This lecture discusses operator methods in quantum mechanics. Some key points:
1. Operators allow quantum mechanics to be formulated without relying on a particular basis. The Hamiltonian operator H acts on state vectors.
2. Dirac notation represents state vectors as "kets" and defines inner products between states. A resolution of identity allows expanding states in a basis.
3. Hermitian operators correspond to physical observables. Their eigenfunctions form a complete basis. The time-evolution operator evolves states forward in time.
4. The uncertainty principle relates the uncertainties of non-commuting operators like position and momentum. Symmetries of the Hamiltonian are represented by unitary operators that commute with it.
Schrodinger equation and its applications: Chapter 2Dr.Pankaj Khirade
Wave function and its physical significance, Schrodinger time dependent equation, Separation in time dependent and time independent parts, Operators in quantum Mechanics, Eigen functions and Eigen values, Particle in one dimensional and three dimensional box (Energy eigen values). Qualitative analysis of potential barrier Tunneling effect). Simple Harmonic Oscillator (Qualitative analysis of Zero point energy)
The presentation is about how to evaluate the probability of finding the system in any particular state at any later time when the simple Hamiltonian was added by time dependent perturbation. So now the wave function will have perturbation-induced time dependence.
The postulates of quantum mechanics have been successfully used for deriving exact solutions to Schrodinger equation for problems like A particle in 1 Dimensional box Harmonic oscillator Rigid rotator Hydrogen atom • However for a multielectron system, the SWE cannot be solved exactly due to inter-electronic repulsion terms.
The SWE is solved by method of seperation of variables.
• However, the inter-electronic repulsion term cannot be solved because the variables cannot be seperated and the SWE cannot be solved. • Approximate methods have helped to generate solutions for such and even more complex real quantum systems. • Approximate methods have been developed for solving Schrodinger equation to find wave function and energy of the complex system under consideration. • Two widely used approximate methods are, 1. Perturbation theory 2. Variation method
Perturbation theory is an approximate method that describes a complex quantum system in terms of a simpler system for which the exact solution is known. • Perturbation theory has been categorized into, i. Time independent perturbation theory, proposed by Erwin Schrodinger, where the perturbation Hamiltonian is static. ii. Time dependent perturbation theory, proposed by Paul Dirac, which studies the effect of time dependent perturbation on a time independent Hamiltonian H0.
PERTURBATION THEOREM
FIRST ORDER PERTURBATION THEORY
FIRST ORDER ENERGY CORRECTION
FIRST ORDER WAVE FUNCTION CORRECTION
APPLICATIONS OF PERTURBATION METHOD
SIGNIFICANCE OF PERTURBATION METHOD
Time Independent Perturbation Theory, 1st order correction, 2nd order correctionJames Salveo Olarve
The presentation is about how to solve the new energy levels and wave functions when the simple Hamiltonian is added by another term due to external effect (can be due to external field) .
The intended reader of this presentation were physics students. The author already assumed that the reader knows dirac braket notation.
7.1 Application of the Schrödinger Equation to the Hydrogen Atom
7.2 Solution of the Schrödinger Equation for Hydrogen
7.3 Quantum Numbers
7.4 Magnetic Effects on Atomic Spectra – The Normal Zeeman Effect
7.5 Intrinsic Spin
7.6 Energy Levels and Electron Probabilities
Schrodinger equation and its applications: Chapter 2Dr.Pankaj Khirade
Wave function and its physical significance, Schrodinger time dependent equation, Separation in time dependent and time independent parts, Operators in quantum Mechanics, Eigen functions and Eigen values, Particle in one dimensional and three dimensional box (Energy eigen values). Qualitative analysis of potential barrier Tunneling effect). Simple Harmonic Oscillator (Qualitative analysis of Zero point energy)
The presentation is about how to evaluate the probability of finding the system in any particular state at any later time when the simple Hamiltonian was added by time dependent perturbation. So now the wave function will have perturbation-induced time dependence.
The postulates of quantum mechanics have been successfully used for deriving exact solutions to Schrodinger equation for problems like A particle in 1 Dimensional box Harmonic oscillator Rigid rotator Hydrogen atom • However for a multielectron system, the SWE cannot be solved exactly due to inter-electronic repulsion terms.
The SWE is solved by method of seperation of variables.
• However, the inter-electronic repulsion term cannot be solved because the variables cannot be seperated and the SWE cannot be solved. • Approximate methods have helped to generate solutions for such and even more complex real quantum systems. • Approximate methods have been developed for solving Schrodinger equation to find wave function and energy of the complex system under consideration. • Two widely used approximate methods are, 1. Perturbation theory 2. Variation method
Perturbation theory is an approximate method that describes a complex quantum system in terms of a simpler system for which the exact solution is known. • Perturbation theory has been categorized into, i. Time independent perturbation theory, proposed by Erwin Schrodinger, where the perturbation Hamiltonian is static. ii. Time dependent perturbation theory, proposed by Paul Dirac, which studies the effect of time dependent perturbation on a time independent Hamiltonian H0.
PERTURBATION THEOREM
FIRST ORDER PERTURBATION THEORY
FIRST ORDER ENERGY CORRECTION
FIRST ORDER WAVE FUNCTION CORRECTION
APPLICATIONS OF PERTURBATION METHOD
SIGNIFICANCE OF PERTURBATION METHOD
Time Independent Perturbation Theory, 1st order correction, 2nd order correctionJames Salveo Olarve
The presentation is about how to solve the new energy levels and wave functions when the simple Hamiltonian is added by another term due to external effect (can be due to external field) .
The intended reader of this presentation were physics students. The author already assumed that the reader knows dirac braket notation.
7.1 Application of the Schrödinger Equation to the Hydrogen Atom
7.2 Solution of the Schrödinger Equation for Hydrogen
7.3 Quantum Numbers
7.4 Magnetic Effects on Atomic Spectra – The Normal Zeeman Effect
7.5 Intrinsic Spin
7.6 Energy Levels and Electron Probabilities
origin of quantum physics -
Inadequacy of classical mechanics and birth of QUANTUM PHYSICS
ref: Quantum mechanics: concepts and applications, N. Zettili
Introduction to Classical Mechanics:
UNIT-I : Elementary survey of Classical Mechanics: Newtonian mechanics for single particle and system of particles, Types of the forces and the single particle system examples, Limitation of Newton’s program, conservation laws viz Linear momentum, Angular Momentum & Total Energy, work-energy theorem; open systems (with variable mass). Principle of Virtual work, D’Alembert’s principle’ applications.
UNIT-II : Constraints; Definition, Types, cause & effects, Need, Justification for realizing constraints on the system
PART V - Continuation of PART III - QM and PART IV - QFT.
I intended to finish with the Hydrogen Atom description and the atomic orbital framework but I deemed the content void of a few important features: the Harmonic Oscillator and an introduction to Electromagnetic Interactions which leads directly to a formulation of the Quantization of the Radiation Field. I could not finish without wrapping it up with a development of Transition Probabilities and Einstein Coefficients which opens up the proof of the Planck distribution law, the photoelectric effect and Higher order electromagnetic interactions. I believe this is the key contribution: making it more understandable up to, but not including, quantum electrodynamics!
This presentation shows a technique of how to solve for the approximate ground state energy using Schrodinger Equation in which the solution for wave function is not on hand
Schrodinger wave equation and its application
a very good animated presentation.
Bs level.
semester 6th.
how to make a very good appreciable presentation.
CONTENTS
INTRODUCTION
CONCEPTS OF WALSH DIAGRAM
APPLICATION IN TRIATOMIC MOLECULES
[IN AH₂ TYPE OF MOLECULES(BeH₂,BH₂,H₂O)]
INTRODUCTION
Arthur Donald Walsh FRS The introducer of walsh diagram (8 August 1916-23 April 1977) was a British chemist, professor of chemistry at the University of Dundee . He was elected FRS in 1964. He was educated at Loughborough Grammar School.
Walsh diagrams were first introduced in a series of ten papers in one issue of the Journal of the Chemical Society . Here, he aimed to rationalize the shapes adopted by polyatomic molecules in the ground state as well as in excited states, by applying theoretical contributions made by Mulliken .
Lecture 8: Introduction to Quantum Chemical Simulation graduate course taught at MIT in Fall 2014 by Heather Kulik. This course covers: wavefunction theory, density functional theory, force fields and molecular dynamics and sampling.
CHAPTER 6 Quantum Mechanics II
6.0 Partial differentials
6.1 The Schrödinger Wave Equation
6.2 Expectation Values
6.3 Infinite Square-Well Potential
6.4 Finite Square-Well Potential
6.5 Three-Dimensional Infinite-Potential Well
6.6 Simple Harmonic Oscillator
6.7 Barriers and Tunneling in some books an extra chapter due to its technical importance
These slides are especially made to understand the postulates of quantum mechanics or chemistry better. easily simplified and at one place you will find each of relevant details about the 5 postulates. so go through it & trust me it will help you a lot if you are chemistry or a science student.
well done
origin of quantum physics -
Inadequacy of classical mechanics and birth of QUANTUM PHYSICS
ref: Quantum mechanics: concepts and applications, N. Zettili
Introduction to Classical Mechanics:
UNIT-I : Elementary survey of Classical Mechanics: Newtonian mechanics for single particle and system of particles, Types of the forces and the single particle system examples, Limitation of Newton’s program, conservation laws viz Linear momentum, Angular Momentum & Total Energy, work-energy theorem; open systems (with variable mass). Principle of Virtual work, D’Alembert’s principle’ applications.
UNIT-II : Constraints; Definition, Types, cause & effects, Need, Justification for realizing constraints on the system
PART V - Continuation of PART III - QM and PART IV - QFT.
I intended to finish with the Hydrogen Atom description and the atomic orbital framework but I deemed the content void of a few important features: the Harmonic Oscillator and an introduction to Electromagnetic Interactions which leads directly to a formulation of the Quantization of the Radiation Field. I could not finish without wrapping it up with a development of Transition Probabilities and Einstein Coefficients which opens up the proof of the Planck distribution law, the photoelectric effect and Higher order electromagnetic interactions. I believe this is the key contribution: making it more understandable up to, but not including, quantum electrodynamics!
This presentation shows a technique of how to solve for the approximate ground state energy using Schrodinger Equation in which the solution for wave function is not on hand
Schrodinger wave equation and its application
a very good animated presentation.
Bs level.
semester 6th.
how to make a very good appreciable presentation.
CONTENTS
INTRODUCTION
CONCEPTS OF WALSH DIAGRAM
APPLICATION IN TRIATOMIC MOLECULES
[IN AH₂ TYPE OF MOLECULES(BeH₂,BH₂,H₂O)]
INTRODUCTION
Arthur Donald Walsh FRS The introducer of walsh diagram (8 August 1916-23 April 1977) was a British chemist, professor of chemistry at the University of Dundee . He was elected FRS in 1964. He was educated at Loughborough Grammar School.
Walsh diagrams were first introduced in a series of ten papers in one issue of the Journal of the Chemical Society . Here, he aimed to rationalize the shapes adopted by polyatomic molecules in the ground state as well as in excited states, by applying theoretical contributions made by Mulliken .
Lecture 8: Introduction to Quantum Chemical Simulation graduate course taught at MIT in Fall 2014 by Heather Kulik. This course covers: wavefunction theory, density functional theory, force fields and molecular dynamics and sampling.
CHAPTER 6 Quantum Mechanics II
6.0 Partial differentials
6.1 The Schrödinger Wave Equation
6.2 Expectation Values
6.3 Infinite Square-Well Potential
6.4 Finite Square-Well Potential
6.5 Three-Dimensional Infinite-Potential Well
6.6 Simple Harmonic Oscillator
6.7 Barriers and Tunneling in some books an extra chapter due to its technical importance
These slides are especially made to understand the postulates of quantum mechanics or chemistry better. easily simplified and at one place you will find each of relevant details about the 5 postulates. so go through it & trust me it will help you a lot if you are chemistry or a science student.
well done
International Refereed Journal of Engineering and Science (IRJES) is a peer reviewed online journal for professionals and researchers in the field of computer science. The main aim is to resolve emerging and outstanding problems revealed by recent social and technological change. IJRES provides the platform for the researchers to present and evaluate their work from both theoretical and technical aspects and to share their views.
Matrix Transformations on Paranormed Sequence Spaces Related To De La Vallée-...inventionjournals
In this paper, we determine the necessary and sufficient conditions to characterize the matrices which transform paranormed sequence spaces into the spaces 푉휎 (휆) and 푉휎 ∞(휆) , where 푉휎 (휆) denotes the space of all (휎, 휆)-convergent sequences and 푉휎 ∞(휆) denotes the space of all (휎, 휆)-bounded sequences defined using the concept of de la Vallée-Pousin mean.
This Presentation describes, in short, Introduction to Time Series and the overall procedure required for Time Series Modelling including general terminologies and algorithms. However the detailed Mathematics is excluded in the slides, this ppt means to give a start to understanding the Time Series Modelling before going into detailed Statistics.
Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
June 3, 2024 Anti-Semitism Letter Sent to MIT President Kornbluth and MIT Cor...Levi Shapiro
Letter from the Congress of the United States regarding Anti-Semitism sent June 3rd to MIT President Sally Kornbluth, MIT Corp Chair, Mark Gorenberg
Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
This Congress will not stand idly by and allow an environment hostile to Jewish students to persist. The House believes that your institution is in violation of Title VI of the Civil Rights Act, and the inability or
unwillingness to rectify this violation through action requires accountability.
Postsecondary education is a unique opportunity for students to learn and have their ideas and beliefs challenged. However, universities receiving hundreds of millions of federal funds annually have denied
students that opportunity and have been hijacked to become venues for the promotion of terrorism, antisemitic harassment and intimidation, unlawful encampments, and in some cases, assaults and riots.
The House of Representatives will not countenance the use of federal funds to indoctrinate students into hateful, antisemitic, anti-American supporters of terrorism. Investigations into campus antisemitism by the Committee on Education and the Workforce and the Committee on Ways and Means have been expanded into a Congress-wide probe across all relevant jurisdictions to address this national crisis. The undersigned Committees will conduct oversight into the use of federal funds at MIT and its learning environment under authorities granted to each Committee.
• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
• The Committee on Oversight and Accountability is investigating the sources of funding and other support flowing to groups espousing pro-Hamas propaganda and engaged in antisemitic harassment and intimidation of students. The Committee on Oversight and Accountability is the principal oversight committee of the US House of Representatives and has broad authority to investigate “any matter” at “any time” under House Rule X.
• The Committee on Ways and Means has been investigating several universities since November 15, 2023, when the Committee held a hearing entitled From Ivory Towers to Dark Corners: Investigating the Nexus Between Antisemitism, Tax-Exempt Universities, and Terror Financing. The Committee followed the hearing with letters to those institutions on January 10, 202
Model Attribute Check Company Auto PropertyCeline George
In Odoo, the multi-company feature allows you to manage multiple companies within a single Odoo database instance. Each company can have its own configurations while still sharing common resources such as products, customers, and suppliers.
Macroeconomics- Movie Location
This will be used as part of your Personal Professional Portfolio once graded.
Objective:
Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
How to Make a Field invisible in Odoo 17Celine George
It is possible to hide or invisible some fields in odoo. Commonly using “invisible” attribute in the field definition to invisible the fields. This slide will show how to make a field invisible in odoo 17.
Welcome to TechSoup New Member Orientation and Q&A (May 2024).pdfTechSoup
In this webinar you will learn how your organization can access TechSoup's wide variety of product discount and donation programs. From hardware to software, we'll give you a tour of the tools available to help your nonprofit with productivity, collaboration, financial management, donor tracking, security, and more.
Introduction to AI for Nonprofits with Tapp NetworkTechSoup
Dive into the world of AI! Experts Jon Hill and Tareq Monaur will guide you through AI's role in enhancing nonprofit websites and basic marketing strategies, making it easy to understand and apply.
The Roman Empire A Historical Colossus.pdfkaushalkr1407
The Roman Empire, a vast and enduring power, stands as one of history's most remarkable civilizations, leaving an indelible imprint on the world. It emerged from the Roman Republic, transitioning into an imperial powerhouse under the leadership of Augustus Caesar in 27 BCE. This transformation marked the beginning of an era defined by unprecedented territorial expansion, architectural marvels, and profound cultural influence.
The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
Under Augustus, the empire experienced the Pax Romana, a 200-year period of relative peace and stability. Augustus reformed the military, established efficient administrative systems, and initiated grand construction projects. The empire's borders expanded, encompassing territories from Britain to Egypt and from Spain to the Euphrates. Roman legions, renowned for their discipline and engineering prowess, secured and maintained these vast territories, building roads, fortifications, and cities that facilitated control and integration.
The Roman Empire’s society was hierarchical, with a rigid class system. At the top were the patricians, wealthy elites who held significant political power. Below them were the plebeians, free citizens with limited political influence, and the vast numbers of slaves who formed the backbone of the economy. The family unit was central, governed by the paterfamilias, the male head who held absolute authority.
Culturally, the Romans were eclectic, absorbing and adapting elements from the civilizations they encountered, particularly the Greeks. Roman art, literature, and philosophy reflected this synthesis, creating a rich cultural tapestry. Latin, the Roman language, became the lingua franca of the Western world, influencing numerous modern languages.
Roman architecture and engineering achievements were monumental. They perfected the arch, vault, and dome, constructing enduring structures like the Colosseum, Pantheon, and aqueducts. These engineering marvels not only showcased Roman ingenuity but also served practical purposes, from public entertainment to water supply.
2. Background
Although wave mechanics is capable of describing quantum
behaviour of bound and unbound particles, some properties can not
be represented this way, e.g. electron spin degree of freedom.
It is therefore convenient to reformulate quantum mechanics in
framework that involves only operators, e.g. ˆH.
Advantage of operator algebra is that it does not rely upon
particular basis, e.g. for ˆH = ˆp2
2m , we can represent ˆp in spatial
coordinate basis, ˆp = −i ∂x , or in the momentum basis, ˆp = p.
Equally, it would be useful to work with a basis for the
wavefunction, ψ, which is coordinate-independent.
3. Operator methods: outline
1 Dirac notation and definition of operators
2 Uncertainty principle for non-commuting operators
3 Time-evolution of expectation values: Ehrenfest theorem
4 Symmetry in quantum mechanics
5 Heisenberg representation
6 Example: Quantum harmonic oscillator
(from ladder operators to coherent states)
4. Dirac notation
Orthogonal set of square integrable functions (such as
wavefunctions) form a vector space (cf. 3d vectors).
In Dirac notation, state vector or wavefunction, ψ, is represented
symbolically as a “ket”, |ψ .
Any wavefunction can be expanded as sum of basis state vectors,
(cf. v = xˆex + yˆey + · · · )
|ψ = λ1|ψ1 + λ2|ψ2 + · · ·
Alongside ket, we can define a “bra”, ψ| which together form the
scalar product,
φ|ψ ≡
∞
−∞
dx φ∗
(x)ψ(x) = ψ|φ ∗
5. Dirac notation
For a complete basis set, φi , we can define the expansion
|ψ =
i
φi |i
where j|ψ =
i
φi j|i
δij
= φj .
For example, in the real space basis, |ψ = dx ψ(x)|x .
Then, since x|x = δ(x − x ),
x |ψ = dx ψ(x) x |x
δ(x−x )
= ψ(x )
In Dirac formulation, real space representation recovered from inner
product, ψ(x) = x|ψ ; equivalently ψ(p) = p|ψ .
6. Operators
An operator ˆA maps one state vector, |ψ , into another, |φ , i.e.
ˆA|ψ = |φ .
If ˆA|ψ = a|ψ with a real, then |ψ is said to be an eigenstate (or
eigenfunction) of ˆA with eigenvalue a.
e.g. plane wave state ψp(x) = x|ψp = A eipx/
is an eigenstate of
the momentum operator, ˆp = −i ∂x , with eigenvalue p.
For every observable A, there is an operator ˆA which acts upon the
wavefunction so that, if a system is in a state described by |ψ , the
expectation value of A is
A = ψ|ˆA|ψ =
∞
−∞
dx ψ∗
(x)ˆAψ(x)
7. Operators
Every operator corresponding to observable is linear and Hermitian,
i.e. for any two wavefunctions |ψ and |φ , linearity implies
ˆA(α|ψ + β|φ ) = α ˆA|ψ + β ˆA|φ
For any linear operator ˆA, the Hermitian conjugate (a.k.a. the
adjoint) is defined by relation
φ|ˆAψ = dx φ∗
(ˆAψ) = dx ψ(ˆA†
φ)∗
= ˆA†
φ|ψ
Hermiticity implies that ˆA†
= ˆA, e.g. ˆp = −i ∂x .
8. Operators
From the definition, ˆA†
φ|ψ = φ|ˆAψ , some useful relations follow:
1 From complex conjugation, ˆA†
φ|ψ ∗
= ψ|ˆA†
φ = ˆAψ|φ ,
i.e. (ˆA†
)†
ψ|φ = ˆAψ|φ , ⇒ (ˆA†
)†
= ˆA
2 From φ|ˆAˆBψ = ˆA†
φ|ˆBψ = ˆB† ˆA†
φ|ψ ,
it follows that (ˆAˆB)†
= ˆB† ˆA†
.
Operators are associative,i.e. (ˆAˆB)ˆC = ˆA(ˆB ˆC),
but not (in general) commutative,
ˆAˆB|ψ = ˆA(ˆB|ψ ) = (ˆAˆB)|ψ = ˆB ˆA|ψ .
9. Operators
A physical variable must have real expectation values (and
eigenvalues) ⇒ physical operators are Hermitian (self-adjoint):
ψ|ˆH|ψ ∗
=
∞
−∞
ψ∗
(x)ˆHψ(x)dx
∗
=
∞
−∞
ψ(x)(ˆHψ(x))∗
dx = ˆHψ|ψ
i.e. ˆHψ|ψ = ψ|ˆHψ = ˆH†
ψ|ψ , and ˆH†
= ˆH.
Eigenfunctions of Hermitian operators ˆH|i = Ei |i form complete
orthonormal basis, i.e. i|j = δij
For complete set of states |i , can expand a state function |ψ as
|ψ =
i
|i i|ψ
In coordinate representation,
ψ(x) = x|ψ =
i
x|i i|ψ =
i
i|ψ φi (x), φi (x) = x|i
10. Resolution of identity
|ψ =
i
|i i|ψ
If we sum over complete set of states, obtain the (useful) resolution
of identity,
i
|i i| = I
i
x |i i|x = x |x
i.e. in coordinate basis, i φ∗
i (x)φi (x ) = δ(x − x ).
As in 3d vector space, expansion |φ = i bi |i and |ψ = i ci |i
allows scalar product to be taken by multiplying components,
φ|ψ = i b∗
i ci .
11. Example: resolution of identity
Basis states can be formed from any complete set of orthogonal
states including position or momentum,
∞
−∞
dx|x x| =
∞
−∞
dp|p p| = I.
From these definitions, can recover Fourier representation,
ψ(x) ≡ x|ψ =
∞
−∞
dp x|p
eipx/
/
√
2π
p|ψ =
1
√
2π
∞
−∞
dp eipx/
ψ(p)
where x|p denotes plane wave state |p expressed in the real space
basis.
12. Time-evolution operator
Formally, we can evolve a wavefunction forward in time by applying
time-evolution operator.
For time-independent Hamiltonian, |ψ(t) = ˆU(t)|ψ(0) , where
time-evolution operator (a.k.a. the “propagator”):
ˆU(t) = e−i ˆHt/
follows from time-dependent Schr¨odinger equation, ˆH|ψ = i ∂t|ψ .
By inserting the resolution of identity, I = i |i i|, where |i are
eigenstates of ˆH with eigenvalue Ei ,
|ψ(t) = e−i ˆHt/
i
|i i|ψ(0) =
i
|i i|ψ(0) e−iEi t/
13. Time-evolution operator
ˆU = e−i ˆHt/
Time-evolution operator is an example of a Unitary operator:
Unitary operators involve transformations of state vectors which
preserve their scalar products, i.e.
φ|ψ = ˆUφ|ˆUψ = φ|ˆU† ˆUψ
!
= φ|ψ
i.e. ˆU† ˆU = I
14. Uncertainty principle for non-commuting operators
For non-commuting Hermitian operators, we can establish a bound
on the uncertainty in the expectation values of ˆA and ˆB:
Given a state |ψ , the mean square uncertainty defined as
(∆A)2
= ψ|(ˆA − ˆA )2
ψ = ψ|ˆU2
ψ
(∆B)2
= ψ|(ˆB − ˆB )2
ψ = ψ| ˆV 2
ψ
where ˆU = ˆA − ˆA , ˆA ≡ ψ|ˆAψ , etc.
Consider then the expansion of the norm ||ˆU|ψ + iλ ˆV |ψ ||2
,
ψ|ˆU2
ψ + λ2
ψ| ˆV 2
ψ + iλ ˆUψ| ˆV ψ − iλ ˆV ψ|ˆUψ ≥ 0
i.e. (∆A)2
+ λ2
(∆B)2
+ iλ ψ|[ˆU, ˆV ]|ψ ≥ 0
Since ˆA and ˆB are just constants, [ˆU, ˆV ] = [ˆA, ˆB].
15. Uncertainty principle for non-commuting operators
(∆A)2
+ λ2
(∆B)2
+ iλ ψ|[ˆA, ˆB]|ψ ≥ 0
Minimizing with respect to λ,
2λ(∆B)2
+ iλ ψ|[ˆA, ˆB]|ψ = 0, iλ =
1
2
ψ|[ˆA, ˆB]|ψ
(∆B)2
and substituting back into the inequality,
(∆A)2
(∆B)2
≥ −
1
4
ψ|[ˆA, ˆB]|ψ 2
i.e., for non-commuting operators,
(∆A)(∆B) ≥
i
2
[ˆA, ˆB]
16. Uncertainty principle for non-commuting operators
(∆A)(∆B) ≥
i
2
[ˆA, ˆB]
For the conjugate operators of momentum and position (i.e.
[ˆp, ˆx] = −i , recover Heisenberg’s uncertainty principle,
(∆p)(∆x) ≥
i
2
[ˆp, x] =
2
Similarly, if we use the conjugate coordinates of time and energy,
[ˆE, t] = i ,
(∆t)(∆E) ≥
i
2
[t, ˆE] =
2
17. Time-evolution of expectation values
For a general (potentially time-dependent) operator ˆA,
∂t ψ|ˆA|ψ = (∂t ψ|)ˆA|ψ + ψ|∂t
ˆA|ψ + ψ|ˆA(∂t|ψ )
Using i ∂t|ψ = ˆH|ψ , −i (∂t ψ|) = ψ|ˆH, and Hermiticity,
∂t ψ|ˆA|ψ =
1
i ˆHψ|ˆA|ψ + ψ|∂t
ˆA|ψ +
1
ψ|ˆA|(−i ˆHψ)
=
i
ψ|ˆH ˆA|ψ − ψ|ˆAˆH|ψ
ψ|[ˆH, ˆA]|ψ
+ ψ|∂t
ˆA|ψ
For time-independent operators, ˆA, obtain Ehrenfest Theorem,
∂t ψ|ˆA|ψ =
i
ψ|[ˆH, ˆA]|ψ .
18. Ehrenfest theorem: example
∂t ψ|ˆA|ψ =
i
ψ|[ˆH, ˆA]|ψ .
For the Schr¨odinger operator, ˆH = ˆp2
2m + V (x),
∂t x =
i
[ˆH, ˆx] =
i
[
ˆp2
2m
, x] =
ˆp
m
Similarly,
∂t ˆp =
i
[ˆH, −i ∂x ] = − (∂x
ˆH) = − ∂x V
i.e. Expectation values follow Hamilton’s classical equations of
motion.
19. Symmetry in quantum mechanics
Symmetry considerations are very important in both low and high
energy quantum theory:
1 Structure of eigenstates and spectrum reflect symmetry of the
underlying Hamiltonian.
2 Transition probabilities between states depend upon
transformation properties of perturbation =⇒ “selection
rules”.
Symmetries can be classified as discrete and continuous,
e.g. mirror symmetry is discrete, while rotation is continuous.
20. Symmetry in quantum mechanics
Formally, symmetry operations can be represented by a group of
(typically) unitary transformations (or operators), ˆU such that
ˆO → ˆU† ˆO ˆU
Such unitary transformations are said to be symmetries of a
general operator ˆO if
ˆU† ˆO ˆU = ˆO
i.e., since ˆU†
= ˆU−1
(unitary), [ ˆO, ˆU] = 0.
If ˆO ≡ ˆH, such unitary transformations are said to be symmetries of
the quantum system.
21. Continuous symmetries: Examples
Operators ˆp and ˆr are generators of space-time transformations:
For a constant vector a, the unitary operator
ˆU(a) = exp −
i
a · ˆp
effects spatial translations, ˆU†
(a)f (r)ˆU(a) = f (r + a).
Proof: Using the Baker-Hausdorff identity (exercise),
e
ˆA ˆBe−ˆA
= ˆB + [ˆA, ˆB] +
1
2!
[ˆA, [ˆA, ˆB]] + · · ·
with e
ˆA
≡ ˆU†
= ea·
and ˆB ≡ f (r), it follows that
ˆU†
(a)f (r)ˆU(a) = f (r) + ai1 ( i1 f (r)) +
1
2!
ai1 ai2 ( i1 i2 f (r)) + · · ·
= f (r + a) by Taylor expansion
22. Continuous symmetries: Examples
Operators ˆp and ˆr are generators of space-time transformations:
For a constant vector a, the unitary operator
ˆU(a) = exp −
i
a · ˆp
effects spatial translations, ˆU†
(a)f (r)ˆU(a) = f (r + a).
Therefore, a quantum system has spatial translation symmetry iff
ˆU(a)ˆH = ˆH ˆU(a), i.e. ˆpˆH = ˆHˆp
i.e. (sensibly) ˆH = ˆH(ˆp) must be independent of position.
Similarly (with ˆL = r × ˆp the angular momemtum operator),
ˆU(b) = exp[− i
b · ˆr]
ˆU(θ) = exp[− i
θˆen · ˆL]
ˆU(t) = exp[− i ˆHt]
effects
momentum translations
spatial rotations
time translations
23. Discrete symmetries: Examples
The parity operator, ˆP, involves a sign reversal of all coordinates,
ˆPψ(r) = ψ(−r)
discreteness follows from identity ˆP2
= 1.
Eigenvalues of parity operation (if such exist) are ±1.
If Hamiltonian is invariant under parity, [ˆP, ˆH] = 0, parity is said to
be conserved.
Time-reversal is another discrete symmetry, but its representation
in quantum mechanics is subtle and beyond the scope of course.
24. Consequences of symmetries: multiplets
Consider a transformation ˆU which is a symmetry of an operator
observable ˆA, i.e. [ˆU, ˆA] = 0.
If ˆA has eigenvector |a , it follows that ˆU|a will be an eigenvector
with the same eigenvalue, i.e.
ˆAU|a = ˆU ˆA|a = aU|a
This means that either:
1 |a is an eigenvector of both ˆA and ˆU (e.g. |p is eigenvector
of ˆH = ˆp2
2m and ˆU = eia·ˆp/
), or
2 eigenvalue a is degenerate: linear space spanned by vectors
ˆUn
|a (n integer) are eigenvectors with same eigenvalue.
e.g. next lecture, we will address central potential where ˆH is
invariant under rotations, ˆU = eiθˆen·ˆL/
– states of angular
momentum, , have 2 + 1-fold degeneracy generated by ˆL±.
25. Heisenberg representation
Schr¨odinger representation: time-dependence of quantum system
carried by wavefunction while operators remain constant.
However, sometimes useful to transfer time-dependence to
operators: For observable ˆB, time-dependence of expectation value,
ψ(t)|ˆB|ψ(t) = e−i ˆHt/
ψ(0)|ˆB|e−i ˆHt/
ψ(0)
= ψ(0)|ei ˆHt/ ˆBe−i ˆHt/
|ψ(0)
Heisenberg representation: if we define ˆB(t) = ei ˆHt/ ˆBe−i ˆHt/
,
time-dependence transferred from wavefunction and
∂t
ˆB(t) =
i
ei ˆHt/
[ˆH, ˆB]e−i ˆHt/
=
i
[ˆH, ˆB(t)]
cf. Ehrenfest’s theorem
26. Quantum harmonic oscillator
The harmonic oscillator holds priviledged position in quantum
mechanics and quantum field theory.
ˆH =
ˆp2
2m
+
1
2
mω2
x2
It also provides a useful platform to illustrate some of the
operator-based formalism developed above.
To obtain eigenstates of ˆH, we could seek solutions of linear second
order differential equation,
−
2
2m
∂2
x +
1
2
mω2
x2
ψ = Eψ
However, complexity of eigenstates (Hermite polynomials) obscure
useful features of system – we therefore develop an alternative
operator-based approach.
27. Quantum harmonic oscillator
ˆH =
ˆp2
2m
+
1
2
mω2
x2
Form of Hamiltonian suggests that it can be recast as the “square
of an operator”: Defining the operators (no hats!)
a =
mω
2
x + i
ˆp
mω
, a†
=
mω
2
x − i
ˆp
mω
we have a†
a =
mω
2
x2
+
ˆp2
2 mω
−
i
2
[ˆp, x]
−i
=
ˆH
ω
−
1
2
Together with aa†
=
ˆH
ω + 1
2 , we find that operators fulfil the
commutation relations
[a, a†
] ≡ aa†
− a†
a = 1
Setting ˆn = a†
a, ˆH = ω(ˆn + 1/2)
28. Quantum harmonic oscillator
ˆH = ω(a†
a + 1/2)
Ground state |0 identified by finding state for which
a|0 =
mω
2
x + i
ˆp
mω
|0 = 0
In coordinate basis,
x|a|0 = 0 = dx x|a|x x |0 = x +
mω
∂x ψ0(x)
i.e. ground state has energy E0 = ω/2 and
ψ0(x) = x|0 =
mω
π
1/4
e−mωx2
/2
N.B. typo in handout!
29. Quantum harmonic oscillator
ˆH = ω(a†
a + 1/2)
Excited states found by acting upon this state with a†
.
Proof: using [a, a†
] ≡ aa†
− a†
a = 1, if ˆn|n = n|n ,
ˆn(a†
|n ) = a†
aa†
a†
a + 1
|n = (a†
a†
a
ˆn
+a†
)|n = (n + 1)a†
|n
equivalently, [ˆn, a†
] = ˆna†
− a†
ˆn = a†
.
Therefore, if |n is eigenstate of ˆn with eigenvalue n, then a†
|n is
eigenstate with eigenvalue n + 1.
Eigenstates form a “tower”; |0 , |1 = C1a†
|0 , |2 = C2(a†
)2
|0 , ...,
with normalization Cn.
30. Quantum harmonic oscillator
ˆH = ω(a†
a + 1/2)
Normalization: If n|n = 1, n|aa†
|n = n|(ˆn + 1)|n = (n + 1),
i.e. with |n + 1 = 1√
n+1
a†
|n , state |n + 1 also normalized.
|n =
1
√
n!
(a†
)n
|0 , n|n = δnn
are eigenstates of ˆH with eigenvalue En = (n + 1/2) ω and
a†
|n =
√
n + 1|n + 1 , a|n =
√
n|n − 1
a and a†
represent ladder operators that lower/raise energy of
state by ω.
31. Quantum harmonic oscillator
In fact, operator representation achieves something remarkable and
far-reaching: the quantum harmonic oscillator describes motion of a
single particle in a confining potential.
Eigenvalues turn out to be equally spaced, cf. ladder of states.
Although we can find a coordinate representation ψn(x) = x|n ,
operator representation affords a second interpretation, one that
lends itself to further generalization in quantum field theory.
Quantum harmonic oscillator can be interpreted as a simple system
involving many fictitious particles, each of energy ω.
32. Quantum harmonic oscillator
In new representation, known as the Fock space representation,
vacuum |0 has no particles, |1 a single particle, |2 has two, etc.
Fictitious particles created and annihilated by raising and lowering
operators, a†
and a with commutation relations, [a, a†
] = 1.
Later in the course, we will find that these commutation relations
are the hallmark of bosonic quantum particles and this
representation, known as second quantization underpins the
quantum field theory of relativistic particles (such as the photon).
33. Quantum harmonic oscillator: “dynamical echo”
How does a general wavepacket |ψ(0) evolve under the action of
the quantum time-evolution operator, ˆU(t) = e−i ˆHt/
?
For a general initial state, |ψ(t) = ˆU(t)|ψ(0) . Inserting the
resolution of identity on the complete set of eigenstates,
|ψ(t) = e−i ˆHt/
n
|n n|ψ(0) =
i
|n n|ψ(0) e−iEnt/
e−iω(n+1/2)t
For the harmonic oscillator, En = ω(n + 1/2).
Therefore, at times t = 2π
ω m, m integer, |ψ(t) = e−iωt/2
|ψ(0)
leading to the coherent reconstruction (echo) of the wavepacket.
At times t = π
ω (2m + 1), the “inverted” wavepacket
ψ(x, t) = e−iωt/2
ψ(−x, 0) is perfectly reconstructed (exercise).
34. Quantum harmonic oscillator: time-dependence
In Heisenberg representation, we have seen that ∂t
ˆB =
i
[ˆH, ˆB].
Therefore, making use of the identity, [ˆH, a] = − ωa (exercise),
∂ta = −iωa, i.e. a(t) = e−iωt
a(0)
Combined with conjugate relation a†
(t) = eiωt
a†
(0), and using
x = 2mω (a†
+ a), ˆp = −i m ω
2 (a − a†
)
ˆp(t) = ˆp(0) cos(ωt) − mωˆx(0) sin(ωt)
ˆx(t) = ˆx(0) cos(ωt) +
ˆp(0)
mω
sin(ωt)
i.e. operators obey equations of motion of the classical harmonic
oscillator.
But how do we use these equations...?
35. Quantum harmonic oscillator: time-dependence
ˆp(t) = ˆp(0) cos(ωt) − mωˆx(0) sin(ωt)
ˆx(t) = ˆx(0) cos(ωt) +
ˆp(0)
mω
sin(ωt)
Consider dynamics of a (real) wavepacket defined by φ(x) at t = 0.
Suppose we know expectation values, p2
0 = φ|ˆp2
|φ , x2
0 = φ|x2
|φ ,
and we want to determine φ(t)|ˆp2
|φ(t) .
In Heisenberg representation, φ(t)|ˆp2
|φ(t) = φ|ˆp2
(t)|φ and
ˆp2
(t) = ˆp2
(0) cos2
(ωt) + (mωx(0))2
sin2
(ωt)
−mω(x(0)ˆp(0) + ˆp(0)x(0))
Since φ|(x(0)ˆp(0) + ˆp(0)x(0))|φ = 0 for φ(x) real, we have
φ|ˆp2
(t)|φ = p2
0 cos2
(ωt) + (mωx0)2
sin2
(ωt)
and similarly φ|ˆx2
(t)|φ = x2
0 cos2
(ωt) +
p2
0
(mω)2 sin2
(ωt)
36. Coherent states
The ladder operators can be used to construct a wavepacket which
most closely resembles a classical particle – the coherent or
Glauber states.
Such states have numerous applications in quantum field theory and
quantum optics.
The coherent state is defined as the eigenstate of the annihilation
operator,
a|β = β|β
Since a is not Hermitian, β can take complex eigenvalues.
The eigenstates are constructed from the harmonic oscillator ground
state the by action of the unitary operator,
|β = ˆU(β)|0 , ˆU(β) = eβa†
−β∗
a
37. Coherent states
|β = ˆU(β)|0 , ˆU(β) = eβa†
−β∗
a
The proof follows from the identity (problem set I),
aˆU(β) = ˆU(β)(a + β)
i.e. ˆU is a translation operator, ˆU†
(β)aˆU(β) = a + β.
By making use of the Baker-Campbell-Hausdorff identity
e
ˆX
e
ˆY
= e
ˆX+ ˆY + 1
2 [ ˆX, ˆY ]
valid if [ˆX, ˆY ] is a c-number, we can show (problem set)
ˆU(β) = eβa†
−β∗
a
= e−|β|2
/2
eβa†
e−β∗
a
i.e., since e−β∗
a
|0 = |0 ,
|β = e−|β|2
/2
eβa†
|0
38. Coherent states
a|β = β|β , |β = e−|β|2
/2
eβa†
|0
Expanding the exponential, and noting that |n = 1√
n!
(a†
)n
|0 , |β
can be represented in number basis,
|β =
∞
n=0
(βa†
)n
n!
|0 =
n
e−|β|2
/2 βn
√
n!
|n
i.e. Probability of observing n excitations is
Pn = | n|β |2
= e−|β|2 |β|2n
n!
a Poisson distribution with average occupation, β|a†
a|β = |β|2
.
39. Coherent states
a|β = β|β , |β = e−|β|2
/2
eβa†
|0
Furthermore, one may show that the coherent state has minimum
uncertainty ∆x ∆p = 2 .
In the real space representation (problem set I),
ψβ(x) = x|β = N exp −
(x − x0)2
4(∆x)2
−
i
p0x
where (∆x)2
= 2mω and
x0 =
2mω
(β∗
+ β) = A cos ϕ
p0 = i
mω
2
(β∗
− β) = mωA sin ϕ
where A = 2
mω and β = |β|eiϕ
.
40. Coherent States: dynamics
a|β = β|β , |β =
n
e−|β|2
/2 βn
√
n!
|n
Using the time-evolution of the stationary states,
|n(t) = e−iEnt/
|n(0) , En = ω(n + 1/2)
it follows that
|β(t) = e−iωt/2
n
e−|β|2
/2 βn
√
n!
e−inωt
|n = e−iωt/2
|e−iωt
β
Therefore, the form of the coherent state wavefunction is preserved
in the time-evolution, while centre of mass and momentum follow
that of the classical oscillator,
x0(t) = A cos(ϕ + ωt), p0(t) = mωA sin(ϕ + ωt)
41. Summary: operator methods
Operator methods provide a powerful formalism in which we may
bypass potentially complex coordinate representations of
wavefunctions.
Operator methods allow us to expose the symmetry content of
quantum systems – providing classification of degenerate
submanifolds and multiplets.
Operator methods can provide insight into dynamical properties of
quantum systems without having to resolve eigenstates.
Quantum harmonic oscillator provides example of
“complementarity” – states of oscillator can be interpreted as a
confined single particle problem or as a system of fictitious
non-interacting quantum particles.