m.sc. quantum mechanics ilnmuacin.in/studentnotice/ddelnmu/2020/m.sc phy. quantum... · 2020. 4....
Post on 06-Mar-2021
2 Views
Preview:
TRANSCRIPT
M.Sc.
QUANTUM MECHANICS I
nwjLFk f”k{kk funs”kky;yfyr ukjk;.k fefFkyk fo”ofo|ky;
dkes”ojuxj] njHkaxk&846008
M.Sc. PhysicsPaper-IIPHY-102
nwjLFk f”k{kk funs”kky;
laiknd eaMy
1. izks- ljnkj vjfoUn flag % funs”kd] nwjLFk f”k{kk funs”kky;] y- uk- fefFkyk fo”ofo|ky;] njHkaxk2. izks- v#.k dqekj flag % lh- ,e- ,l-lh- dkWyst] njHkaxk3. MkW- efgnzk dqekj % HkkSfrdh”kkL= foHkkx] y- uk- fefFkyk fo”ofo|ky;] njHkaxk4. MkW- “kaHkq çlkn % lgk;d funs”kd] nwjLFk f”k{kk funs”kky;] y- uk- fefFkyk fo”ofo|ky;] njHkaxk
ys[kd ,oa leUo;dMkW- efgnzk dqekj
Dr. Shambhu Prasad-Co-ordinator, DDE, LNMU, Darbhanga
fç;&Nk=Xk.k
nwj f”k{kk C;wjks&fo”ofo|ky; vuqnku vk;ksx] ubZ fnYyh ls ekU;rk çkIr ,oa laLrqr ,e- ,l- lh- ¼nwjLFk ekè;e½ ikB~;Øe ds fy, ge gkÆnd vkHkkj izdV djrs gSaA bl Lo&vfèkxe ikB~; lkexzh dks lqyHk djkrs gq, gesa vrho izlUurk gks jgh gS fd Nk=x.k ds fy, ;g lkexzh izkekf.kd vkSj mikns; gksxhA (Quantum Mechanics I)
uked ;g vè;;u lkexzh vc vkids le{k gSA¼çks- ljnkj vjfoUn flag½
funs”kd
izdk”ku o’kZ 2019
nwjLFk ikB~;Øe lEcU/kh lHkh izdkj dh tkudkjh gsrq nwjLFk f”k{kk funs”kky;] y- uk- fEkfFkyk ;wfuoflZVh] dkes”ojuxj] njHkaxk ¼fcgkj½&846008 ls laidZ fd;k tk ldrk gSA bl laLdj.k dk izdk”ku funs”kd] nwjLFk f”k{kk funs”kky;] yfyr ukjk;.k fefFkyk ;wfuoflZVh] njHkaxk gsrq esllZ y{eh ifCyds”kal çk- fy- fnYyh }kjk fd;k x;kA
DLN-2666-195.00-QUANTUM MECH-I PHY-102 C— Typeset at : Atharva Writers, Delhi Printed at :
nwjLFk f'k{kk funs'kky;yfyr ukjk;.k fEkfFkyk fo'ofo|ky;] dkes’ojuxj] njHkaxk&846004 ¼fcgkj½
Qksu ,oa QSSSSDl % 06272 & 246506] osclkbZV % www.ddelnmu.ac.in, bZ&esy % dde@lnmu.ac.in
ekuuh; dqyifr y- uk- fe- fo'ofo|ky;
fç; fo|kFkhZnwjLFk f’k{kk funs’kky;] yfyr ukjk;.k fefFkyk fo’ofo|ky; }kjk fodflr rFkk fofHkUu fudk;ksa ls vuq’kaflr Lo&vf/kxe lkexzh dks lqyHk djkrs gq, vrho çlUurk gks jgh gSA fo’okl gS fd nwjLFk f’k{kk ç.kkyh ds ek/;e ls mikf/k çkIr djus okys fo|kÆFk;ksa dks fcuk fdlh cká lgk;rk ds fo”k;&oLrq dks xzká djus esa fdlh çdkj dh dksbZ dfBukbZ ugha gksxhA ge vk’kk djrs gSa fd ikB~;&lkexzh ds :i esa ;g iqLrd vkids fy, mi;ksxh fl) gksxhA
çks- ¼MkW-½ ,l- ds- flagdqyifr
CONTENTS
Chapters Page No.
1. Origins of Quantum Physics 1
2. Dirac Notation and Hilbert Space 51
3. Postulates of Quantum Mechanics 102
4. One-Dimensional Problems 128
5. Angular Momentum 165
6. Three-Dimensional Problems 191
7. Time Independent Perturbation Theory 225
NOTES
Origins of Quantum Physics
Self-Instructional Material 1
CHAPTER – 1
ORIGINS OF QUANTUM PHYSICSSTRUCTURE
1.1 Learning Objectives 1.2 Introduction 1.3 Particle Aspect of Radiation 1.4 Wave Aspect of Particles 1.5 Particles verses Waves 1.6 Indeterministic Nature of the Microphysical World 1.7 Atomic Transitions and Spectroscopy 1.8 Quantization Rules 1.9 Wave Packets 1.10 Concluding Remarks 1.11 Summary 1.12 Review Questions 1.13 Further Readings
1.1 LEARNING OBJECTIVESAfter studying the chapter, students will be able to:
zz In this chapter are going to review the main physical ideas and experimental facts that defied classical physics and led to the birth of quantum mechanics.
zz The introduction of quantum mechanics was prompted by the failure of classical physics in explaining a number of microphysical phenomena that were observed at the end of the nineteenth and early twentieth centuries.
1.2 INTRODUCTION
,
NOTES
Quantum Mechanics-1
Self-Instructional Material2
NOTES
Origins of Quantum Physics
Self-Instructional Material 3
NOTES
Quantum Mechanics-1
Self-Instructional Material4
1.3 PARTICLE ASPECT OF RADIATION
BlackBody Radiation
T=5000 K
T=4000 K
T=3000 K
T=2000 K
u (10 J m Hz )-16 -3 -1
� (10 Hz)14
P
NOTES
Origins of Quantum Physics
Self-Instructional Material 5
T=5000 K
T=4000 K
T=3000 K
T=2000 K
u (10 J m Hz )-16 -3 -1
� (10 Hz)14
P
T=5000 K
T=4000 K
T=3000 K
T=2000 K
u (10 J m Hz )-16 -3 -1
� (10 Hz)14
P
T=5000 K
T=4000 K
T=3000 K
T=2000 K
u (10 J m Hz )-16 -3 -1
� (10 Hz)14
P
T=4000 K
u (10 J m Hz )-16 -3 -1
� (10 Hz)14
Wien’s Law
Rayleigh-JeansLaw
Planck’s Law
NOTES
Quantum Mechanics-1
Self-Instructional Material6
T=4000 K
u (10 J m Hz )-16 -3 -1
� (10 Hz)14
Wien’s Law
Rayleigh-JeansLaw
Planck’s Law
T=4000 K
u (10 J m Hz )-16 -3 -1
� (10 Hz)14
Wien’s Law
Rayleigh-JeansLaw
Planck’s Law
T=4000 K
u (10 J m Hz )-16 -3 -1
� (10 Hz)14
Wien’s Law
Rayleigh-JeansLaw
Planck’s Law
NOTES
Origins of Quantum Physics
Self-Instructional Material 7
NOTES
Quantum Mechanics-1
Self-Instructional Material8
),
NOTES
Origins of Quantum Physics
Self-Instructional Material 9
P
P
NOTES
Quantum Mechanics-1
Self-Instructional Material10
PhotolectRic effect
P
P
@@@@
@@@@
@@@@
@@@@@R@R@R@R ©©
©©©*
³³³³
³1»»»
»:
-
6
¢¢¢¢¢¢¢¢¢¢¢
¢¢¢¢¢¢¢¢¢¢¢
NOTES
Origins of Quantum Physics
Self-Instructional Material 11
@@@@
@@@@
@@@@
@@@@@R@R@R@R ©©
©©©*
³³³³
³1»»»
»:
-
6
¢¢¢¢¢¢¢¢¢¢¢
¢¢¢¢¢¢¢¢¢¢¢
NOTES
Quantum Mechanics-1
Self-Instructional Material12
comPton effect
NOTES
Origins of Quantum Physics
Self-Instructional Material 13
-
¡¡¡¡¡µ
PPPPPPPPq
NOTES
Quantum Mechanics-1
Self-Instructional Material14
NOTES
Origins of Quantum Physics
Self-Instructional Material 15
-
©©©©
©*
HHHHHj
-
©©©©
©*
HHHHHj
PaiR PRoduction
NOTES
Quantum Mechanics-1
Self-Instructional Material16
1.4 WAVE ASPECT OF PARTICLESde BRoglie's hyPothesis: matteR Waves
NOTES
Origins of Quantum Physics
Self-Instructional Material 17
Experimental Confirmation of de Broglie's Hypothesis
Davission-Germer Experiment
@@@@@R¡
¡¡¡¡µ
@¡
¡@
¡@
@¡
NOTES
Quantum Mechanics-1
Self-Instructional Material18
@@@@@R¡
¡¡¡¡µ
@¡
¡@
¡@
@¡
-------
HH
HHHHHHHHHHHH
HHHHHHHH
HHHHHHHH
XXXXXXXXXXXXXXXXXXXX
XXXXXXXXXXz
»»»»»»
»»»»»»
»»»»»»
»»»
»»»»»»
»»»»:
hhhhhhhhhhhhhhhhhhhh
((((((((
((((((((
((((
NOTES
Origins of Quantum Physics
Self-Instructional Material 19
-------
HH
HHHHHHHHHHHH
HHHHHHHH
HHHHHHHH
XXXXXXXXXXXXXXXXXXXX
XXXXXXXXXXz
»»»»»»
»»»»»»
»»»»»»
»»»
»»»»»»
»»»»:
hhhhhhhhhhhhhhhhhhhh
((((((((
((((((((
((((
Thomson Experiment
matteR Waves foR macRoscoPic oBjects
-------
HH
HHHHHHHHHHHH
HHHHHHHH
HHHHHHHH
XXXXXXXXXXXXXXXXXXXX
XXXXXXXXXXz
»»»»»»
»»»»»»
»»»»»»
»»»
»»»»»»
»»»»:
hhhhhhhhhhhhhhhhhhhh
((((((((
((((((((
((((
-------
HH
HHHHHHHHHHHH
HHHHHHHH
HHHHHHHH
XXXXXXXXXXXXXXXXXXXX
XXXXXXXXXXz
»»»»»»
»»»»»»
»»»»»»
»»»
»»»»»»
»»»»:
hhhhhhhhhhhhhhhhhhhh
((((((((
((((((((
((((
NOTES
Quantum Mechanics-1
Self-Instructional Material20
-------
---
-------
---
-------
------
-------
---
-------
---
-------
------
-------
---
-------
---
-------
------
1.5 PARTICLES VERSES WAVES
classical vieW of PaRticles and Waves
-------
---
-------
---
-------
------
-------
---
-------
---
-------
------
NOTES
Origins of Quantum Physics
Self-Instructional Material 21
-------
---
-------
---
-------
------
Quantum vieW of PaRticles and Waves
ijk
ijk
ijk
ijk
ijk
ijk
ijk
ijk
ijk
NOTES
Quantum Mechanics-1
Self-Instructional Material22
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq{ ¡¡@@
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
NOTES
Origins of Quantum Physics
Self-Instructional Material 23
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq{ ¡¡@@
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq{ ¡¡@@
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq{ ¡¡@@
Wave-PaRticle duality : comPlementaRity
NOTES
Quantum Mechanics-1
Self-Instructional Material24
PRinciPle of lineaR suPeRPosition
NOTES
Origins of Quantum Physics
Self-Instructional Material 25
1.6 INDETERMINISTIC NATURE OF THE MICROPHYSICALWORLD
heisenBeRg's unceRtainty PRinciPle
NOTES
Quantum Mechanics-1
Self-Instructional Material26
NOTES
Origins of Quantum Physics
Self-Instructional Material 27
PRoBaBilistic inteRPRetation
NOTES
Quantum Mechanics-1
Self-Instructional Material28
1.7 ATOMIC TRANSITIONS AND SPECTROSCOPY
RutheRfoRd PlanetaRy model of the atom
BohR model of the hydRogen atom
NOTES
Origins of Quantum Physics
Self-Instructional Material 29
eneRgy levels of the hydRogen atom
R
NOTES
Quantum Mechanics-1
Self-Instructional Material30
R
6
6
??????
????
?
????
?
6?
6
R
R
R
NOTES
Origins of Quantum Physics
Self-Instructional Material 31
6
6
??????
????
?
????
?
6?
6
R
R
R
R
R
Spectroscopy of the Hydrogen Atom
R
R
NOTES
Quantum Mechanics-1
Self-Instructional Material32
R
R
R
R
R
R
R
R
R
R
R
NOTES
Origins of Quantum Physics
Self-Instructional Material 33
R
R
R
R
R
R
R
R
R
R
1.8 QUANTIZATION RULES
NOTES
Quantum Mechanics-1
Self-Instructional Material34
NOTES
Origins of Quantum Physics
Self-Instructional Material 35
1.9 WAVE PACKETS
localized Wave Packets
NOTES
Quantum Mechanics-1
Self-Instructional Material36
-
6
-
6
-
6
-
6
NOTES
Origins of Quantum Physics
Self-Instructional Material 37
-
6
-
6
-
6
-
6
NOTES
Quantum Mechanics-1
Self-Instructional Material38
Wave Packets and the unceRtainty Relations
NOTES
Origins of Quantum Physics
Self-Instructional Material 39
motion of Wave Packets
Propagation of a Wave Packet without Distortion
NOTES
Quantum Mechanics-1
Self-Instructional Material40
-
6
-
-
NOTES
Origins of Quantum Physics
Self-Instructional Material 41
-
6
-
-
NOTES
Quantum Mechanics-1
Self-Instructional Material42
NOTES
Origins of Quantum Physics
Self-Instructional Material 43
Propagation of a Wave Packet with Distortion
NOTES
Quantum Mechanics-1
Self-Instructional Material44
NOTES
Origins of Quantum Physics
Self-Instructional Material 45
-
6
-
-
NOTES
Quantum Mechanics-1
Self-Instructional Material46
-
6
-
-
-
6
-
-
-
6
-
-
NOTES
Origins of Quantum Physics
Self-Instructional Material 47
NOTES
Quantum Mechanics-1
Self-Instructional Material48
-
6
©©©©
©©©©
©©©
HHHH
HHHH
HHH
-
6
©©©©
©©©©
©©©
HHHH
HHHH
HHH
NOTES
Origins of Quantum Physics
Self-Instructional Material 49
-
6
©©©©
©©©©
©©©
HHHH
HHHH
HHH
1.10 CONCLUDING REMARKS
NOTES
Quantum Mechanics-1
Self-Instructional Material50
1.12 REVIEW QUESTIONS
1.13 FURTHER READINGSzz Dass, HK. 2008, Mathematical Physics, New Delhi: S. Chand.
zz Chattopadhyay, PK. 1990, Mathematical Physics, New Delhi: New age International.
zz Hassani, Sadri. 2013, Mathematical Physics: A Modern Introduction to Its Foundations, Berlin: Springer Science & Business Media.
1. What is Particle Aspect of Radiation and Wave Aspect of Particles?2. Explain Particles verses Waves.3. Describe the Indeterministic Nature of the Microphysical World.4. What is Atomic Transitions and Spectroscopy?5. Explain the Quantization Rules.6. What Wave Packets and Concluding Remarks?
1.11 SUMMARYBohr introduced in 1913 his model of the hydrogen atom. In this work, he argued that atoms can be found only in discrete states of energy and that the interaction of atoms with radiation, i.e., the emission or absorption of radiation by atoms, takes place only in discrete amounts of hv because it results from transitions of the atom between its various discrete energy states. This work provided a satisfactory explanation to several outstanding problems such as atomic stability and atomic spectroscopy. Inspired by Planck’s quantization of waves and by Bohr’s model of the hydrogen atom, Heisenberg founded his theory on the notion that the only allowed values of energy exchange between microphysical systems are those that are discrete: quanta. Expressing dynamical quantities such as energy, position, momentum and angular momentum in terms of matrices, he obtained an eigenvalue problem that describes the dynamics of microscopic systems; the diagonalization of the Hamiltonian matrix yields the energy spectrum and the state vectors of the system. Matrix mechanics was very successful in accounting for the discrete quanta of light emitted and absorbed by atoms.
Continuous character of the radiation emitted by a glowing solid object constituted one of the major unsolved problems during the second half of the nineteenth century. All attempts to explain this phenomenon by means of the available theories of classical physics (statistical thermodynamics and classical electromagnetic theory) ended up in miserable failure. This problem consisted in essence of specifying the proper theory of thermodynamics that describes how energy gets exchanged between radiation and matter. All attempts to explain this phenomenon by means of the available theories of classical physics (statistical thermodynamics and classical electromagnetic theory) ended up in miserable failure. This problem consisted in essence of specifying the proper theory of thermodynamics that describes how energy gets exchanged between radiation and matter.
Inspired by Planck’s quantization of electromagnetic radiation, Einstein succeeded in 1905 in giving a theoretical explanation for the dependence of photoelectric emission on the frequency of the incident radiation. He assumed that light is made of corpuscles each carrying an energy hv, called photons.
NOTES
Dirac Notation and Hilbert Space
Self-Instructional Material 51
CHAPTER – 2
DIRAC NOTATION AND HILBERT SPACE
STRUCTURE 2.1 Learning Objectives 2.2 Introduction 2.3 The Hilbert Space and Wave Functions 2.4 Dirac Notation 2.5 Operators 2.6 Representation in Discrete Bases 2.7 Representation in Continuous Bases 2.8 Matrix and Wave Mechanics 2.9 Concluding Remarks 2.10 Summary 2.11 Review Questions 2.12 Further Readings
2.1 LEARNING OBJECTIVESAfter studying the chapter, students will be able to:
zz Understanding Hilbert Space and Wave Functions & Dimension and Basis of a Vector Space.
zz To determine the Physical meaning of the scalar product.
2.2 INTRODUCTION
Dirac Notation and Hilbert Space
NOTES
Quantum Mechanics-1
Self-Instructional Material52
Dirac Notation and Hilbert Space
2.3 THE HILBERT SPACE AND WAVE FUNCTIONSthe lineaR vectoR sPace
Dirac Notation and Hilbert Space
H
H
H
the hilBeRt sPace
H
H
H
NOTES
Dirac Notation and Hilbert Space
Self-Instructional Material 53
H
H
H
H H H
H H H
H
H H H H
dimension and Basis of a vectoR sPace
H H H
H H H
H
H H H H
NOTES
Quantum Mechanics-1
Self-Instructional Material54
H H H
H H H
H
H H H H
NOTES
Dirac Notation and Hilbert Space
Self-Instructional Material 55
sQuaRe-integRaBle functions: Wave functions
NOTES
Quantum Mechanics-1
Self-Instructional Material56
2.4 DIRAC NOTATION
H
H H
NOTES
Dirac Notation and Hilbert Space
Self-Instructional Material 57
H
H H
H
NOTES
Quantum Mechanics-1
Self-Instructional Material58
H
NOTES
Dirac Notation and Hilbert Space
Self-Instructional Material 59
NOTES
Quantum Mechanics-1
Self-Instructional Material60
NOTES
Dirac Notation and Hilbert Space
Self-Instructional Material 61
2.5 OPERATORS
geneRal definitions
P
NOTES
Quantum Mechanics-1
Self-Instructional Material62
P
†
† †
†
†
† †
† †
† †
† †
† † † † †
† ††††
† ††††
†
†
† † †
†
†
†
heRmitian adjoint
),
NOTES
Dirac Notation and Hilbert Space
Self-Instructional Material 63
†
† †
†
†
† †
† †
† †
† †
† † † † †
† ††††
† ††††
†
†
† † †
†
†
†
†
†
† † †
†
† † † †
† †
† † † † †
† † †
† †
†
† †
†
†
†
†
† † †
†
† † † †
† †
† † † † †
† † †
† †
†
† †
†
†
†
†
† † †
†
† † † †
† †
† † † † †
† † †
† †
†
† †
†
†
†
†
† † †
†
† † † †
† †
† † † † †
† † †
† †
†
† †
†
†
†
†
† † †
†
† † † †
† †
† † † † †
† † †
† †
†
† †
†
†
†
†
† † †
†
† † † †
† †
† † † † †
† † †
† †
†
† †
†
†
†
†
† † †
†
† † † †
† †
† † † † †
† † †
† †
†
† †
†
†
NOTES
Quantum Mechanics-1
Self-Instructional Material64
†
†
† † †
†
† † † †
† †
† † † † †
† † †
† †
†
† †
†
†
† † †
†
† ††
†
PRojection oPeRatoRs
commutatoR algeBRa
† † †
†
† ††
†
† † †
†
† ††
†
NOTES
Dirac Notation and Hilbert Space
Self-Instructional Material 65
† † †
† † †
† † †† ††
†
†
†
†
NOTES
Quantum Mechanics-1
Self-Instructional Material66
† † †† ††
†
† † †
† † †† ††
†
† † †
unceRtainty Relation BetWeen tWo oPeRatoRs
† †
NOTES
Dirac Notation and Hilbert Space
Self-Instructional Material 67
† †
† †
† †
† † † † † †
†
†
functions of oPeRatoRs
NOTES
Quantum Mechanics-1
Self-Instructional Material68
† †
† † † † † †
†
†
inveRse and unitaRy oPeRatoRs
†
† † †
† †† † † †
†
† †††† † †††
† †† † †
†
†
NOTES
Dirac Notation and Hilbert Space
Self-Instructional Material 69
†
† † †
† †† † † †
†
† †††† † †††
† †† † †
†
†
eigenvalues and eigenvectoRs of an oPeRatoR
†
† † †
† †† † † †
†
† †††† † †††
† †† † †
†
†
† †
†
†
† †
†
†
NOTES
Quantum Mechanics-1
Self-Instructional Material70
† †
†
†
† †
†
†
†
† †
†
†
† †
†
NOTES
Dirac Notation and Hilbert Space
Self-Instructional Material 71
†
† †
†
†
†
NOTES
Quantum Mechanics-1
Self-Instructional Material72
†
†
infinitesimal and finite unitaRy tRansfoRmations
Unitary Transformations
†
†
†
† † †
† †
† †
† †
† † † †† †
† †
† † † † † †
† † †
NOTES
Dirac Notation and Hilbert Space
Self-Instructional Material 73
Infinitesimal Unitary Transformations
†
† † †
† †
† †
† †
† † † †† †
† †
† † † † † †
† † †
† † † †
†
†
†
† † † † † † †
†
† †
† † † †
† † †
† † † †
†
†
†
† † † † † † †
†
† †
† † † †
† † †
NOTES
Quantum Mechanics-1
Self-Instructional Material74
† † † †
†
†
†
† † † † † † †
†
† †
† † † †
† † †
†
†
Finite Unitary Transformations
2.6 REPRESENTATION IN DISCRETE BASES
†
NOTES
Dirac Notation and Hilbert Space
Self-Instructional Material 75
Matrix Representation of Kets and Bras
matRix RePResentation of kets, BRas, and oPeRatoRs
H
H
NOTES
Quantum Mechanics-1
Self-Instructional Material76
NOTES
Dirac Notation and Hilbert Space
Self-Instructional Material 77
Matrix Representation of Kets and Bras
†
† † †
†
Matrix Representation of Some Other Operators
NOTES
Quantum Mechanics-1
Self-Instructional Material78
†
† † †
†
† †
† †
NOTES
Dirac Notation and Hilbert Space
Self-Instructional Material 79
† †
†
†
†
†
NOTES
Quantum Mechanics-1
Self-Instructional Material80
†
†
NOTES
Dirac Notation and Hilbert Space
Self-Instructional Material 81
matRix RePResentation of seveRal otheR Quantities
NOTES
Quantum Mechanics-1
Self-Instructional Material82
†
†
† † †
Properties of a Matrix A
†
†
† † †
NOTES
Dirac Notation and Hilbert Space
Self-Instructional Material 83
†
†
† † †
† † †
†
†
† †
† † †
†
†
† †
change of Bases and unitaRy tRansfoRmations
NOTES
Quantum Mechanics-1
Self-Instructional Material84
† †
†
†
†
† †
†
† †
†
†
†
† †
†
† †
†
†
†
† †
†
Transformations of Kets, Bras, and Operators
† †
† †
† †
†
†
†
† †
NOTES
Dirac Notation and Hilbert Space
Self-Instructional Material 85
† †
† †
† †
†
†
†
† †
† †
† †
† †
†
†
†
† †
† †
† †
† †
†
†
†
† †
NOTES
Quantum Mechanics-1
Self-Instructional Material86
matRix RePResentation of the eigenvalue PRoBlem
NOTES
Dirac Notation and Hilbert Space
Self-Instructional Material 87
†
NOTES
Quantum Mechanics-1
Self-Instructional Material88
†
†
NOTES
Dirac Notation and Hilbert Space
Self-Instructional Material 89
2.7 REPRESENTATION IN CONTINUOUS BASES
geneRal tReatment
NOTES
Quantum Mechanics-1
Self-Instructional Material90
NOTES
Dirac Notation and Hilbert Space
Self-Instructional Material 91
Position RePResentation
NOTES
Quantum Mechanics-1
Self-Instructional Material92
momentum RePResentation
connecting the Position and momentum RePResentations
NOTES
Dirac Notation and Hilbert Space
Self-Instructional Material 93
Momentum Operator in the Position Representation
NOTES
Quantum Mechanics-1
Self-Instructional Material94
Position Operator in the Momentum Representation
NOTES
Dirac Notation and Hilbert Space
Self-Instructional Material 95
Momentum Operator in the Position Representation
P
P P†
P
P† P
P
P
P P
P
P P P
NOTES
Quantum Mechanics-1
Self-Instructional Material96
P
P P†
P
P† P
P
P
P P
P
P P P
P
P P†
P
P† P
P
P
P P
P
P P P
PaRity oPeRatoR
P† P
P P
P P
P
P
P P
PP
PP
P P
P PP P PP P P PP P PP P
P† P
P P
P P
P
P
P P
PP
PP
P P
P PP P PP P P PP P PP P
NOTES
Dirac Notation and Hilbert Space
Self-Instructional Material 97
P† P
P P
P P
P
P
P P
PP
PP
P P
P PP P PP P P PP P PP P
P P P P
PP† PP†
PP† P
P P
P
PP
PP
PP PP PP PP
PP PP PP PP
NOTES
Quantum Mechanics-1
Self-Instructional Material98
P P P P
PP† PP†
PP† P
P P
P
PP
PP
PP PP PP PP
PP PP PP PP
2.8 MATRIX AND WAVE MECHANICS
matRix mechanics
P P P P
PP† PP†
PP† P
P P
P
PP
PP
PP PP PP PP
PP PP PP PP
P P P P
PP† PP†
PP† P
P P
P
PP
PP
PP PP PP PP
PP PP PP PP
NOTES
Dirac Notation and Hilbert Space
Self-Instructional Material 99
2.9 CONCLUDING REMARKS
Wave mechanics
NOTES
Quantum Mechanics-1
Self-Instructional Material100
2.10 SUMMARYQuantum theory not only works, but works extremely well, and this represents its experimental justification. It has a very penetrating qualitative as well as quantitative prediction power; this prediction power has been verified by a rich collection of experiments. So the accurate prediction power of quantum theory gives irrefutable evidence to the validity of the postulates upon which the theory is built.
An observable is a dynamical variable that can be measured; the dynamical variables encountered most in classical mechanics are the position, linear momentum, angular momentum, and energy. How do we mathematically represent these and other variables in quantum mechanics? In quantum mechanics, however, the measurement process perturbs the system significantly. While carrying out measurements on classical systems, this perturbation does exist, but it is small enough that it can be neglected. In atomic and subatomic systems, however, the act of measurement induces nonnegligible or significant disturbances. Finally, we may state that quantum mechanics is the mechanics applicable to objects for which measurements necessarily interfere with the state of the system. Quantum mechanically, we cannot ignore the effects of the measuring equipment on the system, for they are important. In general, certain measurements cannot be performed without major disturbances to other properties of the quantum system.
Similarly, another measurement of B will yield bn and will leave the system in the same joint eigenstate of A and B. Thus, if two observables A and B are compatible, and if the system is initially in an eigenstate of one of their operators, their measurements not only yield precise values (eigenvalues) but they will not depend on the order in which the measurements were performed. In this case, A and B are said to be simultaneously measurable. So compatible observables can be measured simultaneously with arbitrary accuracy; noncompatible observables cannot.
If the degeneracy persists, we may introduce a fourth operator D that commutes with the previous three and then look for their joint eigenstates which form a complete set. Continuing in this way, we will ultimately exhaust all the operators (that is, there are no more independent operators) which commute with each other. When that happens, we have then obtained a complete set of commuting operators (CSCO). Only then will the state of the system be specified unambiguously, for the joint eigenstates of the CSCO are determined uniquely and will form a complete set (recall that a complete set of eigenvectors of an operator is called a basis). he invariance principles relevant to our study are the time translation invariance and the space translation invariance. We may recall from classical physics that whenever a system is invariant under space translations, its total momentum is conserved; and whenever it is invariant under rotations, its total angular momentum is also conserved.
NOTES
Dirac Notation and Hilbert Space
Self-Instructional Material 101
2.11 REVIEW QUESTIONS
2.12 FURTHER READINGSzz Dass, HK. 2008, Mathematical Physics, New Delhi: S. Chand.
zz Chattopadhyay, PK. 1990, Mathematical Physics, New Delhi: New age International.
zz Hassani, Sadri. 2013, Mathematical Physics: A Modern Introduction to Its Foundations, Berlin: Springer Science & Business Media.
1. Define the Hilbert Space and Wave Functions Dirac Notation.2. What is Operators & Representation in Discrete Bases?3. Describe the Representation in Continuous Bases4. What is Matrix and Wave Mechanics?5. Explain Concluding Remarks.
NOTES
Quantum Mechanics-1
Self-Instructional Material102
CHAPTER – 3
POSTULATES OF QUANTUM MECHANICS
STRUCTURE 3.1 Learning Objectives 3.2 Introduction 3.3 The Basic Postulates of Quantum Mechanics 3.4 The State of a System 3.5 Observables and Operators 3.6 Measurement in Quantum Mechanics 3.7 Time Evolution of the System’s State 3.8 Symmetries and Conservation Laws 3.9 Connecting Quantum to Classical Mechanics 3.10 Summary 3.11 Review Questions 3.12 Further Readings
3.1 LEARNING OBJECTIVESAfter studying the chapter, students will be able to:
zz Understanding the Basic Postulates of Quantum Mechanicszz To determine the Measurement in Quantum Mechanics
3.2 INTRODUCTION
NOTES
Postulates of Quantum Mechanics
Self-Instructional Material 103
H
3.3 THE BASIC POSTULATES OF QUANTUM MECHANICS
NOTES
Quantum Mechanics-1
Self-Instructional Material104
H
H
NOTES
Postulates of Quantum Mechanics
Self-Instructional Material 105
H
3.4 THE STATE OF A SYSTEM
the state of a system
NOTES
Quantum Mechanics-1
Self-Instructional Material106
the suPeRPosition PRinciPle
NOTES
Postulates of Quantum Mechanics
Self-Instructional Material 107
NOTES
Quantum Mechanics-1
Self-Instructional Material108
3.5 OBSERVABLES AND OPERATORS
NOTES
Postulates of Quantum Mechanics
Self-Instructional Material 109
3.6 MEASUREMENT IN QUANTUM MECHANICS
NOTES
Quantum Mechanics-1
Self-Instructional Material110
hoW measuRements distuRB systems
NOTES
Postulates of Quantum Mechanics
Self-Instructional Material 111
exPectation values
NOTES
Quantum Mechanics-1
Self-Instructional Material112
comPlete sets of commuting oPeRatoRs (csco)
NOTES
Postulates of Quantum Mechanics
Self-Instructional Material 113
NOTES
Quantum Mechanics-1
Self-Instructional Material114
measuRement and the unceRtainty Relations
NOTES
Postulates of Quantum Mechanics
Self-Instructional Material 115
3.7 TIME EVOLUTION OF THE SYSTEM’S STATE
time evolution oPeRatoR
NOTES
Quantum Mechanics-1
Self-Instructional Material116
†
†
stationaRy states: time-indePendent Potentials
†
†
NOTES
Postulates of Quantum Mechanics
Self-Instructional Material 117
schRödingeR eQuation and Wave Packets
†
NOTES
Quantum Mechanics-1
Self-Instructional Material118
the conseRvation of PRoBaBility
†
†
†
†
†
NOTES
Postulates of Quantum Mechanics
Self-Instructional Material 119
time evolution of exPectation values
†
†
†
†
†
†
NOTES
Quantum Mechanics-1
Self-Instructional Material120
3.8 SYMMETRIES AND CONSERVATION LAWS
infinitesimal unitaRy tRansfoRmations
NOTES
Postulates of Quantum Mechanics
Self-Instructional Material 121
finite unitaRy tRansfoRmations
NOTES
Quantum Mechanics-1
Self-Instructional Material122
symmetRies and conseRvation laWs
NOTES
Postulates of Quantum Mechanics
Self-Instructional Material 123
P
P
3.9 CONNECTING QUANTUM TO CLASSICAL MECHANICSPoisson BRackets and commutatoRs
P
P
NOTES
Quantum Mechanics-1
Self-Instructional Material124
NOTES
Postulates of Quantum Mechanics
Self-Instructional Material 125
the ehRenfest theoRem
NOTES
Quantum Mechanics-1
Self-Instructional Material126
3.10 SUMMARYDescribe a system in quantum mechanics, we use a mathematical entity (a complex function) belonging to a Hilbert space, the state vector |(t), which contains all the information we need to know about the system and from which all needed physical quantities can be computed. Observable is a dynamical variable that can be measured; the dynamical variables encountered most in classical mechanics are the position, linear momentum, angular momentum, and energy. How do we mathematically represent these and other variables in quantum mechanics? In quantum mechanics, however, the measurement process perturbs the system significantly. While carrying out measurements on classical systems, this perturbation does exist, but it is small enough that it can be neglected. In atomic and subatomic systems, however, the act of measurement induces nonnegligible or significant disturbances. Finally, we may state that quantum mechanics is the mechanics applicable to objects for which measurements necessarily interfere with the state of the system. Quantum mechanically, we cannot ignore the effects of the measuring equipment on the system, for they are important. In general, certain measurements cannot be performed without major disturbances to other properties of the quantum system. Similarly, another measurement of B will yield bn and will leave the system in the same joint eigenstate of A and B. Thus, if two observables A and B are compatible, and if the system is initially in an eigenstate of one of their operators, their measurements not only yield precise values (eigenvalues) but they will not depend on the order in which the measurements were performed. In this case, A and B are said to be simultaneously measurable. So compatible observables can be measured simultaneously with arbitrary accuracy; noncompatible observables cannot. We will show that for each such symmetry there corresponds an observable which is a constant of the motion. The invariance principles relevant to our study are the time translation invariance and the space translation invariance. We may recall from classical physics that whenever a system is invariant under space translations, its total momentum is conserved; and whenever it is invariant under rotations, its total angular momentum is also conserved.
NOTES
Postulates of Quantum Mechanics
Self-Instructional Material 127
3.11 REVIEW QUESTIONS
3.12 FURTHER READINGSzz Dass, HK. 2008, Mathematical Physics, New Delhi: S. Chand.
zz Chattopadhyay, PK. 1990, Mathematical Physics, New Delhi: New age International.
zz Hassani, Sadri. 2013, Mathematical Physics: A Modern Introduction to Its Foundations, Berlin: Springer Science & Business Media.
1. Define the Hilbert Space and Wave Functions Dirac Notation.2. What is Operators & Representation in Discrete Bases?3. Describe the Representation in Continuous Bases4. What is Matrix and Wave Mechanics?5. Explain Concluding Remarks.
NOTES
Quantum Mechanics-1
Self-Instructional Material128
CHAPTER – 4
ONE-DIMENSIONAL PROBLEMS
STRUCTURE 4.1 Learning Objectives 4.2 Introduction 4.3 Properties of One-Dimensional Motion 4.4 The Free Particle: Continuous States 4.5 The Potential Step 4.6 The Potential Barrier and Well 4.7 The Infinite Square Well Potential 4.8 The Finite Square Well Potential 4.9 The Harmonic Oscillator 4.10 The Harmonic Oscillator 4.11 Summary 4.12 Review Questions 4.13 Further Readings
4.1 LEARNING OBJECTIVESAfter studying the chapter, students will be able to:
zz Understanding the Free Particle: Continuous Stateszz To determine the Finite Square Well Potential
4.2 INTRODUCTION
NOTES
One-Dimensional Problems
Self-Instructional Material 129
-
6
?
6
?
6
4.3 PROPERTIES OF ONE-DIMENSIONAL MOTION
-
6
?
6
?
6
discRete sPectRum (Bound states)
-
6
?
6
?
6
NOTES
Quantum Mechanics-1
Self-Instructional Material130
-
6
?
6
?
6
continuous sPectRum (unBound states)
mixed sPectRum
NOTES
One-Dimensional Problems
Self-Instructional Material 131
symmetRic Potentials and PaRity
4.4 THE FREE PARTICLE: CONTINUOUS STATES
NOTES
Quantum Mechanics-1
Self-Instructional Material132
NOTES
One-Dimensional Problems
Self-Instructional Material 133
4.5 THE POTENTIAL STEP
NOTES
Quantum Mechanics-1
Self-Instructional Material134
-
6
-
6
-
6
-
¾-
-
6
-
¾
-
6
-
6
-
6
-
¾-
-
6
-
¾
-
6
-
6
-
6
-
¾-
-
6
-
¾
NOTES
One-Dimensional Problems
Self-Instructional Material 135
K
K
K
K
K
K
NOTES
Quantum Mechanics-1
Self-Instructional Material136
NOTES
One-Dimensional Problems
Self-Instructional Material 137
-
6
-
6
-
6
-
¾
-
¾
-
-
6
-
¾
-
4.6 THE POTENTIAL BARRIER AND WELL
the case e >v0
NOTES
Quantum Mechanics-1
Self-Instructional Material138
-
6
-
6
-
6
-
¾
-
¾
-
-
6
-
¾
-
-
6
6
-
6
6
-
6
-
6
-
6
-
¾
-
¾
-
-
6
-
¾
-
-
6
-
6
-
6
-
¾
-
¾
-
-
6
-
¾
-
-
6
6
-
6
6
NOTES
One-Dimensional Problems
Self-Instructional Material 139
-
6
6
-
6
6
NOTES
Quantum Mechanics-1
Self-Instructional Material140
the case e >v0: tunneling
NOTES
One-Dimensional Problems
Self-Instructional Material 141
NOTES
Quantum Mechanics-1
Self-Instructional Material142
-
6
-
6
the tunneling effect
NOTES
One-Dimensional Problems
Self-Instructional Material 143
-
6
-
6
4.7 THE INFINITE SQUARE WELL POTENTIAL
the asymmetRic sQuaRe Well
NOTES
Quantum Mechanics-1
Self-Instructional Material144
-
6
-
6
NOTES
One-Dimensional Problems
Self-Instructional Material 145
-
6
NOTES
Quantum Mechanics-1
Self-Instructional Material146
the symmetRic Potential Well
NOTES
One-Dimensional Problems
Self-Instructional Material 147
-
6
-
¾
-
¾
-
-
6
4.8 THE FINITE SQUARE WELL POTENTIAL
the scatteRing solutions (e >v0)
-
6
-
¾
-
¾
-
-
6
the Bound state solutions (0 < e < v0)
-
6
-
¾
-
¾
-
-
6
NOTES
Quantum Mechanics-1
Self-Instructional Material148
-
6
-
¾
-
¾
-
-
6
NOTES
One-Dimensional Problems
Self-Instructional Material 149
NOTES
Quantum Mechanics-1
Self-Instructional Material150
-
6
¾
¾
¾
NOTES
One-Dimensional Problems
Self-Instructional Material 151
†
†
†
4.9 THE HARMONIC OSCILLATOR
NOTES
Quantum Mechanics-1
Self-Instructional Material152
†
†
†
†
†
†
†
†
†
†
†
† †
† † † †
† †
NOTES
One-Dimensional Problems
Self-Instructional Material 153
†
†
†
†
†
†
†
†
† †
† † † †
† †
†
† † †
† †
† †
† † †
† † †
† †
† †
eneRgy eigenvalues
†
†
†
†
†
†
†
†
† †
† † † †
† †
NOTES
Quantum Mechanics-1
Self-Instructional Material154
†
† † †
† †
† †
† † †
† † †
† †
† †
†
†
†
†
†
†
†
†
†
†
NOTES
One-Dimensional Problems
Self-Instructional Material 155
†
†
†
†
†
†
†
†
†
†
†
†
†
eneRgy eigenstates
†
†
†
†
†
†
†
†
†
†
eneRgy eigenstates in Position sPace
†
†
†
NOTES
Quantum Mechanics-1
Self-Instructional Material156
†
†
†
†
†
†
†
†
†
†
†
NOTES
One-Dimensional Problems
Self-Instructional Material 157
†
†
†
†
†
†
†
†
NOTES
Quantum Mechanics-1
Self-Instructional Material158
-
6
-
6
-
6
†
† †
†
†
†
†
†
the matRix RePResentation of vaRious oPeRatoRs
-
6
-
6
-
6
†
† †
†
†
†
†
†
NOTES
One-Dimensional Problems
Self-Instructional Material 159
-
6
-
6
-
6
†
† †
†
†
†
†
†
† †
† † †
† †
† † †
† †
NOTES
Quantum Mechanics-1
Self-Instructional Material160
† †
† † †
† †
† † †
† †
† † †
† † †
† † †
† †
† †
exPectation values of vaRious oPeRatoRs
† †
† † †
† †
† † †
† †
NOTES
One-Dimensional Problems
Self-Instructional Material 161
† † †
† † †
† † †
† †
† †
-
6
-
6
4.10 THE HARMONIC OSCILLATOR
numeRical PRoceduRe
† † †
† † †
† † †
† †
† †
NOTES
Quantum Mechanics-1
Self-Instructional Material162
-
6
-
6
algoRithm
NOTES
One-Dimensional Problems
Self-Instructional Material 163
4.11 SUMMARYThe wave packet solution cures and avoids all the subtleties raised above. First, the momentum, the position and the energy of the particle are no longer known exactly; only probabilistic outcomes are possible. Second, as shown in Chapter 1, the wave packet (4.12) and the particle travel with the same speed vg= p/m, called the group speed or the speed of the whole packet. Third, the wave packet (4.12) is normalizable. We are now going to show that the quantum mechanical
NOTES
Quantum Mechanics-1
Self-Instructional Material164
4.12 REVIEW QUESTIONS
4.13 FURTHER READINGSzz Dass, HK. 2008, Mathematical Physics, New Delhi: S. Chand.
zz Chattopadhyay, PK. 1990, Mathematical Physics, New Delhi: New age International.
zz Hassani, Sadri. 2013, Mathematical Physics: A Modern Introduction to Its Foundations, Berlin: Springer Science & Business Media.
1. What is Properties of One-Dimensional Motion?2. Explain the Free Particle: Continuous States.3. Describe the Potential Step 4. What is the Potential Barrier and Well?5. What is the Infinite Square Well Potential & Finite Square Well Potential.6. Describe the Harmonic Oscillator.
predictions differ sharply from their classical counterparts, for the wave function is not zero beyond the barrier. Classically, we would expect total reflection: every particle that arrives at the barrier (x) will be reflected back; no particle can penetrate the barrier, where it would have a negative kinetic energy.
Quantum mechanical effect which is due to the wave aspect of microscopic objects; it is known as the tunneling effect: quantum mechanical objects can tunnel through classically impenetrable barriers. This barrier penetration effect has important applications in various branches of modern physics ranging from particle and nuclear physics to semiconductor devices. For instance, radioactive decays and charge transport in electronic devices are typical examples of the tunneling effect.
This is in sharp contrast to classical mechanics, where the lowest possible energy is equal to the minimum value of the potential energy, with zero kinetic energy. In quantum mechanics, however, the lowest state does not minimize the potential alone, but applies to the sum of the kinetic and potential energies, and this leads to a finite ground state or zero-point energy. This concept has far-reaching physical consequences in the realm of the microscopic world. For instance, without the zero-point motion, atoms would not be stable, for the electrons would fall into the nuclei. Also, it is the zero-point energy which prevents helium from freezing at very low temperatures.
Harmonic oscillator is one of those few problems that are important to all branches of physics. It provides a useful model for a variety of vibrational phenomena that are encountered, for instance, in classical mechanics, electrodynamics, statistical mechanics, solid state, atomic, nuclear, and particle physics.
NOTES
Angular Momentum
Self-Instructional Material 165
CHAPTER – 5
ANGULAR MOMENTUMSTRUCTURE
5.1 Learning Objectives 5.2 Introduction 5.3 Orbital Angular Momentum 5.4 General Formalism of Angular Momentum 5.5 Matrix Representation of Angular Momentum 5.6 Geometrical Representation of Angular Momentum 5.7 Spin Angular Momentum 5.8 Eigenfunctions of Orbital Angular Momentum 5.9 Summary 5.10 Review Questions 5.11 Further Readings
5.1 LEARNING OBJECTIVESAfter studying the chapter, students will be able to:
zz Understanding Geometrical Representation of Angular Momentumzz To determine the general Formalism of Angular Momentum
5.2 INTRODUCTIONAfter treating one-dimensional problems in Chapter 4, we now should deal with three-dimensional problems. However, the study of three-dimensional systems such as atoms cannot be undertaken unless we first cover the formalism of angular momentum.
NOTES
Quantum Mechanics-1
Self-Instructional Material166
5.3 ORBITAL ANGULAR MOMENTUM
NOTES
Angular Momentum
Self-Instructional Material 167
5.4 GENERAL FORMALISM OF ANGULAR MOMENTUM
NOTES
Quantum Mechanics-1
Self-Instructional Material168
NOTES
Angular Momentum
Self-Instructional Material 169
NOTES
Quantum Mechanics-1
Self-Instructional Material170
NOTES
Angular Momentum
Self-Instructional Material 171
†
†
†
†
†
†
†
NOTES
Quantum Mechanics-1
Self-Instructional Material172
5.5 MATRIX REPRESENTATION OF ANGULAR MOMENTUM
NOTES
Angular Momentum
Self-Instructional Material 173
NOTES
Quantum Mechanics-1
Self-Instructional Material174
-©©©©©©¼
6
¡¡¡¡¡¡
@@
@@
@@
¢¢¢¢¢̧
-
6
¡¡¡¡¡¡µ
@@@@@@R
@@
@@
@@I
¡¡
¡¡
¡¡ª
NOTES
Angular Momentum
Self-Instructional Material 175
-©©©©©©¼
6
¡¡¡¡¡¡
@@
@@
@@
¢¢¢¢¢̧
-
6
¡¡¡¡¡¡µ
@@@@@@R
@@
@@
@@I
¡¡
¡¡
¡¡ª
-©©©©©©¼
6
¡¡¡¡¡¡
@@
@@
@@
¢¢¢¢¢̧
-
6
¡¡¡¡¡¡µ
@@@@@@R
@@
@@
@@I
¡¡
¡¡
¡¡ª
5.6 GEOMETRICAL REPRESENTATION OF ANGULARMOMENTUM
-©©©©©©¼
6
¡¡¡¡¡¡
@@
@@
@@
¢¢¢¢¢̧
-
6
¡¡¡¡¡¡µ
@@@@@@R
@@
@@
@@I
¡¡
¡¡
¡¡ª
-
6
AAAAAAAAU
¢¢¢¢¢¢¢¢®
HHHHHHHHj
©©©©©©©©¼
©©©©
©©©©*
HHHH
HHHHY
¢¢¢¢¢¢¢¢̧
AAAAAAAAK
-
6
AAAAAAAAU
HHHHHHHHj
©©©©
©©©©*
¢¢¢¢¢¢¢¢̧
-©©©©©©¼
6
¡¡¡¡¡¡
@@
@@
@@
¢¢¢¢¢̧
-
6
¡¡¡¡¡¡µ
@@@@@@R
@@
@@
@@I
¡¡
¡¡
¡¡ª
-
6
AAAAAAAAU
¢¢¢¢¢¢¢¢®
HHHHHHHHj
©©©©©©©©¼
©©©©
©©©©*
HHHH
HHHHY
¢¢¢¢¢¢¢¢̧
AAAAAAAAK
-
6
AAAAAAAAU
HHHHHHHHj
©©©©
©©©©*
¢¢¢¢¢¢¢¢̧
NOTES
Quantum Mechanics-1
Self-Instructional Material176
-
6
AAAAAAAAU
¢¢¢¢¢¢¢¢®
HHHHHHHHj
©©©©©©©©¼
©©©©
©©©©*
HHHH
HHHHY
¢¢¢¢¢¢¢¢̧
AAAAAAAAK
-
6
AAAAAAAAU
HHHHHHHHj
©©©©
©©©©*
¢¢¢¢¢¢¢¢̧
-©©*
HHj
-
6
@@@@R
¡¡¡¡µ
5.7 SPIN ANGULAR MOMENTUM
-©©*
HHj
-
6
@@@@R
¡¡¡¡µ
exPeRimental evidence of the sPin
NOTES
Angular Momentum
Self-Instructional Material 177
-©©*
HHj
-
6
@@@@R
¡¡¡¡µ
PPPq
6
w³³³1
-©©©©©©¼
6
¡¡¡¡¡¡µ
NOTES
Quantum Mechanics-1
Self-Instructional Material178
PPPq
6
w³³³1
-©©©©©©¼
6
¡¡¡¡¡¡µ
NOTES
Angular Momentum
Self-Instructional Material 179
geneRal theoRy of sPin
sPin 1 / 2 and the Pauli matRice
NOTES
Quantum Mechanics-1
Self-Instructional Material180
NOTES
Angular Momentum
Self-Instructional Material 181
†
NOTES
Quantum Mechanics-1
Self-Instructional Material182
†
5.8 EIGENFUNCTIONS OF ORBITAL ANGULAR MOMENTUM
NOTES
Angular Momentum
Self-Instructional Material 183
eigenfunctions and eigenvalues of
NOTES
Quantum Mechanics-1
Self-Instructional Material184
eigenfunctions of
NOTES
Angular Momentum
Self-Instructional Material 185
NOTES
Quantum Mechanics-1
Self-Instructional Material186
NOTES
Angular Momentum
Self-Instructional Material 187
P
P
PRoPeRties of the sPheRical haRmonics
NOTES
Quantum Mechanics-1
Self-Instructional Material188
P
P
NOTES
Angular Momentum
Self-Instructional Material 189
NOTES
Quantum Mechanics-1
Self-Instructional Material190
5.9 SUMMARYAdditionally, angular momentum plays a critical role in the description of molecular rotations, the motion of electrons in atoms, and the motion of nucleons in nuclei. The quantum theory of angular momentum is thus a prerequisite for studying molecular, atomic, and nuclear systems. There are many ways to represent the angular momentum operators and their eigenstates. In this section we are going to discuss the matrix representation of angular momentum where eigenkets and operators will be represented by column vectors and square matrices, respectively. This is achieved by expanding states and operators in a discrete basis. We will see later how to represent the orbital angular momentum in the position representation.
5.10 REVIEW QUESTIONS
5.11 FURTHER READINGSzz Dass, HK. 2008, Mathematical Physics, New Delhi: S. Chand.
zz Chattopadhyay, PK. 1990, Mathematical Physics, New Delhi: New age International.
zz Hassani, Sadri. 2013, Mathematical Physics: A Modern Introduction to Its Foundations, Berlin: Springer Science & Business Media.
1. What Orbital Angular Momentum? 2. Describe General Formalism of Angular Momentum, 3. Explain Matrix Representation of Angular Momentum. 4. What is Geometrical Representation of Angular Momentum? 5. Define Spin Angular Momentum. 6. What is Eigenfunctions of Orbital Angular Momentum?
NOTES
Three-Dimensional Problems
Self-Instructional Material 191
CHAPTER – 6
THREE-DIMENSIONAL PROBLEMS
STRUCTURE 6.1 Learning Objectives 6.2 Introduction 6.3 3D Problems in Cartesian Coordinates 6.4 3D Problems in Spherical Coordinates 6.5 Concluding Remarks 6.6 Summary 6.7 Review Questions 6.8 Further Readings
6.1 LEARNING OBJECTIVESAfter studying the chapter, students will be able to:
zz Understanding the Problems in Spherical Coordinates & zz To determine the Concluding Remarks
6.2 INTRODUCTION
6.3 3D PROBLEMS IN CARTESIAN COORDINATES
NOTES
Quantum Mechanics-1
Self-Instructional Material192
geneRal tReatment: sePaRation of vaRiaBles
NOTES
Three-Dimensional Problems
Self-Instructional Material 193
the fRee PaRticle
NOTES
Quantum Mechanics-1
Self-Instructional Material194
the Box Potential
NOTES
Three-Dimensional Problems
Self-Instructional Material 195
NOTES
Quantum Mechanics-1
Self-Instructional Material196
the haRmonic oscillatoR
NOTES
Three-Dimensional Problems
Self-Instructional Material 197
NOTES
Quantum Mechanics-1
Self-Instructional Material198
6.4 3D PROBLEMS IN SPHERICAL COORDINATEScentRal Potential: geneRal tReatment
NOTES
Three-Dimensional Problems
Self-Instructional Material 199
NOTES
Quantum Mechanics-1
Self-Instructional Material200
-
6
-
6
NOTES
Three-Dimensional Problems
Self-Instructional Material 201
-
6
the fRee PaRticle in sPheRical cooRdinates
-
6
R
R
R
R R
R
R
R
R
R R
R
NOTES
Quantum Mechanics-1
Self-Instructional Material202
R
R
R
R R
R
-
6
-
6
-
6
-
6
NOTES
Three-Dimensional Problems
Self-Instructional Material 203
-
6
-
6
the sPheRical sQuaRe Well Potential
NOTES
Quantum Mechanics-1
Self-Instructional Material204
NOTES
Three-Dimensional Problems
Self-Instructional Material 205
the isotRoPic haRmonic oscillatoR
NOTES
Quantum Mechanics-1
Self-Instructional Material206
NOTES
Three-Dimensional Problems
Self-Instructional Material 207
NOTES
Quantum Mechanics-1
Self-Instructional Material208
the hydRogen atom
NOTES
Three-Dimensional Problems
Self-Instructional Material 209
NOTES
Quantum Mechanics-1
Self-Instructional Material210
NOTES
Three-Dimensional Problems
Self-Instructional Material 211
NOTES
Quantum Mechanics-1
Self-Instructional Material212
R
R
R
R
R
R
NOTES
Three-Dimensional Problems
Self-Instructional Material 213
R
R
R
R
R
R
?
NOTES
Quantum Mechanics-1
Self-Instructional Material214
NOTES
Three-Dimensional Problems
Self-Instructional Material 215
NOTES
Quantum Mechanics-1
Self-Instructional Material216
-
6
-
6
-
6
-
6
-
6
-
6
-
6
-
6
-
6
-
6
-
6
-
6
NOTES
Three-Dimensional Problems
Self-Instructional Material 217
NOTES
Quantum Mechanics-1
Self-Instructional Material218
NOTES
Three-Dimensional Problems
Self-Instructional Material 219
NOTES
Quantum Mechanics-1
Self-Instructional Material220
effect of magnetic fields on centRal Potentials
NOTES
Three-Dimensional Problems
Self-Instructional Material 221
NOTES
Quantum Mechanics-1
Self-Instructional Material222
ÃÃÃÃÃÃ
ÃÃ```````̀
!!!!
!!!!
ÃÃÃÃÃÃ
ÃÃ```````̀aaaaaaaa
ÃÃÃÃÃÃ
ÃÃ```````̀
!!!!
!!!!
ÃÃÃÃÃÃ
ÃÃ```````̀aaaaaaaa
NOTES
Three-Dimensional Problems
Self-Instructional Material 223
ÃÃÃÃÃÃ
ÃÃ```````̀
!!!!
!!!!
ÃÃÃÃÃÃ
ÃÃ```````̀aaaaaaaa
6.5 CONCLUDING REMARKS
6.6 SUMMARYThus, the zero-point energy for a particle in a three-dimensional box is three times that in a one-dimensional box. The factor 3 can be viewed as originating from the fact that we are confining the particle symmetrically in all three dimensions. Degeneracy occurs only when there is a symmetry in the problem. For the present case of a particle in a cubic box, there is a great deal of symmetry, since all three dimensions are equivalent. Note that for the rectangular box, there
NOTES
Quantum Mechanics-1
Self-Instructional Material224
6.7 REVIEW QUESTIONS
6.8 FURTHER READINGSzz Dass, HK. 2008, Mathematical Physics, New Delhi: S. Chand.
zz Chattopadhyay, PK. 1990, Mathematical Physics, New Delhi: New age International.
zz Hassani, Sadri. 2013, Mathematical Physics: A Modern Introduction to Its Foundations, Berlin: Springer Science & Business Media.
1. What is 3D Problems in Cartesian Coordinates>2. Describe the 3D Problems in Spherical Coordinates.3. What is the Concluding Remarks?
is no degeneracy since the three dimensions are not equivalent. Moreover, degeneracy did not exist when we treated one-dimensional problems in Chapter 4, for they give rise to only one quantum number. Since the energy depends on the sum of nx, ny, nz, any set of quantum numbers having the same sum will represent states of equal energy.
Hydrogen
NOTES
Time Independent Perturbation Theory
Self-Instructional Material 225
CHAPTER – 7
TIME INDEPENDENT PERTURBATION THEORY
STRUCTURE 7.1 Learning Objectives 7.2 Introduction 7.3 Time-Independent Perturbation Theory 7.4 The Variational Method 7.5 Summary 7.6 Review Questions 7.7 Further Readings
7.1 LEARNING OBJECTIVESAfter studying the chapter, students will be able to:
zz Understanding the Degenerate Perturbation Theory.zz To determine Nondegenerate Perturbation Theory.
7.2 INTRODUCTION
Time Independent Perturbation Theory
7
Time Independent Perturbation Theory
7
NOTES
Quantum Mechanics-1
Self-Instructional Material226
Time Independent Perturbation Theory
7
7.3 TIME-INDEPENDENT PERTURBATION THEORY
Time Independent Perturbation Theory
7
nondegeneRate PeRtuRBation theoRy
NOTES
Time Independent Perturbation Theory
Self-Instructional Material 227
NOTES
Quantum Mechanics-1
Self-Instructional Material228
E
E
E
E
E
E
E
E
NOTES
Time Independent Perturbation Theory
Self-Instructional Material 229
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E E
NOTES
Quantum Mechanics-1
Self-Instructional Material230
E
E
E
E
E
E E
E
E
E
E
E
E E
E E
E E
E
E
E
NOTES
Time Independent Perturbation Theory
Self-Instructional Material 231
E E
E E
E
E
E
NOTES
Quantum Mechanics-1
Self-Instructional Material232
degeneRate PeRtuRBation theoRy
NOTES
Time Independent Perturbation Theory
Self-Instructional Material 233
E E
E E E
E
E E
E E E
E
NOTES
Quantum Mechanics-1
Self-Instructional Material234
E E
E E E
E
E
E
E
E
E
E
E
E
E
E
fine stRuctuRe and the anomalous zeeman effect
NOTES
Time Independent Perturbation Theory
Self-Instructional Material 235
E
E
E
E
E
E
E
E
E
E
E
E
E
E
E
PPPq
6
w t
³³³1 PP
Pi
6
t
w
³³³1
NOTES
Quantum Mechanics-1
Self-Instructional Material236
PPPq
6
w t
³³³1 PP
Pi
6
t
w
³³³1
NOTES
Time Independent Perturbation Theory
Self-Instructional Material 237
NOTES
Quantum Mechanics-1
Self-Instructional Material238
NOTES
Time Independent Perturbation Theory
Self-Instructional Material 239
NOTES
Quantum Mechanics-1
Self-Instructional Material240
NOTES
Time Independent Perturbation Theory
Self-Instructional Material 241
ÃÃÃÃÃÃ
ÃÃ```````̀
!!!!
!!!!
ÃÃÃÃÃÃ
ÃÃ```````̀aaaaaaaa
ÃÃÃÃÃÃ
ÃÃ```````̀
!!!!
!!!!
ÃÃÃÃÃÃ
ÃÃ```````̀aaaaaaaa
NOTES
Quantum Mechanics-1
Self-Instructional Material242
9.4 THE VARIATIONAL METHOD
qqq
qqq
qqq
qqq
qqq
qqq
¡¡¡¡¡
@@@@@ ¡
¡¡
©©©
HHH@@@
¡¡¡
©©©
HHH@@@
6?
6?
6?
6?
NOTES
Time Independent Perturbation Theory
Self-Instructional Material 243
qqq
qqq
qqq
qqq
qqq
qqq
¡¡¡¡¡
@@@@@ ¡
¡¡
©©©
HHH@@@
¡¡¡
©©©
HHH@@@
6?
6?
6?
6?
qqq
qqq
qqq
qqq
qqq
qqq
¡¡¡¡¡
@@@@@ ¡
¡¡
©©©
HHH@@@
¡¡¡
©©©
HHH@@@
6?
6?
6?
6?
qqq
qqq
qqq
qqq
qqq
qqq
¡¡¡¡¡
@@@@@ ¡
¡¡
©©©
HHH@@@
¡¡¡
©©©
HHH@@@
6?
6?
6?
6?
qqq
qqq
qqq
qqq
qqq
qqq
¡¡¡¡¡
@@@@@ ¡
¡¡
©©©
HHH@@@
¡¡¡
©©©
HHH@@@
6?
6?
6?
6?
NOTES
Quantum Mechanics-1
Self-Instructional Material244
NOTES
Time Independent Perturbation Theory
Self-Instructional Material 245
NOTES
Quantum Mechanics-1
Self-Instructional Material246
-
6
NOTES
Time Independent Perturbation Theory
Self-Instructional Material 247
-
6
NOTES
Quantum Mechanics-1
Self-Instructional Material248
NOTES
Time Independent Perturbation Theory
Self-Instructional Material 249
7.5 SUMMARYPerturbation theory is based on the assumption that the problem we wish to solve is, in some sense, only slightly different from a problem that can be solved exactly. In the case where The WKB method is useful for finding the energy eigenvalues and wave functions of systems for which the classical limit is valid. Unlike perturbation theory, the variational and WKB methods do not require the existence of a closely related Hamiltonian that can be solved exactly.
Hyperfine corrections are included, they would split each of the fine structure levels into a series of hyperfine levels. For instance, when the hyperfine coupling is taken into account in the ground state of hydrogen, it would split the 1S1/2 level into two hyperfine levels separated by an energy of 5.89 × 10–6 eV. This corresponds, when the atom makes a spontaneous transition from the higher hyperfine level to the lower one, to a radiation of 1.42 × 109 Hz frequency and 21 cm wavelength. We should note that most of the information we possess about interstellar hydrogen clouds had its origin in the radioastronomy study of this 21 cm line.
NOTES
Quantum Mechanics-1
Self-Instructional Material250
7.6 REVIEW QUESTIONS
7.7 FURTHER READINGSzz Dass, HK. 2008, Mathematical Physics, New Delhi: S. Chand.
zz Chattopadhyay, PK. 1990, Mathematical Physics, New Delhi: New age International.
zz Hassani, Sadri. 2013, Mathematical Physics: A Modern Introduction to Its Foundations, Berlin: Springer Science & Business Media.
1. Explain the Time-Independent Perturbation Theory.2. What is the The Variational Method?3. Describe the Fine Structure and the Anomalous Zeeman Effect.
top related