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
Partition function indicates the mode of distribution of particles in various energy states. It plays a role similar to the wave function of the quantum mechanics,which contains all the dynamical information about the system.
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
Partition function indicates the mode of distribution of particles in various energy states. It plays a role similar to the wave function of the quantum mechanics,which contains all the dynamical information about the system.
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
This is an introduction to modern quantum mechanics – albeit for those already familiar with vector calculus and modern physics – based on my personal understanding of the subject that emphasizes the concepts from first principles. Nothing of this is new or even developed first hand but the content (or maybe its clarity) is original in the fact that it displays an abridged yet concise and straightforward mathematical development that provides for a solid foundation in the tools and techniques to better understand and have a good appreciation for the physics involved in quantum theory and in an atom!
content-
Chemistry & Chemical Engineering
History of Catalysis
Catalysis
Recent trends in Catalysis
Future trends in Catalysis
Summary
role-
24% of GDP from Products made using catalysts (Food, Fuels, Clothes, Polymers, Drug, Agro-chemicals)
> 90 % of petro refining & petrochemicals processes use catalysts
90 % of processes & 60 % of products in the chemical industry
> 95% of pollution control technologies
Catalysis in the production/use of alternate fuels (NG,DME, H2, Fuel Cells, biofuels…)
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 .
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
This is an introduction to modern quantum mechanics – albeit for those already familiar with vector calculus and modern physics – based on my personal understanding of the subject that emphasizes the concepts from first principles. Nothing of this is new or even developed first hand but the content (or maybe its clarity) is original in the fact that it displays an abridged yet concise and straightforward mathematical development that provides for a solid foundation in the tools and techniques to better understand and have a good appreciation for the physics involved in quantum theory and in an atom!
content-
Chemistry & Chemical Engineering
History of Catalysis
Catalysis
Recent trends in Catalysis
Future trends in Catalysis
Summary
role-
24% of GDP from Products made using catalysts (Food, Fuels, Clothes, Polymers, Drug, Agro-chemicals)
> 90 % of petro refining & petrochemicals processes use catalysts
90 % of processes & 60 % of products in the chemical industry
> 95% of pollution control technologies
Catalysis in the production/use of alternate fuels (NG,DME, H2, Fuel Cells, biofuels…)
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 .
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.
Deep Behavioral Phenotyping in Systems Neuroscience for Functional Atlasing a...Ana Luísa Pinho
Functional Magnetic Resonance Imaging (fMRI) provides means to characterize brain activations in response to behavior. However, cognitive neuroscience has been limited to group-level effects referring to the performance of specific tasks. To obtain the functional profile of elementary cognitive mechanisms, the combination of brain responses to many tasks is required. Yet, to date, both structural atlases and parcellation-based activations do not fully account for cognitive function and still present several limitations. Further, they do not adapt overall to individual characteristics. In this talk, I will give an account of deep-behavioral phenotyping strategies, namely data-driven methods in large task-fMRI datasets, to optimize functional brain-data collection and improve inference of effects-of-interest related to mental processes. Key to this approach is the employment of fast multi-functional paradigms rich on features that can be well parametrized and, consequently, facilitate the creation of psycho-physiological constructs to be modelled with imaging data. Particular emphasis will be given to music stimuli when studying high-order cognitive mechanisms, due to their ecological nature and quality to enable complex behavior compounded by discrete entities. I will also discuss how deep-behavioral phenotyping and individualized models applied to neuroimaging data can better account for the subject-specific organization of domain-general cognitive systems in the human brain. Finally, the accumulation of functional brain signatures brings the possibility to clarify relationships among tasks and create a univocal link between brain systems and mental functions through: (1) the development of ontologies proposing an organization of cognitive processes; and (2) brain-network taxonomies describing functional specialization. To this end, tools to improve commensurability in cognitive science are necessary, such as public repositories, ontology-based platforms and automated meta-analysis tools. I will thus discuss some brain-atlasing resources currently under development, and their applicability in cognitive as well as clinical neuroscience.
Comparing Evolved Extractive Text Summary Scores of Bidirectional Encoder Rep...University of Maribor
Slides from:
11th International Conference on Electrical, Electronics and Computer Engineering (IcETRAN), Niš, 3-6 June 2024
Track: Artificial Intelligence
https://www.etran.rs/2024/en/home-english/
Earliest Galaxies in the JADES Origins Field: Luminosity Function and Cosmic ...Sérgio Sacani
We characterize the earliest galaxy population in the JADES Origins Field (JOF), the deepest
imaging field observed with JWST. We make use of the ancillary Hubble optical images (5 filters
spanning 0.4−0.9µm) and novel JWST images with 14 filters spanning 0.8−5µm, including 7 mediumband filters, and reaching total exposure times of up to 46 hours per filter. We combine all our data
at > 2.3µm to construct an ultradeep image, reaching as deep as ≈ 31.4 AB mag in the stack and
30.3-31.0 AB mag (5σ, r = 0.1” circular aperture) in individual filters. We measure photometric
redshifts and use robust selection criteria to identify a sample of eight galaxy candidates at redshifts
z = 11.5 − 15. These objects show compact half-light radii of R1/2 ∼ 50 − 200pc, stellar masses of
M⋆ ∼ 107−108M⊙, and star-formation rates of SFR ∼ 0.1−1 M⊙ yr−1
. Our search finds no candidates
at 15 < z < 20, placing upper limits at these redshifts. We develop a forward modeling approach to
infer the properties of the evolving luminosity function without binning in redshift or luminosity that
marginalizes over the photometric redshift uncertainty of our candidate galaxies and incorporates the
impact of non-detections. We find a z = 12 luminosity function in good agreement with prior results,
and that the luminosity function normalization and UV luminosity density decline by a factor of ∼ 2.5
from z = 12 to z = 14. We discuss the possible implications of our results in the context of theoretical
models for evolution of the dark matter halo mass function.
DERIVATION OF MODIFIED BERNOULLI EQUATION WITH VISCOUS EFFECTS AND TERMINAL V...Wasswaderrick3
In this book, we use conservation of energy techniques on a fluid element to derive the Modified Bernoulli equation of flow with viscous or friction effects. We derive the general equation of flow/ velocity and then from this we derive the Pouiselle flow equation, the transition flow equation and the turbulent flow equation. In the situations where there are no viscous effects , the equation reduces to the Bernoulli equation. From experimental results, we are able to include other terms in the Bernoulli equation. We also look at cases where pressure gradients exist. We use the Modified Bernoulli equation to derive equations of flow rate for pipes of different cross sectional areas connected together. We also extend our techniques of energy conservation to a sphere falling in a viscous medium under the effect of gravity. We demonstrate Stokes equation of terminal velocity and turbulent flow equation. We look at a way of calculating the time taken for a body to fall in a viscous medium. We also look at the general equation of terminal velocity.
Slide 1: Title Slide
Extrachromosomal Inheritance
Slide 2: Introduction to Extrachromosomal Inheritance
Definition: Extrachromosomal inheritance refers to the transmission of genetic material that is not found within the nucleus.
Key Components: Involves genes located in mitochondria, chloroplasts, and plasmids.
Slide 3: Mitochondrial Inheritance
Mitochondria: Organelles responsible for energy production.
Mitochondrial DNA (mtDNA): Circular DNA molecule found in mitochondria.
Inheritance Pattern: Maternally inherited, meaning it is passed from mothers to all their offspring.
Diseases: Examples include Leber’s hereditary optic neuropathy (LHON) and mitochondrial myopathy.
Slide 4: Chloroplast Inheritance
Chloroplasts: Organelles responsible for photosynthesis in plants.
Chloroplast DNA (cpDNA): Circular DNA molecule found in chloroplasts.
Inheritance Pattern: Often maternally inherited in most plants, but can vary in some species.
Examples: Variegation in plants, where leaf color patterns are determined by chloroplast DNA.
Slide 5: Plasmid Inheritance
Plasmids: Small, circular DNA molecules found in bacteria and some eukaryotes.
Features: Can carry antibiotic resistance genes and can be transferred between cells through processes like conjugation.
Significance: Important in biotechnology for gene cloning and genetic engineering.
Slide 6: Mechanisms of Extrachromosomal Inheritance
Non-Mendelian Patterns: Do not follow Mendel’s laws of inheritance.
Cytoplasmic Segregation: During cell division, organelles like mitochondria and chloroplasts are randomly distributed to daughter cells.
Heteroplasmy: Presence of more than one type of organellar genome within a cell, leading to variation in expression.
Slide 7: Examples of Extrachromosomal Inheritance
Four O’clock Plant (Mirabilis jalapa): Shows variegated leaves due to different cpDNA in leaf cells.
Petite Mutants in Yeast: Result from mutations in mitochondrial DNA affecting respiration.
Slide 8: Importance of Extrachromosomal Inheritance
Evolution: Provides insight into the evolution of eukaryotic cells.
Medicine: Understanding mitochondrial inheritance helps in diagnosing and treating mitochondrial diseases.
Agriculture: Chloroplast inheritance can be used in plant breeding and genetic modification.
Slide 9: Recent Research and Advances
Gene Editing: Techniques like CRISPR-Cas9 are being used to edit mitochondrial and chloroplast DNA.
Therapies: Development of mitochondrial replacement therapy (MRT) for preventing mitochondrial diseases.
Slide 10: Conclusion
Summary: Extrachromosomal inheritance involves the transmission of genetic material outside the nucleus and plays a crucial role in genetics, medicine, and biotechnology.
Future Directions: Continued research and technological advancements hold promise for new treatments and applications.
Slide 11: Questions and Discussion
Invite Audience: Open the floor for any questions or further discussion on the topic.
This presentation explores a brief idea about the structural and functional attributes of nucleotides, the structure and function of genetic materials along with the impact of UV rays and pH upon them.
The ability to recreate computational results with minimal effort and actionable metrics provides a solid foundation for scientific research and software development. When people can replicate an analysis at the touch of a button using open-source software, open data, and methods to assess and compare proposals, it significantly eases verification of results, engagement with a diverse range of contributors, and progress. However, we have yet to fully achieve this; there are still many sociotechnical frictions.
Inspired by David Donoho's vision, this talk aims to revisit the three crucial pillars of frictionless reproducibility (data sharing, code sharing, and competitive challenges) with the perspective of deep software variability.
Our observation is that multiple layers — hardware, operating systems, third-party libraries, software versions, input data, compile-time options, and parameters — are subject to variability that exacerbates frictions but is also essential for achieving robust, generalizable results and fostering innovation. I will first review the literature, providing evidence of how the complex variability interactions across these layers affect qualitative and quantitative software properties, thereby complicating the reproduction and replication of scientific studies in various fields.
I will then present some software engineering and AI techniques that can support the strategic exploration of variability spaces. These include the use of abstractions and models (e.g., feature models), sampling strategies (e.g., uniform, random), cost-effective measurements (e.g., incremental build of software configurations), and dimensionality reduction methods (e.g., transfer learning, feature selection, software debloating).
I will finally argue that deep variability is both the problem and solution of frictionless reproducibility, calling the software science community to develop new methods and tools to manage variability and foster reproducibility in software systems.
Exposé invité Journées Nationales du GDR GPL 2024
Salas, V. (2024) "John of St. Thomas (Poinsot) on the Science of Sacred Theol...Studia Poinsotiana
I Introduction
II Subalternation and Theology
III Theology and Dogmatic Declarations
IV The Mixed Principles of Theology
V Virtual Revelation: The Unity of Theology
VI Theology as a Natural Science
VII Theology’s Certitude
VIII Conclusion
Notes
Bibliography
All the contents are fully attributable to the author, Doctor Victor Salas. Should you wish to get this text republished, get in touch with the author or the editorial committee of the Studia Poinsotiana. Insofar as possible, we will be happy to broker your contact.
THE IMPORTANCE OF MARTIAN ATMOSPHERE SAMPLE RETURN.Sérgio Sacani
The return of a sample of near-surface atmosphere from Mars would facilitate answers to several first-order science questions surrounding the formation and evolution of the planet. One of the important aspects of terrestrial planet formation in general is the role that primary atmospheres played in influencing the chemistry and structure of the planets and their antecedents. Studies of the martian atmosphere can be used to investigate the role of a primary atmosphere in its history. Atmosphere samples would also inform our understanding of the near-surface chemistry of the planet, and ultimately the prospects for life. High-precision isotopic analyses of constituent gases are needed to address these questions, requiring that the analyses are made on returned samples rather than in situ.
(May 29th, 2024) Advancements in Intravital Microscopy- Insights for Preclini...Scintica Instrumentation
Intravital microscopy (IVM) is a powerful tool utilized to study cellular behavior over time and space in vivo. Much of our understanding of cell biology has been accomplished using various in vitro and ex vivo methods; however, these studies do not necessarily reflect the natural dynamics of biological processes. Unlike traditional cell culture or fixed tissue imaging, IVM allows for the ultra-fast high-resolution imaging of cellular processes over time and space and were studied in its natural environment. Real-time visualization of biological processes in the context of an intact organism helps maintain physiological relevance and provide insights into the progression of disease, response to treatments or developmental processes.
In this webinar we give an overview of advanced applications of the IVM system in preclinical research. IVIM technology is a provider of all-in-one intravital microscopy systems and solutions optimized for in vivo imaging of live animal models at sub-micron resolution. The system’s unique features and user-friendly software enables researchers to probe fast dynamic biological processes such as immune cell tracking, cell-cell interaction as well as vascularization and tumor metastasis with exceptional detail. This webinar will also give an overview of IVM being utilized in drug development, offering a view into the intricate interaction between drugs/nanoparticles and tissues in vivo and allows for the evaluation of therapeutic intervention in a variety of tissues and organs. This interdisciplinary collaboration continues to drive the advancements of novel therapeutic strategies.
1. Ladder operator technique for solving Schr¨odinger
equation for a particle exhibiting simple harmonic
motion in one-dimension1
1 Simple harmonic motion in 1D
For a particle exhibiting simple harmonic motion, Hooke’s law is applicable,
which is given as
F = −k(x − xeq), (1)
where F is the force acting on the particle, k is called force constant, x is
the displacement of the particle from the equilibrium position (xeq). The
negative sign represents that the direction of force on the particle works
in the direction opposite to the displacement. Considering the equilibrium
position at the origin of the Co-ordinate system, Eq. 1 becomes
F = −kx (2)
The force constant k is related to the mass (m) of the particle and angular
velocity of oscillation (ω) by the following relation
k = mω2
(3)
The Hamiltonian for this system is given by
ˆH =
ˆp2
x
2m
+ ˆV (x), (4)
where ˆpx = −i ∂
∂x
, is the linear momentum operator and ˆV (x) is the potential
energy operator for the system. ˆV (x) can be obtained from the definition of
force i.e.
F = −
dV
dx
=⇒ dV = − Fdx
V = k xdx =
1
2
kx2
+ C,
(5)
where C is the constant of integration. Since, at x = 0 (i.e. at equilibrium
position), the total energy of the particle is kinetic in nature. The potential
1
This notes is highly based on the derivations given in the book “Quantum Mechanics”
by David Griffiths. However, you’ll find the derivations and descriptions given in this
notes more detailed.
1
2. energy is zero i.e. V = 0. This means C = 0. Thus,
V =
1
2
kˆx2
=
1
2
mω2
ˆx2
(6)
Using Eq. 6 in Eq. 4, we get
ˆH =
ˆp2
x
2m
+
1
2
mω2
ˆx2
=
1
2m
ˆp2
x + m2
ω2
ˆx2
(7)
Therefore, the Schr¨odinger equation for the particle exhibiting simple har-
monic motion in one-dimension becomes
1
2m
ˆp2
x + m2
ω2
ˆx2
ψ(x) = Eψ(x) (8)
ψ(x) is the eigenstate and E is the eigenvalue of ˆH. Our target is to solve
Eq. 8. This looks a simple second order differential equation. However, be-
cause of the presence of x2
term, it is not really trivial to solve it.
In this notes, we are going to explore the ladder operator technique for
solving Eq. 8.
1.1 Factorization of the Hamiltonian
In ladder operator technique, our first aim will be to factorize the Hamil-
tonian of the system i.e. to express ˆH in Eq. 7 as a product of two or
more operators. If px and x were simple numbers then the task of factor-
ization would be very easy. For example, we could write px + m2
ω2
x2
=
(ipx + mωx) (−ipx + mωx). But, since ˆx and ˆpx are operators and these do
not commute with each other, we can not really get perfect factorization like
this. In spite of this fact, let us try this kind of factorization. We assume
two operators2
,
a+ =
1
√
2 mω
(−iˆpx + mωˆx)
a− =
1
√
2 mω
(iˆpx + mωˆx)
(9)
In short, we can write a± = 1√
2 mω
( iˆpx + mωˆx). Now, with these two new
operators, we can write
a±a =
1
2 mω
( iˆpx + mωˆx) (±iˆpx + mωˆx)
2
Later on, we’ll see why these operators are called ladder operators.
2
3. =
1
2 mω
ˆp2
x iˆpxmωˆx ± imωˆxˆpx + m2
ω2
ˆx2
=
1
2 mω
ˆp2
x + m2
ω2
ˆx2
± imω (ˆxˆpx − ˆpx ˆx) (10)
Since (ˆxˆpx − ˆpx ˆx) represents the commutator [ˆx, ˆpx] and it’s value is i , we
can write
a±a =
1
2 mω
ˆp2
x + m2
ω2
ˆx2
mω
=
1
ω
ˆH
1
2
=⇒ ˆH = ω a±a ±
1
2
(11)
Thus, we see that complete factorization of the Hamiltonian cann not achieved
by using the a+ and a− operators. However, it’s not a big issue. we can sur-
vive with this partial factorization. However, before we proceed, let us first
study some of the important properties of a+ and a− operators, which will
be used in our discussion.
1.2 Some properties of a+ and a− operators
Property 1: Commutation relation between a+ and a−
Let us first find out the commutation relation between a+, a− and ˆH.
[a+, a−] = a+a− − a−a+
=
1
ω
ˆH −
1
2
−
1
ω
ˆH +
1
2
(using Eq. 11)
=⇒ [a+, a−] = 1
(12)
Thus, a+ and a− operators do not commute with each other.
Property 2: Commutation relation between a± and ˆH
a+, ˆH =a+
ˆH − ˆHa+
=
1
√
2 mω
(−iˆpx + mωˆx)
1
2m
ˆp2
x + m2
ω2
ˆx2
−
1
2m
ˆp2
x + m2
ω2
ˆx2 1
√
2 mω
(−iˆpx + mωˆx)
=
1
2m
√
2 mω
−iˆp3
x + mωˆxˆp2
x − im2
ω2
ˆpx ˆx2
+ m3
ω3
ˆx3
− −iˆp3
x − im2
ω2
ˆx2
ˆpx + mωˆp2
x ˆx + m3
ω3
ˆx3
3
4. =
1
2m
√
2 mω
mω ˆx, ˆp2
x + im2
ω2
ˆx2
, ˆpx (13)
Now, we can write
ˆx, ˆp2
x = ˆxˆpx ˆpx − ˆpx ˆpx ˆx
= (i + ˆpx ˆx) ˆpx − ˆpx ˆpx ˆx (Using ˆxˆpx − ˆpx ˆx = i )
= i ˆpx + ˆpx ˆxˆpx − ˆpx ˆpx ˆx
= i ˆpx + ˆpx (ˆxˆpx − ˆpx ˆx)
= i ˆpx + ˆpxi
= 2i ˆpx
(14)
Similarly, we can write
ˆx2
, ˆpx = ˆxˆxˆpx − ˆpx ˆxˆx
= ˆx (i + ˆpx ˆx) − ˆpx ˆxˆx (Using ˆxˆpx − ˆpx ˆx = i )
= i ˆx + ˆxˆpx ˆx − ˆpx ˆxˆx
= i ˆx + (ˆxˆpx − ˆpx ˆx) ˆx
= i ˆx + i ˆx
= 2i ˆx
(15)
Using Eq. 14 and 15 in Eq. 13, we get
a+, ˆH =
1
2m
√
2 mω
mω × 2i ˆpx + im2
ω2
× 2i ˆx
= −
2 mω
2m
√
2 mω
(−iˆpx + mωˆx)
= − ωa+
(16)
Similarly,
a−, ˆH =a−
ˆH − ˆHa−
=
1
√
2 mω
(iˆpx + mωˆx)
1
2m
ˆp2
x + m2
ω2
ˆx2
−
1
2m
ˆp2
x + m2
ω2
ˆx2 1
√
2 mω
(iˆpx + mωˆx)
=
1
2m
√
2 mω
iˆp3
x + mωˆxˆp2
x + im2
ω2
ˆpx ˆx2
+ m3
ω3
ˆx3
− iˆp3
x + im2
ω2
ˆx2
ˆpx + mωˆp2
x ˆx + m3
ω3
ˆx3
4
5. =
1
2m
√
2 mω
mω ˆx, ˆp2
x − im2
ω2
ˆx2
, ˆpx
=
1
2m
√
2 mω
mω × 2i ˆpx − im2
ω2
× 2i ˆx
=
2 mω
2m
√
2 mω
{iˆpx + mωˆx} = ωa−
Thus,
a±, ˆH = ωa± (17)
Property 3: a+ and a− behave as raising and lowering operators
respectively
First lets prove that a+ acts as a raising operator. Let ψ(x) be an eigenfunc-
tion of the Hamiltonian ˆH and E be the corresponding eigenvalue.
ˆHψ(x) = Eψ(x) (18)
To prove that a+ is a raising operator, we need to show that when it operates
over some eigenfunction of ˆH then it yields the next higher eigenfunction of
ˆH. Thus, we consider a+ψ(x) and check whether this is the next higher
eigenfunction of ˆH or not.
ˆHa+ψ(x) = ω a+a− +
1
2
a+ψ(x) (Using Eq. 11)
= ω a+a−a+ +
1
2
a+ ψ(x)
= ωa+ a−a+ +
1
2
ψ(x)
= a+ ω 1 + a+a− +
1
2
ψ(x) (Using Eq. 12)
= a+ ω a+a− +
1
2
ψ(x) + ωψ(x)
= a+
ˆHψ(x) + ωψ(x) (Using Eq. 11)
= a+ [Eψ(x) + ωψ(x)]
ˆHa+ψ(x) = (E + ω) a+ψ(x) (19)
Thus, if ψ(x) be an eigenfunction of ˆH with eigenvalue E, then a+ψ(x) will
also be an eigenfunction of ˆH with eigenvalue (E + ω). Therefore, a+ when
operates on an eigenfunction of ˆH, it generates another eigenfunction of ˆH,
with eigenvalue (E + ω), which proves that a+ is a raising operator. If we
5
6. apply a+ once again i.e. a2
+ψ(x) then this will generate the second higher
eigenfunction with eigenvalues (E + 2 ω).
Now, to prove that a− is a lowering operator, we need to show that
when it operates over some eigenfunction of ˆH then it yields the next lower
eigenfunction of ˆH. Thus, we consider a−ψ(x) and check whether this is the
next lower eigenfunction of ˆH or not.
ˆHa−ψ(x) = ω a−a+ −
1
2
a−ψ(x) (Using Eq. 11)
= ω a−a+a− −
1
2
a− ψ(x)
= ωa− a+a− −
1
2
ψ(x)
= a− ω a−a+ − 1 −
1
2
ψ(x) (Using Eq. 12)
= a− ω a−a+ −
1
2
ψ(x) − ωψ(x)
= a−
ˆHψ(x) − ωψ(x) (Using Eq. 11)
= a− [Eψ(x) − ωψ(x)]
ˆHa−ψ(x) = (E − ω) a−ψ(x) (20)
Thus, if ψ(x) be an eigenfunction of ˆH with eigenvalue E, then a−ψ(x) will
also be an eigenfunction of ˆH with eigenvalue (E − ω). Therefore, a− when
operates on an eigenfunction of ˆH, it generates another eigenfunction of ˆH,
with eigenvalue (E − ω), which proves that a− is a lowering operator. If
we apply a− once again i.e. a2
−ψ(x) then this will generate the second lower
eigenfunction with eigenvalues (E − 2 ω).
We’ll study some more properties related to our raising and lowering
operators. However, before that we need to find out the ground state (lowest
energy state for our simple harmonic oscillator.
1.3 Finding out the wavefunction of the lowest energ
state of the simple harmonic oscillator
We have seen above that a− behaves as a lowering operator. That means,
if we apply a− operator on some eigenfunction of ˆH then the result will
be another eigenfunction of ˆH with eigenvalue ω less than the previous
eigenfunction. If we keep on applying a− on the eigenfunction then we’ll
keep on getting lower and lower energy eigenfunctions. However, in reality,
6
7. this lowering should stop at some point. There should some minimum energy
state (called ground state) below which no other energy eigenfunction should
exist. Keeping this fact in mind, we can say that the operation of a− operator
on the minimum energy state or ground state should result in zero.
Property 4: Effect of a+a− and a−a+ on some eigenfunction of ˆH
We know,
ˆH = ω a±a ±
1
2
=⇒ ˆHψ = Eψ
ω a±a ±
1
2
ψ = Eψ
a±a ψ =
E
ω
1
2
ψ (21)
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