Complete Introduction to Quantum Mechanics-Zagoskin

418 Pages·2016·37.46 MB·English

by  Alexandre Zagoskin

. ®
Teach
Yourself
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•
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QUANTUM
MECHAN.I.CS
A C.omplete rfltroduction
.
.
Arexandre
Zagos~in
..
Contents·
..
Acknowledgements
XVII
.
Introduction
XIX
A fair warning
1 Familiar physics in strange spaces 1
Particles, coordinates, vectors and trajectories
Velocity, acceleration, derivatives and differential equations
Phase space .
· Harmonic phase trajectories, energy
o~cillator,
conservation and action
. Hamilton, Hamiltonian, Hamiltons equations
and state vectors ·
Generalized coordinates, canonical momenta,
and configuration space
Statistical mechanics, statistical ensemble and
probability distribution function
Liouville equation*
2 Less familiar physics in stranger spaces 34
Quantum oscillator and adiabatic invariants
Action quantization, phase space and the uncertainty principle
Zero-point energy and complex numbers
State vector and Hilbert space
Operators, commutators, observables and expectation values
Eigenstates, eigenvalues, intrinsic randomness of
Nature and Borns rule
Quantization and Heisenberg uncertainty relations
3 Equations of quantum mechanics 66
Poisson brackets, Heisenberg equations of motion
and the Hamiltonian
.
"
Matrices and mP..trfx elements:ot:g~lK4tR£%; .., .
Schrodinger equation anc! wave functlori" , .,:.: ~r.}!r~Eii'':x:.<·-.,~,: · ·
. .
.,
Quantum propagation '<·"':"-~'·· :·-
A particle in a box and on a ring: energy and
momentum quantization .·
Quantum superposition p-rinciple . • . ,.
. Quantum statistical mechanics, quantu.m ·ensembles
and density matrix*
The Von Neumann equation and the master equation*
.....
4 Qubits and pieces 96
Qubits and other two-level quantum systems
_Qubit's state vector, Hilbert space, Bloch
vector and Bloch sphere
Qubit Hamiltonian and Schrodinger equation
· Energy and quantum beats
eigen~tates
Bloch equation, quantum beats {revisited) and qubit control
Qubit_observables, Pauli matrices and expectation values
Qubit density matrix, von Neumann and Bloch
equations and NMR*
Vintage charge qubits
5 Obserying the observables 134
Measurement, projection postulate and Born's rule {revisited)
The 'collapse of the wave function ·, measurement
problem and quantum-classical transition
Observing a charge qubit _
Bloch vect.or and density matrix; Bloch equations (revisited);
dephasing; and weak continuous measurement*
Measurements and Heisenberg uncertainty relations:
Heisenberg microscope
Standard quantum limit .
Quantum non-demolition {QND) measurements
and energy-time uncertainty relation
QuantufT) Zeno paradox
6 Strange and unusual 165
Quantum tunnelling: a-decay
Quantum tunnelling: scanning tunnelling microscope
Waves like particles: photoelectric effect and the single-photon
double slit experiment
Particles like waves: diffraction and fnterference of massive
'
·-
particles; de Broglie wavelength
7 The game of numbers 191
Electron in an atom
Electrons in an atom and Mendeleev periodic table
Pauli exclusion principle, fermions and bosons, and spin
Quantum many-body systems, Fock states, space and
Foe~
creation and annihilation operators
Quantum oscillator, Heisenberg equations and energy
quanta~
Quantum fields and second quantization
Why c(assicallight is a wave and a piece of metal is not
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8 The virtual reality 22-1
Electrons in a jellium, charge screening, electron
gas and Fermi-Dirac distribution ·
Electrons in a crystal: quasiparticles•
Electrons in a crystal: Bloch functions; quasimomentum;
Brillouin zones; energy bands; and Fermi surface•
Quasiparticles and Greens functions
Perturbation theory
Feynman diagrams and virtual particles
Summing Feynman diagrams and charge screening
Dressing the quasip articles
9 The path of extremal action 254
Variational approach to classical mechanics
Variational approach to quantum mechanics
s
Feynmah path integrals
Charged quantum particles in an electromagnetic field, path
integrals, the Aharonov-Bohm effect and the Berry phase"'
10 Order! Order! 281
Classica(magnets, second order phase transitions
and the order parameter
Landaus theory of s.econd order phase transitions and
. spontaneous symmetry breaking
Superconductors, superconducting phase transition
and the superconducting order parameter
BCS theory, Bose-Einstein condensate, Cooper pairs
and the superconducting energy gap
Josephson effect•
11 Curiouser and curiouser · ·311
Einstein-Podolsky-Rosen paradox
Entanglement and faster-than-light communications
Spooks acting at a distance, no-cloning
and quantum tomography
Schrodingers cat and Wjgner's friend
Bells inequality• · ··: · .
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G)
. </' Contents
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12 Schrodinger's elephants and quantum
slide rules 339
The need and promise of quantum computers
Digital computers, circuits and gates
Q1:1aritum gates
Quantum parallelism and the Deutsch algorithm*
The Shor algorithm, code-breaking and the promise of an
exponential speed-up
Sc/lriidingers elephants, quantum error correction and
DiVincenzo criteria
Qu_antum slide rules: adiabatic quantum computing and
quantum optimizers
13 History and philosophy 374
Sturm und Orang: the old quantum theory
When all roads led to Copenhagen: creation of quantum
• mechanics
What the FAPP?
'It was a warm summer evening in ancient Greece·
The proof of the pudding
Tl)e final.spell
Fact-check Answers 393
•
Taking it further 395
Figure credits 397
Index 398
•
Introduction
·A fair warning
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·~ :~ about magic and books of magic. And the second thing he :
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: Learns is that a perfectly respectable example of the former may :
•
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• be had for two or three guineas at a good bookseller. and that :
•
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• the value of the latter is above rubies. :
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: Susanna Clarke, Jonathan Strange & Mr Norrell ·:
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This book will help you teach yourself about quantum
mechanics, which is not quite the same as teaching yourself
quantum mechanics. Here is the difference. ·
To learn quantum mechanics means being able to do something
with quantum mechanics as a matter of course (solving equations,
mostly. H you are an experimentalist, this also entails conducting
experiments, the planning and of which require solving
analy~is
equations), and to have been doing this long enough to acquire
a certain intuition about these equations and experiments- and
therefore, about quantum mechanics. At the very least, one would
..
have to go through a good undergraduate textbook on quantum
mechanics and solve aU the problems (which means also going
through a couple of textbooks on mathematics and solving the
problems from them too), and discuss all the questions you may
have with somebody who already has the above-mentioned
intuition. In other words, to take a regular introductory course ·
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of quantum mechanics at a university, and this is not always
an option. And anyway, you,may not ,be planning to become a
·professional physicist (which is a plty). .· . . . -. ,_ _ .
.
. . .
You can instead learn about quantum mechanics. This is a·
book written exactly for this purpose- to help you 'teach .
yourself about quantum mechanics. This means that in the end
you will not become a magician, that is, physicist -
pra~sing
but hopefully you will have a better understanding of what
physicists do and think when they do research in quantum
@
· .. ihtroduction.
mechanics, and what this research tells us about the world
we live in. Besides, much of our technology uses quantum ·
mechanical effects routinely {like lasers and semiconductpr
microchips, which were produced by the 'first quantum
revolution' qack in the 20th century), and if the current trends
hold, we will be soon enough· playing with the technological
results of the 'second quantum revolution', using even more
.
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subtle and bizarre quantum effects. It is therefore wise to be
prepared. As a bonus, you will learn about·q uite a lot of things
are not taught in introductory quantum· mechanics courses
tha~
to physics undergraduates.
This book contains quite a number of equations {though much
less than a book of quantum mechanics would), so brushing
up your school maths would be a good idea. I have tried to
introduce all the new mathematical things, and some things
you may already know. Still, this is just a book about quantuni .
So why equations?
mechan~cs.
.
Quantum mechanics describes our world with an unbelievable
precision. But the scale of phenomena, where this description
prinlarily applies, is so far removed from our everyday eXperience
that our htiman intUition and language, however rich, are totally
inadequate f~r the task of expressing quantwri mechanics· directly.
In such straits it was common to use poetry with its allusions,
allegories and similes. This is indeed the only way one can write
. about quantum mechanics' using only words - even if the poetry
lacks ~hymes'and rhythm. This is a nice and.often inspiring way
of doing things,. but poetry is too imprecise, too .emotional and
too individual.
-It is therefore:.desirable to use the language in which quantum ·
ean
mechanics be expressed: mathematics, a language more
remote from our common speech than any Elvish dialect.
Fortunately, this is ·also the language in which all physics is
most naturally expressed, and it applies to such areas -like
mecha.ilies ~:~h.~re,\fe.do have an. inborn and daily trained
intuition. I did therefore try to relate the one to the other - ·you
·will be the judge of how well this approach succeeded.l ·
- -__.,.;.-,:,-.-. . '. ...., ,.;·.;;, -.~ '.-.. ;., . . . . .
,;·;,.; ..;.., :.,
. 1 The more 'ini'the'fuati.~ai:;sections, which can be skipped at a ·first reading,
are marked with an asi:ei:'isk.
:< • < • ' •• •
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Familiar physics in
strange spaces
To an umiccustomed ear, the very language of quantum
mechanics may seem mtimida~gly impenetrable. his full of
commutators, operators, state vectors, Hilbert spaces and other
mathematical horrors. At any rate quantum physics seems
totally different from the clarity and simplicity of Newtonian
mechanics, where (ootballs, cars, satellites, stars, planets,
barstools, bees, birds and butterflies move along well-defined
trajectories, accelerate or decelerate when acted upon by forces,
and always have a definite position and velocity. All this can
be described by the simple mathematical laws discovered by
Sir Isaac Newton and taught at school- or just intuitively
comprehended based on our everyday experience. Quantum
mechanics, on the contrary, is best left to somebody else.
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Philosophy is written in that great book which lies before :
evt~r
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our eyes - I mean the u_niverse - but we cannot understand it :
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• if we do not first learn the language and grasp the symbols, :
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• in which it is written. This book is written in the mathematical :
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• language, and the symbols are triangles, circles and other :
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• geometrical figures, without whose help it is impossible to :
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comprehend.-a single word of it; without which one :
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• wanders .in vain through a dark labyrinth. :
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• Galileo Galilei ( 1564- 1642) :
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•.• ••••••••••••••••••••••••. ••• • •• •••••••••••••••••••••••••••••••••••••••
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Spotlight: Rene Descartes 11596-1650)
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As the tradition has it. Descartes discovered the Cartesian
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coordinates when serving as a military officer. He had a lazy day
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indoors. observed a fly wandering on the ceiling and realized that
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• its position can be determined by its distance from the walls. •
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• Why ·cartesian'? Descartes wrote some of his works in Latin. and •
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.• his Latinized name is Cartesius. •
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Actually, this impression is false. We are not saying that quantum
mechanics does not differ from classical mechanics - of course
it does, and it indeed presents a counterintuitive, puzzling, very
beautiful and very precise view of the known Universe. But to see
these differences in the proper .light, it is necessary first to cast a
better look at the stiuct\ire of classical mechanics and realize -to
our surprise- that in many important respects it is very similar to
quantum mechanics.
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Key·idea: Coordinate transformation . :
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• Switching between different coordinate systems is called •
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• coordinate transformation. Here is a simple example. Suppose •
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•• a point .4-has. .c oordi-nates (x.y,zl. Let us now introduce a new •• •
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• coordinate system (O'x'y'z'l with the origin in the point with •
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Y.Zl.
• coordinates (X. •
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