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.
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
Mathematical description of Legendre Functions.
Presentation at Undergraduate in Science (math, physics, engineering) level.
Please send any comments or suggestions to improve to solo.hermelin@gmail.com.
More presentations can be found on my website at http://www.solohermelin.com.
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.
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
Mathematical description of Legendre Functions.
Presentation at Undergraduate in Science (math, physics, engineering) level.
Please send any comments or suggestions to improve to solo.hermelin@gmail.com.
More presentations can be found on my website at http://www.solohermelin.com.
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
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.
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.
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
Dealing with Notations and conventions in tensor analysis-Einstein's summation convention covariant and contravariant and mixed tensors-algebraic operations in tensor symmetric and skew symmetric tensors-tensor calculus-Christoffel symbols-kinematics in Riemann space-Riemann-Christoffel tensor.
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
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.
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.
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
Dealing with Notations and conventions in tensor analysis-Einstein's summation convention covariant and contravariant and mixed tensors-algebraic operations in tensor symmetric and skew symmetric tensors-tensor calculus-Christoffel symbols-kinematics in Riemann space-Riemann-Christoffel tensor.
First-order cosmological perturbations produced by point-like masses: all sca...Maxim Eingorn
This presentation based on the paper http://arxiv.org/abs/1509.03835 was made at Institute of Cosmology, Tufts University, on November 12, 2015. The abstract follows:
In the framework of the concordance cosmological model the first-order scalar and vector perturbations of the homogeneous background are derived without any supplementary approximations in addition to the weak gravitational field limit. The sources of these perturbations (inhomogeneities) are presented in the discrete form of a system of separate point-like gravitating masses. The obtained expressions for the metric corrections are valid at all (sub-horizon and super-horizon) scales and converge in all points except the locations of the sources, and their average values are zero (thus, first-order backreaction effects are absent). Both the Minkowski background limit and the Newtonian cosmological approximation are reached under certain well-defined conditions. An important feature of the velocity-independent part of the scalar perturbation is revealed: up to an additive constant it represents a sum of Yukawa potentials produced by inhomogeneities with the same finite time-dependent Yukawa interaction range. The suggesting itself connection between this range and the homogeneity scale is briefly discussed along with other possible physical implications.
Slides from my presentation at the Joint CoEPP-CAASTRO Workshop (http://www.caastro.org/event/2013/coepp), 28 February 2013. Brief overview of the evidence for dark matter in the Universe, plus discussion of challenges, hints of possible signals, and some references for further reading.
The presentation time-slot was 30 minutes + 20 minutes discussion.
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
On the Seidel’s Method, a Stronger Contraction Fixed Point Iterative Method o...BRNSS Publication Hub
In the solution of a system of linear equations, there exist many methods most of which are not fixed point iterative methods. However, this method of Sidel’s iteration ensures that the given system of the equation must be contractive after satisfying diagonal dominance. The theory behind this was discussed in sections one and two and the end; the application was extensively discussed in the last section.
On Application of Power Series Solution of Bessel Problems to the Problems of...BRNSS Publication Hub
One of the most powerful techniques available for studying functions defined by differential equations is to produce power series expansions of their solutions when such expansions exist. This is the technique I now investigated, in particular, its feasibility in the solution of an engineering problem known as the problem of strut of variable moment of inertia. In this work, I explored the basic theory of the Bessel’s function and its power series solution. Then, a model of the problem of strut of variable moment of inertia was developed into a differential equation of the Bessel’s form, and finally, the Bessel’s equation so formed was solved and result obtained.
impulse(GreensFn), Principle of SuperpositionSc Pattar
Impulse superposition
Green’s function for underdamped oscillator
Exponential driving force
Green’s function for an undamped oscillator
Solution for constant force
Step function method
I am Grey N. I am a Physical Chemistry Assignment Expert at eduassignmenthelp.com. I hold a Ph.D. in Physical Chemistry, from Calgary, Canada. I have been helping students with their homework for the past 6 years. I solve assignments related to Physical Chemistry.
Visit eduassignmenthelp.com or email info@eduassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Physical Chemistry Assignments.
Z Transform And Inverse Z Transform - Signal And SystemsMr. RahüL YøGi
The z-transform is the most general concept for the transformation of discrete-time series.
The Laplace transform is the more general concept for the transformation of continuous time processes.
For example, the Laplace transform allows you to transform a differential equation, and its corresponding initial and boundary value problems, into a space in which the equation can be solved by ordinary algebra.
The switching of spaces to transform calculus problems into algebraic operations on transforms is called operational calculus. The Laplace and z transforms are the most important methods for this purpose.
This presentation is intended for undergraduate students in physics and engineering.
Please send comments to solo.hermelin@gmail.com.
For more presentations on different subjects please visit my homepage at http://www.solohermelin.com.
This presentation is in the Physics folder.
"Impact of front-end architecture on development cost", Viktor TurskyiFwdays
I have heard many times that architecture is not important for the front-end. Also, many times I have seen how developers implement features on the front-end just following the standard rules for a framework and think that this is enough to successfully launch the project, and then the project fails. How to prevent this and what approach to choose? I have launched dozens of complex projects and during the talk we will analyze which approaches have worked for me and which have not.
LF Energy Webinar: Electrical Grid Modelling and Simulation Through PowSyBl -...DanBrown980551
Do you want to learn how to model and simulate an electrical network from scratch in under an hour?
Then welcome to this PowSyBl workshop, hosted by Rte, the French Transmission System Operator (TSO)!
During the webinar, you will discover the PowSyBl ecosystem as well as handle and study an electrical network through an interactive Python notebook.
PowSyBl is an open source project hosted by LF Energy, which offers a comprehensive set of features for electrical grid modelling and simulation. Among other advanced features, PowSyBl provides:
- A fully editable and extendable library for grid component modelling;
- Visualization tools to display your network;
- Grid simulation tools, such as power flows, security analyses (with or without remedial actions) and sensitivity analyses;
The framework is mostly written in Java, with a Python binding so that Python developers can access PowSyBl functionalities as well.
What you will learn during the webinar:
- For beginners: discover PowSyBl's functionalities through a quick general presentation and the notebook, without needing any expert coding skills;
- For advanced developers: master the skills to efficiently apply PowSyBl functionalities to your real-world scenarios.
Essentials of Automations: Optimizing FME Workflows with ParametersSafe Software
Are you looking to streamline your workflows and boost your projects’ efficiency? Do you find yourself searching for ways to add flexibility and control over your FME workflows? If so, you’re in the right place.
Join us for an insightful dive into the world of FME parameters, a critical element in optimizing workflow efficiency. This webinar marks the beginning of our three-part “Essentials of Automation” series. This first webinar is designed to equip you with the knowledge and skills to utilize parameters effectively: enhancing the flexibility, maintainability, and user control of your FME projects.
Here’s what you’ll gain:
- Essentials of FME Parameters: Understand the pivotal role of parameters, including Reader/Writer, Transformer, User, and FME Flow categories. Discover how they are the key to unlocking automation and optimization within your workflows.
- Practical Applications in FME Form: Delve into key user parameter types including choice, connections, and file URLs. Allow users to control how a workflow runs, making your workflows more reusable. Learn to import values and deliver the best user experience for your workflows while enhancing accuracy.
- Optimization Strategies in FME Flow: Explore the creation and strategic deployment of parameters in FME Flow, including the use of deployment and geometry parameters, to maximize workflow efficiency.
- Pro Tips for Success: Gain insights on parameterizing connections and leveraging new features like Conditional Visibility for clarity and simplicity.
We’ll wrap up with a glimpse into future webinars, followed by a Q&A session to address your specific questions surrounding this topic.
Don’t miss this opportunity to elevate your FME expertise and drive your projects to new heights of efficiency.
Transcript: Selling digital books in 2024: Insights from industry leaders - T...BookNet Canada
The publishing industry has been selling digital audiobooks and ebooks for over a decade and has found its groove. What’s changed? What has stayed the same? Where do we go from here? Join a group of leading sales peers from across the industry for a conversation about the lessons learned since the popularization of digital books, best practices, digital book supply chain management, and more.
Link to video recording: https://bnctechforum.ca/sessions/selling-digital-books-in-2024-insights-from-industry-leaders/
Presented by BookNet Canada on May 28, 2024, with support from the Department of Canadian Heritage.
Accelerate your Kubernetes clusters with Varnish CachingThijs Feryn
A presentation about the usage and availability of Varnish on Kubernetes. This talk explores the capabilities of Varnish caching and shows how to use the Varnish Helm chart to deploy it to Kubernetes.
This presentation was delivered at K8SUG Singapore. See https://feryn.eu/presentations/accelerate-your-kubernetes-clusters-with-varnish-caching-k8sug-singapore-28-2024 for more details.
Slack (or Teams) Automation for Bonterra Impact Management (fka Social Soluti...Jeffrey Haguewood
Sidekick Solutions uses Bonterra Impact Management (fka Social Solutions Apricot) and automation solutions to integrate data for business workflows.
We believe integration and automation are essential to user experience and the promise of efficient work through technology. Automation is the critical ingredient to realizing that full vision. We develop integration products and services for Bonterra Case Management software to support the deployment of automations for a variety of use cases.
This video focuses on the notifications, alerts, and approval requests using Slack for Bonterra Impact Management. The solutions covered in this webinar can also be deployed for Microsoft Teams.
Interested in deploying notification automations for Bonterra Impact Management? Contact us at sales@sidekicksolutionsllc.com to discuss next steps.
UiPath Test Automation using UiPath Test Suite series, part 3DianaGray10
Welcome to UiPath Test Automation using UiPath Test Suite series part 3. In this session, we will cover desktop automation along with UI automation.
Topics covered:
UI automation Introduction,
UI automation Sample
Desktop automation flow
Pradeep Chinnala, Senior Consultant Automation Developer @WonderBotz and UiPath MVP
Deepak Rai, Automation Practice Lead, Boundaryless Group and UiPath MVP
Key Trends Shaping the Future of Infrastructure.pdfCheryl Hung
Keynote at DIGIT West Expo, Glasgow on 29 May 2024.
Cheryl Hung, ochery.com
Sr Director, Infrastructure Ecosystem, Arm.
The key trends across hardware, cloud and open-source; exploring how these areas are likely to mature and develop over the short and long-term, and then considering how organisations can position themselves to adapt and thrive.
GDG Cloud Southlake #33: Boule & Rebala: Effective AppSec in SDLC using Deplo...James Anderson
Effective Application Security in Software Delivery lifecycle using Deployment Firewall and DBOM
The modern software delivery process (or the CI/CD process) includes many tools, distributed teams, open-source code, and cloud platforms. Constant focus on speed to release software to market, along with the traditional slow and manual security checks has caused gaps in continuous security as an important piece in the software supply chain. Today organizations feel more susceptible to external and internal cyber threats due to the vast attack surface in their applications supply chain and the lack of end-to-end governance and risk management.
The software team must secure its software delivery process to avoid vulnerability and security breaches. This needs to be achieved with existing tool chains and without extensive rework of the delivery processes. This talk will present strategies and techniques for providing visibility into the true risk of the existing vulnerabilities, preventing the introduction of security issues in the software, resolving vulnerabilities in production environments quickly, and capturing the deployment bill of materials (DBOM).
Speakers:
Bob Boule
Robert Boule is a technology enthusiast with PASSION for technology and making things work along with a knack for helping others understand how things work. He comes with around 20 years of solution engineering experience in application security, software continuous delivery, and SaaS platforms. He is known for his dynamic presentations in CI/CD and application security integrated in software delivery lifecycle.
Gopinath Rebala
Gopinath Rebala is the CTO of OpsMx, where he has overall responsibility for the machine learning and data processing architectures for Secure Software Delivery. Gopi also has a strong connection with our customers, leading design and architecture for strategic implementations. Gopi is a frequent speaker and well-known leader in continuous delivery and integrating security into software delivery.
UiPath Test Automation using UiPath Test Suite series, part 4DianaGray10
Welcome to UiPath Test Automation using UiPath Test Suite series part 4. In this session, we will cover Test Manager overview along with SAP heatmap.
The UiPath Test Manager overview with SAP heatmap webinar offers a concise yet comprehensive exploration of the role of a Test Manager within SAP environments, coupled with the utilization of heatmaps for effective testing strategies.
Participants will gain insights into the responsibilities, challenges, and best practices associated with test management in SAP projects. Additionally, the webinar delves into the significance of heatmaps as a visual aid for identifying testing priorities, areas of risk, and resource allocation within SAP landscapes. Through this session, attendees can expect to enhance their understanding of test management principles while learning practical approaches to optimize testing processes in SAP environments using heatmap visualization techniques
What will you get from this session?
1. Insights into SAP testing best practices
2. Heatmap utilization for testing
3. Optimization of testing processes
4. Demo
Topics covered:
Execution from the test manager
Orchestrator execution result
Defect reporting
SAP heatmap example with demo
Speaker:
Deepak Rai, Automation Practice Lead, Boundaryless Group and UiPath MVP
To Graph or Not to Graph Knowledge Graph Architectures and LLMs
Perturbation
1. WELCOME TO YOU ALL
PHYSICAL
CHEMISTRY
PRESENTATION
M.SC.-II –
(SEM. III )
2013–
2014
PERTURBATION
Presented by :–
Dharmendra R. Prajapati
RAMNIRANJAN JHUNJHUNWALA
COLLEGE
3. Perturbation
Original Equation
X 2 − 25 = 0
Y = X 2 + ε X − 25
Perturbed equation
X 2 + ε X − 25 = 0
Y vs X
15
10
0 ≤ ε <1
5
0
-8
-6
-4
-2
0
2
4
6
8
-5
Y
Epsilon=0.8
-10
Epsilon=0.5
Epsilon=0.0
-15
-20
-25
-30
X
4. Perturbation
Perturbed equation
X 2 + ε X − 25 = 0
Change in result (absolute values) vs Change in equation
Perturbation in the result
Perturbation in the result (root)
Root -1
-2
-0.15
-0.15
Simple (Regular)
Perturbation
0.06
ε=0.1
ε=0.1
0.05
ε=-0.1
ε=-0.1
0.04
0.03
Answer can be
in the form
0.02
-0.1
-0.1
0.01
0.01
ε=-0.01
ε=-0.01
0
0
-0.05
0
-0.05
0
ε=0.01
ε=0.01
0.05
0.05
Epsilon (perturbation)
Epsilon (perturbation)
0.1
0.1
0.15
0.15
X = X 0 + ε φ ( X 0 ) + ε 2 ...
5. Y = εX + X − 5
2
Y vs X
Perturbation
Original Equation
X −5 = 0
Perturbed equation
ε X 2 + X −5 = 0
40
35
30
25
Epsilon=0
Two roots instead
15
Y
20
Epsilon=0.8
one
10
Epsilon=1
5
0
-6
-4
-2
-5 0
-10
X
2
4
6
Roots are not
close to the
original root
of
6. Perturbation
Y = εX + X − 5
Change in result (absolute values) vs Change in equation
2
Other root varies
from the original
root dramatically,
as epsilon
approaches zero!
Root-1
Root-2
3.5
1200
Perturbation
Perturbation in Resultin Result
3
2.5
1000
2
800
1.5
600
1
0.5
400
0
200
0
0.2
0.4
0.6
0.8
1
1.2
Epsilon
0
0
0.2
0.4
0.6
Epsilon
0.8
1
1.2
Singular
perturbation
Answer may NOT
be in the form
X = X 0 + ε φ ( X 0 ) + ε 2 ...
7. dy
=a
dx
Solution
Differential Equations
a <1
y =ax
Perturbation-1
Solution
y = 0, at x = 0
dy
= a +ε y
dx
(
y = 0, at x = 0
)
a εx
a
ε 2 x2
y = e −1
= 1 + ε x +
+ ... − 1
ε
ε
2
a
ε 2 x2
= ε x +
+ ...
ε
2
ε → 0, y → ax
εx
= ax1 + + ...
2
Regular Perturbation
8. § Time-Independent Perturbation
Theory
FIRST ORDER PERTURBATION:•
We are often interested in systems for which we
could solve the Schreodinger equation if the
potential energy were slightly different.
•
Consider a one-dimensional example for which
we can write the actual potential energy as
Vactual(x) = V(x) + υ(x)
(12.1)
• where υ(x) is a small perturbation added to the
unperturbed potential V(x).
9. •
The Hamiltonian operator of the “unperturbed”
(i.e. exactly solvable) system is
• and the Schreodinger equation of that system
therefore is
H0ψl = Elψl (12.3)
with a known set of eigenvalues El and
eigenfunctions ψl.
•
The Schreodinger equation of the “perturbed”
system contains the additional term υ(x) in the
Hamiltonian:
10. [H0 + υ(x)]ψn’ = En’ψn’
(12.4)
•
• which may make it impossible to solve directly for
the eigenfunctions ψn’ and eigenvalues En’.
•
However, Postulate 3 tells us that any
acceptable wave function may be expanded in a
series of eigenfunctions of the unperturbed
Hamiltonian.
Therefore we may write each function ψn’ as a
•
series of the functions ψl with constant coefficients:
• (The letter l is simply an index, having nothing to do
with the angular momentum quantum number; this
11. •
Substitution into Eq.(12.4) gives
•
The technique used in finding coefficients in a
Fourier series may be employed here.
We multiply each side of Eq.(12.6) by ψm*—the
•
complex conjugate of a particular unperturbed
eigenfunction ψm. We then integrate both sides over
all x, to obtain
∞
∞
∞
∞
−∞
l =1
−∞
l =1
*
'
*
ψ m [ H 0 + v( x)] ∑ anlψ l dx = ∫ Enψ m ∑ anlψ l dx
∫
12. •
Integrating each side of Eq.(12.7) term by term
and removing the space-independent factors anl and
En’ from the integrals. we obtain
•
Because the functions ψ are normalized and
orthogonal, the right-hand side of Eq.(12.8) reduces
to the single term anmEn’ for which l = m, and we
have
•
We can rewrite the left-hand side of Eq.(12.8) by
using the fact that H0ψl = Elψl [Eq.(12.3)]; this
13. •
•
∞
∞
l =1
−∞
*
'
anl ∫ψ m [ El + v( x)]ψ l dx = anm En
∑
or
∞
∞
∞
∞
*
*
'
anl El ∫ψ mψ l dx + ∑ anl ∫ψ m [ v( x)]ψ l dx = anm En
∑
•
•
(12.9)
Again, because the functions ψ are normalized
and orthogonal, the first term on the left reduces to
anmEm.
•
The second term can be written in abbreviated
form as ∑∞l=1 anlυml, where υml is an abbreviation for the
integral ∫∞-∞ψ*mυ(x)ψl dx, called the matrix element of
the perturbing potential υ(x) between the states m
and l.
l =1
−∞
l =1
−∞
14. •
(This term can also be written in Dirac notation as
<m|υ(x)|l>.)
• Substitution into Eq.(12.9) now yields
• and after rearranging terms,
•
Equation (12.10) is exact. No approximations
have been used in deriving it. However, it contains
too many unknown quantities to permit an exact
solution in most cases.
15. •
The most important of the unknown quantities is
the perturbed energy of the nth level, En’,or more
precisely,Δ En = En’ - En . So we let m=n in Eq.(12.10)
to obtain
∞
'
anl vnl = ann ( En − En )
•
∑
(12.10a)
l =0
•
To find a first approximation to this energy
difference, we make the arbitrary assumption that
anl = 1 if l = n and anl = 0 if l ≠ n. In that case, the lefthand side of Eq.(12.10a) collapses to a single term:
vnn.
• Thus Eq.(12.10) is reduced to the approximation
16. •
This is a first-order approximation to the
perturbed energy En’. You can recognize this
integral from the formula for the expectation value
of an operator.
•
This formula is not exact because the integral
contains the wave function of the unperturbed
system rather than the actual wave function.
17. Applications of perturbation theory
Perturbation theory is an important tool for describing
real quantum systems, as it turns out to be very difficult
to find exact solutions to the Schrödinger equation
for Hamiltonians of even moderate complexity.
The Hamiltonians to which we know exact solutions, such
as the hydrogen atom, the quantum harmonic
oscillator and the particle in a box, are too idealized to
adequately describe most systems.
Using perturbation theory, we can use the known
solutions of these simple Hamiltonians to generate
solutions for a range of more complicated systems.
We will start with the mathematical background. And here is the reason.
Many books start with the Navier Stokes equation and then say, ‘based on order of magnitude analysis, we throw away these terms and keep these terms. Then using the following transformation, we can solve this equation’.
Why do you have to do that way? If you are given an equation, can you figure out this is what you have to do , out of the blue?
What are the methods that are normally tried for these kind of problems and why do we choose this method? I would like you to get some idea of these issues and hence we start with some mathematical background.
A perturbation is a ‘disturbance’, usually a small disturbance. Consider an equation, or a system, and say you know the solution of the system. If the equation changes very slightly (perturbed), perhaps the solution will also change slightly. In that case, you can find the solution of the ‘perturbed equation’ to be very close to the solution of the unperturbed equation. Look at the example above.
In the case considered here, if the perturbation is smaller and smaller (i.e. equation becomes closer to the original eqn), then the solution approaches the original solution. These kind of perturbations are called simple perturbations or regular perturbations.
If the perturbation parameter (in the original equation) is epsilon, then the solution can be written as “original solution + epsilon * something…”
Let us consider another system, where the situation is not so simple. Here a small perturbation alters the solution drastically.
Even if the perturbation tends to zero, the solution does NOT tend to the original solution. We cannot write the solution like we did in the previous case. These are called ‘singular perturbation’.
We can look at similar examples in differential equations. An example of regular perturbation is given here.