- three excited states, with { nx,ny,nz} equal to either {2,1,1}, or {1,2,1}, or {1,1,2}, with equal energies 211= 121 = 112 = (22+12+12)0 = 60.
According to the above simple formulation of the Pauli principle, each of these orbital energy levels can accommodate up to two fermions. Hence the lowest-energy (ground) state of the five-fermion system is achieved by placing two of them on the ground level 111 = 30, and the remaining three particles, in any of the degenerate “excited” states of energy 60, so that the ground-state energy of the system is 12 2 2
E 2 3 3 6 24
.
(8.57)
g
0
0
0
2
ma
Moreover, in many cases, relatively weak interaction between fermions does not blow up such a simple quantum state classification scheme qualitatively, and the Pauli principle allows tracing the order of single-particle state filling. This is exactly the simple approach that has been used in our discussion of atoms in Sec. 3.7. Unfortunately, it does not allow for a more specific characterization of the ground states of most atoms, in particular the evaluation of the corresponding values of the quantum numbers S, L, and J that characterize the net angular momenta of the atom, and hence its response to an external magnetic field. These numbers are defined by relations similar to Eqs. (48), each for the corresponding vector operator of the net angular momenta:
N
N
N
ˆS ˆ
ˆ
s ,
L
ˆ
ˆ
ˆ
l ,
J
j ;
(8.58)
k
k
k
k 1
k 1
k 1
note that these definitions are consistent with Eq. (5.170) applied both to the angular momenta s k, l k, and j k of each particle, and to the full vectors S, L, and J. When the numbers S, L, and J for a state are known, they are traditionally recorded in the form of the so-called Russell-Saunders symbols:20
2 S 1
L ,
(8.59)
J
where S and J are the corresponding values of these quantum numbers, while L is a capital letter, encoding the quantum number L – via the same spectroscopic notation as for single particles (see Sec.
3.6): L = S for L = 0, L = P for L = 1, L = D for L = 2, etc. (The reason why the front superscript of the Russel-Saunders symbol lists 2 S + 1 rather than just S, is that according to the last of Eqs. (48), it 20 Named after H. Russell and F. Saunders, whose pioneering (circa 1925) processing of experimental spectral-line data has established the very idea of vector addition of the electron spins, described by the first of Eqs. (58).
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shows the number of possible values of the quantum number MS, which characterizes the state’s spin degeneracy, and is called its multiplicity.)
For example, for the simplest, hydrogen atom ( Z = 1), with its single electron in the ground 1 s state, L = l = 0, S = s = ½, and J = S = ½, so that its Russell-Saunders symbol is 2 S 1/2. Next, the discussion of the helium atom ( Z = 2) in the previous section has shown that in its ground state L = 0
(because of the 1 s orbital state of both electrons), and S = 0 (because of the singlet spin state), so that the total angular momentum also vanishes: J = 0. As a result, the Russell-Saunders symbol is 1 S 0. The structure of the next atom, lithium ( Z = 3) is also easy to predict, because, as was discussed in Sec. 3.7, its ground-state electron configuration is 1 s 22 s 1, i.e. includes two electrons in the “helium shell”, i.e. on the 1 s orbitals (now we know that they are actually in a singlet spin state), and one electron in the 2 s state, of higher energy, also with zero orbital momentum, l = 0. As a result, the total L in this state is evidently equal to 0, and S is equal to ½, so that J = ½, meaning that the Russell-Saunders symbol of lithium is 2 P 1/2. Even in the next atom, beryllium ( Z = 4), with the ground state configuration 1 s 22 s 2, the symbol is readily predictable, because none of its electrons has non-zero orbital momentum, giving L =
0. Also, each electron pair is in the singlet spin state, i.e. we have S = 0, so that J = 0 – the quantum number set described by the Russell-Saunders symbol 1 S 0 – just as for helium.
However, for the next, boron atom ( Z = 5), with its ground-state electron configuration 1 s 22 s 22 p 1
(see, e.g., Fig. 3.24), there is no obvious way to predict the result. Indeed, this atom has two pairs of electrons, with opposite spins, on its two lowest s-orbitals, giving zero contributions to the net S, L, and J. Hence these total quantum numbers may be only contributed by the last, fifth electron with s = ½ and l = 1, giving S = ½, L = 1. As was discussed in Sec. 5.7 for the single-particle case, the vector addition of the angular momenta S and L enables two values of the quantum number J: either L + S = ³/2 or L – S
= ½. Experiment shows that the difference between the energies of these two states is very small (~2
meV), so that at room temperature (with k B T 26 meV) they are both occupied, with the genuine ground state having J = ½, so that its Russell-Saunders symbol is 2 P 1/2.
Such energy differences, which become larger for heavier atoms, are determined both by the Coulomb and spin-orbit21 interactions between the electrons. Their quantitative analysis is rather involved (see below), but the results tend to follow simple phenomenological Hund rules, with the following hierarchy:
Rule 1. For a given electron configuration, the ground state has the largest possible S, and hence the largest multiplicity 2 S + 1.
Rule 2. For a given S, the ground state has the largest possible L.
Rule 3. For given S and L, J has its smallest possible value, L – S , if the given sub-shell { n, l}
is filled not more than by half, while in the opposite case, J has its largest possible value, L + S.
Let us see how these rules work for the boron atom we have just discussed. For it, the Hund Rules 1 and 2 are satisfied automatically, while the sub-shell { n = 2, l = 1}, which can house up to 2(2 l
+ 1) = 6 electrons, is filled with just one 2 p electron, i.e. by less than a half of the maximum value. As a result, the Hund Rule 3 predicts the ground state’s value J = ½, in agreement with experiment.
21 In light atoms, the spin-orbit interaction is so weak that it may be reasonably well described as an interaction of the total momenta L and S of the system – the so-called LS (or “Russell-Saunders”) coupling. On the other hand, in very heavy atoms, the interaction is effectively between the net momenta j k = l k + s k of the individual electrons
– the so-called jj coupling. This is the reason why in such atoms the Hund rule 3 may be violated.
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Generally, for lighter atoms, the Hund rules are well obeyed. However, the lower down the Hund rule hierarchy, the less “powerful” the rules are, i.e. the more often they are violated in heavier atoms.
Now let us discuss possible approaches to a quantitative theory of multiparticle systems – not only atoms. As was discussed in Sec. 1, if fermions do not interact directly, the stationary states of the system have to be the antisymmetric eigenstates of the permutation operator, i.e. satisfy Eq. (55). To understand how such states may be formed from the single-electron ones, let us return for a minute to the case of two electrons, and rewrite Eq. (11) in the following compact form:
2
state
1
state
(8.60a)
1
β '
'
'
1
1,
number
particle
2
2
β '
2,
number
particle
where the direct product signs are just implied. In this way, the Pauli principle is mapped on the well-known property of matrix determinants: if any of two columns of a matrix coincide, its determinant vanishes. This Slater determinant approach22 may be readily generalized to N fermions occupying any N
(not necessarily the lowest-energy) single-particle states , ’, ’’, etc:
list
state
β'
"
particle
Slater
1
β'
"
determinant
N
list
(8.60b)
N 1/2
!
β'
"
N
The Slater determinant form is extremely nice and compact – in comparison with direct writing of a sum of N! products, each of N ket factors. However, there are two major problems with using it for practical calculations:
(i) For the calculation of any bra-ket product (say, within the perturbation theory) we still need to spell out each bra- and ket-vector as a sum of component terms. Even for a limited number of electrons (say N ~ 102 in a typical atom), the number N! ~ 10160 of terms in such a sum is impracticably large for any analytical or numerical calculation.
(ii) In the case of interacting fermions, the Slater determinant does not describe the eigenvectors of the system; rather the stationary state is a superposition of such basis functions, i.e. of the Slater determinants – each for a specific selection of N states from the full set of single-particle states – that is generally larger than N.
For atoms and simple molecules, whose filled-shell electrons may be excluded from an explicit analysis (by describing their effects, approximately, with effective pseudo-potentials), the effective number N may be reduced to a smaller number N ef of the order of 10, so that N ef! < 106, and the Slater determinants may be used for numerical calculations – for example, in the Hartree-Fock theory – see the next section. However, for condensed-matter systems, such as metals and semiconductors, with the 22 It was suggested in 1929 by John C. Slater.
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number of free electrons is of the order of 1023 per cm3, this approach is generally unacceptable, though with some smart tricks (such as using the crystal’s periodicity) it may be still used for some approximate (also mostly numerical) calculations.
These challenges make the development of a more general theory that would not use particle numbers (which are superficial for indistinguishable particles to start with) a must for getting any final analytical results for multiparticle systems. The most effective formalism for this purpose, which avoids particle numbering at all, is called the second quantization.23 Actually, we have already discussed a particular version of this formalism, for the case of the 1D harmonic oscillator, in Sec. 5.4. As a reminder, after the definition (5.65) of the “creation” and “annihilation” operators via those of the particle’s coordinate and momentum, we have derived their key properties (5.89),
1/ 2
†
1/ 2
ˆ
a n n
n 1 ,
ˆ a n n
1
n 1 ,
(8.61)
where n are the stationary (Fock) states of the oscillator. This property allows an interpretation of the operators’ actions as the creation/annihilation of a single excitation with the energy 0 – thus justifying the operator names. In the next chapter, we will show that such excitation of an electromagnetic field mode may be interpreted as a massless boson with s = 1, called the photon.
In order to generalize this approach to arbitrary bosons, not appealing to a specific system, we may use relations similar to Eq. (61) to define the creation and annihilation operators. The definitions look simple in the language of the so-called Dirac states, described by ket-vectors Dirac
N , N ,... N ,... ,
(8.62)
1
2
j
state
where Nj is the state occupancy, i.e. the number of bosons in the single-particle state j. Let me emphasize that here the indices 1, 2, … j,… number single-particle states (including their spin parts) rather than particles. Thus the very notion of an individual particle’s number is completely (and for indistinguishable particles, very relevantly) absent from this formalism. Generally, the set of single-particle states participating in the Dirac state may be selected arbitrarily, provided that it is full and orthonormal in the sense
'
'
N ,N
, '
... N , ... N ,N
,
... N , ...
...
...,
(8.63)
1
2
j'
1
2
j
N N'
N N'
N N'
1
1
2
2
j
j
though for systems of non- (or weakly) interacting bosons, using the stationary states of individual particles in the system under analysis is almost always the best choice.
Now we can define the particle annihilation operator as follows:
Boson
ˆ a N , N ,... N ,...
1/ 2
N
N , N ,... N ,...
1
.
(8.64) annihilation
j
1
2
j
j
1
2
j
operator
Note that the pre-ket coefficient, similar to that in the first of Eqs. (61), guarantees that any attempt to annihilate a particle in an initially unpopulated state gives the non-existing (“null”) state: ˆ a N , N ,...0 ,... 0 ,
(8.65)
j
1
2
j
23 It was invented (first for photons and then for arbitrary bosons) by P. Dirac in 1927, and then modified in 1928
for fermions by E. Wigner and P. Jordan. Note that the term “second quantization” is rather misleading for the non-relativistic applications we are discussing here, but finds certain justification in the quantum field theory.
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where the symbol 0 j means zero occupancy of the j th state. According to Eq. (63), an equivalent way to write Eq. (64) is
'
N , '
N ,..., '
N ,... ˆ a . N , N ,.., N ,...
1/ 2
N
...
...
(8.66)
1
2
j
j
1
2
j
j
N N' N N'
N' , N 1
1
1
2
2
j
j
According to the general Eq. (4.65), the matrix element of the Hermitian conjugate operator †
ˆ a is
j
'
'
'
†
'
'
'
*
N ,N ,..., N ,... ˆ a N , N ,... N ,... N , N ,..., N ,... ˆ a N ,N ,..., N ,...
1
2
j
j
1
2
j
1
2
j
j
1
2
j
N , N ,..., N ,...
'
'
'
'
'
N
N ,N
N
N
...
(8.67)
1
2
j
j1/2
,...,
,...
1
1
2
j
j1/2
...
N N' N N'
N , N' 1
j
j
N
j
1
1
2
2
1 1/2
...
...,
N N' N N'
N ,
1 N'
1
1
2
2
j
j
meaning that
Boson
creation
ˆ†
a N , N ,..., N ,... N
N N
N
(8.68)
j
1
2
j
j 11/2 , ,...,
,...
1
,
1
2
j
operator
in total compliance with the second of Eqs. (61). In particular, this particle creation operator allows the description of the generation of a single particle from the vacuum (not null!) state 0, 0, …: ˆ†
a
,
0 ,...,
0
0 ,...,0 ,
0
1
,...,
0
,...0 ,
(8.69)
j
j
j
and hence a product of such operators may create, from the vacuum, a multiparticle state with an arbitrary set of occupancies: 24
ˆ†
a ˆ†
a ... ˆ†
a ˆ†
a ˆ†
a ... ˆ†
a
... ,
0 ,...
0
N N
N N
(8.70)
1
1
1
2
2
2
!1 2
!... 1/2
,
,... .
1
2
N
N
1 times
2 times
Next, combining Eqs. (64) and (68), we get
ˆ†
a ˆ a N , N ,... N ,... N N , N ,..., N ,... , (8.71)
j
j
1
2
j
j
1
2
j
so that, just as for the particular case of harmonic oscillator excitations, the operator Number-counting
N ˆ
a ˆ†
a ˆ
(8.72)
j
j
j
operator
“counts” the number of particles in the j th single-particle state, while preserving the whole multiparticle state. Acting on a state by the creation-annihilation operators in the reverse order, we get ˆ a ˆ†
a N , N ,..., N ,... N
N N
N
(8.73)
j
j
1
2
j
j 1 , ,..., ,... .
1
2
j
Eqs. (71) and (73) show that for any state of a multiparticle system (which may be represented as a linear superposition of Dirac states with all possible sets of numbers Nj), we may write ˆ a ˆ†
a ˆ†
a ˆ a
ˆ a , ˆ†
a I ,
ˆ
(8.74)
j
j
j
j
j
j
24 The resulting Dirac state is not an eigenstate of every multiparticle Hamiltonian. However, we will see below that for a set of non-interacting particles it is a stationary state, so that the full set of such states may be used as a good basis in perturbation theories of systems of weakly interacting particles.
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again in agreement with what we had for the 1D oscillator – cf. Eq. (5.68). According to Eqs. (63), (64), and (68), the creation and annihilation operators corresponding to different single-particle states do commute, so that Eq. (74) may be generalized as
ˆ a , ˆ†
ˆ
a
I ,
(8.75)
j
j'
jj'
Bosonic
operators:
while the similar operators commute, regardless of which states do they act upon:
commutation
relations
ˆ†
a , ˆ†
a
ˆ a ,
a
0ˆ
ˆ
.
(8.76)
j
j'
j j'
As was mentioned earlier, a major challenge in the Dirac approach is to rewrite the Hamiltonian of a multiparticle system, that naturally carries particle numbers k (see, e.g., Eq. (22) for k = 1, 2), in the second quantization language, in which there are no these numbers. Let us start with single- particle components of such Hamiltonians, i.e. operators of the type
N
Single-
F ˆ f ˆ .
(8.77) particle
k
operator
k 1
where all N operators f âre similar, besides that each of them acts on one specific ( k th) particle, and N
k
is the total number of particles in the system, which is evidently equal to the sum of single-particle state occupancies:
N N .
(8.78)
j
j
The most important examples of such operators are the kinetic energy of N similar single particles, and their potential energy in an external field:
N
p 2
ˆ
N
T ˆ k , ˆ U
ˆ
u(r .)
(8.79)
k
k 1 2 m
k 1
For bosons, instead of the Slater determinant (60), we have to write a similar expression, but without the sign alternation at permutations:
N
N
1/ 2
!... !...
1
j
N ,... N ,...
... ' " ... ,
(8.80)
1
j
N!
P
N operands
sometimes called the permanent. Note again that the left-hand side of this relation is written in the Dirac notation (that does not use particle numbering), while on its right-hand side, just in relations of Secs. 1
and 2, the particle numbers are coded with the positions of the single-particle states inside the state vectors, and the summation is over all different permutations of the states in the ket – cf. Eq. (10).
(According to the basic combinatorics,25 there are N!/( N 1!... Nj!...) such permutations, so that the front coefficient in Eq. (80) ensures the normalization of the Dirac state, provided that the single-particle states , ’, …are normalized.) Let us use Eq. (80) to spell out the following matrix element for a system with ( N –1) particles:
25 See, e.g., MA Eq. (2.3).
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ˆ
... N ,... N ,...
1
F ... N ,...
1 N ,...
j
j'
j
j'
N !...( N )!...(
1
N )!...
1
N
(8.81)
1
j
j'
N N
' "
f
' "
j
j
'
1
1/ 2
ˆ
...
...
...
... ,
( N )!
1
k
P N 1
P N 1
k 1
where all non-specified occupation numbers in the corresponding positions of the bra- and ket-vectors are equal to each other. Each single-particle operator f ˆ participating in the operator sum, acts on the k
bra- and ket-vectors of states only in one ( k th) position, giving the following result, independent of the position number:
f ˆ
f ˆ
f .
(8.82)
j
th
k
j'
th
j
j'
jj'
i k
n
position
k
in
position
Since in both permutation sets participating in Eq. (81), with ( N – 1) state vectors each, all positions are equivalent, we can fix the position (say, take the first one) and replace the sum over k with the multiplication by of the bracket by ( N – 1). The fraction of permutations with the necessary bra-vector (with number j) in that position is Nj/( N – 1), while that with the necessary ket-vector (with number j’) in the same position is Nj’/( N – 1). As the result, the permutation sum in Eq. (81) reduces to N
N
( N )
1
j
j'
f
' "
' "
(8.83)
jj
...
... ...
... ,
N 1 N 1
'
P N 2 P N 2
where our specific position k is now excluded from both the bra- and ket-vector permutations. Each of these permutations now includes only ( Nj – 1) states j and ( Nj’ – 1) states j’, so that, using the state orthonormality, we finally arrive at a very simple result:
ˆ
... N ,... N ,...
1
F ... N ,...
1 N ,...
j
j'
j
j'
N !...( N )!...(
1
N )!...
1
N
N
1
j
j'
1/ 2
j
j'
( N 2)!
N N
N
f
(8.84)
j
j'
(
)
1
( N )!
1
N 1 N 1 jj' N !...( N )!...(
1
N )!...
1
1
j
j'
N N
f
j
j' 1/ 2
.
jj'
On the other hand, let us calculate matrix elements of the following operator:
†
f
ˆ a ˆ a .
(8.85)
jj'
j
j'
j, j'
A direct application of Eqs. (64) and (68) shows that the only non-vanishing of the elements are
†
... N ,... N ,...
1
f ˆ a ˆ a ... N ,...,
1
N ,... N N
f .
(8.86)
j
j'
jj '
j
j'
j
j'
j j'1/2 jj'
But this is exactly the last form of Eq. (84), so that in the basis of Dirac states, the operator (77) may be represented as
Single-
particle
operator
F ˆ f a ˆ† a ˆ .
(8.87)
jj'
j
j'
in Dirac
j, j'
language
This beautifully simple relation is the key formula of the second quantization theory, and is essentially the Dirac-language analog of Eq. (4.59) of the single-particle quantum mechanics. Each term of the sum (87) may be described by a very simple mnemonic rule: for each pair of single-particle states Chapter 8
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j and j’, annihilate a particle in the state j’, create one in the state j, and weigh the result with the corresponding single-particle matrix element. One of the corollaries of Eq. (87) is that the expectation value of an operator whose eigenstates coincide with the Dirac states is
ˆ
F ... N ,... F ... N ,...
f N
(8.88)
j
j
,
jj
j
j
with an evident physical interpretation as the sum of single-particle expectation values over all states, weighed by the occupancy of each state.
Proceeding to fermions, which have to obey the Pauli principle, we immediately notice that any occupation number Nj may only take two values, 0 or 1. To account for that, and also make the key relation (87) valid for fermions as well, the creation-annihilation operators are defined by the following relations:
ˆ a N , N ,...,0 ,... ,
0
ˆ a N , N
1
,..., ,... ( )
1 ,
1
( j )
1 N , N ,...,0 ,... ,
(8.89) Fermion
j
1
2
j
j
1
2
j
1
2
j
creation-
annihilation
ˆ†
a N , N ,...,0 ,... ( )
1 ,
1
( j )
1 N , N
1
,..., ,... ,
ˆ†
a N , N
1
,..., ,... ,
0
(8.90) operators
j
1
2
j
1
2
j
j
1
2
j
where the symbol ( J, J’) means the sum of all occupancy numbers in the states with numbers from J to J’, including the border points:
J'
( J , J' ) N ,
(8.91)
j
j J
so that the sum participating in Eqs. (89)-(90) is the total occupancy of all states with the numbers below j. (The states are supposed to be numbered in a fixed albeit arbitrary order.) As a result, these relations may be conveniently summarized in the following verbal form: if an operator replaces the j th state’s occupancy with the opposite one (either 1 with 0, or vice versa), it also changes the sign before the result if (and only if) the total number of particles in the states with j’ < j is odd.
Let us use this (perhaps somewhat counter-intuitive) sign alternation rule to spell out the ket-vector 11 of a completely filled two-state system, formed from the vacuum state 00 in two different ways. If we start by creating a fermion in the state 1, we get
ˆ†
a
,
0 0 ( )
1 0 ,
1 0 ,
1 0 ,
ˆ†
a ˆ†
a
,
0 0 ˆ†
a
,
1 0 ( )
1 1 ,
1 1 ,
1 1 ,
(8.92a)
1
2
1
2
while if the operator order is different, the result is
ˆ†
a
,
0 0 ( )
1 0 ,
0 1 ,
0 1 ,
ˆ†
a ˆ†
a
,
0 0 ˆ†
a
,
0 1 ( )
1 0 ,
1 1 ,
1 1 ,
(8.92b)
2
1
2
1
so that
ˆ†
a ˆ†
a ˆ†
a ˆ†
a
,
0 0 0 .
(8.93)
1
2
2
1
Since the action of any of these operator products on any initial state rather than the vacuum one also gives the null ket, we may write the following operator equality:
ˆ†
a ˆ†
a ˆ†
a ˆ†
a
ˆ†
a , †
a
.
0ˆ
ˆ
(8.94)
1
2
2
1
1
2
It is straightforward to check that this result is valid for Dirac vectors of an arbitrary length, and does not depend on the occupancy of other states, so that we may generalize it as
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ˆ†
a , ˆ†
a
a a
;
(8.95)
j
j'
ˆ ,
j
j' 0
ˆ
ˆ
Fermionic
operators: these equalities hold for j = j’ as well. On the other hand, an absolutely similar calculation shows that commutation the mixed creation-annihilation commutators do depend on whether the states are different or not:26
relations
a ˆ , a ˆ† I
ˆ
.
(8.96)
j
j'
jj'
These equations look very much like Eqs. (75)-(76) for bosons, “only” with the replacement of commutators with anticommutators. Since the core laws of quantum mechanics, including the operator compatibility (Sec. 4.5) and the Heisenberg equation (4.199) of operator evolution in time, involve commutators rather than anticommutators, one might think that all the behavior of bosonic and fermionic multiparticle systems should be dramatically different. However, the difference is not as big as one could expect; indeed, a straightforward check shows that the sign factors in Eqs. (89)-(90) just compensate those in the Slater determinant, and thus make the key relation (87) valid for the fermions as well. (Indeed, this is the very goal of the introduction of these factors.)
To illustrate this fact on the simplest example, let us examine what does the second quantization formalism say about the dynamics of non-interacting particles in the system whose single-particle properties we have discussed repeatedly, namely two nearly-similar potential wells, coupled by tunneling through the separating potential barrier – see, e.g., Figs. 2.21 or 7.4. If the coupling is so small that the states localized in the wells are only weakly perturbed, then in the basis of these states, the single-particle Hamiltonian of the system may be represented by the 22 matrix (5.3). With the energy reference selected at the middle between the energies of unperturbed states, the coefficient b vanishes, this matrix is reduced to
c
c
h
z
c σ
,
with c c ic ,
(8.97)
x
y
c
c
z
and its eigenvalues to
c,
with c c
(8.98)
2 2 2
c c c
x
y
z 1/ 2 .
Now following the recipe (87), we can use Eq. (97) to represent the Hamiltonian of the whole system of particles in terms of the creation-annihilation operators:
†
†
†
†
ˆ
H c ˆ a ˆ a c ˆ a ˆ a c ˆ a ˆ a c ˆ a ˆ a , (8.99)
z 1
1
1
2
2
1
z
2
2
where †
ˆ a and ˆ a are the operators of creation and annihilation of a particle in the corresponding
,
1 2
,
1 2
potential well. (Again, in the second quantization approach the particles are not numbered at all!) As Eq. (72) shows, the first and the last terms of the right-hand side of Eq. (99) describe the particle energies 1,2 = c z in uncoupled wells,
c ˆ†
a ˆ
ˆ
ˆ
a c N N ,
c ˆ†
a ˆ
ˆ
ˆ
a c N N ,
(8.100)
z 1
1
z
1
1
1
z
2
2
z
2
2
2
26 A by-product of this calculation is proof that the operator defined by Eq. (72) counts the number of particles Nj (now equal to either 1 or 0), just at it does for bosons.
Chapter 8
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while the sum of the middle two terms is the second-quantization description of tunneling between the wells.
Now we can use the general Eq. (4.199) of the Heisenberg picture to spell out the equations of motion of the creation-annihilation operators. For example,
i ˆ
a
a H c a a a c a a a c a a a c a a
(8.101)
1
ˆ ˆ,
1
†
†
†
†
ˆ , ˆ ˆ
ˆ , ˆ ˆ
ˆ , ˆ ˆ
ˆ , ˆ ˆ .
z
1
1
1
a
1
1
2
1
2
1
z
1
2
2
Since the Bose and Fermi operators satisfy different commutation relations, one could expect the right-hand side of this equation to be different for bosons and fermions. However, it is not so. Indeed, all commutators on the right-hand side of Eq. (101) have the following form:
ˆ a , ˆ†
a ˆ a ˆ a ˆ†
a ˆ a ˆ†
a ˆ a ˆ a .
(8.102)
j
j'
j"
j
j'
j"
j'
j"
j
As Eqs. (74) and (94) show, the first pair product of operators on the right-hand side may be recast as ˆ a ˆ†
ˆ
a I ˆ†
a ˆ a ,
(8.103)
j
j'
jj'
j'
j
where the upper sign pertains to bosons and the lower one to fermions, while according to Eqs. (76) and (95), the very last pair product in Eq. (102) is
ˆ a ˆ a ˆ a ˆ a ,
(8.104)
j"
j
j
j"
with the same sign convention. Plugging these expressions into Eq. (102), we see that regardless of the particle type, there is a universal (and generally very useful) commutation relation
a ˆ , a ˆ† a ˆ a ˆ ,
(8.105)
j
j'
j"
j"
jj'
valid for both bosons and fermions. As a result, the Heisenberg equation of motion for the operator ˆ a , 1
and the equation for ˆ a (which may be obtained absolutely similarly), are also universal:27
2
i ˆ a
c ˆ a c ˆ a ,
1
z 1
2
(8.106)
i ˆ a
c ˆ a c ˆ a .
2
1
z
2
This is a system of two coupled, linear differential equations, which is similar to the equations for the c-number probability amplitudes of single-particle wavefunctions of a two-level system – see, e.g., Eq. (2.201) and the model solution of Problem 4.25. Their general solution is a linear superposition ˆ a ( t)
ˆ ()
t
(8.107)
,
1 2
,
1 2
exp .
As usual, to find the exponents , it is sufficient to plug in a particular solution a ˆ
t
( ) ˆ
,
1 2
,
1 2
exp t
into Eq. (106) and require that the determinant of the resulting homogeneous, linear system for the
“coefficients” (actually, time-independent operators) ˆ
equals zero. This gives us the following
,
1 2
characteristic equation
†
27 Equations of motion for the creation operators ˆ a are just the Hermitian-conjugates of Eqs. (106), and do not
,
1 2
add any new information about the system’s dynamics.
Chapter 8
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c i
z
c
0 ,
(8.108)
c
c i
z
with two roots = i/2, where 2 c/ – cf. Eq. (5.20). Now plugging each of the roots, one by one, into the system of equations for ˆ
, we can find these operators, and hence the general solution of
,
1 2
system (98) for arbitrary initial conditions.
Let us consider the simple case cy = cz = 0 (meaning in particular that the wells are exactly aligned, see Fig. 2.21), so that /2 c = cx; then the solution of Eq. (106) is
t
t
t
t
ˆ a ( t) ˆ a (0) cos
i ˆ a (0)sin
,
ˆ a ( t) i ˆ a (0)sin
ˆ a (0) cos
.
(8.109)
1
1
2
2
2
2
1
2
2
2
Multiplying the first of these relations by its Hermitian conjugate, and ensemble-averaging the result, we get
t
t
Quantum
N ˆ†
a ( t) ˆ a ( t) ˆ†
a ( )
0 ˆ a ( )
0 cos2
ˆ†
a ( )
0 ˆ a ( )
0 sin 2
1
1
1
1
1
2
2
oscillations:
second
2
2
(8.110)
quantization
t
t
i ˆ†
a ( )
0 ˆ a ( )
0 ˆ†
a ( )
0 ˆ a ( )
0 sin
cos
.
form
1
2
2
1
2
2
Let the initial state of the system be a single Dirac state, i.e. have a definite number of particles in each well; in this case, only the two first terms on the right-hand side of Eq. (110) are different from zero, giving:28
t
t
N N ( )
0 cos2
N ( )
0 sin 2
.
(8.111)
1
1
2
2
2
For one particle, initially placed in either well, this gives us our old result (2.181) describing the usual quantum oscillations of the particle between two wells with the frequency . However, Eq. (111) is valid for any set of initial occupancies; let us use this fact. For example, starting from two particles, with initially one particle in each well, we get N 1 = 1, regardless of time. So, the occupancies do not oscillate, and no experiment may detect the quantum oscillations, though their frequency is still formally present in the time evolution equations. This fact may be interpreted as the simultaneous quantum oscillations of two particles between the wells, exactly in anti-phase. For bosons, we can go on to even larger occupancies by preparing the system, for example, in the state with N 1(0) = N, N 2(0) = 0.
The result (111) says that in this case, we see that the quantum oscillation amplitude increases N-fold; this is a particular manifestation of the general fact that bosons can be (and evolve in time) in the same quantum state. On the other hand, for fermions we cannot increase the initial occupancies beyond 1, so that the largest oscillation amplitude we can get is if we initially fill just one well.
The Dirac approach may be readily generalized to more complex systems. For example, Eq. (99) implies that an arbitrary system of potential wells with weak tunneling coupling between the adjacent wells may be described by the Hamiltonian
H ˆ a† a ˆ
a† a ˆ
,
h.c.
(8.112)
j
j
j
jj'
j
j'
j
j, j'
28 For the second well’s occupancy, the result is complementary, N 2( t) = N 1(0)sin2 t + N 2(0)cos2 t , giving in particular a good sanity check: N 1( t) + N 2( t) = N 1(0) + N 2(0) = const.
Chapter 8
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where the symbol { j, j’} means that the second sum is restricted to pairs of next-neighbor wells – see, e.g., Eq. (2.203) and its discussion. Note that this Hamiltonian is still a quadratic form of the creation-annihilation operators, so the Heisenberg-picture equations of motion of these operators are still linear, and its exact solutions, though possibly cumbersome, may be studied in detail. Due to this fact, the Hamiltonian (112) is widely used for the study of some phenomena, for example, the very interesting Anderson localization effects, in which a random distribution of the localized-site energies j prevents tunneling particles, within a certain energy range, from spreading to unlimited distances.29
8.4. Perturbative approaches
The situation becomes much more difficult if we need to account for direct interactions between the particles. Let us assume that the interaction may be reduced to that between their pairs (as it is the case at their Coulomb interaction and most other interactions30), so that it may be described by the following “pair-interaction” Hamiltonian
1 N
ˆ
U
ˆ
u (r , r ,)
(8.113)
int
2
int
k
k'
k , k ' 1
k k '
with the front factor of ½ compensating the double-counting of each particle pair. The translation of this Pair-interaction
operator to the second-quantization form may be done absolutely similarly to the derivation of Eq. (87), Hamiltonian: and gives a similar (though naturally more involved) result
two forms
1
ˆ
U
u
ˆ†
a ˆ†
a ˆ a ˆ
a ,
(8.114)
int
2
jj'll'
j
j' l' l
j, j' , l, l'
where the two-particle matrix elements are defined similarly to Eq. (82):
u
ˆ u .
(8.115)
jj'll'
j
j'
int
l
l'
The only new feature of Eq. (114) is a specific order of the indices of the creation operators. Note the mnemonic rule of writing this expression, similar to that for Eq. (87): each term corresponds to moving a pair of particles from states l and l’ to states j’ and j (in this order!) factored with the corresponding two-particle matrix element (115).
However, with the account of such term, the resulting Heisenberg equations of the time evolution of the creation/annihilation operators are nonlinear, so that solving them and calculating observables from the results is usually impossible, at least analytically. The only case when some general results may be obtained is the weak interaction limit. In this case, the unperturbed Hamiltonian contains only single-particle terms such as (79), and we can always (at least conceptually :-) find such a basis of orthonormal single-particle states j in which that Hamiltonian is diagonal in the Dirac representation:
29 For a review of the 1D version of this problem, see, e.g., J. Pendry, Adv. Phys. 43, 461 (1994).
30 A simple but important example from the condensed matter theory is the so-called Hubbard model, in which particle repulsion limits their number on each of localized sites to either 0, or 1, or 2, with negligible interaction of the particles on different sites – though the next-neighbor sites are still connected by tunneling – as in Eq. (112).
Chapter 8
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H ˆ (0) (0)
a ˆ† a ˆ .
(8.116)
j
j
j
j
Now we can use Eq. (6.14), in this basis, to calculate the interaction energy as a first-order perturbation:
)
1
(
1
ˆ
E
N , N ,... U N , N ,...
N , N ,... u ˆ†
a ˆ†
a ˆ a ˆ a N , N ,...
int
1
2
int
1
2
2
1
2
jj'll'
j
j'
l'
l
1
2
j, j' , l, l'
(8.117)
1
u N , N ,... ˆ†
a ˆ†
a ˆ a ˆ a N , N ,... .
2
jj'll'
1
2
j
j'
l'
l
1
2
j, j' , l, l'
Since, according to Eq. (63), the Dirac states with different occupancies are orthogonal, the last long bracket is different from zero only for three particular subsets of its indices:
(i) j j’, l = j, and l’ = j’. In this case, the four-operator product in Eq. (117) is equal to ˆ†
a ˆ†
a ˆ a ˆ a , and applying the commutation rules twice, we can bring it to the so-called normal ordering, j
j'
j'
j
with each creation operator standing to the right of the corresponding annihilation operator, thus forming the particle number operator (72):
a ˆ† a ˆ† a ˆ a ˆ a ˆ† a ˆ† a ˆ a ˆ a ˆ† a ˆ a ˆ†
a ˆ a ˆ† a ˆ a ˆ† a ˆ N ˆ N ˆ , (8.118)
j
j'
j'
j
j
j'
j
j'
j
j
j'
j'
j
j
j'
j'
j
j'
with a similar sign of the final result for bosons and fermions.
(ii) j j’, l = j’, and l’ = j. In this case, the four-operator product is equal to a ˆ† a ˆ† a ˆ a ˆ , and j
j'
j
j'
bringing it to the form N ˆ N ˆ requires only one commutation:
j
j'
a ˆ† a ˆ† a ˆ a ˆ a ˆ† a ˆ a ˆ†
a ˆ a ˆ† a ˆ a ˆ† a ˆ N ˆ N ˆ , (8.119)
j
j'
j
j'
j
j
j'
j'
j
j
j'
j'
j
j'
with the upper sign for bosons and the lower sign for fermions.
(iii) All indices are equal to each other, giving a ˆ† a ˆ† a ˆ a ˆ a ˆ† a ˆ† a ˆ a ˆ . For fermions, such an j
j' l' l
j
j
j
j
operator (that “tries” to create or to kill two particles in a row, in the same state) immediately gives the null-vector. In the case of bosons, we may use Eq. (74) to commute the internal pair of operators, getting ˆ†
a ˆ†
a ˆ a ˆ a ˆ†
a ˆ a ˆ†
ˆ
a I
ˆ
ˆ
ˆ
ˆ
a N ( N I ) .
(8.120)
j
j
j
j
j
j
j
j
j
j
Note, however, that this expression formally covers the fermion case as well (always giving zero). As a result, Eq. (117) may be rewritten in the following universal form:
Particle
interaction:
)
1
(
1
1
1st-order
E
N N u u
N N u
(8.121)
int
j
j' jj'jj'
jj'j'j
(
)
1
.
energy
2
j
j
jjjj
j, j'
2 j
j
correction
j'
The corollaries of this important result are very different for bosons and fermions. In the former case, the last term usually dominates, because the matrix elements (115) are typically the largest when all basis functions coincide. Note that this term allows a very simple interpretation: the number of the diagonal matrix elements it sums up for each state ( j) is just the number of interacting particle pairs residing in that state.
Chapter 8
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In contrast, for fermions the last term is zero, and the interaction energy is proportional to the difference of the two terms inside the first parentheses. To spell them out, let us consider the case when there is no direct spin-orbit interaction. Then the vectors j of the single-particle state basis may be represented as direct products o j m j of their orbital and spin-orientation parts. (Here, for the brevity of notation, I am using m instead of ms.) For spin-½ particles, including electrons, mj may equal only either +½ or –½; in this case the spin part of the first matrix element, proportional to ujj’jj’, equals m m' m m' ,
(8.122)
where, as in the general Eq. (115), the position of a particular state vector in each direct product is encoding the particle’s number. Since the spins of different particles are defined in different Hilbert spaces, we may move their state vectors around to get
m m' m m' m m m' m' 1,
(8.123)
1
2
for any pair of j and j’. On the other hand, the second matrix element, ujj’j’j, is factored by m m' m' m m m' m' m
.
(8.124)
1
2
mm'
In this case, it is convenient to rewrite Eq. (121) in the coordinate representation, using single-particle wavefunctions called spin-orbitals
r
( ) r
r o m .
(8.125) Spin-
j
j
j
orbital
They differ from the spatial parts of the usual orbital wavefunctions of the type (4.233) only in that their index j should be understood as the set of the orbital-state and the spin-orientation indices.31 Also, due to the Pauli-principle restriction of numbers Nj to either 0 or 1, Eq. (121) may be also rewritten without the explicit occupancy numbers, with the understanding that the summation is extended only over the pairs of occupied states. As a result, it becomes
*
(r) *
( ' r) u (r, ' r) (r) ( ' r)
Energy
)
1
(
1
j
j'
int
j
j'
correction
3
3
E
d r d r'
.
int
(8.126) due to
2
*
j, j'
*
(r) ( '
r ) u (r, '
r ) (r) ( '
r )
fermion
j j'
j
j'
int
j'
j
interaction
In particular, for a system of two electrons, we may limit the summation to just two states ( j, j’ =
1, 2). As a result, we return to Eqs. (39)-(41), with the bottom (minus) sign in Eq. (39), corresponding to the triplet spin states. Hence, Eq. (126) may be considered as the generalization of the direct and exchange interaction balance picture to an arbitrary number of orbitals and an arbitrary total number N
of electrons. Note, however, that this formula cannot correctly describe the energy of the singlet spin states, corresponding to the plus sign in Eq. (39), and also of the entangled triplet states.32 The reason is 31 The spin-orbitals (125) are also close to spinors (13), besides that the former definition takes into account that the spin s of a single particle is fixed, so that the spin-orbital may be indexed by the spin’s orientation m ms only. Also, if an orbital index is used, it should be clearly distinguished from j, i.e. the set of the orbital and spin indices. This is why I believe that the frequently met notation of spin-orbitals as j,s(r) may lead to confusion.
32 Indeed, due to the condition j’ j, and Eq. (124), the calculated negative exchange interaction is limited to electron state pairs with the same spin direction – such as the factorable triplet states ( and ) of a two-electron system, in which the contribution of E ex, given by Eq. (41), to the total energy is also negative.
Chapter 8
Page 26 of 52
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that the description of entangled spin states, given in particular by Eqs. (18) and (20), requires linear superpositions of different Dirac states. (A proof of this fact is left for the reader’s exercise.) Now comes a very important fact: the approximate result (126), added to the sum of unperturbed energies (0)
j
, equals the sum of exact eigenenergies of the so-called Hartree-Fock equation:33
2
2
u(r) (r)
Hartree-
j
Fock
2 m
(8.127)
equation
*
(r ' ) u (r, r ' ) (r) (r ' )
*
(r ' ) u (r, r ' ) (r) (r) 3
d r' (r),
j'
int
j
j'
j'
int
j'
j
j
j
j' j
where u(r) is the external-field potential acting on each particle separately – see the second of Eqs. (79).
An advantage of this equation in comparison with Eq. (126) is that it allows the (approximate) calculation of not only the energy spectrum of the system, but also the corresponding spin-orbitals, taking into account their electron-electron interaction. Of course, Eq. (127) is an integro-differential rather than just a differential equation. There are, however, efficient methods of numerical solution of such equations, typically based on iterative methods. One more important practical trick is the exclusion of the filled internal electron shells (see Sec. 3.7) from the explicit calculations, because the shell states are virtually unperturbed by the valence electron effects involved in typical atomic phenomena and chemical reactions. In this approach, the Coulomb field of the shells, described by fixed, pre-calculated, and tabulated pseudo-potentials, is added to that of the nuclei. This approach dramatically cuts the computing resources necessary for systems of relatively heavy atoms, enabling a pretty accurate simulation of electronic and chemical properties of rather complex molecules, with thousands of electrons.34 As a result, the Hartree-Fock approximation has become the de-facto baseline of all so-called ab-initio (“first-principle”) calculations in the very important field of quantum chemistry.35
In departures from this baseline, there are two opposite trends. For larger accuracy (and typically smaller systems), several “post-Hartree-Fock methods”, notably including the configuration interaction method , 36 that are more complex but may provide higher accuracy, have been developed.
There is also a strong opposite trend of extending such ab initio (“first-principle”) methods to larger systems while sacrificing some of the results’ accuracy and reliability. The ultimate limit of this trend is applicable when the single-particle wavefunction overlaps are small and hence the exchange interaction is negligible. In this limit, the last term in the square brackets in Eq. (127) may be ignored, and the multiplier j(r) taken out of the integral, which is thus reduced to a differential equation –
formally just the Schrödinger equation for a single particle in the following self-consistent effective potential:
33 This equation was suggested in 1929 by Douglas Hartree for the direct interaction and extended to the exchange interaction by Vladimir Fock in 1930. To verify its compliance with Eq. (126), it is sufficient to multiply all terms of Eq. (127) by * j(r), integrate them over all r-space (so that the right-hand side would give
j), and then sum these single-particle energies over all occupied states j.
34 For condensed-matter systems, this and other computational methods are applied to single elementary spatial cells, with a limited number of electrons in them, using cyclic boundary conditions.
35 See, e.g., A. Szabo and N. Ostlund, Modern Quantum Chemistry, Revised ed., Dover, 1996.
36 That method, in particular, allows the calculation of proper linear superpositions of the Dirac states (such as the entangled states for N = 2, discussed above) which are missing in the generic Hartree-Fock approach – see, e.g., the just-cited monograph by Szabo and Ostlund.
Chapter 8
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u (r) u(r) u (r), u (r)
*
(r )
(r, r ) (r )
.
(8.128) Hartree
ef
dir
dir
' u
'
' d 3 r'
j'
int
j'
approximation
j' j
This is the so-called Hartree approximation – that gives reasonable results for some systems,37
especially those with low electron density.
However, in dense electron systems (such as typical atoms, molecules, and condensed matter) the exchange interaction, described by the second term in the square brackets of Eqs. (126)-(127), may be as high as ~30% of the direct interaction, and frequently cannot be ignored. The tendency of taking this interaction in the simplest possible form is currently dominated by the Density Functional Theory,38
universally known by its acronym DFT. In this approach, the equation solved for each eigenfunction
j(r) is a differential, Schrödinger-like Kohn-Sham equation
2
Kohn-
2
u(r)
KS
u (r) u (r) (r) (r)
,
(8.129) Sham
2
dir
xc
j
j
j
m
equation
where
KS
1
3
(r ' )
u (r) e(r),
(r)
d r'
,
(r) en(r),
(8.130)
dir
4
r r '
0
and n(r) is the total electron density in a particular point, calculated as n(r)
*
(r) (r .)
(8.131)
j
j
j
The most important feature of the Kohn-Sham Hamiltonian is the simplified description of the exchange and correlation effects by the effective exchange-correlation potential u xc(r). This potential is calculated in various approximations, most of them valid only in the limit when the number of electrons in the system is very high. The simplest of them (proposed by Kohn et al. in the 1960s) is the Local Density Approximation (LDA) in which the effective exchange potential at each point is a function only of the electron density n at the same point r, taken from the theory of a uniform gas of free electrons.39
However, for many tasks of quantum chemistry, the accuracy given by the LDA is insufficient, because inside molecules the density n typically changes very fast. As a result, DFT has become widely accepted in that field only after the introduction, in the 1980s, of more accurate, though more cumbersome models for u xc(r), notably the so-called Generalized Gradient Approximations (GGAs). Due to its relative simplicity, DFT enables the calculation, with the same computing resources and reasonable precision, some properties of much larger systems than the methods based on the Hartree-Fock theory.
As the result, is has become a very popular tool of ab initio calculations. This popularity is enhanced by the availability of several advanced DFT software packages, some of them in the public domain.
37 An extreme example of the Hartree approximation is the Thomas-Fermi model of heavy atoms (with Z >> 1), in which atomic electrons, at each distance r from the nucleus, are treated as an ideal, uniform Fermi gas, with a certain density n( r) corresponding to the local value u ef( r), but a global value of their highest full single-particle energy, = 0, to ensure the equilibrium. (The analysis of this model is left for the reader’s exercise.) 38 It had been developed by Walter Kohn and his associates (notably Pierre Hohenberg) in 1965-66, and eventually (in 1998) was marked with a Nobel Prize in Chemistry for W. Kohn.
39 Just for the reader’s reference: for a uniform, degenerate Fermi-gas of electrons (with the Fermi energy F >> k B T), the most important, exchange part u x of u xc may be calculated analytically: u x = -(3/4) e 2 k F/40, where the Fermi momentum k
3
3
F = (2 m eF)1/2/ is defined by the electron density: n = 2(4/3) k F /(2)3 k F /32.
Chapter 8
Page 28 of 52
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Please note, however, that despite this undisputable success, this approach has its problems.
From my personal point of view, the most offensive of them is the implicit assumption of the unphysical Coulomb interaction of an electron with itself (by dropping, on the way from Eq. (128) to Eq. (130), the condition j’ j at the calculation of KS
u ). As a result, for a reasonable description of some effects, the
dir
available DFT packages are either inapplicable at all or require substantial artificial tinkering.40
Unfortunately, because of lack of time/space, for details I have to refer the interested reader to specialized literature.41
8.5. Quantum computation and cryptography
Now I have to review the emerging fields of quantum computation and encryption. (Since these fields are much related, they are often referred to under the common title of “quantum information science”, though this term is somewhat misleading, de-emphasizing physical aspects of the topic.) These fields are currently the subject of intensive research and development efforts, which has already brought, besides an enormous body of hype, some results of general importance. My coverage, by necessity short, will focus on these results, referring the reader interested in details to special literature.42 Because of the very active stage of the fields, I will also provide quite a few references to recent publications, making the style of this section closer to a brief literature review than to a textbook’s section.
Presently, most work on quantum computation and encryption is based on systems of spatially separated (and hence distinguishable) two-level systems – in this context, universally called qubits.43
Due to this distinguishability, the issues that were the focus of the first sections of this chapter, including the second quantization approach, are irrelevant here. On the other hand, systems of qubits have some interesting properties that have not been discussed in this course yet.
First of all, a system of N >> 1 qubits may contain much more information than the same number of N classical bits. Indeed, according to the discussions in Chapter 4 and Sec. 5.1, an arbitrary pure state of a single qubit may be represented by its ket vector (4.37) – see also Eq. (5.1):
u u ,
(8.132)
1
1
2
2
N 1
where { uj} is any orthonormal two-state basis. It is natural and common to employ, as uj, the eigenstates aj of the observable A that is eventually measured in the particular physical implementation of the qubit
– say, a certain Cartesian component of spin-½. It is also common to write the kets of these base states as 0 and 1, so that Eq. (132) takes the form
40 As just a few examples, see N. Simonian et al., J. Appl. Phys. 113, 044504 (2013); M. Medvedev et al., Science 335, 49 (2017); A. Hutama et al., J. Phys. Chem. C 121, 14888 (2017).
41 See, e.g., either the monograph by R. Parr and W. Yang, Density-Functional Theory of Atoms and Molecules, Oxford U. Press, 1994, or the later textbook J. A. Steckel and D. Sholl, Density Functional Theory: Practical Introduction, Wiley, 2009. For a popular review and references to more recent work in this still-developing field, see A. Zangwill, Phys. Today 68, 34 (July 2015).
42 Despite the recent flood of new books on the field, one of its first surveys, by M. Nielsen and I. Chuang, Quantum Computation and Quantum Information, Cambridge U. Press, 2000, is perhaps still the best one.
43 In some texts, the term qubit (or “Qbit”, or “Q-bit”) is used instead for the information contents of a two-level system – very much like the classical bit of information (in this context, frequently called “Cbit” or “C-bit”) describes the information contents of a classical bistable system – see, e.g., SM Sec. 2.2.
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Single qubit
a 0 a 1
a j .
(8.133) state’:
0
1
1
j
N
j0 1
,
representation
(Here, and in the balance of this section, the letter j is used to denote an integer equal to either 0 or 1.) According to this relation, any state of a qubit is completely defined by two complex c-numbers aj, i.e. by 4 real numbers. Moreover, due to the normalization condition a 12 + a 22 = 1, we need just 3
independent real numbers – say, the Bloch sphere coordinates and (see Fig. 5.3), plus the common phase , which becomes important only when we consider coherent states of a several-qubit system.
This is a good time to note that a qubit is very much different from any classical bistable system used to store single bits of information – such as two possible voltage states of the usual SRAM cell (essentially, a positive-feedback loop of two transistor-based inverters). Namely, the stationary states of a classical bistable system, due to its nonlinearity, are stable with respect to small perturbations, so that they may be very robust to unintentional interaction with their environment. In contrast, the qubit’s state may be disturbed (i.e. its representation point on the Bloch sphere shifted) by even minor perturbations, because it does not have such an internal state stabilization mechanism.44 Due to this reason, qubit-based systems are rather vulnerable to environment-induced drifts, including the dephasing and relaxation discussed in the previous chapter, creating major experimental challenges – see below.
Now, if we have a system of 2 qubits, the vectors of its arbitrary pure state may be represented as a sum of 22 = 4 terms,45
Two-qubit
a 00 a 01 a 10 a 11
a
j j ,
(8.134) state’s
N
00
01
10
11
2
j j 1 2
j , j 1
,
0
1 2
representation
1 2
with four complex coefficients, i.e. eight real numbers, subject to just one normalization condition, which follows from the requirement = 1:
2
a
1
j j
.
(8.135)
1 2
j 1
,
0
,
1 2
The evident generalization of Eqs. (133)-(134) to an arbitrary pure state of an N-qubit system is a sum of 2 N terms:
a
j j
j
,
(8.136)
N
j j ...
...
j
1 2
N
j ,
1 2
N
j ,.. j 1
,
0
1 2
N
including all possible combinations of 0s and 1s for indices j, so that the state is fully described by 2 N
complex numbers, i.e. 22 N 2 N+1 real numbers, with only one constraint, similar to Eq. (135), imposed by the normalization condition. Let me emphasize that this exponential growth of the information contents would not be possible without the qubit state entanglement. Indeed, in the particular case when qubit states are not entangled, i.e. are factorable:
44 In this aspect as well, the information processing systems based on qubits are closer to classical analog computers (which were popular once, but nowadays are used for a few special applications only) rather than classical digital ones.
45 Here and in most instances below I use the same shorthand notation as was used at the beginning of this chapter
– cf. Eq. (1b). In this short form, qubit’s number is coded by the order of its state index inside a full ket-vector, while in the long form, such as in Eq. (137), it is coded by the order of its single-qubit vector in a full direct product.
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...
,
(8.137)
1
2
N
N
where each n is described by an equality similar to Eq. (133) with its individual expansion coefficients, the system state description requires only 3 N – 1 real numbers – e.g., N sets {, , } less one common phase.
However, it would be wrong to project this exponential growth of information contents directly on the capabilities of quantum computation, because this process has to include the output information readout, i.e. qubit state measurements. Due to the fundamental intrinsic uncertainty of quantum systems, the measurement of a single qubit even in a pure state (133) generally may give either of two results, with probabilities W 0 = a 02 and W 1 = a 12. To comply with the general notion of computation, any quantum computer has to provide certain (or virtually certain) results, and hence the probabilities Wj have to be very close to either 0 or 1, so that before the measurement, each measured qubit has to be in a basis state – either 0 or 1. This means that the computational system with N output qubits, just before the final readout, has to be in one of the factorable states
j j ... j
j j ... j ,
(8.138)
1
2
N
1 2
N
N
which is a very small subset even of the set of all unentangled states (137), and whose maximum information contents is just N classical bits.
Now the reader may start thinking that this constraint strips quantum computations of any advantages over their classical counterparts, but such a view is also superficial. To show that, let us consider the scheme of the most actively explored type of quantum computation, shown in Fig. 3.46
j
j
1 in
j
1 out
1 in
1
qubit
j 1
out
classical
classical
bits
j
j
2 in
2 out
bits
of the
j
j 2
2
2
qubit
in
U
out
of the
input
output
number
number
j
j
N in
N
j
out
jN
N
qubit N
in
out
qubit state
unitary
qubit state
in
out
preparation
transform
measurement
Fig. 8.3. The baseline scheme of quantum computation.
Here each horizontal line (sometimes called a “wire”47) corresponds to a single qubit, tracing its time evolution in the same direction as at the usual time function plots: from left to right. This means 46 Numerous modifications of this “baseline” scheme have been suggested, for example with the number of output qubits different from that of input qubits, etc. Some other options are discussed at the end of this section.
47 The notion of “wires” stems from the similarity between such quantum schemes and the drawings describing classical computation circuits – see, e.g., Fig. 4a below. In the classical case, the lines may be indeed understood as physical wires connecting physical devices: logic gates and/or memory cells. In this context, note that classical computer components also have non-zero time delays, so that even in this case the left-to-right device ordering is useful to indicate the timing of (and frequently the causal relation between) the signals.
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that the left column in of ket-vectors describes the initial state of the qubits,48 while the right column
out describes their final (but pre-measurement) state. The box labeled U represents the qubit evolution in time due to their specially arranged interactions between each other and/or external drive “forces”.
Besides these forces, during this evolution the system is supposed to be ideally isolated from the dephasing and energy-dissipating environment, so that the process may be described by a unitary operator defined in the 2 N-dimensional Hilbert space of N qubits:
ˆ
U .
(8.139)
out
in
With the condition that the input and output states have the simple form (138), this equality reads
ˆ
j
j
... j
U j
j
... j
.
(8.140)
1
out
2
N
out
out
1
in
2
N
in
in
The art of quantum computer design consists of selecting such unitary operators U ˆ that would:
- satisfy Eq. (140),
- be physically implementable, and
-enable substantial performance advantages of the quantum computation over its classical counterparts with similar functionality, at least for some digital functions (algorithms).
I will have time/space to demonstrate the possibility of such advantages on just one, perhaps the simplest example – the so-called Deutsch problem,49 on the way discussing several common notions and issues of this field. Let us consider the family of single-bit classical Boolean functions j out = f( j in). Since both j are Boolean variables, i.e. may take only values 0 and 1, there are evidently only 4 such functions
– see the first four columns of the following table:
f
f(0)
f(1)
class
F
f(1)- f(0)