Key
commutation
L ˆ , r ˆ i r ˆ
L ˆ
,
, p ˆ i p ˆ
L ˆ
,
L ˆ
,
i L ˆ ;
(5.149)
j
j'
j"
jj'j"
j j'
j"
jj'j"
j j'
j"
jj'j"
relations
the last of them shows that the commutator of two different Cartesian components of the vector-operator Lîs proportional to its complementary component.
Also introducing, in a natural way, the (scalar!) operator of the observable L 2 L2, 3
Operator
ˆ2
ˆ2
ˆ2
ˆ2
2
L L L L
L
(5.150)
of L 2
x
y
z
, j
j 1
it is straightforward to check that this operator commutes with each of the Cartesian components:
ˆ2 ˆ
L , L
(5.151)
j
.
0
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This result, at the first sight, may seem to contradict the last of Eqs. (149). Indeed, haven’t we learned in Sec. 4.5 that commuting operators (e.g., 2
ˆ L and any of L ˆ ) share their eigenstate sets? If yes, shouldn’t
j
this set has to be common for all four angular momentum operators? The resolution in this paradox may be found in the condition that was mentioned just after Eq. (4.138), but (sorry!) was not sufficiently emphasized there. According to that relation, if an operator has degenerate eigenstates (i.e. if some Aj =
Aj’ even for j j’), they should not be necessarily all shared by another compatible operator.
This is exactly the situation with the orbital angular momentum operators, which may be schematically shown on a Venn diagram (Fig. 10):45 the eigenstates of the operator 2
ˆ L are highly
degenerate,46 and their set is broader than those of any component operator L ˆ (that, as will be shown j
below, are non-degenerate – until we consider particle’s spin).
Fig. 5.10. The Venn diagram showing the partitioning of
the set of eigenstates of the operator 2
ˆ L . Each inner sector
L ˆ
corresponds to the states shared with one of the Cartesian
z
component operators L ˆ , while the outer (shaded) ring
j
L ˆ
L ˆ
x
y
represents the eigenstates of 2
ˆ L that are not shared with
either of L ˆ – for example, all linear combinations of the
j
eigenstates of different component operators.
Let us focus on just one of these three joint sets of eigenstates – by tradition, of the operators 2
ˆ L
and L ˆ . (This tradition stems from the canonical form of the spherical coordinates, in which the polar z
angle is measured from the z-axis . Indeed, in the coordinate representation we may write
L ˆ ˆ p
x
ˆ p
y
x i
.
(5.152)
z
y
x
y i
i
y
x
Writing the standard eigenproblem for the operator in this representation, L ˆ L , we see that it z
m
z
m
is satisfied by the eigenfunctions (146), with eigenvalues Lz = m – which was already conjectured in Sec. 3.5.) More specifically, let us consider a set of eigenstates { l, m} corresponding to a certain degenerate eigenvalue of the operator 2
ˆ L , and all possible eigenvalues of the operator L ˆ , i.e. all z
possible quantum numbers m. (At this point, l is just some label of the eigenvalue of the operator 2
ˆ L ; it
will be defined more explicitly in a minute.) To analyze this set, it is instrumental to introduce the so-called ladder (also called, respectively, “raising” and “lowering”) operators 47
45 This is just a particular example of the Venn diagrams (introduced in the 1880s by John Venn) that show possible relations (such as intersections, unions, complements, etc.) between various sets of objects, and are very useful tool in the general set theory.
46 Note that this particular result is consistent with the classical picture of the angular momentum vector: even when its length is fixed, the vector may be oriented in various directions, corresponding to different values of its Cartesian components. However, in the classical picture, all these components may be fixed simultaneously, while in the quantum picture this is not true.
47 Note a substantial similarity between this definition and Eqs. (65) for the creation/annihilation operators.
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Ladder
L ˆ L ˆ iL ˆ .
(5.153)
x
y
operators
It is simple (and hence left for the reader’s exercise) to use this definition and the last of Eqs. (149) to calculate the following commutators:
Important
commutation
L ˆ L ˆ
,
L ˆ
2
,
L ˆ
and
L ˆ
,
L ˆ ,
(5.154)
z
z
relations
and also to use Eqs. (149)-(150) to prove two other important operator relations:
L ˆ2 L ˆ2 L ˆ L ˆ L ˆ
.
(5.155)
L ˆ
,
2 L ˆ2 L ˆ L ˆ L ˆ
z
z
z
z
Now let us rewrite the last of Eqs. (154) as
L ˆ L ˆ L ˆ L ˆ L ˆ ,
(5.156)
z
z
and act by its both sides upon the ket-vector l, m of an arbitrary common eigenstate: ˆ ˆ
ˆ ˆ
ˆ
L L l, m L L l, m L l, m .
(5.157)
z
z
Since the eigenvalues of the operator L âre equal to
z
m, in the first term of the right-hand side of Eq.
(157) we may write
ˆ L l, m m l, m .
(5.158)
z
With that, Eq. (157) may be recast as
L ˆ ˆ
ˆ
,
1
,
.
(5.159)
z L
l m
m L l m
In a spectacular similarity with Eqs. (78)-(79) for the harmonic oscillator, Eq. (159) means that the states L ˆ l, m are also eigenstates of the operator L ˆ , corresponding to eigenvalues
z
( m 1). Thus
the ladder operators act exactly as the creation and annihilation operators of a harmonic oscillator, moving the system up or down a ladder of eigenstates – see Fig. 11.
eigenket
L ôf
eigenvalue
z
l, l
l
L ˆ
ˆ
L
…
L ˆ l, m
m 1
L ˆ
ˆ
L
l, m
m
L ˆ
ˆ
L
L ˆ l, m
m 1
…
Fig. 5.11. The ladder diagram of the common
L ˆ
ˆ
eigenstates of the operators 2
ˆ L and L ˆ .
L
z
l, l
l
The most significant difference is that now the state ladder must end in both directions, because an infinite increase of m, with whichever sign of m, would cause the expectation values of the operator Chapter 5
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2
2
2
2
ˆ
ˆ
ˆ
ˆ
L L L L ,
(5.160)
x
y
z
which corresponds to a non-negative observable, to become negative. Hence there have to be two states at both ends of the ladder, with such ket-vectors l, m max and l, m min that ˆ
ˆ
L l, m
,
0
L l, m
.
0
(5.161)
max
min
Due to the symmetry of the whole problem with respect to the replacement m – m, we should have m min = – m max. This m max is exactly the quantum number traditionally called l, so that Relation
l m l.
(5.162) between
m and l
Evidently, this relation of quantum numbers m and l is semi-quantitatively compatible with the classical image of the angular momentum vector L, of the same length L, pointing in various directions, thus affecting the value of its component Lz. In this classical picture, however, L 2 would be equal to the square of ( Lz)max, i.e. to ( l)2; however, in quantum mechanics, this is not so. Indeed, applying both parts of the second of the operator equalities (155) to the top state’s vector l, m max l, l, we get ˆ2
ˆ
ˆ2
ˆ ˆ
L l, l L l, l L l, l L L l, 2
l l l,
2 2
l l l, l 0
z
z
(5.163) Eigenvalues
2
of L 2
l l
1 l, l .
Since by our initial assumption, all eigenvectors l, m correspond to the same eigenvalue of 2
ˆ L , this
result means that all these eigenvalues are equal to 2 l( l + 1). Just as in the case of the spin-½ vector operators discussed in Sec. 4.5, the deviation of this result from 2 l 2 may be interpreted as the result of unavoidable uncertainties (“fluctuations”) of the x- and y-components of the angular momentum, which give non-zero positive contributions to L 2
2
x and Ly , and hence to L 2, even if the angular momentum
vector is aligned with the z-axis in the best possible way.
(For various applications of the ladder operators (153), one more relation is convenient: ˆ L l, m l l
m m
l m
.
(5.164)
1 11/2 , 1
This equality, valid to the multiplier ei with an arbitrary real phase , may be readily proved from the above relations in the same way as the parallel Eqs. (89) for the harmonic-oscillator operators (65) were proved in Sec. 4; due to this similarity, the proof is also left for the reader’s exercise.48) Now let us compare our results with those of Sec. 3.6. Using the expression of Cartesian coordinates via the spherical ones exactly as this was done in Eq. (152), we get the following expressions for the ladder operators (153) in the coordinate representation:
48 The reader is also challenged to use the commutation relations discussed above to prove one more important property of the common eigenstates of L ând 2
ˆ L :
z
l, m r ˆ l' , m' ,
0
l'
unless l and
1
m'
m
either
m
or
1
.
j
This property gives the selection rule for the orbital electric-dipole quantum transitions, to be discussed later in the course, especially in Sec. 9.3. (The final selection rules at these transitions may be affected by the particle’s spin – see the next section.)
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i
ˆ L
e
i
cotan
.
(5.165)
Angular
momentum
operators:
Now plugging this relation, together with Eq. (152), into any of Eqs. (155), we get
coordinate
representation
1
1
2
2
2
ˆ L
sin
.
(5.166)
2
2
sin
sin
But this is exactly the operator (besides its division by the constant parameter 2m R 2) that stands on the left-hand side of Eq. (3.156). Hence that equation, which was explored by the “brute-force” (wave-mechanical) approach in Sec. 3.6, may be understood as the eigenproblem for the operator 2
ˆ L in the
coordinate representation, with the eigenfunctions Y m
l (,) corresponding to the eigenkets l, m, and the
eigenvalues L 2 = 2m R 2 E. As a reminder, the main result of that, rather involved analysis was expressed by Eq. (3.163), which now may be rewritten as
2
L 2
2
2
R
m
E l( l )
1 ,
(5.167)
l
l
in full agreement with Eq. (163), which was obtained by much more efficient means based on the bra-ket formalism. In particular, it is fascinating to see how easy it is to operate with the eigenvectors l, m, while the coordinate representations of these ket-vectors, the spherical harmonics Y m l (,), may be only
expressed by rather complicated functions – please have one more look at Eq. (3.171) and Fig. 3.20.
Note that all relations discussed in this section are not conditioned by any particular Hamiltonian of the system under analysis, though they (as well as those discussed in the next section) are especially important for particles moving in spherically-symmetric potentials.
5.7. Spin and its addition to orbital angular momentum
The theory described in the last section is useful for much more than orbital motion analysis. In particular, it helps to generalize the spin-½ results discussed in Chapter 4 to other values of spin s – the parameter still to be quantitatively defined. For that, let us notice that the commutation relations (4.155) for spin-½, which were derived from the Pauli matrix properties, may be rewritten in exactly the same form as Eqs. (149) and (151) for the orbital momentum:
Spin
operators:
commutation
ˆ ˆ
S , S
i S
S S
(5.168)
j
j'
ˆ
,
j"
jj'j"
ˆ2 ˆ, j 0
relations
It had been postulated (and then confirmed by numerous experiments) that these relations hold for quantum particles with any spin. Now notice that all the calculations of the last section have been based almost exclusively on such relations – the only exception will be discussed imminently. Hence, we may repeat them for the spin operators, and get the relations similar to Eqs. (158) and (163): Spin
operators:
eigenstates
S ˆ s, m m
and
s, m
S ˆ
,
2 s, m 2
s( s )
1 s, m , 0 s, s m s ,
(5.169)
z
s
s
s
s
s
s
eigenvalues
where ms is a quantum number parallel to the orbital magnetic number m, and the non-negative constant s is defined as the maximum value of ms . The c-number s is exactly what is called the particle’s spin.
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Now let us return to the only part of our orbital moment calculations that has not been derived from the commutation relations. This was the fact, based on the solution (146) of the orbital motion problems, that the quantum number m (the analog of ms) may be only an integer. For spin, we do not have such a solution, so that the spectrum of numbers ms (and hence its limits s) should be found from the more loose requirement that the eigenstate ladder, extending from – s to + s, has an integer number of steps. Hence, 2 s has to be an integer, i.e. the spin s of a quantum particle may be either integer (as it is, for example, for photons, gluons, and massive bosons W and Z0), or half-integer (e.g., for all quarks and leptons, notably including electrons).49 For s = ½, this picture yields all the properties of the spin-½
that were derived in Chapter 4 from Eqs. (4.115)-(4.117). In particular, the operators 2
ˆ S and S ˆ have
z
two common eigenstates ( and ), with Sz = ms = /2, both with S 2= s( s +1)2 = (3/4)2.
Note that this analogy with the angular momentum sheds new light on the symmetry properties of spin-½. Indeed, the fact that m in Eq. (146) is integer was derived in Sec. 3.5 from the requirement that making a full circle around axis z, we should find a similar value of wavefunction m, which differs from the initial one by an inconsequential factor exp{2 im} = +1. With the replacement m ms = ½, such operation would multiply the wavefunction by exp{ i} = –1, i.e. reverse its sign. Of course, spin properties cannot be described by a usual wavefunction, but this odd parity of electrons, shared by all other spin-½ particles, is clearly revealed in properties of multiparticle systems (see Chapter 8 below), and as a result, in their statistics (see, e.g., SM Chapter 2).
Now we are sufficiently equipped to analyze the situations in which a particle has both the orbital momentum and the spin – as an electron in an atom. In classical mechanics, such an object, with the spin S interpreted as the angular moment of its internal rotation, would be characterized by the total angular momentum vector J = L + S. Following the correspondence principle, we may assume that quantum-mechanical properties of this observable may be described by the similarly defined vector operator:
Total
Jˆ Lˆ Sˆ ,
(5.170) angular
momentum
with Cartesian components
ˆ
ˆ
ˆ
J L S , etc. ,
(5.171)
z
z
z
and the magnitude squared equal to
ˆ 2
ˆ 2
ˆ 2
ˆ 2
J J J J .
(5.172)
x
y
z
Let us examine the properties of this vector operator. Since its two components (170) describe different degrees of freedom of the particle, i.e. belong to different Hilbert spaces, they have to be completely commuting:
ˆ ˆ
L , S
L S
L S
L S
(5.173)
j
j'
,
0
ˆ2 ˆ,2 ,0 ˆ ˆ,2
j
,0 ˆ2 ˆ, j 0.
The above equalities are sufficient to derive the commutation relations for the operator Jˆ , and unsurprisingly, they turn out to be absolutely similar to those of its components:
49 As a reminder, in the Standard Model of particle physics, such hadrons as mesons and baryons (notably including protons and neutrons) are essentially composite particles. However, at non-relativistic energies, protons and neutrons may be considered fundamental particles with s = ½.
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Total
momentum:
commutation
ˆ ˆ
J , J
i J
J J
.
(5.174)
j
j'
ˆ
,
j"
jj'j"
ˆ2 ˆ, j 0
relations
Now repeating all the arguments of the last section, we may derive the following expressions for the common eigenstates of the operators 2
ˆ J and J ˆ :
z
Total
momentum:
eigenstates,
ˆ
J j, m m
j, m ,
2
J j,
2
m j( j )
1 j, m , 0 j, j m j,
(5.175)
z
j
j
j
j
j
j
and
eigenvalues
where j and mj are new quantum numbers.50 Repeating the arguments just made for s and ms, we may conclude that j and mj may be either integer or half-integer.
Before we proceed, one remark on notation: it is very convenient to use the same letter m for numbering eigenstates of all momentum components participating in Eq. (171), with corresponding indices ( j, l, and s), in particular, to replace what we called m with ml. With this replacement, the main results of the last section may be summarized in a form similar to Eqs. (168), (169), (174), and (175): Orbital
momentum:
ˆ ˆ
L , L
i L
L L
,
(5.176)
j
j'
ˆ
,
j"
jj'j"
ˆ2 ˆ, j 0
basic
properties
ˆ
ˆ
(new notation)
L l, m m
l, m ,
2
L l,
2
m l( l )
1 l, m , 0 l, l m l.
(5.177)
z
l
l
l
l
l
l
In order to understand which eigenstates participating in Eqs. (169), (175), and (177) are compatible with each other, it is straightforward to use Eq. (172), together with Eqs. (168), (173), (174), and (176) to get the following relations:
ˆ 2 ˆ
J , 2
L 0, ˆ 2 ˆ
J , 2
S 0,
(5.178)
ˆ 2 ˆ
J , L
J S
(5.179)
z 0,
ˆ2 ˆ, z 0.
This result is represented schematically on the Venn diagram shown in Fig. 12, in which the crossed arrows indicate the only non-commuting pairs of operators.
operators
2
ˆ L
2
ˆ S
2
ˆ J
diagonal in
the coupled
representation
operators
diagonal in
the uncoupled
L ˆ
S ˆ
J ˆ
z
z
z
Fig. 5.12. The Venn diagram of angular momentum
representation
operators, and their mutually-commuting groups.
This means that there are eigenstates shared by two operator groups encircled with colored lines in Fig. 12. The first group (encircled red), consists of all these operators but ˆ 2
J . Hence there are
eigenstates shared by the five remaining operators, and these states correspond to definite values of the corresponding quantum numbers: l, ml, s, ms, and mj. Actually, only four of these numbers are 50 Let me hope that the difference between the quantum number j, and the indices j, j’, j” numbering the Cartesian components in the relations like Eqs. (168) or (174), is absolutely clear from the context.
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independent, because due to Eq. (171) for these compatible operators, for each eigenstate of this group, their “magnetic” quantum numbers m have to satisfy the following relation:
m m m .
(5.180)
j
l
s
Hence the common eigenstates of the operators of this group are fully defined by just four quantum numbers, for example, l, ml, s, and ms. For some calculations, especially those for the systems whose Hamiltonians include only the operators of this group, it is convenient51 to use this set of eigenstates as the basis; frequently this approach is called the uncoupled representation.
However, in some situations we cannot ignore interactions between the orbital and spin degrees of freedom (in the common jargon, the spin-orbit coupling), which leads in particular to splitting (called the fine structure) of the atomic energy levels even in the absence of external magnetic field. I will discuss these effects in detail in the next chapter, and now will only note that they may be described by a term proportional to the product Lˆ Sˆ , in the system’s Hamiltonian. If this term is substantial, the uncoupled representation becomes inconvenient. Indeed, writing
2
2
2
2
2
2
2
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ ˆ
ˆ ˆ
ˆ
ˆ
ˆ
J (L S) L S 2L S,
2
that
so
L S J L S ,
(5.181)
and looking at Fig. 12 again, we see that operator Lˆ Sˆ , describing the spin-orbit coupling, does not commute with operators L ând S ˆ . This means that stationary states of the system with such term in z
z
the Hamiltonian do not belong to the uncoupled representation’s basis. On the other hand, Eq. (181) shows that the operator Lˆ Sˆ does commute with all four operators of another group, encircled blue in Fig. 12. According to Eqs. (178), (179), and (181), all operators of that group also commute with each other, so that they have common eigenstates, described by the quantum numbers, l, s, j, and mj. This group is the basis for the so-called coupled representation of particle states.
Excluding, for the notation briefness, the quantum numbers l and s, common for both groups, it is convenient to denote the common ket-vectors of each group as, respectively,
m , m ,
tion'
representa
uncolpled
for the
basis,
s
l
s
(5.182) Coupled and
uncoupled
j, m ,
tion'
representa
coupled
for the
basis.
s
j
bases
As we will see in the next chapter, for the solution of some important problems (e.g., the fine structure of atomic spectra and the Zeeman effect), we will need the relation between the kets j, mj and the kets
ml, ms. This relation may be represented as the usual linear superposition, Clebsch-
j, m m , m m , m j, m .
(5.183) Jordan
j
l
s
l
s
j
coefficients:
m , m
definition
l
s
The short brackets in this relation, essentially the elements of the unitary matrix of the transformation between two eigenstate bases (182), are called the Clebsch-Gordan coefficients.
The best (though imperfect) classical interpretation of Eq. (183) I can offer is as follows. If the lengths of the vectors L and S (in quantum mechanics associated with the numbers l and s, respectively), 51 This is especially true for motion in spherically-symmetric potentials, whose stationary states correspond to definite l and ml; however, the relations discussed in this section are important for some other problems as well.
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and also their scalar product LS, are all fixed, then so is the length of the vector J = L + S – whose length in quantum mechanics is described by the number j. Hence, the classical image of a specific eigenket j, mj, in which l, s, j, and mj are all fixed, is a state in which L 2, S 2, J 2, and Jz are fixed.
However, this fixation still allows for arbitrary rotation of the pair of vectors L and S (with a fixed angle between them, and hence fixed LS and J 2) about the direction of the vector J – see Fig. 13.
Hence the components Lz and Sz in these conditions are not fixed, and in classical mechanics may take a continuum of values, two of which (with the largest and the smallest possible values of Sz) are shown in Fig. 13. In quantum mechanics, these components are quantized, with their states represented by eigenkets ml, ms, so that a linear combination of such kets is necessary to represent a ket
j, mj. This is exactly what Eq. (183) does.
z
z
J
J
J
J
z
z
L
L
z
Lz
L
S
Fig. 5.13. A classical image of two
S
different quantum states with the
z
Sz
S
same quantum numbers l, s, j, and
0
mj, but different ml and ms.
0
Some properties of the Clebsch-Gordan coefficients ml, ms j, mj may be readily established.
For example, the coefficients do not vanish only if the involved magnetic quantum numbers satisfy Eq.
(180). In our current case, this relation is not an elementary corollary of Eq. (171), because in the Clebsch-Gordan coefficients, with the quantum numbers ml, ms in one state vector, and mj in the other state vector, characterize the relation between different groups of the basis states, so we need to prove this fact. All matrix elements of the null-operator
ˆ
ˆ
ˆ
J ( L S 0ˆ
)
(5.184)
z
z
z
should equal zero in any basis; in particular
ˆ
ˆ
ˆ
j, m J ( L S ) m , m .
0
(5.185)
j
z
z
z
l
s
Acting by the operator J ûpon the bra-vector, and by the sum ˆ
ˆ
( L S ) upon the ket-vector, we get
z
z
z
m ( m m ) j m m m
(5.186)
j
l
s
,
,
,
0
j
l
s
thus proving that
m , m j, m j, m m , m * ,
0
if m
m m .
(5.187)
l
s
s
s
l
s
j
l
s
For the most important case of spin-½ particles (with s = ½, and hence ms = ½), whose uncoupled representation basis includes 2(2 l + 1) states, the restriction (187) enables the representation of all non-zero Clebsch-Gordan coefficients on the simple “rectangular” diagram shown in Fig. 14.
Indeed, each coupled-representation eigenket j, mj, with mj = ml + ms = ml ½, may be related by nonzero Clebsch-Gordan coefficients to at most two uncoupled-representation eigenstates ml, ms. Since ml Chapter 5
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may only take integer values from – l to + l, mj may only take semi-integer values on the interval [- l – ½, l + ½]. Hence, by the definition of j as ( mj)max, its maximum value has to be l + ½, and for mj = l + ½, this is the only possible value with this j. This means that the uncoupled state with ml = l and ms = ½
should be identical to the coupled-representation state with j = l + ½ and mj = l + ½: j l ½, m l ½ m m ½, m ½
.
(5.188)
j
l
j
s
In Fig. 14, these two identical states are represented with the top-rightmost point (the uncoupled representation) and the sloped line passing through it (the coupled representation).
m
s
m l ½ m l 3 2
m l 3 2 m l ½
j
j
j
j
j l ½
j l ½
j l ½
j l ½
m l ½
j
½
j l ½
l
l 1
l 2
0
l 2
l 1
l
ml
m l ½
j
j l ½
½
Fig. 5.14. A graphical representation of possible basis states of a spin-½ particle with a fixed l. Each dot corresponds to an uncoupled-representation ket-vector ml, ms, while each sloped line corresponds to one coupled-representation ket-vector j, mj, related by Eq. (183) to the kets ml, ms whose dots it connects.
However, already the next value of this quantum number, mj = l – ½, is compatible with two values of j, so that each ml, ms ket has to be related to two j, mj kets by two Clebsch-Gordan coefficients. Since j changes in unit steps, these values of j have to be l ½. This choice, j l ½ ,
(5.189)
evidently satisfies all lower values of mj as well – see Fig. 14.52 (Again, only one value, j = l + ½, is necessary to represent the state with the lowest mj = – l – ½ – see the bottom-leftmost point of that diagram.) Note that the total number of the coupled-representation states is 1 + 22 l + 1 2(2 l + 1), i.e.
is the same as those in the uncoupled representation. So, for spin-½ systems, each sum (183), for fixed j and mj (plus the fixed common parameter l, plus the common s = ½), has at most two terms, i.e. involves at most two Clebsch-Gordan coefficients.
These coefficients may be calculated in a few steps, all but the last one rather simple even for an arbitrary spin s. First, the similarity of the vector operators Jˆ
Sˆ
and
to the operator Lˆ , expressed by
Eqs. (169), (175), and (177), may be used to argue that the matrix elements of the operators S ˆ
J ˆ
and
,
defined similarly to L ˆ , have the matrix elements similar to those given by Eq. (164). Next, acting by
the operator J ˆ L ˆ
S ˆ
upon both parts of Eq. (183), and then inner-multiplying the result by the bra
52 Eq. (5.189) allows a semi-qualitative classical interpretation in terms of the vector diagrams shown in Fig. 13: since, according to Eq. (169), s gives the scale of the length of the vector S, if it is small ( s = ½), the length of vector J (similarly scaled by j) cannot deviate much from the length of the vector L (scaled by l) for any spatial orientation of these vectors, so that j cannot differ from l too much. Note also that for a fixed mj, the alternating sign in Eq. (189) is independent of the sign of ms – see also Eqs. (190).
Chapter 5
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vector ml, ms and using the above matrix elements, we may get recurrence relations for the Clebsch-Gordan coefficients with adjacent values of ml, ms, and mj. Finally, these relations may be sequentially applied to the adjacent states in both representations, starting from any of the two states common for them – for example, from the state with the ket-vector (188), corresponding to the top right point in Fig.
14.
Let me leave these straightforward but a bit tedious calculations for the reader’s exercise, and just cite the final result of this procedure for s = ½:53
l m ½ 1/2
j
m m ½, m ½ j l ½, m
,
l
j
s
j
Clebsch –
2 l 1
Gordan
(5.190)
coefficients
l m ½ 1/2
j
for s = ½
m m ½, m ½ j l ½, m
.
l
j
s
j
2 l 1
In this course, these relations will be used mostly in Sec. 6.4 for an analysis of the anomalous Zeeman effect. Moreover, the angular momentum addition theory described above is also valid for the addition of angular momenta of multiparticle system components, so we will revisit it in Chapter 8.
To conclude this section, I have to note that the Clebsch-Gordan coefficients (for arbitrary s) participate also in the so-called Wigner-Eckart theorem that expresses the matrix elements of spherical tensor operators, in the coupled-representation basis j, mj, via a reduced set of matrix elements. This theorem may be useful, for example, for the calculation of the rate of quantum transitions to/from high- n states in spherically-symmetric potentials. Unfortunately, a discussion of this theorem and its applications would require a higher mathematical background than I can expect from my readers, and more time/space than I can afford.54
5.8. Exercise problems
5.1. Use the discussion in Sec. 1 to find an alternative solution of Problem 4.18.
5.2. A spin-½ is placed into an external magnetic field, with a time-independent orientation, its magnitude B( t) being an arbitrary function of time. Find explicit expressions for the Heisenberg operators and the expectation values of all three Cartesian components of the spin, as functions of time, in a coordinate system of your choice.
5.3. A two-level system is in the quantum state described by the ket-vector = +
, with given (generally, complex) c-number coefficients . Prove that we can always select such a geometric c-number vector c = { cx, cy, cz}, that is an eigenstate of c σˆ , where σˆ is the Pauli vector operator. Find all possible values of c satisfying this condition, and the second eigenstate (orthogonal to
) of the operator c σˆ . Give a Bloch-sphere interpretation of your result.
53 For arbitrary spin s, the calculations and even the final expressions for the Clebsch-Gordan coefficients are rather bulky. They may be found, typically in a table form, mostly in special monographs – see, e.g., A.
Edmonds, Angular Momentum in Quantum Mechanics, Princeton U. Press, 1957.
54 For the interested reader, I can recommend either Sec. 17.7 in E. Merzbacher, Quantum Mechanics, 3rd ed., Wiley, 1998, or Sec. 3.10 in J. Sakurai, Modern Quantum Mechanics, Addison-Wesley, 1994.
Chapter 5
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5.4.* Analyze statistics of the spacing S E+ – E- between energy levels of a two-level system, assuming that all elements Hjj’ of its Hamiltonian matrix (2) are independent random numbers, with equal and constant probability densities within the energy interval of interest. Compare the result with that for a purely diagonal Hamiltonian matrix, with the similar probability distribution of its random diagonal elements.
5.5. For a periodic motion of a single particle in a confining potential U(r), the virial theorem of non-relativistic classical mechanics55 is reduced to the following equality:
1
T r U ,
2
where T is the particle’s kinetic energy, and the top bar means averaging over the time period of motion.
Prove the following quantum-mechanical version of the theorem for an arbitrary stationary quantum state, in the absence of spin effects:
1
T
r U ,
2
where the angular brackets mean the expectation values of the observables.
Hint: Mimicking the proof of the classical virial theorem, consider the time evolution of the following operator:
ˆ
G rˆ pˆ .
5.6. Calculate, in the WKB approximation, the transparency T of the following saddle-shaped potential barrier:
xy
U ( x, y) U 1
,
0
2
a
where U 0 > 0 and a are real constants, for tunneling of a 2D particle with energy E < U 0.
5.7. Calculate the so-called Gamow factor 56 for the alpha decay of atomic nuclei, i.e. the exponential factor in the transparency of the potential barrier resulting from the following simple model for the alpha-particle’s potential energy as a function of its distance from the nuclear center:
U 0,
for
r R,
U r 0
2
ZZ'e
,
for R r,
4 r
0
(where Ze = 2 e > 0 is the charge of the particle, Z’e > 0 is that of the nucleus after the decay, and R is the nucleus’ radius), in the WKB approximation.
5.8. Use the WKB approximation to calculate the average time of ionization of a hydrogen atom, initially in its ground state, made metastable by application of an additional weak, uniform, time-independent electric field E. Formulate the conditions of validity of your result.
55 See, e.g., CM Problem 1.12.
56 Named after G. Gamow, who made this calculation as early as in 1928.
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5.9. For a 1D harmonic oscillator with mass m and frequency 0, calculate: (i) all matrix elements n x 3
ˆ n' , and
(ii) the diagonal matrix elements n x 4
ˆ n ,
where n and n’ are arbitrary Fock states.
5.10. Calculate the sum (over all n > 0) of the so-called oscillator strengths, 2 m
f
,
n
E E
n
2
n ˆ x 0
2
0
(i) for a 1D harmonic oscillator, and
(ii) for a 1D particle confined in an arbitrary stationary potential.
5.11. Prove the so-called Bethe sum rule,
2
2
2
E E
ˆ
,
n'
n
x
ik
k
n e
n'
n'
2 m
valid for a 1D particle moving in an arbitrary time-independent potential U( x), and discuss its relation with the Thomas-Reiche-Kuhn sum rule whose derivation was the subject of the previous problem.
Hint: Calculate the expectation value, in a stationary state n, of the following double commutator,
D ˆ ˆ
x
ik
H, e ˆ x
ik
, e
ˆ ,
in two ways: first, just spelling out both commutators, and, second, using the commutation relations between operators p ând
x
ik
e ˆ , and compare the results.
x
5.12. Given Eq. (116), prove Eq. (117), using the hint given in the accompanying footnote.
5.13. Use Eqs. (116)-(117) to simplify the following operators:
(i)
exp ia ˆ
x p êxp ˆ , and
x
ia
x
(ii)
exp iap ˆ x êxp iap ˆ ,
x
x
where a is a c-number.
5.14. For a 1D harmonic oscillator, calculate:
(i) the expectation value of energy, and
(ii) the time evolution of the expectation values of the coordinate and momentum,
provided that in the initial moment ( t = 0) it was in the state described by the following ket-vector: 1
31 32 ,
2
where n are the ket-vectors of the stationary (Fock) states of the oscillator.
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5.15.* Re-derive the London dispersion force’s potential of the interaction of two isotropic 3D
harmonic oscillators (already calculated in Problem 3.16), using the language of mutually-induced polarization.
5.16. An external force pulse F( t), of a finite time duration T, has been exerted on a 1D harmonic oscillator, initially in its ground state. Use the Heisenberg-picture equations of motion to calculate the expectation value of the oscillator’s energy at the end of the pulse.
5.17. Use Eqs. (144)-(145) to calculate the uncertainties x and p for a harmonic oscillator in its squeezed ground state, and in particular, to prove Eqs. (143) for the case = 0.
5.18. Calculate the energy of a harmonic oscillator in the squeezed ground state .
5.19.* Prove that the squeezed ground state, described by Eqs. (142) and (144)-(145), may be sustained by a sinusoidal modulation of a harmonic oscillator’s parameter, and calculate the squeezing factor r as a function of the parameter modulation depth, assuming that the depth is small, and the oscillator’s damping is negligible.
5.20. Use Eqs. (148) to prove that the operators L ând 2
ˆ L commute with the Hamiltonian of a
j
spinless particle placed in any central potential field.
5.21. Use Eqs. (149)-(150) and (153) to prove Eqs. (155).
5.22. Derive Eq. (164), using any of the prior formulas.
5.23. In the basis of common eigenstates of the operators L ând 2
ˆ L , described by kets l, m:
z
(i) calculate the matrix elements
ˆ
l, m L l, m and
2
ˆ
l, m L l, m ;
1
x
2
1
x
2
(ii) spell out your results for diagonal matrix elements (with m 1 = m 2) and their y-axis counterparts; and
(iii) calculate the diagonal matrix elements l m L ˆ
,
L ˆ l, m and l m L ˆ
,
L ˆ l, m .
x
y
y
x
5.24. For the state described by the common eigenket l, m of the operators L ând 2
ˆ L in a
z
reference frame { x, y, z}, calculate the expectation values L
2
z’ and Lz’ in the reference frame whose
z’-axis forms angle with the z-axis.
5.25. Write down the matrices of the following angular momentum operators:
L ˆ L ˆ
,
L ˆ
,
L ˆ
and
,
, in the z-basis of the { l, m} states with l = 1.
x
y
z
5.26. Calculate the angular factor of the orbital wavefunction of a particle with a definite value of L 2, equal to 62, and the largest possible value of Lx. What is this value?
Chapter 5
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5.27. For the state with the wavefunction = Cxye- r, with a real, positive , calculate: (i) the expectation values of the observables Lx, Ly, Lz, and L 2, and (ii) the normalization constant C.
5.28. An angular state of a spinless particle is described by the following ket-vector:
1
l ,3 m 0 l ,3 m 1 .
2
Calculate the expectation values of the x- and y-components of its angular momentum. Is the result sensitive to a possible phase shift between the component eigenkets?
5.29. A particle is in a quantum state with the orbital wavefunction proportional to the spherical harmonic 1
Y ( ,). Find the angular dependence of the wavefunctions corresponding to the 1
following ket-vectors:
(i) L ˆ ,
(ii) L ˆ ,
(iii) L ˆ , (iv) L ˆ L ˆ , and (v) 2
ˆ L .
x
y
z
5.30. A charged, spinless 2D particle of mass m is trapped in a soft potential well U( x, y) =
m 2
0 ( x 2 + y 2)/2. Calculate its energy spectrum in the presence of a uniform magnetic field B, normal to the [ x, y]-plane of particle’s motion.
5.31. Solve the previous problem for a spinless 3D particle, placed (in addition to a uniform magnetic field B) into a spherically-symmetric potential well U(r) = m 2
0 r 2/2.
5.32. Calculate the spectrum of rotational energies of an axially-symmetric, rigid body.
5.33. Simplify the following double commutator:
ˆ r L r .
j 2
ˆ
,
, ˆ j'
5.34. Prove the following commutation relation:
L ˆ2 L ˆ
, 2 , r ˆ
2 2
r ˆ L ˆ2 L ˆ2 r ˆ .
j
j
j
5.35. Use the commutation relation proved in the previous problem, and Eq. (148), to prove the orbital electric-dipole selection rules mentioned in Sec. 5.6 of the lecture notes.
5.36. Express the commutators listed in Eq. (179), J ˆ 2 L ˆ
,
and J ˆ 2 S ˆ
,
, via L ând S ˆ .
z
z
j
j
5.37. Find the operator T ˆ describing a quantum state’s rotation by angle about a certain axis, using the similarity of this operation with the shift of a Cartesian coordinate, discussed in Sec. 5. Then use this operator to calculate the probabilities of measurements of spin-½ components of a beam of Chapter 5
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QM: Quantum Mechanics
particles with z-polarized spin, by a Stern-Gerlach instrument turned by angle within the [ z, x] plane, where y is the axis of particle propagation – see Fig. 4.1.57
5.38. The rotation (“angular translation”) operator T ˆ analyzed in the previous problem, and the linear translation operator T ˆ discussed in Sec. 5, have a similar structure:
X
ˆ
T
êxp iC / ,
where is a real c-number, characterizing the shift, and C îs a Hermitian operator, which does not explicitly depend on time.
(i) Prove that such operators T ˆ are unitary.
(ii) Prove that if the shift by , induced by the operator T ˆ , leaves the Hamiltonian of some system unchanged for any , then C is a constant of motion for any initial state of the system.
(iii) Discuss what the last conclusion means for the particular operators T ând .
X
T ˆ
5.39. A particle with spin s is in a state with definite quantum numbers l and j. Prove that the observable LS also has a definite value, and calculate it.
5.40. For a spin-½ particle in a state with definite quantum numbers l, ml, and ms, calculate the expectation value of the observable J 2, and the probabilities of all its possible values. Interpret your results in the terms of the Clebsh-Gordan coefficients (190).
5.41. Derive general recurrence relations for the Clebsh-Gordan coefficients.
Hint: Using the similarity of the commutation relations discussed in Sec. 7, write the relations similar to Eqs. (164) for other components of the angular momentum, and apply them to Eq. (170).
5.42. Use the recurrence relations derived in the previous problem to prove Eqs. (190) for the spin-½ Clebsh-Gordan coefficients.
5.43. A spin-½ particle is in a state with definite values of L 2, J 2, and Jz. Find all possible values of the observables S 2, Sz, and Lz, the probability of each listed value, and the expectation value for each of these observables.
5.44. Re-solve the Landau-level problem discussed in Sec. 3.2, for a spin-½ particle. Discuss the result for the particular case of an electron, with the g-factor equal to 2.
5.45. In the Heisenberg picture of quantum dynamics, find an explicit relation between the operators of velocity ˆv d ˆr / dt and acceleration ˆa d ˆv / dt of a spin-½ particle with electric charge q, moving in an arbitrary external electromagnetic field. Compare the result with the corresponding classical expression.
Hint: For the orbital motion’s description, you may use Eq. (3.26).
57 Note that the last task is just a particular case of Problem 4.18 (see also Problem 1).
Chapter 5
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5.46. A byproduct of the solution of Problem 41 is the following relation for the spin operators (valid for any spin s):
ˆ
m 1 S m
.
s m s m
s
s
s
1
s 1/ 2
Use this result to spell out the matrices S x, S y, S z, and S2 of a particle with s = 1, in the z-basis – defined as the basis in which the matrix S z is diagonal.
5.47.* For a particle with an arbitrary spin s, moving in a spherically-symmetric field, find the ranges of the quantum numbers mj and j that are necessary to describe, in the coupled-representation basis:
(i) all states with a definite quantum number l, and
(ii) a state with definite values of not only l, but also ml and ms.
Give an interpretation of your results in terms of the classical geometric vector diagram (Fig. 13).
5.48. A particle of mass m, with electric charge q and spin s, free to move along a plane ring of a radius R, is placed into a constant, uniform magnetic field B, directed normally to the ring’s plane.
Calculate the energy spectrum of the system. Explore and interpret the particular form the result takes when the particle is an electron with the g-factor g e = 2.
Chapter 5
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Chapter 5
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Chapter 6. Perturbative Approaches
This chapter discusses several perturbative approaches to problems of quantum mechanics, and their simplest but important applications including the fine structure of atomic energy levels, and the effects of external dc and ac electric and magnetic fields on these levels. It continues with a discussion of the perturbation theory of transitions to continuous spectrum and the Golden Rule of quantum mechanics, which naturally brings us to the issue of open quantum systems – to be discussed in the next chapter.
6.1. Time-independent perturbations
Unfortunately, only a few problems of quantum mechanics may be solved exactly in an analytical form. Actually, in the previous chapters we have solved a substantial part of such problems for a single particle, while for multiparticle systems, the exactly solvable cases are even more rare.
However, most practical problems of physics feature a certain small parameter, and this smallness may be exploited by various approximate analytical methods giving asymptotically correct results – i.e. the results whose error tends to zero at the reduction of the small parameter(s). Earlier in the course, we have explored one of them, the WKB approximation, which is adequate for a particle moving through a soft potential profile. In this chapter, we will discuss other techniques that are more suitable for other cases. The historic name for these techniques is the perturbation theory, though it is more fair to speak about several perturbative approaches, because they are substantially different for different situations.
The simplest version of the perturbation theory addresses the problem of stationary states and energy levels of systems described by time-independent Hamiltonians of the type
ˆ
ˆ (0)
ˆ )1
(
H H
H ,
(6.1)
where the operator
)
1
(
ˆ
H , describing the system’s “perturbation”, is relatively small – in the sense that its addition to the unperturbed operator
(0)
ˆ
H results in a relatively small change of the eigenenergies En of
the system, and the corresponding eigenstates. A typical problem of this type is the 1D weakly anharmonic oscillator (Fig. 1), described by the Hamiltonian (1) with
Weakly
ˆ 2
2
p
m ˆ 2
x
anharmonic
ˆ (0)
0
ˆ
H
,
)
1
(
H
ˆ3
x ˆ 4
x ...
(6.2)
oscillator
2 m
2
with sufficiently small coefficients , , ….
U ( x)
2
2
m x
(0)
)
1
(
(0)
0
E
U
H
U
n
2
(0)
E
E
2
Fig. 6.1. The simplest application of
2
the perturbation theory: a weakly
(0)
E 1
anharmonic 1D oscillator. (Dashed
E 1
lines characterize the unperturbed,
harmonic oscillator.)
x
0
© K. Likharev
QM: Quantum Mechanics
I will use this system as our first example, but let me start by describing the perturbative approach to the general time-independent Hamiltonian (1). In the bra-ket formalism, the eigenproblem (4.68) for the perturbed Hamiltonian, i.e. the stationary Schrödinger equation of the system, is
H (0)
ˆ
H )1
(
ˆ n E n .
(6.3)
n
Let the eigenstates and eigenvalues of the unperturbed Hamiltonian, which satisfy the equation (0) (0)
(0)
(0)
ˆ
H
n
E
n
,
(6.4)
n
be considered as known. In this case, the solution of problem (3) means finding, first, its perturbed eigenvalues En and, second, the coefficients n’ (0) n of the expansion of the perturbed state’s vectors n
in the following series over the unperturbed ones, n’ (0):
(0)
(0)
n n'
n'
n .
(6.5)
n'
Let us plug Eq. (5), with the summation index n’ replaced with n” (just to have a more compact notation in our forthcoming result), into both sides of Eq. (3):
n" (0) n H (0)
ˆ
n" ( 0) n" (0) n H )1(
ˆ
n" (0) n" (0) n E n" (0) .
(6.6)
n
n"
n"
n"
and then inner-multiply all terms by an arbitrary unperturbed bra-vector n’ (0) of the system. Assuming that the unperturbed eigenstates are orthonormal, n’ (0) n” (0) = n’n” , and using Eq. (4) in the first term on the left-hand side, we get the following system of linear equations
(0)
)
1
(
(0)
n" n H n' n E E ,
(6.7)
n'n"
(0)
n
n'
n"
where the matrix elements of the perturbation are calculated, by definition, in the unperturbed brackets: Perturbation’s
)
1
(
(0)
)
1
(
(0)
ˆ
H
n'
H
n"
.
(6.8)
matrix
n'n"
elements
The linear equation system (7) is still exact,1 and is frequently used for numerical calculations.
(Since the matrix coefficients (8) typically decrease when n’ and/or n” become sufficiently large, the sum on the left-hand side of Eq. (7) may usually be truncated, still giving an acceptable accuracy of the solution.) To get analytical results, we need to make approximations. In the simple perturbation theory we are discussing now, this is achieved by the expansion of both the eigenenergies and the expansion coefficients into the Taylor series in a certain small parameter of the problem:
(0)
)
1
(
(2)
E E
E E ...,
(6.9)
n
n
n
n
(0)
)
1
(
(2)
(0)
(0)
(0)
(0)
n'
n n'
n
n'
n
n"
n
...,
(6.10)
where
1 Please note the similarity of Eq. (7) with Eq. (2.215) of the 1D band theory. Indeed, the latter equation is not much more than a particular form of Eq. (7) for the 1D wave mechanics, and a specific (periodic) potential U( x) considered as the perturbation Hamiltonian. Moreover, the whole approximate treatment of the weak-potential limit in Sec. 2.7 is essentially a particular case of the perturbation theory we are discussing now (in its 1st order).
Chapter 6
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( k )
( k)
(0)
k
E
n'
n
.
(6.11)
n
In order to explore the 1st-order approximation, which ignores all terms O(2) and higher, let us plug only the two first terms of the expansions (9) and (10) into the basic equation (7):
)
1
(
)
1
(
)
1
(
(0)
(0)
H n" n n' n
E
E E
.
(6.12)
n'n"
n"n
n'n
(0)
)
1
(
(0)
n
n
n'
n"
Now let us open the parentheses, and disregard all the remaining terms O(2). The result is
)
1
(
)
1
(
)
1
(
(0)
H
E n'
n
( (0)
(0)
E
E ),
(6.13)
n'n
n'n
n
n
n'
This relation is valid for any set of indices n and n’; let us start from the case n = n’, immediately getting a very simple (and practically, the most important!) result:
Energy:
1st-order
)
1
(
)
1
(
(0)
)
1
(
(0)
ˆ
E
H
n
H
n
.
(6.14)
correction
n
nn
For example, let us see what this result gives for two first perturbation terms in the weakly anharmonic oscillator (2):
)
1
(
(0)
3
(0)
(0)
4
(0)
E
n
ˆ x n
n
ˆ x n
.
(6.15)
n
As the reader knows (or should know :-) from the solution of Problem 5.9, the first bracket equals zero, while the second one yields2
)
1
(
3
4
E
x
n n .
(6.16)
n
0 2 2
2
1
4
Naturally, there should be some non-vanishing contribution to the energies from the (typically, larger) perturbation proportional to , so that for its calculation we need to explore the 2nd order of the theory.
However, before doing that, let us complete our discussion of its 1st order.
For n’ n, Eq. (13) may be used to calculate the eigenstates rather than the eigenvalues:
)
1
(
)
1
(
H
(0)
n'
n
n'n
,
for n' .
n
(6.17)
(0)
(0)
E
E
n
n'
This means that the eigenket’s expansion (5), in the 1st order, may be represented as
)
1
(
States:
H
1st-order
)
1
(
(0)
n'n
(0)
n
C n
result
n'
.
(6.18)
(0)
(0)
n'
n E
E
n
n'
The coefficient C n(0) n(1) cannot be found from Eq. (17); however, requiring the final state n to be normalized, we see that other terms may provide only corrections O(2), so that in the 1st order we should take C = 1. The most important feature of Eq. (18) is its denominators: the closer are the unperturbed eigenenergies of two states, the larger is their mutual “interaction” due to the perturbation.
2 A useful exercise for the reader: analyze the relation between Eq. (16) and the result of the classical theory of such weakly anharmonic (”nonlinear”) oscillator – see, e.g., CM Sec. 5.2, in particular, Eq. (5.49).
Chapter 6
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This feature also affects the 1st-order’s validity condition, which may be quantified using Eq.
(17): the magnitudes of the brackets it describes have to be much less than the unperturbed bracket
n n(0) = 1, so that all elements of the perturbation matrix have to be much less than the difference between the corresponding unperturbed energies. For the anharmonic oscillator’s energy corrections (16), this requirement is reduced to E (1)
n
<< 0.
Now we are ready for going after the 2nd-order approximation to Eq. (7). Let us focus on the case n’ = n, because as we already know, only this term will give us a correction to the eigenenergies.
Moreover, since the left-hand side of Eq. (7) already has a small factor H(1) n’n” , the bracket coefficients in that part may be taken from the 1st-order result (17). As a result, we get
)
1
(
)
1
(
)
1
(
H H
(2)
(0)
)
1
(
E
n"
n
H
(6.19)
n
nn"
n"n nn" .
(0)
(0)
n"
n" n E
E
n
n"
Since
)
1
(
ˆ
H has to be Hermitian, we may rewrite this expression as
2
2
H )1
(
n' (0) H )1
(
ˆ
n(0)
Energy:
E (2)
.
(6.20)
n
n'n
2nd-order
(0)
(0)
(0)
(0)
n' n E
E
E
E
correction
n
n'
n' n
n
n'
This is the much-celebrated 2nd-order perturbation result, which frequently (in sufficiently symmetric problems) is the first non-vanishing correction to the state energy – for example, from the cubic term (proportional to ) in our weakly anharmonic oscillator problem (2). To calculate the corresponding correction, we may use another result of the solution of Problem 5.9:
3
x
n' ˆ3
0
x n
2
(6.21)
n( n )(
1 n )
2 1/2
3 3/2
n
(
3 n )
1 3/ 2
n n
n
n' , n3
n' , n 1
n' , n 1
(
)(
1
)(
2
)
3 1/2 n' , n3.
So, according to Eq. (20), we need to calculate
6
x
(2)
2
0
E
n
2
(6.22)
2
n( n )(
1 n )
2 1/2
3 3/2
n
(
3 n )
1 3/ 2
n n
n
n' , n3
n' , n 1
n' , n 1
(
)(
1
)(
2
)
3 1/2 n' , n3
.
n' n
( n n' )
0
The summation is not as cumbersome as may look, because at the curly brackets’ squaring, all mixed products are proportional to the products of different Kronecker deltas and hence vanish, so that we need to sum up only the squares of each term, finally getting
x
(2)
15 2 60 2
11
E
.
(6.23)
n
n n
4 0
30
This formula shows that all energy level corrections are negative, regardless of the sign of .3 On the contrary, the 1st order correction E (1)
n
, given by Eq. (16), does depend on the sign of , so that the net
correction, E (1)
(2)
n
+ En , may be of any sign.
3 Note that this is correct for the ground-state energy correction E (2) g
of any system, because for this state, the
denominators of all terms of the sum (20) are negative, while their numerators are always non-negative.
Chapter 6
Page 4 of 36
QM: Quantum Mechanics
The results (18) and (20) are clearly inapplicable to the degenerate case where, in the absence of perturbation, several states correspond to the same energy level, because of the divergence of their denominators.4 This divergence hints that in this case, the largest effect of the perturbation is the degeneracy lifting, e.g., some splitting of the initially degenerate energy level E(0) (Fig. 2), and that for the analysis of this case we can, in the first approximation, ignore the effect of all other energy levels.
(A careful analysis shows that this is indeed the case until the level splitting becomes comparable with the distance to other energy levels.)
E 1
(0)
1
2(0)
...
(0)
N
(0)
E
E 2
...
Fig. 5.2. Lifting the energy
E
N
level degeneracy by a
(0)
(0)
)
1
(
perturbation (schematically).
ˆ
ˆ
ˆ
ˆ
ˆ
H H
H H
H
Limiting the summation in Eq. (7) to the group of N degenerate states with equal E (0) n’
E(0), we
reduce it to
N
(0)
)
1
(
(0)
n" n H n' n
,
(6.24)
n'n"
(0)
E
E
n
n" 1
where now n’ and n” number the N states of the degenerate group.5 For n = n’, Eq. (24) may be rewritten as
N
)
1
(
1
H E
n"
n'
E
E E
(6.25)
n'n"
n"
n'n"
(0)
,
0
where
)