Basic Theoretical Physics: A Concise Overview P28 pdf

278 35 Cooper Pairs; Superconductors and Superfluids
This means that below this temperature there is a component of the liquid
which flows without energy dissipation.
He
4
-atoms (2 protons, 2 neutrons, 2 electrons) as composite particles are
bosons. Thus, even though there is a strong interaction between the bo-
son particles, they can condense without pairing, and the transition to
superfluidity consists in the formation of such a condensate.
The case of He
3
is fundamentally different. This atom consists of two
protons, only one neutron, and again two electrons; thus as a composite
particle it is a fermion. Originally one did not expect superfluidity at all
in He
3
. All the greater was the surprise when in 1972 Osheroff et al., [26],
found that also a He
3
-liquid becomes a superfluid, but admittedly at much
lower temperatures, i.e. below 2.6 mK. In fact at these low temperatures
(not yet ultralow, see Part IV) also in this liquid Cooper pairs of He
3
-
atoms are formed, where, however, this time the interaction favors triplet
pairing, i.e. now the position factor of a Cooper pair is antisymmetric, e.g.,
with l = 1, and the spin function is symmetric, i.e., a triplet function.As
a consequence, in the superfluid state of He
3
both the expectation value of
ˆ

L and that of
ˆ
S can be different from zero at the same time, which makes
the theory very complex. In fact in 2003 a physics Nobel prize was awarded
to Anthony Legget,
2
who was involved in the theory of He
3
-superfluidity.
iii) Finally, some words on high-temperature superconductors.Thisisacer-
tain class of non-metallic superconductors with many CuO
2
-planes, and
values of T
s
around 120 K, i.e., six times higher than usual. In these sys-
tems the electrons also seem to form pairs; the “elementary charge” of
the carriers of superconductivity is again 2e, but the underlying mech-
anism is not yet clear, although the systems have been under scrutiny
since 1986/87. Perhaps one is dealing with Bose-Einstein condensation
(see Part IV) of electron-pair aggregates
3
. The “phononic mechanism”
of conventional metallic superconductivity (see above) is apparently re-
placed here by electronic degrees of freedom themselves. Moreover it is
also not accidental that the planar radius of the respective Cooper pairs
is much smaller than usual.
The electronic degrees of freedom also include antiferromagnetic spin cor-
relations, which seem to play an essential role. Furthermore, the position-
space factor seems to correspond to d

x
2
−y
2
and/or d
xy
pairing, i.e., to
attractive and repulsive interactions, respectively, at four alternating pla-
nar axes distinguished by 90
◦
.
2
Nobel prize winners in 2003: Alexej A. Abrikosov, Vitalij L. Ginzburg, Anthony
J. Legget, see below.
3
There may be a relation to the crossover from Bardeen-Cooper-Schrieffer behav-
ior of (weakly coupled) ultracold molecules of fermions to Bose-Einstein conden-
sation of (strongly coupled) preformed fermion pairs, when a so-called Feshbach
resonance is crossed. See PartIV.
36 On the Interpretation
of Quantum Mechanics
(Reality?, Locality?, Retardation?)
36.1 Einstein-Podolski-Rosen Experiments
In a famous paper published in 1935 [29] Einstein, Podolski and Rosen dis-
cussed the implications of quantum mechanics, and correctly pointed out
several consequences that do not agree with common sense. For the following
two decades in Princeton, Einstein and his group tried to obtain an “ac-
ceptable” theory by augmenting quantum mechanics with so-called “hidden
variables”. Only a few years after Einstein’s death, however, John Bell at
CERN showed that this is not possible (see below).

The phenomena related to the “unacceptable consequences” involve the
notion of entangled states. In the paper of Einstein, Podolski and Rosen
entangled states are constructed from eigenstates of the position and mo-
mentum operators ˆx and ˆp
x
. However, it is more obvious to form entangled
states from spin operators. For example, in the singlet state |S =0,M =0
of two interacting electrons, see (34.5), the spin states of the single particles
are entangled, which means that the result cannot be written as a product
of single-particle functions.Forthetriplet states, in (34.4), two of them,
|S =1,M = ±1,arenot entangled, while the third state, |S =1,M =0,is
as entangled as the singlet state.
1
Thus, if one has a two-fermion s-wave decay of a system (the “source”),
with an original singlet state of that system, where two identical fermi-
ons leave the source in diametrically opposed directions, and if one finds
by measurements on these particles (each performed with a single-particle
measuring apparatus at large distances from the source) a well-defined z-
component
ˆ
S
z
of the first particle (e.g., the positive value +

2
), then one
always finds simultaneously(!), with a similar apparatus applied to the sec-
ond diametrically-opposed particle, the negative value −

2

,andvice versa,if
quantum mechanics is correct (which it is, according to all experience; see
Fig. 36.1).
However, since the same singlet wave function |S =0,M =0diagonalizes
not only the z-component of the total spin,
1
For three particles, there are two different classes of entangled states, viz |ψ
1
∝
|↑↑↑+|↓↓↓ and |ψ
2
∝↑↑↓ + ↓↑↑ + ↑↓↑,andforfourormoreparticles,there
are even more classes, which have not yet been exploited.
280 36 On the Interpretation of Quantum Mechanics
S
z
:= S
z
(1) + S
z
(2) ,
but also (as can easily be shown, → exercises) the x-component
S
x
:= S
x
(1) + S
x
(2) ,
with the same eigenvalue  ·M (= 0), identical statements can also be made

for a measurement of
ˆ
S
x
instead of
ˆ
S
z
(this is true for a singlet state although
the operators do not commute).
These consequences of quantum mechanics seemed “unacceptable” to Ein-
stein, since they violate three important postulates, which according to Ein-
stein an “acceptable” theory should always satisfy:
a) the postulate of reality, i.e., that even before the measurement the state
of the system is already determined (i.e., it appears unacceptable that
by figuratively switching a button on the measuring equipment from
ˆ
S

z
to
ˆ
S

x
the observed spins apparently switch from alignment (or anti-
alignment) along the z-direction to alignment (or anti-alignment) along
the x-direction;
b) the locality of the measuring process (i.e., a measurement at position 1
should not determine another property at a different location 2), and

c) retardation of the propagation of information (i.e., information from a po-
sition 1 should reach a position 2 only after a finite time (as in electrody-
namics, see Part II, with a minimum delay of Δt :=
|r
1
−r
2
|
c
).
Fig. 36.1. Singlet decay processes (schematically). A quantum system has decayed
from a state without any angular momentum. We assume that the decay products
are two particles, which travel diametrically away from each other in opposite di-
rections, and which are always correlated despite their separation. In particular |ψ
always has even symmetry w.r.t. exchange of the spatial positions of the two par-
ticles, whereas the spin function is a singlet; i.e., if a measuring apparatus on the
r.h.s. prepares a spin state with a value

2
in a certain direction, then a simultanous
measurement on the opposite side, performed with a similar apparatus, prepares
the value −

2
. (This is stressed by the small line segments below the r.h.s. and
above the l.h.s. of the diagram, where we purposely do not present arrows in the
z or x direction, since any direction is possible. Purposely also no origin has been
drawn to stress that this correlation is always present.)
36.2 The Aharonov-Bohm Effect; Berry Phases 281
In contrast, in quantum mechanics the individual state of the system

is determined (i.e., prepared) just by the measurement
2
. This concerns the
problem of reality.
Above all, however, quantum mechanics is non-local, i.e., |ψ already con-
tains all the information for all places, so that no additional information must
be transferred from place 1 to 2 or vice versa
3
. This simultaneously concerns
problems 2 and 3.
Of course, Einstein’s objections must be taken very seriously, the more so
since they really pinpoint the basis of the formalism. The same applies to the
efforts, although inappropriate after all (see below), to modify Schr¨odinger’s
theory by adding hidden variables to obtain an acceptable theory (i.e. “real-
istic”, “local”and “retarded”). Ultimately, nine years after Einstein’s death,
it was shown by Bell, [30], a theorist at CERN, that the situation is more
complex than Einstein assumed. Bell proved that Einstein’s “acceptability”
postulates imply certain inequalities for the outcome of certain experiments
(the so-called “Bell inequalities” for certain four-site correlations), which sig-
nificantly differ from the prediction of quantum mechanics. As a consequence,
predictions of either quantum mechanics or (alternatively) the quasi-classical
“acceptability” postulates of Einstein, Podolski and Rosen, can now be veri-
fied in well-defined experiments. Up to the present time many such Bell ex-
periments have been performed, and quantum mechanics has always “won”
the competition.
36.2 The Aharonov-Bohm Effect; Berry Phases
In the following, both the non-locality of quantum mechanics and wave-
particle duality appear explicitly. We shall consider a magnetized straight
wire along the z-axis. Since we assume that the magnetization in the wire
points in the z-direction, the magnetic induction outside the wire vanishes

everywhere,
B
outside
≡ 0 .
Thus, an electron moving outside should not take any notice of the wire, as
it “passes by” according to the classical equation of motions
m ·
dv
dt
= e · v ×B .
2
For example, a singlet two-spin function contains a coherent superposition of
both cases (m
s
)(1) = +/2and(m
s
)(1) = −/2; they only become mutually
exclusive by the (classical) measurement equipment.
3
It is perhaps not accidental that difficulties are encountered (e.g., one has to
perform renormalizations) if one attempts to unify quantum mechanics with
special relativity, which is a local theory (in contrast to general relativity,which
is, however, not even renormalizable).
282 36 On the Interpretation of Quantum Mechanics
However, quantum mechanically, electrons are represented by a wave func-
tion ψ(r,t), i.e., due to this fact they also “see” the vector potential A(r)
of the magnetic field, and this is different from zero outside the wire, giving
rise to a nontrivial magnetic flux Φ
0
(see below), a gauge invariant quantity,

which causes a visible interference effect. In detail: For a so-called “symmet-
rical gauge” we have
A ≡ e
ϕ
A
ϕ
(r
⊥
)
where the azimuthal component
A
ϕ
(r
⊥
)=
Φ
0
2πr
⊥
, and r
⊥
:=

x
2
+ y
2
, where Φ
0
= BπR

2
is the magnetic flux through the cross-section of the wire.
As already mentioned, the magnetic field vanishes identically outside of
the wire, i.e.,
B
z
=
d(r
⊥
·A
ϕ
)
r
⊥
dr
⊥
=0.
However, it is not B, but A, which enters the equation of motion for the
quantum-mechanical probability amplitude
i
dψ
dt
= Hψ =

∇
i
− eA

2
ψ/(2m) .

Hence the probability amplitude is influenced by A, and finally one ob-
serves interferences in the counting rate of electrons, which pass either side
of the wire and enter a counter behind the wire.
The results are gauge invariant, taking into account the set of three equa-
tions (24.16). The decisive fact is that the closed-loop integral

W
A · dr = Φ
0
(=0),
which determines the interference is gauge-invariant. The wave function ef-
fectively takes the wire fully into account, including its interior, although the
integration path W is completely outside.
Aharonov-Bohm interferences have indeed been observed, e.g., by B¨orsch
and coworkers, [27], in 1961, thus verifying that quantum mechanics is non-
local, in contrast to classical mechanics.
In this connection we should mention the general notion of a so-called
Berry phase; this is the (position dependent) phase difference
4
that results
in an experiment if the wave function not only depends on r but also on
parameters α (e.g., on an inhomogeneous magnetic field), such that these
parameters change adiabatically slowly along a closed loop. This formulation
is consciously rather lax, since the concept of a Berry phase is based on
general topological relations (e.g., parallel transport in manifolds), see [28].
4
We remind ourselves that wavefunctions ψ(r), which differ only by a global
complex factor, describe the same state.
36.3 Quantum Computing 283
36.3 Quantum Computing

The subject of quantum computing is based on the fact that quantum-
mechanical wave functions are superposable and can interfere with each other:
ψ = ψ
1
+ ψ
2
, |ψ|
2
= |ψ
1
|
2
+ |ψ
2
|
2
+2·Re(ψ
∗
1
· ψ
2
) .
5
This is a field of research which has seen great progress in the last few
years.
Firstly we should recall that classical computing is based on binary digits;
e.g., the decimal number “9” is given by 1 × 2
3
+1×2
0

, or the bit sequence
1001; as elementary “bits” one only has zero and unity; hence N-digit binary
numbers are the vertices of a 2
N
-dimensional hypercube. In contrast, quan-
tum computing is based on the ability to superimpose quantum mechanical
wave functions. One considers N-factor product states of the form
|Ψ =
N

ν=1

c
(ν)
0



ψ
(ν)
0

+ c
(ν)
1



ψ
(ν)

1

;
i.e., as in Heisenberg spin systems with S =
1
2
one uses the Hilbert space
H
N
“2-level”
,
a (tensor) product of N “2-level systems”, where the orthonormalized basis
states



ψ
(ν)
0

and



ψ
(ν)
1

are called “quantum bits” or qubits.Ofcourse|Ψ is, as usual, only defined
up to a complex factor.

The way that large amounts of computation velocity can be gained is
illustrated by the following example.
Assume that (i) the unitary operator
ˆ
U, i.e., a complex rotation in Hilbert-
space, is applied to the initial state |Ψ
0
 to generate the intermediate state
|Ψ
1
 =
ˆ
U|Ψ
0
 ,
before (ii) another unitary operator
ˆ
V is applied to this intermediate state
|Ψ
1
 to generate the end state
|Ψ
2
 =
ˆ
V |Ψ
1
 .
Now, the unitary product operator
ˆ

W :=
ˆ
V
ˆ
U
5
Several comprehensive reviews on quantum computing and quantum cryptogra-
phy can be found in the November issue 2005 of the German “Physik Journal”.
284 36 On the Interpretation of Quantum Mechanics
directly transforms the initial state |Ψ
0
 into the final state |Ψ
2
. In general
a classical computation of the product matrix
ˆ
W is very complex; it costs
many additions, and (above all!) multiplication processes, since
W
i,k
=

j
V
i,j
U
j,k
.
In contrast, sequential execution by some kind of “experiment”,
|Ψ

0
→|Ψ
1
→|Ψ
2
 ,
may under certain circumstances be a relatively easy and fast computation
for a skilled experimenter.
One could well imagine that in this way in special cases (in particular for
N →∞) even an exponentially difficult task (i.e., growing exponentially with
the number of digits, N, or with the size of the system, L) can be transformed
to a much less difficult problem, which does not grow exponentially but only
polynomially with increasing N and L.
In fact there are computation which are classically very difficult, e.g., the
decomposition of a very large number into prime factors. One can easily see
that 15 = 3 × 5and91=7×13; but even for a medium-sized number, e.g.
437, factorization is not easy, and for very large numbers the task, although
systematically solvable, can take days, weeks, or months, even on modern
computers. In contrast, the same problem treated by a quantum computer
with a special algorithm (the Shor algorithm), tailored for such computers
and this problem, would be solved in a much shorter time.
This is by no means without importance in daily life: it touches on the
basis of the encoding principles used by present-day personal computers for
secure messages on the internet, i.e., so-called PGP encoding (PGP ˆ=“pretty
good privacy”).
According to this encoding, every user has two “keys”, one of which, the
“public key” of the receiver, is used by the sender for encoding the message.
This “public key” corresponds to the afore-mentioned “large number” Z.
But for fast decoding of the message the receiver also needs a private key,
which corresponds to the decomposition of Z into prime factors, and this key

remains known only to the receiver
6
.
A “spy”, knowing the computer algorithms involved and the “public key”
of the receiver, e.g., his (or her) “large number”, can thus in principle calculate
the corresponding “private key”, although this may take weeks or months and
would not matter for a lot of short-term transactions by the receiver until the
“spy” has finished his computation. But if the “spy” could use a “quantum
computer”, this would be a different matter.
6
According to this so-called PGP-concept (PGP ˆ= “Pretty Good Privacy”) the
private key is only “effectively private”, see below, similar to a very large integer
being factorized by a finite-time computation.
36.4 2d Quantum Dots 285
Fortunately, even now (i.e., in our age of classical computers), quantum
mechanics offers a totally different way of encoding, called quantum crypto-
graphy (see below) which, in contrast to the classical encoding schemes is ab-
solutely secure. However, because of the “coherency” requirement
7
of quan-
tum mechanics the method suffers from the problem of range, so that at
present it can only be applied over distances smaller than typically ∼ 10 to
100 km
8
.
After these preliminary remarks on quantum cryptography, we return to
quantum computing.
There are other relevant examples where quantum computing would be
much more effective than classical computing, e.g., in the task of sorting an
extremely large set of data, where on a quantum computer the Grover algo-

rithm would be much faster than comparable algorithms on classical com-
puters.
One may ask oneself why under these circumstances have quantum com-
puters not yet been realized, even though intense work has been going on in
this field for many years. The difficulties are hidden in the notion of “a skilled
experimentalist”, used above, because it is necessary to ensure that, (i), dur-
ing the preparation of the initial state |ψ
0
; then, (ii), during the execution of
all operations on this state; and finally, (iii), during all measurements of the
results the coherence and superposability of all signals should be essentially
undisturbed. This implies inter alia that errors (due to the necessary auto-
matic correction) should only happen with an extremely small probability,
e.g., < 10
−4
, and that all experiments performed in the course of a quantum
computation should be extremely well-controlled. Additionally, the system
should be “scalable” to N  1.
Many different suggestions have indeed been made for producing a quan-
tum computer, one of which will be outlined in the next section.
36.4 2d Quantum Dots
Two-dimensional quantum dots can be regarded as artificial atoms with di-
ameters in the region of ∼ 100
˚
A(= 10 nm) in a two-dimensional electron (or
hole) gas [“2DEG” (or “2DHG”)]. A two-dimensional electron gas is formed,
e.g., at the planar interface between a GaAs-semiconductor region (for z<0)
and an Al
1−x
Ga

x
As-region (for z>0). At the interface there is a deflexion
of the energy bands, and as a consequence an attractive potential trench
V (x, y, z) forms, which is, however, attractive only w.r.t. the z-coordinate,
7
The coherency demands do not allow, for quantum mechanical purposes, the
usual amplification of the signals, which is necessary for the transmission of
electromagnetic signals over hundreds or thousands of kilometers.
8
A quantum cryptographical encoding/decoding software has been commercially
available since the winter of 2003/2004.
286 36 On the Interpretation of Quantum Mechanics
whereas parallel to the interface (in the x-andy-directions) the potential
is constant. Although the electrons are thus in a bound state w.r.t. the z
coordinate, they can move freely parallel to the interface, however, with an
effective mass m
∗
that can be much smaller than the electron mass in vacuo,
m
e
(e.g., m
∗
=0.067m
e
).
Similarly to the case of a transistor, where one can fine tune the current
between the source and drain very sensitively via the gate voltage, it is ex-
perimentally possible to generate a local depression region of the potential
energy V (r). Often the local region of this depression is roughly circular and
can be described by a parabolic confining potential, as for a 1d-harmonic os-

cillator, but modified to 2d (the range of the confined region is x
2
+ y
2
≤ R
2
,
with R ≈ 50
˚
A(= 5 nm), and the magnitude of the depression is typically
−3.5meV).
As a consequence, in the resulting two-dimensional potential well one has
a finite number of bound states of the “2DEG” (so-called confined electrons).
For simplicity the confinement potential is (as already mentioned) usually
described by a parabolic potential (see below). In this way one obtains a kind
of artificial 2d-atom, or so-called 2d-quantum dot
9
, with a characteristic
radius R in the region of 5 nm, which, on the one hand, is microscopically
small, but conversely (in spite of the suggestive name “quantum dot”) very
large compared to a normal atom, which has a diameter in the range of
0.1nm.
The electrons of the 2d-quantum dot are described by the following single-
particle Hamiltonian (where the spin is neglected, which makes sense
10
):
H =
p
2
r

⊥
2m
∗
+
(p
ϕ
− e · A
ϕ
)
2
2m
∗
+
m
∗
ω
2
0
2
r
2
⊥
, (36.1)
where
r
⊥
:=

x
2

+ y
2
and ϕ =arctan
y
x
are the usual planar polar coordinates. The azimuthal quantity A
ϕ
is the
only component of our vector potential belonging to the constant magnetic
induction B (= curlA
11
),whichpointsinthez-direction.
If more electrons are confined to the dot one must write down a sum of
such terms, and additionally the Coulomb repulsion of the electrons and the
Pauli principle must be taken into account.
9
There are also 3d-quantum dots.
10
Theeffectivemassm
∗
is much smaller than the free-electron mass m
e
,which
enters into the Bohr magneton μ
B
=
μ
0
e
2m

e
. Thus in the considered “artificial
atoms” the (orbital) second term on the r.h.s. of (36.1) dominates the spin term
−g
∗
·μ
B
ˆ
S ·(B/μ
0
). Here g
∗
= −0.44 is the effective Land´e factor of the system,
and the other quantities have their usual meaning (see above).
11
We remind ourselves that A is not unique.
36.5 Interaction-free Quantum Measurement; “Which Path?” Experiments 287
An intermediate calculation results in an even simpler expression for the
magnetic induction: with the ansatz
A
ϕ
≡
B
0
2
· r
⊥
and the formula
B
z

=
d(r
⊥
· A
ϕ
)
r
⊥
·dr
⊥
(B
r
⊥
and B
ϕ
vanish) ,
one obtains
B ≡ e
z
·B
0
.
Again a product ansatz is helpful:
ψ(r
⊥
,ϕ)
!
= R(r
⊥
) · e

imϕ
.
With this ansatz one obtains an explicit solution of the Schr¨odinger equation
corresponding to (36.1). Due to the 2π-periodicity w.r.t. ϕ the variable m
must of course be an integer. However, for B
z
= 0 one obtains only chiral
cylindrical symmetry (i.e., m and −m are not equivalent). This implies that
with B = 0 there is of course a preferred chirality, i.e., clockwise or counter-
clockwise.
The equations around (36.1) are known as Fock-Darwin theory. This yields
one of the few explicit solutions of nontrivial quantum mechanical problems,
and in 2001 there was even a set of written examination questions on this
theory for teacher students (file 6 of U.K.’s exercises in winter 2003).
Finally, from such artificial atoms (i) one can create artificial molecules;
(ii) the spin state of these molecules can be calculated in a kind of Heitler-
London theory, see above; and (iii) these states can be fine-tuned in a very
controlled way as is necessary for quantum computing. In fact, one of the most
promising suggestions of quantum computation is based on these systems. (In
this context we should mention the theoretical work of Loss and DiVicenzo,
[33], plus the experiments of Vandersypen and Kouwenhoven, [34].)
36.5 Interaction-free Quantum Measurement;
“Which Path?” Experiments
In principle the Aharonov-Bohm experiment can already be considered as
a form of interaction-free quantum measurement, since the electron “mea-
sures” the presence of the magnetized wire without coming into direct contact
with it. But there is an indirect interaction via the vector potential, which is
influenced by the wire.
Interaction-free quantum measurements in another sense (see below) have
been performed more recently by Anton Zeilinger’s group at Innsbruck

12
in
12
Now at the university of Vienna.