportional to
1
1
* ˆ )1
(
1
d Y
H Y sin d sin cos sin cos
d
(6.33)
1
1
1 cos2 (cos ),
0
1
and hence are equal to zero – as any limit-symmetric integral of an odd function. Finally, for the states 2 s and 2 p with m = 0, the diagonal elements H 11 and H 22 are also killed by the polar-angle integration:
1
0* ˆ )
1
(
0
d Y H Y sin d cos cos d(cos ) 0
,
(6.34)
0
0
0
1
1
1* ˆ )
1
(
1
d Y H Y sin d cos3 cos3 d(cos ) 0.
(6.35)
0
0
0
1
Hence, the only non-zero elements of the matrix (30) are two off-diagonal elements H 12 and H 21, which relate two states with the same m = 0, but different l = {0, 1}, because they are proportional to
0*
0
3 2
2
1
d Y
cos Y
d sin d cos
0.
(6.36)
0
1
4 0
0
3
What remains is to use Eqs. (3.209) for the radial parts of these functions to complete the calculation of those two matrix elements:
H
q
H
E r drR ( r) rR ( r).
(6.37)
12
2
21
2,0
2 1
,
3 0
Due to the additive structure of the function R 2,0( r), the integral falls into a sum of two table integrals, both of the type MA Eq. (6.7d), finally giving
H H 3 q r
E ,
(6.38)
12
21
0
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where r 0 is the spatial scale (3.192); for the hydrogen atom, it is just the Bohr radius r B – see Eq. (1.10).
Thus, the perturbation matrix (30) is reduced to
0
3 q r
E
0 0
0
3 q r
E
0
0 0
H )1
(
0
,
(6.39)
0
0
0 0
0
0
0 0
so that the condition (26) of self-consistency of the system (25),
1
E
3 q r
E
0
0
2
0
1
3 q r
E
E
0
0
0
2
(6.40)
1
,
0
0
0
E
0
2
1
0
0
0
E 2
gives a very simple characteristic equation
E
E
q r
E
.
(6.41)
2 2
1
2
1
2
3
2
0
0
with four roots:
Linear
)1
(
E
E
q r
(6.42) Stark
2
,
0
,
1 2
)1(2
3 E .
3,4
0
effect
for n = 2
so that the degeneracy is only partly lifted – see the levels in Fig. 3.7
1
2 s 2 p
2
m 0
3 q E r 0
(0)
E
m 1
2
3 q E r 0
m 0
Fig. 6.3. The linear Stark effect for the
1
level n = 2 of a hydrogen-like atom.
2 s 2 p
2
Generally, in order to understand the nature of states corresponding to these levels, we should return to Eq. (25) with each calculated value of E (1)
2
, and find the corresponding expansion coefficients
n” (0) n’ that describe the perturbed states. However, in our simple case, the outcome of this procedure is clear in advance. Indeed, since the states with { l = 1, m = 1} are not affected by the perturbation at all (in the linear approximation in the electric field), their degeneracy is not lifted, and energy is not affected – see the middle line in Fig. 3. On the other hand, the partial perturbation matrix connecting the states 2 s and 2 p, i.e. the top left 22 part of the full matrix (39), is proportional to the Pauli matrix x, and we already know the result of its diagonalization – see Eqs. (4.113)-(4.114). This means that the 7 The proportionality of this splitting to the small field is responsible for the qualifier “linear” in the name of this effect. If observable effects grow only as 2
E (see, e.g., Problem 9), the term quadratic Stark effect is used instead.
Chapter 6
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upper and lower split levels correspond to very simple linear combinations of the previously degenerate states with m = 0,
1
2 s 2 p .
(6.43)
2
Finally, let us estimate the magnitude of the linear Stark effect for a hydrogen atom. For a very high electric field of E = 3106 V/m,8 q = e 1.610-19 C, and r 0 = r B 0.510-10 m, we get a level splitting of 3 q E r 0 0.810-22 J 0.5 meV. This number is much lower than the unperturbed energy of the level, E 2 = – E H/(222) –3.4 eV, so that the perturbative result is quite applicable. On the other hand, the calculated splitting is much larger than the resolution limit imposed by the line’s natural width (~10-7 E 2, see Chapter 9), so that the effect is quite observable even in substantially lower electric fields.
Note, however, that our simple results are quantitatively correct only when the Stark splitting (42) is much larger that the fine-structure splitting of the same level in the absence of the field– see the next section.
6.3. Fine structure of atomic levels
Now let us use the same perturbation theory to analyze, also for the simplest case of a hydrogen-like atom/ion, the so-called fine structure of atomic levels – their degeneracy lifting even in the absence of external fields. Since the effective speed v of the electron motion in atoms is much smaller than the speed of light c, the fine structure may be analyzed as a sum of two independent relativistic effects. To analyze the first of them, let us expand the well-known classical relativistic expression9 for the kinetic energy T = E – m c 2 of a free particle with the rest mass m,10
1/ 2
2
1/ 2
p
T 2 4
m c 2 2
p c
2
m c
2
c
m 1
1 ,
(6.44)
2 2
m c
into the Taylor series with respect to the small ratio ( p/ mc)2 ( v/ c)2:
1
2
p
p
2
1
4
2
4
p
p
T c
m
1
... 1
...,
(6.45)
2 m c
8
m c
2m 8 3 2
m c
and drop all the terms besides the two spelled-out terms. Of them, the first term is non-relativistic, while the second one represents the first relativistic correction to T.
Following the correspondence principle, the quantum-mechanical problem in this approximation may be described by the perturbative Hamiltonian (1), whose unperturbed part (whose eigenstates and eigenenergies were discussed in Sec. 3.5) is
(0
p ˆ2
)
C
H ˆ
U ˆ ( r
U ˆ
),
( r)
,
(6.46)
2m
r
while the kinetic-relativistic perturbation
8 This value approximately corresponds to the threshold of electric breakdown in air at ambient conditions, due to the impact ionization. As a result, experiments with higher dc fields are rather difficult.
9 See, e.g., EM Eq. (9.78) – or any undergraduate text on special relativity.
10 This fancy font is used, as in Secs. 3.5-3.8, to distinguish the mass m from the magnetic quantum number m.
Chapter 6
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2
4
2
Kinetic-
ˆ p
1 ˆ p
)
1
(
ˆ
H
.
(6.47) relativistic
3 2
2
8m c
2
m c
2
m
perturbation
Using Eq. (46), we may rewrite the last formula as
1
)
1
(
ˆ
ˆ
ˆ
H
H
U ( r) ,
(6.48)
2
(0
2
)
2 c
m
so that its matrix elements participating in the characteristic equation (25) for a given degenerate energy level (3.201), i.e. a given principal quantum number n, are
)
1
(
1
nlm H ˆ
nl'm'
nlm
,
(6.49)
2
H ˆ(0) U ˆ( r H ˆ
)
(0)
U ˆ ( r) nl'm'
2 c
m
where the bra- and ket-vectors describe the unperturbed eigenstates, whose eigenfunctions (in the coordinate representation) are given by Eq. (3.200):
m
n,l,m = Rn,l( r) Yl (,).
It is straightforward (and hence left for the reader :-) to prove that all off-diagonal elements of the set (49) are equal to 0. Thus we may use Eq. (27) for each set of the quantum numbers{ n, l, m}:
)
1
(
(0)
)
1
(
1
ˆ
E
E
E
nlm H
nlm
H
U r
n, l, m
n, l, m
n
2
ˆ (0 ˆ 2
)
( )
2 c
m
n, l, m
(6.50)
1
E
E
2
2
1
2
0
0
1
2
1
ˆ
ˆ
E 2 E U
U
C
C
n
n
,
2
2
c
n, l
n l
m
2
2
,
c
m
4 4
2
2
n
n
r
r
n, l
n, l
where the index m has been dropped, because the radial wavefunctions Rn,l( r), which affect these expectation values, do not depend on that quantum number. Now using Eqs. (3.191), (3.201) and the first two of Eqs. (3.211), we finally get
Kinetic-
2
m C
n
E
)
1
(
3
2 2 n n
3
E
.
(6.51) relativistic
n, l
energy
2 2 2 4
c n l ½ 4
2
m c l ½
4
correction
Let us discuss this result. First of all, its last form confirms that the correction (51) is indeed much smaller than the unperturbed energy En (and hence the perturbation theory is valid) if the latter is much smaller than the relativistic rest energy m c 2 of the particle – as it is for the hydrogen atom. Next, since in the Bohr problem’s solution n l + 1, the first fraction in the parentheses of Eq. (51) is always larger than 1, and hence than 3/4, so that the kinetic relativistic correction to energy is negative for all n and l. (Actually, this fact could be predicted already from Eq. (47), which shows that the perturbation’s Hamiltonian is a negatively defined form.) Finally, for a fixed principal number n, the negative correction’s magnitude decreases with the growth of l. This fact may be interpreted using the second of Eqs. (3.211): the larger is l (at fixed n), the larger is the particle’s effective distance from the center, and hence the smaller is its effective velocity, i.e. the smaller is the magnitude of the quantum-mechanical average of the negative relativistic correction (47) to the kinetic energy.
The result (51) is valid for the Coulomb interaction U( r) = – C/ r of any physical nature. However, if we speak specifically about hydrogen-like atoms/ions, there is also another relativistic correction to energy, due to the so-called spin-orbit interaction (alternatively called the “spin-orbit coupling”). Its physics may be understood from the following semi-quantitative classical reasoning: from the “the point of view” of an electron rotating about the nucleus at distance r with velocity v, it is the nucleus, of the Chapter 6
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electric charge Ze, that rotates about the electron with the velocity (-v) and hence the time period T =
2 r/ v. From the point of view of magnetostatics, such circular motion of the electric charge Q = Ze , is equivalent to a circular dc electric current I = Q/ T = ( Ze)( v/2 r), which creates, at the electron’s location, i.e. in the center of the current loop, the magnetic field with the following magnitude:11
Zev Zev
0
0
0
B
I
.
(6.52)
a
2
2 r
2 r 2 r
4 r
The field’s direction n is perpendicular to the apparent plane of the nucleus’ rotation (i.e. that of the real rotation of the electron), and hence its vector may be readily expressed via the similarly directed vector L = m e vrn of the electron’s angular (orbital) momentum:
Zev
Ze
Ze
Ze
0
B
n
0
m
n
0
vr
L
L ,
(6.53)
a
2
3
e
3
3
2
4 r
4 r
m
4 r
m
4 r m c
e
e
0
e
where the last step used the basic relation between the SI-unit constants : 0 1/ c 20.
A more careful (but still classical) analysis of the problem12 brings both good and bad news. The bad news is that the result (53) is wrong by the so-called Thomas factor of two even for the circular motion, because the electron moves with acceleration, and the reference frame bound to it cannot be inertial (as was implied in the above reasoning), so that the effective magnetic field felt by the electron is actually
Ze
B
L .
(6.54)
3
2
8 r m c
0
e
The good news is that this result is valid not only for circular but an arbitrary orbital motion in the Coulomb field U( r). Hence from the discussion in Sec. 4.1 and Sec. 4.4 we may expect that the quantum-mechanical description of the interaction between this effective magnetic field and the electron’s spin moment (4.115) is given by the following perturbation Hamiltonian13
Ze
)
1
(
1
2
Ze
1
ˆ
H
m
ˆ ˆ
B Sˆ
Lˆ
Sˆ Lˆ ,
(6.55)
e
8
3
2
r m c
2 2 2
m c 4
3
r
0
e
e
0
where at spelling out the electron’s gyromagnetic ratio e – g e e/2 m e, the small correction to the value g e
= 2 of the electron’s g-factor (see Sec. 4.4) is ignored, because Eq. (55) is already a small correction.
This expectation is confirmed by the fully-relativistic Dirac theory, to be discussed in Sec. 9.7 below: it yields, for an arbitrary central potential U( r), the following spin-orbit coupling Hamiltonian: Spin-
)
1
(
1
1 dU ( r)
orbit
ˆ
H
Sˆ Lˆ .
(6.56)
coupling
2 2 2
m c r
dr
e
For the Coulomb potential U( r) = – Ze 2/40 r, this formula is reduced to Eq. (55).
11 See, e.g., EM Sec. 5.1, in particular, Eq. (5.24). Note that such effective magnetic field is induced by any motion of electrons, in particular that in solids, leading to a variety of spin-orbit effects there – see, e.g., a concise review by R. Winkler et al., in B. Kramer (ed.), Advances in Solid State Physics 41, 211 (2001).
12 It was carried out first by Llewellyn Thomas in 1926; for a simple review see, e.g., R. Harr and L. Curtis, Am.
J. Phys. 55, 1044 (1987).
13 In the Gaussian units, Eq. (55) is valid without the factor 40 in the denominator; while Eq. (56), “as is”.
Chapter 6
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As we already know from the discussion in Sec. 5.7, the angular factor of this Hamiltonian commutes with all the operators of the coupled-representation group (inside the blue line in Fig. 5.12): 2
ˆ L , 2
ˆ S , 2
ˆ J , and J ˆ , and hence is diagonal in the coupled-representation basis with definite quantum z
numbers l, j, and mj (and of course s = ½). Hence, using Eq. (5.181) to rewrite Eq. (56) as 2
1
Ze
1 1
)
1
(
ˆ
ˆ
ˆ
ˆ
H
J L S ,
(6.57)
2 2
3
2 2 2
2 m c 4 r 2
e
0
we may again use Eq. (27) for each set { s, l, j, mj}, with common n:
)
1
(
1
Ze 2
1
1
E
J 2
ˆ L 2ˆ S 2
ˆ
,
(6.58)
n, j, l
2 m 2 c 2 4
r 3
2
j, s
e
0
n l,
where the indices irrelevant for each particular factor have been dropped. Now using the last of Eqs.
(3.211), and similar expressions (5.169), (5.175), and (5.177) for eigenvalues of the involved operators, we get an explicit expression for the spin-orbit corrections14
Spin-
Ze
j j l l
E
orbit
n
j j l l
)
1
(
1
2
2
(
)
1
(
)
1
¾
2
(
)
1
(
)
1
¾
E
n
,
(6.59)
n, j, l
energy
2 2 2
m c 4 2 3
3
r
n l( l ½)( l )
1
2
m c
l( l ½)( l )
1
correction
e
0
0
e
with l and j related by Eq. (5.189): j = l ½.
The last form of its result shows clearly that this correction has the same scale as the kinetic correction (51).15 In the 1st order of the perturbation theory, they may be just added (with m = m e), giving a surprisingly simple formula for the net fine structure of the n th energy level: 2
E
Fine
)
1
(
n
4 n
E
.
(6.60) structure
fine
3
2
2
m c
j
of atomic
e
½
levels
This simplicity, as well as the independence of the result of the orbital quantum number l, will become less surprising when (in Sec. 9.7) we see that this formula follows in one shot from the Dirac theory, in which the Bohr atom’s energy spectrum in numbered only with n and j, but not l. Let us recall that for an electron ( s = ½), according to Eq. (5.189) with 0 l n – 1, the quantum number j may take n positive half-integer values, from ½ to n – ½. Hence, Eq. (60) shows that the fine structure of the n th Bohr’s energy level has n sub-levels – see Fig. 4.
Please note that according to Eq. (5.175), each of these sub-levels is still (2 j + 1)-times degenerate in the quantum number mj. This degeneracy is very natural, because in the absence of an external field the system is still isotropic. Moreover, on each fine-structure level (besides the highest one with j = n – ½), each of the mj-states is doubly-degenerate in the orbital quantum number l = j ½ – see the labels of l in Fig. 4. (According to Eq. (5.190), each of these states, with fixed j and mj, may be 14 The factor l in the denominator does not give a divergence at l = 0, because in this case j = s = ½, so that j( j +
1) = ¾, and the numerator turns into 0 as well. A careful analysis of this case (which may be found, e.g., in G.
Woolgate, Elementary Atomic Structure, 2nd ed., Oxford, 1983), as well as the exact analysis of the hydrogen atom using the Dirac theory (see Sec. 9.7), show that Eq. (60), which does not include l, is valid even in this case.
15 This is natural, because the magnetic interaction of charged particles is essentially a relativistic effect, of the same order (~ v 2/ c 2) as the kinetic correction (47) – see, e.g., EM Sec. 5.1, in particular Eq. (5.3).
Chapter 6
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represented as a linear combination of two states with adjacent values of l, and hence different electron spin orientations, ms = ½, weighed with the Clebsch-Gordan coefficients.)
En
j n ½
l n 1
...
...
j
5 / 2
l ,
2 3
j 3/ 2
l ,
1 2
Fig. 6.4. The fine structure of a
j
hydrogen-like atom’s level.
½
l ,
0 1
These details aside, one may crudely say that the relativistic corrections combined make the total eigenenergy grow with l, contributing to the effect already mentioned at our analysis of the periodic table of elements in Sec. 3.7. The relative scale of this increase may be scaled by the largest deviation from the unperturbed energy En, reached for s-states (with l = 0, j = ½):
)
1
(
2
E max
En
3
2
Ze
1
3
2
2 1
3
2 n
Z
.
(6.61)
2
E
m c
2
4 c
n
4 n
n
4 n
n
e
0
2
2
where is the fine-structure (“Sommerfeld’s”) constant,
2
e
1
,
(6.62)
4 c 137
0
(already mentioned in Sec. 4.4), which characterizes the relative strength (or rather weakness :-) of the electromagnetic effects in quantum mechanics – which in particular makes the perturbative quantum electrodynamics possible.16 These expressions show that the fine-structure splitting is a very small effect (~2 ~ 10-6) for the hydrogen atom, but it rapidly grows (as Z 2) with the nuclear charge (i.e. the atomic number) Z, and becomes rather substantial for the heaviest stable atoms with Z ~ 102.
6.4. The Zeeman effect
Now, we are ready to review the Zeeman effect – the atomic level splitting by an external magnetic field.17 Using Eq. (3.26), with q = – e, for the description of the electron’s orbital motion in the field, and the Pauli Hamiltonian (4.163), with = – e/ m e, for the electron spin’s interaction with the field, we see that even for a hydrogen-like (i.e. single-electron) atom/ion, neglecting the relativistic effects, the full Hamiltonian is rather involved:
1
2
2
Ze
e
ˆ
H
ˆ ˆ
p A
e
ˆ
B .
S
(6.63)
2 m
4 r
m
e
0
e
16 The expression 2 = E H/ m e c 2, where E H is the Hartree energy (1.13), i.e. the scale of energies En, is also very revealing.
17 It was discovered experimentally in 1896 by Pieter Zeeman who, amazingly, was fired from the University of Leiden for unauthorized use of lab equipment for this work – just to receive a Nobel Prize for it in a few years!
Chapter 6
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There are several simplifications we may make. First, let us assume that the external field is spatial-uniform on the atomic scale (which is a very good approximation for most cases), so that we can take its vector potential in an axially-symmetric gauge – cf. Eq. (3.132):
1
A B .
r
(6.64)
2
Second, let us neglect the terms proportional to B2, which are small in practical magnetic fields of the order of a few teslas.18 The remaining term in the effective kinetic energy, describing the interaction with the magnetic field, is linear in the momentum operator, so that we may repeat the standard classical calculation19 to reduce it to the product of B by the orbital magnetic moment’s component mz = –
eLz/2 m e – besides that both mz and Lz should be understood as operators now. As a result, the Hamiltonian (63) reduces to Eq. (1), ˆ (0)
ˆ )1
(
H
H , where (0)
ˆ
H is that of the atom at B = 0, and
e
ˆ )1
(
ˆ
ˆ
H
B ( L 2 S ).
(6.65) Zeeman
2
z
z
m
effect’s
e
perturbation
This expression immediately reveals the major complication with the Zeeman effect’s analysis.
Namely, in comparison with the equal orbital and spin contributions to the total angular momentum (5.170) of the electron, its spin produces a twice larger contribution to the magnetic moment, so that the right-hand side of Eq. (65) is not proportional to the total angular moment J. As a result, the effect’s description is simple only in two limits.
If the magnetic field is so high that its effects are much stronger than the relativistic (fine-structure) effects discussed in the previous section, we may treat the two terms in Eq. (65) as independent perturbations of different (orbital and spin) degrees of freedom. Since each of the perturbation matrices is diagonal in its own z-basis, we can again use Eq. (27) to write e B
e
Paschen-
(0)
E E
ˆ
ˆ
,
n l, m L
,
n l, m 2 m S m
B
m
m
B m
(6.66) Back
l
z
l
s
z
s
2
l
s
(
).
1
2 m
2
B
l
m
effect
e
e
This result describes splitting of each 2(2 l + 1)-degenerate energy level, with certain n and l, into (2 l
+3) levels (Fig. 5), with the adjacent level distance of BB , of the order of 10-4 eV per tesla.
. . .
m ,
2 m 1/ 2
l
s
m ,
0 m 1/ 2
l
s
BB
m ,
1 m 1/ 2
(0)
l
s
E
n, l
m ,
1 m 1/ 2
l
s
BB
. . .
m ,
0 m 1/ 2
l
s
Fig. 6.5. The Paschen-Back effect.
m ,2 m 1/2
l
s
18 Despite its smallness, the quadratic term is necessary for a description of the negative contribution of the orbital motion to the magnetic susceptibility m (the so-called orbital diamagnetism, see EM Sec. 5.5), whose analysis, using Eq. (63), is left for the reader’s exercise.
19 See, e.g., EM Sec. 5.4, in particular Eqs. (5.95) and (5.100).
Chapter 6
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Note that all the levels, these besides the top and bottom ones, remain doubly degenerate. This limit of the Zeeman effect is sometimes called the Paschen-Back effect – whose simplicity was recognized only in the 1920s, due to the need in very high magnetic fields for its observation.
In the opposite limit of relatively low magnetic fields, the Zeeman effect takes place on the background of the much larger fine-structure splitting. As was discussed in Sec. 3, at B = 0 each split sub-level has a 2(2 j + 1)-fold degeneracy corresponding to (2 j + 1) different values of the half-integer quantum number mj, ranging from – j to + j, and two values of the integer l = j ½ – see Fig. 4.20 The magnetic field lifts this degeneracy. Indeed, in the coupled representation discussed in Sec. 5.7, the perturbation (65) is described by the matrix with elements
)
1
(
e B
ˆ
ˆ
e B
ˆ
ˆ
H
j, m L 2 S j' , m
j, m J S j' , m
2
j
z
z
j'
m
2
j
z
z
j'
m
e
(6.67)
e
B
e
ˆ
m
j, m S j' , m
j
m m
j
z
j' .
2
j
j'
m e
To spell out the second term, let us use the general expansion (5.183) for the particular case s =
½, when (as was discussed in the end of Sec. 5.7) it has at most two non-vanishing terms, with the Clebsh-Gordan coefficients (5.190):
j l ½, m j
l m ½ 1/2
l m
(6.68)
j
½ 1/2
j
m m ½, m ½
m m ½, m ½
.
2 l 1
l
j
s
2 l 1
l
j
s
Taking into account that the operator S ˆ gives non-zero brackets only for m z
s = ms’, the 22 matrix of
elements
ˆ
m m ½, m ½
is diagonal, so we may use Eq. (27) to get
l
j
S m m ½, m ½
s
z
l
j
s
e B
l m ½
j
l m ½
j
(0)
Anomalous
E E
m
Zeeman
2 m
j
2
2 l 1
2
2 l 1
effect
e
(6.69)
for s = 1/2
e B
1
1
m 1
B m 1
, for j m j,
2 m
j
2 l 1
B
j
2 l 1
j
e
where the two signs correspond to the two possible values of l = j
. . .
½ – see Fig. 6.
. . .
m 3 / 2
j
m 3 / 2
j
m 1/ 2
m
j
1/ 2
(0)
j
E
(0)
E
n, j
n, j
l j ½
m 1/ 2
l j ½
m 1/ 2
j
j
. . .
m 3 / 2
j
. . .
m 3 / 2
j
Fig. 6.6. The anomalous Zeeman effect in a hydrogen-like atom/ion.
20 In the almost-hydrogen-like, but more complex atoms (such as those of alkali metals), the degeneracy in l may be lifted by electron-electron Coulomb interaction even in the absence of external magnetic field.
Chapter 6
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We see that the magnetic field splits each sub-level of the fine structure, with a given l, into 2 j +
1 equidistant levels, with the distance between the levels depending on l. In the late 1890s, when the Zeeman effect was first observed, there was no notion of spin at all, so that this puzzling result was called the anomalous Zeeman effect. (In this terminology, the normal Zeeman effect is the one with no spin splitting, i.e. without the second terms in the parentheses of Eqs. (66), (67), and (69); it was first observed in 1898 by Preston Thomas in atoms with zero net spin.)
The strict quantum-mechanical analysis of the anomalous Zeeman effect for arbitrary s (which is important for applications to multi-electron atoms) is conceptually not complex, but requires explicit expressions for the corresponding Clebsch-Gordan coefficients, which are rather bulky. Let me just cite the unexpectedly simple result of this analysis:
Δ E
m
B
g,
(6.70a)
B
j
Anomalous
Zeeman
where g is the so-called Lande factor:21
effect
for arbitrary s
j( j )
1 s( s )
1 l( l )
1
g 1
.
(6.70b)
2 j( j )
1
For s = ½ (and hence j = l ½), this factor is reduced to the parentheses in the last forms of Eq. (69).
It is remarkable that Eqs. (70) may be readily derived using very plausible classical arguments, similar to those used in Sec. 5.7 – see Fig. 5.13 and its discussion. As was discussed in Sec. 5.6, in the absence of spin, the quantization of the observable Lz is an extension of the classical picture of the torque-induced precession of the vector L about the magnetic field’s direction, so that the interaction energy, proportional to B Lz = B L, remains constant – see Fig. 7a. On the other hand, at the spin-orbit interaction without an external magnetic field, the Hamiltonian function of the system includes the product SL, so that in the stationary state it has to be constant, together with J 2, L 2, and S 2. Hence, this system’s classical image is a joint precession of the vectors S and L about the direction of the vector J =
L + S, in such a manner that the spin-orbit interaction energy, proportional to the product LS, remains constant (Fig. 7b). On this backdrop, the anomalous Zeeman effect in a relatively weak magnetic field B
= Bn z corresponds to a much slower precession of the vector J about the z-axis, “dragging” with it the vectors L and S, rapidly rotating around it.
z
(a)
z
(b)
J
Lz
L
(L )
J z
L J
Fig. 6.7. Classical images of (a)
B
L
the orbital angular momentum’s
S
quantization in a magnetic field,
(S )
J z
and (b) the fine-structure level
S J
B L const
L S const
splitting.
0
0
21 This formula is frequently used with capital letters J, S, and L, which denote the quantum numbers of the atom as a whole.
Chapter 6
Page 16 of 36
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This physical picture allows us to conjecture that what is important for the slow precession rate are only the vectors L and S averaged over the period of their much faster precession around vector J –
in other words, only their components L J and S J along the vector J. Classically, these components may be calculated as
L J
S J
L
J,
and S
.
J
(6.71)
J
2
2
J
J
J
The scalar products participating in these expressions may be readily expressed via the squared lengths of the vectors, using the following geometric formulas:
2
S (J L)2
2
2
J L 2L J,
2
L (J S)2
2
2
J S 2J .
S
(6.72)
As a result, we get the following time average:
L J
S J
L
J
2 S L
S
J
J
L J
S J
z
z
2
J
J
z
2
z 2
2
2
2
J
J
J
z
(6.73)
( 2
2
2
J L S ) 2( 2
2
2
J S L )
2
2
2
J S L
J
J 1
.
z
2 2
z
J
2 2
J
The last move is to smuggle in some quantum mechanics by using, instead of the vector lengths squared and the z-component of Jz, their eigenvalues given by Eqs. (5.169), (5.175), and (5.177). As a result, we immediately arrive at the exact Eqs. (70). This coincidence encourages thinking about quantum mechanics of angular momenta in the classical terms of torque-induced precession, which turns out to be very fruitful in some more complex problems of atomic and molecular physics.
The high-field limit and low-field limits of the Zeeman effect, described respectively by Eqs.
(66) and (69), are separated by a medium field range, in which the Zeeman splitting is of the order of the fine-structure splitting analyzed in Sec. 3. There is no time in this course for a quantitative analysis of this crossover.22
6.5. Time-dependent perturbations
Now let us proceed to the case when the perturbation
)
1
(
ˆ
H in Eq. (1) is a function of time, while
(0)
ˆ
H is time-independent. The adequate perturbative approach to this problem, and its results, depend critically on the relation between the characteristic frequency of the perturbation and the distance between the initial system’s energy levels:
E E .
(6.74)
n
n'
In the case when all essential frequencies of a perturbation are very small in the sense of Eq.
(74), we are dealing with the so-called adiabatic change of parameters, that may be treated essentially as a time-independent perturbation – see the previous sections of this chapter). The most interesting observation here is that the adiabatic perturbation does not allow any significant transfer of system’s 22 For a more complete discussion of the Stark, Zeeman, and fine-structure effects in atoms, I can recommend, for example, either the monograph by G. Woolgate cited above, or the one by I. Sobelman, Theory of Atomic Spectra, Alpha Science, 2006.
Chapter 6
Page 17 of 36
QM: Quantum Mechanics
probability from one eigenstate to another. For example, in the WKB limit of the orbital motion, the Bohr quantization rule and its Wilson-Sommerfeld modification (2.110) guarantee that the integral
p dr ,
(6.75)
C
taken along the particle’s classical trajectory, is an adiabatic invariant, i.e. does not change at a slow change of system’s parameters. (It is curious that classical mechanics also guarantees the invariance of the integral (75), but its proof there23 is much harder than the quantum-mechanical derivation of this fact, carried out in Sec. 2.4.) This is why even if the perturbation becomes large with time (while changing sufficiently slowly), we can expect the classification of eigenstates and eigenvalues to persist.
Let us proceed to the harder case when both sides of Eq. (74) are comparable, using for this discussion the Schrödinger picture of quantum dynamics, given by Eq. (4.158). Combining it with Eq.
(1), we get the Schrödinger equation in the form
i
( t ˆ
)
(0)
)
1
(
H
H ( t) ( t) .
(6.76)
t
Very much in the spirit of our treatment of the time-independent case in Sec. 1, let us represent the time-dependent ket-vector of the system with its expansion,
( t) n n ( t) ,
(6.77)
n
over the full and orthonormal set of the unperturbed, stationary ket-vectors defined by equation H (0)
ˆ
n E n .
(6.78)
n
(Note that these kets n are exactly what was called n(0) in Sec. 1; we may afford a less bulky notation in this section, because only the lowest orders of the perturbation theory will be discussed.) Plugging the expansion (77), with n replaced with n’, into both sides of Eq. (76), and then inner-multiplying both its sides by the bra-vector n of another unperturbed (and hence time-independent) state of the system, we get the following set of linear, ordinary differential equations for the expansion coefficients: d
i
n ( t) E n ( t)
)
1
(
H
t n' t
(6.79)
n
( )
( ) ,
nn'
dt
n'
where the matrix elements of the perturbation, in the unperturbed state basis, defined similarly to Eq.
(8), are now functions of time:
H )1
(
t
( n H ˆ
)
)
1
(
t
( ) n' .
(6.80)
nn'
The set of differential equations (79), which are still exact, may be useful for numerical calculations.24 However, it has a certain technical inconvenience, which becomes clear if we consider its (evident) solution in the absence of perturbation:25
23 See, e.g., CM Sec. 10.2.
24 Even if the problem under analysis may be described by the wave-mechanics Schrödinger equation (1.25), a direct numerical integration of that partial differential equation is typically less convenient than that of the ordinary differential equations (79).
25 This is of course just a more general form of Eq. (1.62) of the wave mechanics of time-independent systems.
Chapter 6
Page 18 of 36
QM: Quantum Mechanics
En
n t
( ) n ( )
0 exp i
t.
(6.81)
We see that these solutions oscillate very fast, and their numerical modeling may represent a challenge for even the fastest computers. These spurious oscillations (whose frequency, in particular, depends on the energy reference level) may be partly tamed by looking for the general solution of Eqs. (79) in a form inspired by Eq. (81):
En
n t
( ) a t
( ) exp
.
(6.82)
n
i
t
Here an( t) are new functions of time (essentially, the stationary states’ probability amplitudes), which may be used, in particular, to calculate the time-dependent level occupancies, i.e. the probabilities Wn to find the perturbed system on the corresponding energy levels of the unperturbed system: 2
W ( t) n ( t)
a
.
(6.83)
n
n t 2
Plugging Eq. (82) into Eq. (79), for these functions we readily get a slightly modified system of equations:
Probability
amplitudes:
i a
)
1
(
i
t
a H
t
( ) e
,
(6.84)
n
nn'
n'
nn'
evolution
n'
where the factors nn’, defined by the relation
Quantum
transition
frequencies
E E ,
(6.85)
nn'
n
n'
have the physical sense of frequencies of potential quantum transitions between the n th and n’ th energy levels of the unperturbed system. (The conditions when such transitions indeed take place will be clear soon.) The advantages of Eq. (84) over Eq. (79), for both analytical and numerical calculations, is their independence of the energy reference, and lower frequencies of oscillations of the right-hand side terms, especially when the energy levels of interest are close to each other.26
In order to continue our analytical treatment, let us focus on a particular but very important problem of a sinusoidal perturbation turned on at some moment – which may be taken for t = 0: Turning on
,
0
for
t ,
0
sinusoidal
ˆ )1
(
H ( t)
(6.86)
perturbation
ˆ i t
ˆ†
e
A
A e i t for
,
t ,
0
where the perturbation amplitude operators A ând †
ˆ A ,27 and hence their matrix elements,
26 Note that the relation of Eq. (84) to the initial Eq. (79) is very close to the relation of the interaction picture of quantum dynamics, discussed at the end of Sec. 4.6, to its Schrödinger picture, with the perturbation Hamiltonian playing the role of the interaction one – compare Eqs. (1) and Eq. (4.206). Indeed, Eq. (84) could be readily obtained from the interaction picture, and I did not do this just to avoid using this heavy bra-ket artillery for our current (relatively) simple problem, and hence to keep its physics more transparent.
27 The notation of the amplitude operators in Eq. (86) is justified by the fact that the perturbation Hamiltonian has to be self-adjoint (Hermitian), and hence each term on the right-hand side of that relation has to be a Hermitian conjugate of its counterpart, which is evidently true only if the amplitude operators are also the Hermitian conjugates of each other. Note, however, that each of these amplitude operators is generally not Hermitian.
Chapter 6
Page 19 of 36
QM: Quantum Mechanics
ˆ
ˆ
n A n' A ,
†
*
n A n' A ,
(6.87)
nn'
n'n
are time-independent after the turn-on moment. In this case, Eq. (84) yields
i t
i
(
)
t
nn
*
(
)
'
nn'
i a
.
(6.88)
n
a A e
A e
,
for t 0
n'
nn'
n'
n
n'
This is, generally, still a nontrivial system of coupled differential equations; however, it allows simple and explicit solutions in two very important limits. First, let us assume that our system initially was definitely in one eigenstate n’ (usually, though not necessarily, in the ground state), and that the occupancies Wn of all other levels stay very low all the time. (We will find the condition when the second assumption is valid a posteriori – from the solution.) With these assumptions, a ;
1
a ,
1 for n n',
(6.89)
n'
n
Eq. (88) may be readily integrated, giving
*
A
nn'
i
t
A
(
)
nn
n'n
i(
)
a
t
(6.90)
n
e
e
nn
n n'
nn'
'
1
nn'
'
1 ,
for
.
This expression describes what is colloquially called the ac excitation of (other) energy levels.
Qualitatively, it shows that the probability Wn (83) of finding the system in each state (“on each energy level”) of the system does not tend to any constant value but rather oscillates in time. It also shows that that the ac-field-induced transfer of the system from one state to the other one has a clearly resonant character: the maximum occupancy Wn of a level number n n’ grows infinitely when the corresponding detuning 28
,
(6.91)
nn'
nn'
tends to zero. This conclusion is clearly unrealistic, and is an artifact of our initial assumption (89); according to Eq. (90), it is satisfied only if29
A
,
(6.92)
'
nn
nn'
and hence which does not allow a more deep analysis of the resonant excitation.
In order to overcome this limitation, we may perform the following trick – very similar to the one we used for the transfer to the degenerate case in Sec. 1. Let us assume that for a certain level n,
, ,
,
n"
all
for
n, n'
(6.93)
nn'
n"n
n"n'
– the condition illustrated in Fig. 8. Then, according to Eq. (90), we may ignore the occupancy of all but two levels, n and n’, and also the second, non-resonant term with frequency nn’ + 2 >> nn’ in Eqs. (88),30 now written for two probability amplitudes, an and an’.
28 The notion of detuning is also very useful in the classical theory of oscillations (see, e.g., CM Chapter 5), where the role of nn’ is played by the own frequency 0 of the oscillator.
29 Strictly speaking, one more condition is that the number of “resonance” levels is also not too high – see Sec. 6.
30 The second assumption, i.e. the omission of non-resonant terms in the equations for amplitudes is called the Rotating Wave Approximation (RWA); the same idea in the classical theory of oscillations is the basis of what is usually called the van der Pol method, and its result, the reduced equations – see, e.g., CM Secs. 5.3-5.5.
Chapter 6
Page 20 of 36
QM: Quantum Mechanics
En
0
nn '
E
Fig. 6.8. The resonant excitation of
n"
an energy level.
E
n'
The result is the following system of two linear equations:
i t
i a a Ae
,
* i t
i a a A e ,
(6.94)
n
n'
n'
n
which uses the shorthand notation A Ann’ and nn’. (I will use this notation for a while – until other energy levels become involved, at the beginning of the next section). This system may be readily reduced to a form without explicit time dependence of the right-hand parts – for example, by introducing the following new probability amplitudes, with the same moduli:
i t
/ 2
i t / 2
b a e
,
b a e
,
(6.95)
n
n
n'
n'
so that
i t
/ 2
i / 2
a b e
,
t
a b e
.
(6.96)
n
n
n'
n'
Plugging these relations into Eq. (94), we get two usual linear first-order differential equations:
i b
b Ab ,
i b A b
*
b .
(6.97)
n
n
n'
n'
n
n'
2
2
As the reader knows very well by now, the general solution of such a system is a linear combination of two exponential functions, exp{ t}, with the exponents that may be found by plugging any of these functions into Eq. (97), and requiring the consistency of the two resulting linear algebraic equations. In our case, the consistency condition (i.e. the characteristic equation of the system) is
/ 2 i
A
0
*
,
(6.98)
A
/ 2 i
and has two solutions = i, where
1/ 2
1/ 2
2
2
2
Rabi
Δ
A
A
oscillations:
2
Ω
,
2
i.e. 4
.
(6.99)
2
2
frequency
4
The coefficients at the exponents are determined by initial conditions. If, as was assumed before, the system was completely on the level n’ initially (at t = 0), i.e. if an’ (0) = 1, an(0) = 0, so that bn’ (0) =
1, bn(0) = 0 as well, then Eqs. (97) yield, in particular:
A
b ( t) i
sin t
,
(6.100)
n
so that the n th level occupancy is
Chapter 6
Page 21 of 36
QM: Quantum Mechanics
A 2
2
A 2
W b
2
sin
t
2
sin .
(6.101) Rabi
n
n
2
2
2
formula
/ 2
t
A
2
This is the famous Rabi oscillation formula.31 If the detuning is large in comparison with A /, though still small in the sense of Eq. (93), the frequency 2 of the Rabi oscillations is completely determined by the detuning, and their amplitude is small:
2
A
t
W ( t) 4
sin
,
1
for 2
2
A (
(6.102)
n
)2 ,
2
2
2
– the result which could be obtained directly from Eq. (90), just neglecting the second term on its right-hand side. However, now we may also analyze the results of an increase of the perturbation amplitude: it leads not only to an increase of the amplitude of the probability oscillations, but also of their frequency
– see Fig. 9. Ultimately, at A >> (for example, at the exact resonance, = 0., i.e. nn’ = , so that En = En’ + ), Eqs. (101)-(102) give = A / and ( Wn)max = 1, i.e. describe a periodic, full
“repumping” of the system from one level to another and back, with a frequency proportional to the perturbation amplitude.32
A 3
0.8
0.6
W
1
n
0.4
0.3
Fig. 6.9. The Rabi oscillations
0.2
for several values of the
0.1
normalized amplitude of ac
0
perturbation.
0