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Файл:Introduction to superfluidity and superconductivity. Учебное пособие
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Министерство образования и науки Российской Федерации
Федеральное государственное бюджетное образовательное
учреждение высшего образования
«Московский педагогический государственный университет»
С. А. Рябчун
INTRODUCTION TO SUPERFLUIDITY
AND SUPERCONDUCTIVITY
Учебное посо
2-е издание, стереотипное
электронное
бие
МПГУ
Москва • 2024
УДК 538.945
ББК 22.268.3
Р985
Рецензенты:
Г. М. Чулкова, доктор физико-математических наук, профессор
кафедры общей и экспериментальной физики факультета физики
и информационных технологий МПГУ
В. А. Ильин, доктор физико-математических наук, профессор кафедры
общей и экспериментальной физики факультета физики
и информационных технологий МПГУ
Рябчун, Сергей Александрович.
Р985 Introduction to superfluidity and superconductivity: учебное пособие /
С. А. Рябчун. – 2-е изд., стер. электрон. –Москва: МПГУ, 2024. – 72 с.
– Текст: электронный.
ISBN 978-5-4263-0572-4
These notes have appeared as a result of a one-term course in
superfluidity and superconductivity given by the author to fourth-year
undergraduate students and first-year graduate students of the Department of
Physics, Moscow State University of Education. The goal was not to give a
detailed picture of these two macroscopic quantum phenomena with an
extensive coverage of the experimental background and all the modern
developments, but rather to show how the knowledge of undergraduate
quantum mechanics and statistical physics could be used to discuss the basic
concepts and simple problems, and draw parallels between superconductivity
and superfluidity.
УДК 538.945
ББК 22.268.3
ISBN 978-5-4263-0572-4 © МПГУ, 2017
© Рябчун С. А., 2017
3
CONTENTS
Preface ................................................................................................... 5
1. Second quantisation ........................................................................... 7
1.1. General remarks .......................................................................... 7
1.2. Second quantisation for fermions ............................................... 8
1.3. Second quantisation for bosons ................................................ 10
1.4. Operators in second-quantised form ......................................... 11
Exercises ........................................................................................... 16
2. The uniform weakly interacting Bose gas ....................................... 17
2.1. The ideal Bose gas .................................................................... 17
2.2. Elements of scattering theory.................................................... 21
2.3. Costructing the Hamiltonian ..................................................... 24
2.4. Bogoliubov transformations...................................................... 25
2.5. The two-fluid picture ................................................................ 27
Exercises ........................................................................................... 29
3. The non-uniform weakly interacting Bose gas ............................... 31
3.1. The time-independent Gross-Pitaevskii equation ..................... 31
3.2. The coherence length ................................................................ 33
3.3. The time-dependent Gross-Pitaevskii equation ........................ 33
3.4. Bogoliubov equations ............................................................... 35
3.5. Vortex states .............................................................................. 36
Exercises ........................................................................................... 40
4. The uniform superconducting state ................................................. 42
4.1. The ideal Fermi gas ................................................................... 42
4.2. Constructing the Hamiltonian ................................................... 43
4.3. Excitations and the energy gap ................................................. 47
4.4. The critical temperature ............................................................ 49
4.5. The discontinuity of the electronic heat capacity ..................... 50
4.6. The Meissner effect ................................................................... 51
4.7. Tunnelling ................................................................................. 54
4.7.1. A simple problem ................................................................ 54
4.7.2. NN-contact.......................................................................... 55
4.7.3. NS-contact .......................................................................... 55
Exercises ........................................................................................... 57
4
5. The non-uniform superconducting state near TC ............................ 58
5.1. Thermodynamics of superconductors ....................................... 58
5.2. The Ginzburg-Landau equations ............................................... 59
5.3. Characteristic lengths of the Ginzburg-Landau theory............. 61
5.4. Flux quantisation ....................................................................... 62
5.5. Energy of the SN interface ........................................................ 63
5.6. Critical field of a thin film ........................................................ 65
5.7. Critical current of a thin film .................................................... 66
5.8. Vortex solutions ......................................................................... 67
Exercises ........................................................................................... 71
5
PREFACE
These notes have appeared as a result of a one-term course in superfluidity
and superconductivity given by the author to fourth-year undergraduate students
and first-year graduate students of the Department of Physics, Moscow State
University of Education. The goal was not to give a detailed picture of these two
macroscopic quantum phenomena with an extensive coverage of the experimental
background and all the modern developments, but rather to show how the
knowledge of undergraduate quantum mechanics and statistical physics could be
used to discuss the basic concepts and simple problems, and draw parallels
between superconductivity and superfluidity.
Superconductivity and superfluidity are two phenomena where quantum
mechanics, typically constrained to the microscopic realm, shows itself on the
macroscopic level. Conceptually and mathematically, these phenomena are
related very closely, and some results obtained for one can, with a few
modifications, be immediately carried over to the other. However, the student of
these notes should be aware of important differences between superconductivity
and superfluidity that stem mainly from two facts: (1) electrons in a
superconductor carry a charge, therefore one has to take into account interaction
with electromagnetic radiation; (2) electrons move in a lattice, therefore phonons
play a role not only a mediators of attractive interaction between pairs of
electrons, but also as scatterers of charge carriers.
Although these are notes on superfluidity and superconductivity, and there
are a few cross-references, the two subjects can be studied independently with,
perhaps, a little extra work by the student to fill in the gaps resulting from such
study. The material of Chapter 1 introduces the method of second quantisation
that is commonly used to discuss systems with many interacting particles. It is
then applied in Chaper 2 to treat the uniform weakly interacting Bose gas within
the approach by N. Bogoliubov, and in Chapter 4 to formulate the theory of the
uniform superconducting state put forth by J. Bardeen, L. Cooper and R.
Schrieffer. Chapter 3 presents the theory proposed independently by E. Gross and
L. Pitaevskii of a non-uniform weakly interacting Bose gas, with a discussion of
vortices, rotation of the condensate, and the Bogoliubov equations. In Chapter 5
we discuss the Ginzburd-Landau theory of a non-uniform superconductor near the
critical temperature and apply it to a few simple problems such as the surface
energy of the boundary between a normal metal and a superconductor, critical
current and critical magnetic field, and vortices.
A lovely introduction to both subjects, without formidable mathematics and
6
with a good coverage of experiment is presented in the book Superconductivity
and superfluidity by D.R. Tilley and J. Tilley, Institute of Physics Publishing, 3rd
edition, 1990.
A good presentation of second quantisation with applications to many
topics of condensed matter can be found in the book by P. L. Taylor and O.
Heinonen, A Quantum Approach to Condensed Matter Physics, Cambridge
University Press, 2002.
The student wishing to focus of superfluidity can study Chapters 1 – 3 of
these notes and then consult the following books:
• C. J. Pethick and Harry Smith, Bose-Einstein Condensation in
Dilute Gases, 2nd edition, Cambridge University Press, 2008.
• P. Nozieres, D. Pines, Theory Of Quantum Liquids: Superfluid
Bose Liquids, Advanced Books Classics, 1994.
• I.M. Khalatnikov, An introduction to the theory of
superfluidity, Advanced Books Classics, 2000.
If, however, the student is more inclined to take up the subject of
superconductivity, then, after studying Caps. 1, 4 and 5 of the notes, he may find
useful the following sources:
• J. B. Ketterson and S. N. Song, Superconductivity, Cambridge
University Press, 1999.
• M. Tinkham, Introduction to Superconductivity, Dover
Publications, 2nd edition, 2004.
• P.G. de Gennes, Superconductivity of metals and alloys,
Advanced Books Classics, 1999.
This list of books is by no means extensive and merely reflects the tastes of
the author of these notes.
This course was taught within the project No. 14.B25.31.0007 funded
by the Ministry of Education and Science of the Russian Federation.
.
7
1. SECOND QUANTISATION
1.1. General remarks
When we deal with a system of many identical particles in quantum
mechanics, we should make sure that the wave function is properly normalised to
take account of the fact that identical particles are indistinguishable. Take for
example two non-interacting particles. The properly symmetrised wave function
is
x
,x
xxxx
,
(1)
with the plus sign for bosons and the minus sign for fermions. In the
language of state vectors
(2)
If the particles interact the state vector can no longer be represented
by (2) and is a linear combination of a complete set of properly symmetrised
vectors corresponding to non-interacting particles. We define the following basis
kets
the vacuum state, i.e. a state with no particles;
{
}
a complete set of one-particle states;
{
}
a complete set of properly symmetrised two-
particle states;
. . . . . . . . . . . . . . . .
{
}
a complete set of n-particle states.
In these definitions the letters α, β etc. stand for quantum numbers or a
collection of quantum numbers that specify the state occupied by a particle.
Now define the permutation operator, which interchanges two
particles:
(3)
A permutation of two particles cannot produce to a new physical state, so
the state vector
can only differ from the state vector
by at most a phase
factor with a unit modulus:
,
(4)
which means that the original state is an eigenstate of the permutation
operator. Applying the permutation operator to the state
should, on the one
hand, return us to the original state. One the other hand, in view of (4), the state
vector will have acquired another phase factor :
8
(5)
Therefore there are two possibilities for the phase factor, or two eigenvalues
of the permutation operator:
,bosons;
,fermions
(6)
As one can see from the above brief discussion, working with properly
symmetrised wave functions (or, in general, state vectors) is not very convenient,
and a language is needed that will automatically take the symmetrisation
requirement into account. This language is developed in the subsequent sections.
1.2. Second quantisation for fermions
Consider fermions first. Define creation operators as follows
,
(7)
For convenience, let's agree on the following convention: the symbol
representing the newly created state is placed immediately after the vertical line.
The operators only act on the state represented by the symbol immediately after
the vertical line. We have two corollaries of (7):
,
, for
(8)
The creation operators thus anticommute. We see that the creation operators
automatically take into account the symmetry of a state with two or more
fermions, and ensure that a given state can only be occupied by one fermion (the
Pauli principle).
To proceed further we need more elaborate notation:
,
(9)
with the tilde indicating that the state α had not been occupied before the
creation operator acted, and the dots referring to other states that differ from α
(occupied or not as the case may be). With this, we can define an operator
conjugate to the creation operator:
(10)
Although we do not know the meaning of the Hermitian-conjugate operator
yet, we can nevertheless write
,
(11)
9
where the first identity is a manifestation of the Pauli principle that no state
can be occupied by more that one fermion. With what we have established so far,
we can write down the following identities:
a)
;
b)
;
c)
;
d)
if;
e)
if
(12)
Putting
in (12) c) and e), we shall have that the ket
is orthogonal
to any bra regardless of whether the state α is occupied or not, i.e. it is orthogonal
to any bra. This can only be if
if we now choose
,then from (12)
c) and e) we shall conclude that
is too orthogonal to any bra regardless of
whether the state α is occupied or not, i.e. it is too orthogonal to any bra. Thus we
may conclude
;
(13)
The operator
is called the annihilation operator. Taking the Hermitian
conjugate of the first line of (8), we write
(14)
We now build two operators
and
,and let them act in turn on a
ket. This ket must represent a situation with the state α occupied and the state β
free, otherwise we shall get zero:
;
(15)
The number-of-particles operator
(16)
acting on a ket with the state α occupied reproduces that ket:
(17)
If we introduce the anticommutator of two operators
,,
(18)
We can neatly summarise our main achievements:
,
,
,
,
(19)
where δ
αβ
is the Kronecker delta.
10
1.3. Second quantisation for bosons
Since any number of bosons can occupy a given state, we shall be using
different notation for a ket-vector of a many-boson system:
,, ,,
,
(20)
where the occupation numbers nα show how many particles there are in the
state α. A creation operator acting on such a vector will add one particle to the
appropriate state, and when acting on the vacuum ket it will create one particle in
the appropriate state:
,
,
(21)
where in the second line we have allowed for the possibility of a numerical
factor. Similarly, a destruction operator can be defined:
,
(22)
To fix the numerical factors we define the number-of-particles
operator as the operator whose eigenvectors are kets (20) corresponding to
eigenvalues that are numbers of particles in a certain state:
(23)
This definition gives
,
(24)
which fixes the numerical factor in the definition of the annihilation
operator:
(25)
For the creation operator write
,
(26)
where x is a yet unknown numerical factor. Consider the following chain:
(27)
Comparing the two lines we write
(28)
The following commutation relations follow immediately from the
definitions of the boson creation and annihilation operators:
,
,
;
,
(29)
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