Path integrals and Hamiltonians: principles and methods
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Cambridge [u.a.]
Cambridge Univ. Press
2014
|
| Ausgabe: | 1. publ. |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis Klappentext |
| Beschreibung: | XVII, 417 S. graph. Darst. |
| ISBN: | 9781107009790 |
Internformat
MARC
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| 245 | 1 | 0 | |a Path integrals and Hamiltonians |b principles and methods |c Belal E. Baaquie |
| 250 | |a 1. publ. | ||
| 264 | 1 | |a Cambridge [u.a.] |b Cambridge Univ. Press |c 2014 | |
| 300 | |a XVII, 417 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 650 | 4 | |a Differential operators | |
| 650 | 4 | |a Differential equations | |
| 650 | 4 | |a Hamiltonian operator | |
| 650 | 4 | |a Path integrals | |
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| 689 | 1 | |5 DE-604 | |
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Datensatz im Suchindex
| _version_ | 1804152135196082176 |
|---|---|
| adam_text | Contents
Preface
page
xv
Acknowledgements
xviii
1
Synopsis
1
Part one Fundamental principles
5
2
The mathematical structure of quantum mechanics
7
2.1
The Copenhagen quantum postulate
7
2.2
The superstructure of quantum mechanics
10
2.3
Degree of freedom space J7
10
2.4
State space V(T)
11
2.4.1
Hubert space
14
2.5
Operators OCF)
14
2.6
The process of measurement
18
2.7
The
Schrödinger
differential equation
19
2.8 Heisenberg
operator approach
22
2.9
Dirac-Feynman path integral formulation
23
2.10
Three formulations of quantum mechanics
25
2.11
Quantum entity
26
2.12
Summary
27
3
Operators
30
3.1
Continuous degree of freedom
30
3.2
Basis states for state space
35
3.3
Hermitian operators
36
3.3.1
Eigenfunctions; completeness
37
3.3.2
Hamiltonian for a periodic degree of freedom
39
3.4
Position and momentum operators
χ
and
ρ
40
3.4.1
Momentum operator/?
41
viii Contents
3.5
Weyl
operators
43
3.6
Quantum
numbers; commuting operator
46
3.7 Heisenberg
commutation equation
47
3.8
Unitary representation of
Heisenberg
algebra
48
3.9
Density matrix: pure and mixed states
50
3.10
Self-adjoint operators
51
3.10.1
Momentum operator on finite interval
52
3.11
Self-adjoint domain
54
3.11.1
Real eigenvalues
54
3.12
Hamiltonian s self-adjoint extension
55
3.12.1
Delta function potential
57
3.13
Fermi pseudo-potential
59
3.14
Summary
60
4
The Feynman path integral
61
4.1
Probability amplitude and time evolution
61
4.2
Evolution kernel
63
4.3
Superposition: indeterminate paths
65
4.4
The Dirac-Feynman formula
67
4.5
The Lagrangian
69
4.5.1
Infinite divisibility of quantum paths
70
4.6
The Feynman path integral
70
4.7
Path integral for evolution kernel
73
4.8
Composition rule for probability amplitudes
76
4.9
Summary
79
5
Hamiltonian mechanics
80
5.1
Canonical equations
80
5.2
Symmetries and conservation laws
82
5.3
Euclidean Lagrangian and Hamiltonian
84
5.4
Phase space path integrals
85
5.5
Poisson
bracket
87
5.6
Commutation equations
88
5.7
Dirac bracket and constrained quantization
90
5.7.1
Dirac bracket for two constraints
91
5.8
Free particle evolution kernel
93
5.9
Hamiltonian and path integral
94
5.10
Coherent states
95
5.11
Coherent state vector
96
5.12
Completeness equation: over-complete
98
5.13
Operators; normal ordering
98
Contents ix
5.14
Path
integral
for coherent states
99
5.14.1
Simple harmonic oscillator
101
5.15
Forced harmonic oscillator
102
5.16
Summary
103
6
Path integral quantization
105
6.1
Hamiltonian from Lagrangian
106
6.2
Path integral s classical limit
ћ
—> 0 109
6.2.1
Nonclassical paths and free particle 111
6.3
Fermat s principle of least time
112
6.4
Functional differentiation
115
6.4.1
Chain rule
115
6.5
Equations of motion
116
6.6
Correlation functions
117
6.7 Heisenberg
commutation equation
118
6.7.1
Euclidean commutation equation
121
6.8
Summary
122
Part two Stochastic processes
123
7
Stochastic systems
125
7.1
Classical probability: objective reality
127
7.1.1
Joint, marginal and conditional probabilities
128
7.2
Review of Gaussian integration
129
7.3
Gaussian white noise
132
7.3.1
Integrals of white noise
134
7.4
Ito
calculus
136
7.4.1
Stock price
137
7.5
Wilson expansion
138
7.6
Linear
Langevin
equation
140
7.6.1
Random paths
142
7.7
Langevin
equation with potential
143
7.7.1
Correlation functions
144
7.8
Nonlinear
Langevin
equation
145
7.9
Stochastic quantization
148
7.9.1
Linear
Langevin
path integral
149
7.10
Fokker-Planck Hamiltonian
151
7.11
Pseudo-Hermitian
Fokker—
Planck Hamiltonian
153
7.12
Fokker-Planck path integral
156
7.13
Summary
158
x
Contents
Part three Discrete degrees of freedom
159
8
Ising model
8.1
Ising degree of freedom and state space
161
8.1.1
Ising spin s state space V
163
8.1.2
Bloch sphere
164
8.2
Transfer matrix
165
8.3
Correlators
167
8.3.1
Periodic lattice
168
8.4
Correlator for periodic boundary conditions
169
8.4.1
Correlator as vacuum expectation values
171
8.5
Ising model
s
path integral
171
8.5.1
Ising partition function
172
8.5.2
Path integral calculation of Cr
173
8.6
Spin decimation
175
8.7
Ising model on
2
χ
TV lattice
176
8.8
Summary
179
9
Ising model: magnetic field
180
9.1
Periodic Ising model in a magnetic field
180
9.2
Ising model s evolution kernel
182
9.3
Magnetization
183
9.3.1
Correlator
184
9.4
Linear regression
185
9.5
Open chain Ising model in a magnetic field
189
9.5.1
Open chain magnetization
190
9.6
Block spin renormalization
191
9.6.1
Block spin renormalization: magnetic field
195
9.7
Summary
196
10
Fermions
198
10.1
Fermionic variables
199
10.2
Fermion integration
200
10.3
Fermion Hubert space
201
10.3.1
Fermionic completeness equation
203
10.3.2
Fermionic momentum operator
204
10.4
Antifermion state space
204
10.5
Fermion and antifermion Hubert space
206
10.6
Real and complex fenmons: Gaussian integration
207
10.6.1
Complex Gaussian fermion
209
10.7
Fermionic operators
211
Contents xi
10.8 Fermionic
path
integral 211
10.9 Fennion—antifermion Hamiltonian 214
10.9.1
Orthogonality and completeness
216
10.10
Fermi
on-antifermion Lagrangian 217
10.11 Fermionic
transition probability amplitude
219
10.12 Quark
confinement
220
10.13
Summary
222
Part four Quadratic path integrals
223
11
Simple harmonic oscillator
225
11.1
Oscillator Hamiltonian
226
11.2
The propagator
226
11.2.1
Finite time propagator
227
11.3
Infinite time oscillator
230
11.4
Harmonic oscillator s evolution kernel
230
11.5
Normalization
233
11.6
Generating functional for the oscillator
234
11.6.1
Classical solution with source
234
11.6.2
Source free classical solution
236
11.7
Harmonic oscillator s conditional probability
239
11.8
Free particle path integral
240
11.9
Finite lattice path integral
241
11.9.1
Coordinate and momentum basis
243
11.10
Lattice free energy
243
11.11
Lattice propagator
245
11.12
Lattice transfer matrix and propagator
246
11.13
Eigenfunctions from evolution kernel
249
11.14
Summary
250
12
Gaussian path integrals
251
12.1
Exponential operators
252
12.2
Periodic path integral
253
12.3
Oscillator normalization
254
12.4
Evolution kernel for indeterminate final position
256
12.5
Free degree of freedom: constant external source
260
12.6
Evolution kernel for indeterminate positions
261
12.7
Simple harmonic oscillator: Fourier expansion
264
12.8
Evolution kernel for a magnetic field
267
12.9
Summary
270
xii Contents
Part five Action with acceleration
271
13
Acceleration Lagrangian
273
13.1
Lagrangian
273
13.2
Quadratic potential: the classical solution
275
13.3
Propagator: path integral
277
13.4
Dirac constraints and acceleration Hamiltonian
280
13.5
Phase space path integral and Hamiltoman operator
283
13.6
Acceleration path integral
286
13.7
Change of path integral boundary conditions
289
13.8
Evolution kernel
291
13.9
Summary
293
14
Pseudo
-Hermi
tian
Euclidean Hamiltonian
294
14.1
Pseudo-Hermitian Hamiltonian; similarity transformation
295
14.2
Equivalent Hermitian Hamiltonian Ho
297
14.3
The matrix elements of e~xQ
298
14.4
e~~xQ and similarity transformations
301
14.5
Eigenfunctions of oscillator Hamiltoman Ho
304
14.6
Eigenfunctions of
Я
and
1Ѓ
305
14.6.1
Dual energy eigenstates
307
14.7
Vacuum state; eQf2
309
14.8
Vacuum state and classical action
312
14.9
Excited states of
Я
313
14.9.1
Energy
ω
eigenstate
4>ю(х,
ν)
314
14.9.2
Energy a>i eigenstate
Ψοι(*, ν)
315
14.10
Complex
α>], α>2
317
14.11
State space V of Euclidean Hamiltonian
318
14.11.1
Operators acting on V
320
14.11.2 Heisenberg
operator equations
322
14.12
Propagator: operators
323
14.13
Propagator: state space
324
14.14
Many degrees of freedom
327
14.15
Summary
329
15
Non-
Hermitian Hamiltonian: Jordan blocks
330
15.1
Hamiltonian: equal frequency limit
331
15.2
Propagator and states for equal frequency
331
15.3
State vectors for equal frequency
334
15.3.1
State
vectorii
(τ)}
334
15.3.2
State vector |
ψ2(τ))
335
Contents xiii
15.4
Completeness equation for
2
χ
2
block
336
15.5
Equal frequency propagator
337
15.6
Hamiltonian: Jordan block structure
339
15.7 2x2
Jordan block
340
15.7.1
Hamiltonian
342
15.7.2 Schrödinger
equation for Jordan block
343
15.7.3
Time evolution
344
15.8
Jordan block propagator
344
15.9
Summary
347
Part six Nonlinear path integrals
349
16
The quartic potential:
instantons
351
16.1
Semi-classical approximation
352
16.2
A one-dimensional integral
353
16.3
Instantons
in quantum mechanics
355
16.4
Instanton
zero mode
362
16.5
Instanton
zero mode: Faddeev-Popov analysis
364
16.5.1
Instanton
coefficient
M
368
16.6
Multi-instantons
370
16.7
Instanton
transition amplitude
371
16.7.1
Lowest energy states
372
16.8
Instanton
correlation function
373
16.9
The dilute gas approximation
374
16.10
Ising model and the double well potential
376
16.11
Nonlocal Ising model
377
16.12
Spontaneous symmetry breaking
380
16.12.1
Infinite well
381
16.12.2
Double well
381
16.13
Restoration of symmetry
381
16.14
Multiple wells
383
16.15
Summary
383
17
Compact degrees of freedom
385
17.1
Degree of freedom: a circle
386
17.1.1
Poisson
summation formula
387
17.1.2
The Sl Lagrangian
388
17.2
Multiple classical solutions
388
17.2.1
Large radius limit
391
17.3
Degree of freedom: a sphere
391
17.4
Lagrangian for the rigid rotor
393
xiv Contents
395
397
399
401
403
18
Conclusions
405
409
413
17.5
Cancellation of divergence
17.6
Conformation of
DNA
17.7
DNA
extension
17.8
DNA
persistence length
17.9
Summary
Conclusions
References
Index
PATH INTEGRALS AND HAMILTONIANS
Providing a pedagogical introduction to the essential principles of path integrals
and Hamiltonians, this book describes cutting-edge quantum mathematical tech¬
niques applicable to a vast range of fields, from quantum mechanics, solid state
physics, statistical mechanics, quantum field theory, and superstring theory to fi¬
nancial modeling, polymers, biology, chemistry, and quantum finance.
Eschewing use of the
Schrödinger
equation, the powerful and flexible combina¬
tion of Hamiltonian operators and path integrals is used to study a range of differ¬
ent quantum and classical random systems, succinctly demonstrating the interplay
between a system s path integral, state space, and Hamiltonian. With a practical
emphasis on the methodological and mathematical aspects of each derivation, this
is a perfect introduction to these versatile mathematical methods, suitable for re¬
searchers and graduate students in physics, mathematical finance, and engineering.
Belal E. Baaquie is a Professor of Physics at the National University of
Singapore, specializing in quantum field theory, quantum mathematics, and quan¬
tum finance. He is the author of Quantum Finance
(2004),
Interest Rates and
Coupon Bonds in Quantum Finance
(2009),
and The Theoretical Foundations of
Quantum Mechanics
(2013),
and co-author of Exploring Integrated Science
(2010).
|
| any_adam_object | 1 |
| author | Baaquie, Belal E. |
| author_GND | (DE-588)171927850 |
| author_facet | Baaquie, Belal E. |
| author_role | aut |
| author_sort | Baaquie, Belal E. |
| author_variant | b e b be beb |
| building | Verbundindex |
| bvnumber | BV041810199 |
| callnumber-first | Q - Science |
| callnumber-label | QC174 |
| callnumber-raw | QC174.12 |
| callnumber-search | QC174.12 |
| callnumber-sort | QC 3174.12 |
| callnumber-subject | QC - Physics |
| classification_rvk | UK 4500 |
| ctrlnum | (OCoLC)884046688 (DE-599)BVBBV041810199 |
| dewey-full | 530.1201/51539 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 530 - Physics |
| dewey-raw | 530.1201/51539 |
| dewey-search | 530.1201/51539 |
| dewey-sort | 3530.1201 551539 |
| dewey-tens | 530 - Physics |
| discipline | Physik |
| edition | 1. publ. |
| format | Book |
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| id | DE-604.BV041810199 |
| illustrated | Illustrated |
| indexdate | 2024-07-10T01:05:54Z |
| institution | BVB |
| isbn | 9781107009790 |
| language | English |
| lccn | 013044323 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-027255568 |
| oclc_num | 884046688 |
| open_access_boolean | |
| owner | DE-11 DE-703 DE-20 DE-355 DE-BY-UBR |
| owner_facet | DE-11 DE-703 DE-20 DE-355 DE-BY-UBR |
| physical | XVII, 417 S. graph. Darst. |
| publishDate | 2014 |
| publishDateSearch | 2014 |
| publishDateSort | 2014 |
| publisher | Cambridge Univ. Press |
| record_format | marc |
| spelling | Baaquie, Belal E. Verfasser (DE-588)171927850 aut Path integrals and Hamiltonians principles and methods Belal E. Baaquie 1. publ. Cambridge [u.a.] Cambridge Univ. Press 2014 XVII, 417 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Differential operators Differential equations Hamiltonian operator Path integrals Hamilton-Operator (DE-588)4072278-8 gnd rswk-swf Pfadintegral (DE-588)4173973-5 gnd rswk-swf Pfadintegral (DE-588)4173973-5 s DE-604 Hamilton-Operator (DE-588)4072278-8 s Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027255568&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027255568&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Baaquie, Belal E. Path integrals and Hamiltonians principles and methods Differential operators Differential equations Hamiltonian operator Path integrals Hamilton-Operator (DE-588)4072278-8 gnd Pfadintegral (DE-588)4173973-5 gnd |
| subject_GND | (DE-588)4072278-8 (DE-588)4173973-5 |
| title | Path integrals and Hamiltonians principles and methods |
| title_auth | Path integrals and Hamiltonians principles and methods |
| title_exact_search | Path integrals and Hamiltonians principles and methods |
| title_full | Path integrals and Hamiltonians principles and methods Belal E. Baaquie |
| title_fullStr | Path integrals and Hamiltonians principles and methods Belal E. Baaquie |
| title_full_unstemmed | Path integrals and Hamiltonians principles and methods Belal E. Baaquie |
| title_short | Path integrals and Hamiltonians |
| title_sort | path integrals and hamiltonians principles and methods |
| title_sub | principles and methods |
| topic | Differential operators Differential equations Hamiltonian operator Path integrals Hamilton-Operator (DE-588)4072278-8 gnd Pfadintegral (DE-588)4173973-5 gnd |
| topic_facet | Differential operators Differential equations Hamiltonian operator Path integrals Hamilton-Operator Pfadintegral |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027255568&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027255568&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT baaquiebelale pathintegralsandhamiltoniansprinciplesandmethods |