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Math Intuition for Quantum Mechanics & Quantum Field Theory

Emanuele Pesaresi

This course aims to mathematically motivate both Quantum Mechanics (QM) and Quantum field Theory (QFT). The first part is devoted to the most important concepts and equations of QM, whereas the second part deals with QFT.

Due to the conceptual and mathematical difficulty of these subjects, some prerequisites to this course are unavoidably required. The student should be familiar with:

1) the Fourier Series and Transform;

2) Multivariable Calculus;

3) Probability theory and random variables;

4)  Classical Physics;

Read more

This course aims to mathematically motivate both Quantum Mechanics (QM) and Quantum field Theory (QFT). The first part is devoted to the most important concepts and equations of QM, whereas the second part deals with QFT.

Due to the conceptual and mathematical difficulty of these subjects, some prerequisites to this course are unavoidably required. The student should be familiar with:

1) the Fourier Series and Transform;

2) Multivariable Calculus;

3) Probability theory and random variables;

4)  Classical Physics;

5) Complex Calculus (especially residues and calculation of integrals on a contour), although this is necessary only for some parts of the course devoted to QFT;

6) Special Relativity and tensors for QFT.

Note 1: the first few prerequisites might be enough if you are interested only in the first part of the course, which is related to QM (consider that this course has tens of hours' worth of material, you might be interested only in some parts);

Note 2: I'm more than willing to reply if you have doubts/need clarifications, or -why not- have any recommendations to improve the quality of the course.

Note 3: I'll still keep editing the videos (for example by adding notes) to make the video-lectures as clear as possible.

The references for the part on QFT are the following:

- Quantum Field Theory, M.Srednicki

- Quantum Field Theory, Itzykson & Zuber

- QFT by Mandl & Shaw

- QFT in a nutshell, A.Zee

- QFT by Ryder, Ramand

- The Quantum Theory of Fields, S.Weinberg

- Gauge Theories in Particle Physics, Aitchison & Z.Hey

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What's inside

Learning objectives

  • The mathematical intuition for quantum mechanics and quantum field theory
  • How to (intuitively) derive the schrodinger's equation from the classical theory
  • Quantum operators
  • Quantum states
  • Importance of commutators
  • Derivation of heisenberg uncertainty principle
  • Unitary operators
  • Quantum tunneling
  • Energy spectrum of the hydrogen atom
  • How to quantize a classical field theory
  • Klein gordon equation
  • Wick's theorem
  • Time ordering
  • Normal ordering
  • Noether's theorem
  • Properties of the infinitesimal lorentz transformation
  • Spectrum of the hamiltonian
  • Scattering cross-section
  • Annihilation and creation operators
  • Causality in quantum field theories
  • Ground state
  • Green functions
  • Schrodinger's picture
  • Heisenberg's picture
  • Interaction picture
  • Theory of fermions
  • Theory of bosons
  • Dirac equation
  • Interacting field theory
  • Feynman diagrams
  • Anomalous magnetic moment
  • Show more
  • Show less

Syllabus

Why Operators are needed in Quantum Mechanics
Derivation of the energy operator in QM
Derivation of the momentum operator in QM
How to derive (intuitively) the Schrodinger equation, how to define States, Eigenstates, and the importance of Hermitian operators
Read more
Construction of the Schrodinger equation in QM
Eigenstates and normalized states
Importance of Hermitian operators
the Commutator and its relationship with the Uncertainty principle. Unitary operators and time evolution of states.
Commutator and uncertainty principle
Uncertainty principle for position and momentum
Unitary operators and time evolution of a system
Understand how the concept of quantum tunneling arises in QM from the Schrodinger equation. This section is independent of the others, if you wish you can move to the next one without any problems.
intro to quantum tunneling
Rectangular potential barrier (classical physics)
Solving the QM problem with the method of separation of variables
Solution for x less than 0
Solution for x in the interval [0, L]
Solution for x greater than L
Constraints on the solutions and definition of the transmission coefficient
Derivation of the transmission coefficient
How to derive the discrete energy spectrum of a hydrogen-like atom from the Schrodinger equation. This section is independent of the others, if you wish you can move to the next section.
Intro to this section
Hydrogen-like atoms
Hamiltonian of a hydrogen-like atom
More on potential energy and how to find the spectrum
Separation of variables in the Schrodinger equation
Time independent Schrodinger equation in spherical coordinates
Separating the variables in the time-independent Schrodinger equation
Radial Schrodinger equation
Working on the radial Schrodinger equation
Solution to the radial Schrodinger equation
Derivation of the discrete energy spectrum
Why Quantum field theory is needed. The Classical theory of fields will be revised, and the Klein-Gordon equation will be derived from the theory.
Reconciling Quantum Mechanics with Special Relativity
Review of Classical Field Theory part 1
Review of Classical Field Theory part 2
Understand important concepts of Classical field theory, which will be useful for Quantum Field theory
Klein Gordon equation derived from Classical field theory
Noether's theorem
Example of complex scalar field & more on Noether's theorem
Symmetry with respect to coordinate and field transformation
Energy momentum tensor, orbital angular momentum, intrinsic angular momentum
Derivation of important properties of the infinitesimal Lorentz transformation
How to: quantize a Classical Field theory, derive the Spectrum of the Hamiltonian, learn about causality. Real and complex Scalar fields are introduced here.
Quantization of a Classical Field part 1
Quantization of a Classical Field part 2
Derivation of the spectrum of the Hamiltonian
Annihilation and creation operators to determine the spectrum of the Hamiltonian
Ground state and construction of states with a certain number of quanta
Definition of the number operator
Number operator acting on a two-particle state
Causality of space-like events part 1
Causality of space-like events part 2
Causality of space-like events part 3, an important property of Dirac delta
Causality of space-like events part 4
Green function of the Klein Gordon field part 1
Green function of the Klein Gordon field part 2
Vacuum expectation, Propagator, Time Ordering
Complex Klein Gordon field
How to extend the free Field theory to take interactions into account. The Heisenberg's picture in quantum mechanics wll be introduced and compared with Heisenberg's. A new picture is also required.
Interacting Classical Field theory part 1
Interacting Classical Field Theory part 2
Heisenberg's picture and Schrodinger's picture
Interaction picture
Interacting field theory part 1
Interacting field theory part 2
Interacting Field theory, S matrix
Rewriting time ordering in the S matrix part 1, proof of Hausdorff formula
Rewriting time ordering in the S matrix part 2
Relation between Normal and Time ordered products
Wick's theorem
How Scattering cross sections are defined and calculated. The so-called LSZ formula will be introduced.
Scattering cross section part 1
Scattering cross section part 2
Scattering cross section part 3
LSZ formula part 1
LSZ formula part 2
LSZ formula part 3
LSZ formula part 4
LSZ formula part 5
LSZ formula part 6
LSZ formula part 7
How to quantize the electromagnetic field
Quantization of the E.M. field part 1
Quantization of the E.M. field part 2
Quantization of the E.M. field part 3
Quantization of the E.M. field part 4
Quantization of the E.M. field part 5
Quantization of the E.M. field part 6
Understand why quantum systems have instrinsic angular momentum. The concepts of : Pauli matrices, spinors, will be introduced. The relation between Lorentz transformations and spin will be emphasized
Stern-Gerlach experiment
Intrinsic Angular Momentum in QM (spin) part 1
Intrinsic Angular Momentum in QM (spin) part 2
Pauli matrices, spinors
Unitary operators, Lorentz transformations, angular momenta in QFT part 1
Unitary operators, Lorentz transformations, angular momenta in QFT part 2
Unitary operators, Lorentz transformations, angular momenta in QFT part 3
Unitary operators, Lorentz transformations, angular momenta in QFT part 4
Commutator between a scalar field and angular momentum
Commutator between a vector field and angular momentum

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Activities

Be better prepared before your course. Deepen your understanding during and after it. Supplement your coursework and achieve mastery of the topics covered in Math Intuition for Quantum Mechanics & Quantum Field Theory with these activities:
Review Multivariable Calculus
Strengthen your understanding of multivariable calculus, which is essential for understanding concepts in both Quantum Mechanics and Quantum Field Theory.
Browse courses on Multivariable Calculus
Show steps
  • Review partial derivatives and gradients.
  • Practice calculating line and surface integrals.
  • Study vector fields and their properties.
Brush up on Fourier Transforms
Revisit Fourier Series and Transforms, as they are fundamental tools used throughout Quantum Mechanics and Quantum Field Theory.
Browse courses on Fourier Transform
Show steps
  • Review the definition of Fourier Series.
  • Practice calculating Fourier Transforms of common functions.
  • Understand the properties of Fourier Transforms.
Read 'Quantum Mechanics' by David Griffiths
Supplement your learning with a comprehensive textbook on Quantum Mechanics to gain a deeper understanding of the core concepts.
View Introduction to Electrodynamics on Amazon
Show steps
  • Read assigned chapters before each corresponding lecture.
  • Work through the example problems in the book.
  • Attempt the end-of-chapter problems for practice.
Four other activities
Expand to see all activities and additional details
Show all seven activities
Participate in weekly study groups
Collaborate with peers to discuss challenging concepts and problem-solving strategies in Quantum Mechanics and Quantum Field Theory.
Show steps
  • Form a study group with classmates.
  • Review lecture notes and discuss key concepts.
  • Work through practice problems together.
Solve Quantum Mechanics Problems
Practice applying theoretical knowledge by solving a variety of problems related to quantum operators, states, and tunneling.
Show steps
  • Find problem sets online or in textbooks.
  • Attempt to solve problems independently.
  • Compare solutions with peers or consult solution manuals.
Study 'Quantum Field Theory in a Nutshell' by A. Zee
Deepen your understanding of Quantum Field Theory with a book that emphasizes physical intuition and provides a broader perspective.
View Einstein Gravity in a Nutshell on Amazon
Show steps
  • Read assigned chapters after each corresponding lecture.
  • Focus on understanding the physical interpretations.
  • Work through the exercises to test your understanding.
Create Feynman Diagrams
Solidify your understanding of interacting field theory by drawing Feynman diagrams for various processes.
Show steps
  • Review the rules for drawing Feynman diagrams.
  • Choose a specific interaction to analyze.
  • Draw the corresponding Feynman diagrams.
  • Explain the physical meaning of each diagram.

Career center

Learners who complete Math Intuition for Quantum Mechanics & Quantum Field Theory will develop knowledge and skills that may be useful to these careers:
Theoretical Physicist
Theoretical physicists develop mathematical models and theories to explain the physical world. This course, *Math Intuition for Quantum Mechanics & Quantum Field Theory*, helps deepen your understanding of the mathematical concepts underpinning quantum mechanics and quantum field theory, which is essential for success as a theoretical physicist. The course explores the derivation of essential equations, quantum operators, quantum states, and the importance of commutators. By learning about the Klein Gordon equation, Wick's theorem, time ordering, normal ordering, and the properties of Lorentz transformation, you will be better able to tackle complex problems in theoretical physics, propose new theories, and contribute to our understanding of the universe at its most fundamental level. Theoretical physicists may find the syllabus useful, as it provides a framework for deriving the spectrum of the Hamiltonian.
See salaries and explore the career path for Theoretical Physicist
Computational Physicist
Computational physicists use computer simulations to solve complex physics problems. This course, *Math Intuition for Quantum Mechanics & Quantum Field Theory*, provides a basis of the mathematical and theoretical knowledge necessary to develop and interpret these simulations. Proficiency with quantum operators, quantum states, and the Schrödinger equation, as covered in this course, is useful for modeling quantum systems. Furthermore, the exploration of Quantum Field Theory concepts, such as Green's functions and Feynman diagrams, enables computational physicists to simulate particle interactions and other complex phenomena. The theory of Bosons and Theory of Fermions are especially helpful.
See salaries and explore the career path for Computational Physicist
Quantum Computing Researcher
A quantum computing researcher works on developing new quantum algorithms and hardware. This is a career where understanding the underlying mathematical principles is paramount. This course, *Math Intuition for Quantum Mechanics & Quantum Field Theory*, helps build the crucial mathematical foundation needed to grasp the concepts of quantum mechanics, quantum operators, and quantum states. The course offers a review of linear algebra, quantum mechanics, quantum field theory, classical physics, special relativity, and tensors for Quantum Field Theory, equipping you with the necessary tools to model and analyze quantum systems, develop quantum algorithms, and contribute to the advancement of quantum computing technologies. The learning objectives, like Heisenberg's picture and Schrodinger's picture, may be useful for researchers.
See salaries and explore the career path for Quantum Computing Researcher
Mathematical Modeler
Mathematical modelers create and analyze mathematical models to solve problems in various fields, including physics, engineering, and finance. This course, *Math Intuition for Quantum Mechanics & Quantum Field Theory*, directly applies to the skills needed to model physical systems at the quantum level. A mathematical modeler may find the knowledge of quantum operators, unitary operators, and the Schrödinger equation as a starting point. The advanced topics in quantum field theory, such as Wick's theorem and Feynman diagrams, allows for even more sophisticated models of particle interactions. As a mathematical modeler, the learning objectives from the course provide a framework for solving a wide range of problems.
See salaries and explore the career path for Mathematical Modeler
Quantum Engineer
Quantum engineers apply the principles of quantum mechanics to develop new technologies and devices. This course, *Math Intuition for Quantum Mechanics & Quantum Field Theory,* helps build a strong grasp of the mathematical concepts underpinning quantum mechanics needed to design and implement practical quantum systems. Quantum engineers will benefit from the course's coverage of quantum operators, quantum states, and the Heisenberg Uncertainty Principle, which are foundational to quantum engineering. The exploration of quantum field theory, including topics such as annihilation and creation operators, may be helpful for engineers working on advanced quantum technologies. By studying scattering cross-sections from the syllabus, one can apply the knowledge to real-world problems.
See salaries and explore the career path for Quantum Engineer
Research Scientist
Research scientists design and conduct experiments, analyze data, and publish findings in various fields, including physics and materials science. This course, *Math Intuition for Quantum Mechanics & Quantum Field Theory*, helps to develop the strong analytical and mathematical skills necessary to succeed in this role. The course covers topics such as quantum operators, quantum states, the Heisenberg Uncertainty Principle, and quantum tunneling. Furthermore, the deep dive into quantum field theory, including Wick's theorem, time ordering, and Feynman diagrams, may be useful for research scientists working on cutting-edge experiments involving quantum phenomena. The course may prepare scientists to interpret experimental results, develop new research methodologies, and contribute to scientific breakthroughs.
See salaries and explore the career path for Research Scientist
Mathematics Professor
A mathematics professor teaches mathematics courses at the university level and performs research in a specific area of mathematics. This course, *Math Intuition for Quantum Mechanics & Quantum Field Theory*, helps to strengthen understanding of the mathematical foundations of quantum mechanics and quantum field theory. This will be valuable both in teaching related courses and in pursuing research in areas where mathematics intersects with physics. The course's coverage of topics like quantum operators, unitary operators, and the derivation of the Schrödinger equation helps provide a deeper and more intuitive understanding of the material, enabling a mathematics professor to convey these complex topics more effectively to students. The professor can also use these skills in his or her original research.
See salaries and explore the career path for Mathematics Professor
Physics Teacher
Physics teachers educate students in the principles of physics at the high school or college level. This course, *Math Intuition for Quantum Mechanics & Quantum Field Theory*, may be useful to enhance your understanding of advanced physics concepts, enabling you to teach more effectively and inspire students. The course's coverage of quantum mechanics, quantum field theory, and related mathematical techniques may be useful for teachers. Physics teachers may find the derivation of essential equations, quantum operators, and the Heisenberg Uncertainty Principle to be particularly beneficial, providing them with a deeper understanding of the subject matter.
See salaries and explore the career path for Physics Teacher
Optical Engineer
Optical engineers design and develop optical systems and devices, such as lasers, sensors, and imaging systems. This course, *Math Intuition for Quantum Mechanics & Quantum Field Theory,* helps build a foundation in the mathematical and physical principles underlying optics and photonics. The course's exploration of quantum mechanics and quantum field theory helps to understand the behavior of light at a fundamental level. The study of quantum operators, quantum states, and the interaction of light with matter may be useful for optical engineers. The course may be helpful to design and optimize optical systems, develop new optical materials, and create innovative optical devices.
See salaries and explore the career path for Optical Engineer
Data Scientist
Data scientists analyze large datasets to extract meaningful insights and develop predictive models. While seemingly disparate, the mathematical rigor developed in this course, *Math Intuition for Quantum Mechanics & Quantum Field Theory*, may be beneficial for a data science career, especially in fields dealing with complex systems or simulations. Reviewing the Fourier series and transform, multivariable calculus, probability theory, and complex calculus may help to formulate and test hypotheses, build sophisticated models, and effectively communicate findings to stakeholders. Knowledge of Quantum Mechanics could be applicable for specialized roles in areas like financial modeling or materials discovery.
See salaries and explore the career path for Data Scientist
Aerospace Engineer
Aerospace engineers design, develop, and test aircraft and spacecraft. This course, *Math Intuition for Quantum Mechanics & Quantum Field Theory,* may not be directly applicable to the day-to-day tasks of most aerospace engineers, it helps develop a strong foundation in mathematical and physical principles that may be valuable in specialized areas. For example, the knowledge of quantum mechanics could be relevant in developing advanced sensors or propulsion systems. The course's coverage of topics like the Schrödinger equation and quantum tunneling may be useful to solve complex engineering challenges. Furthermore, the acquired mathematics from the syllabus may be beneficial in other aspects of the job.
See salaries and explore the career path for Aerospace Engineer
Software Developer
Software developers design and build software applications. Although this course, *Math Intuition for Quantum Mechanics & Quantum Field Theory*, may not be directly applicable to most software development roles, it helps develop analytical and problem-solving skills that might be useful in specialized areas such as developing simulation software for scientific research or optimizing algorithms for complex calculations. The course's focus on mathematical rigor and theoretical understanding of quantum mechanics may be useful to learn new technologies. Software Developers may benefit from a review of multivariable calculus and complex calculus.
See salaries and explore the career path for Software Developer
Machine Learning Engineer
Machine learning engineers develop and implement machine learning algorithms and systems. While this course, *Math Intuition for Quantum Mechanics & Quantum Field Theory,* is not directly related to machine learning, it may help cultivate a strong mathematical foundation that may be useful in certain specialized areas of machine learning research. In particular, the course's coverage of linear algebra, calculus, and probability theory are relevant to the mathematical underpinnings of many machine learning algorithms. The course may be useful to explore advanced topics such as quantum machine learning. Developers may find reviewing the Fourier series beneficial.
See salaries and explore the career path for Machine Learning Engineer
Financial Analyst
Financial analysts evaluate financial data, provide investment recommendations, and manage financial risk. While this course, *Math Intuition for Quantum Mechanics & Quantum Field Theory*, is not directly related to finance, it may help develop strong quantitative and analytical skills. A background in mathematics and physics, as provided by this course, may be useful in developing sophisticated financial models or analyzing complex market dynamics. However, most financial analysis roles require specific knowledge of finance and economics, which are not covered in this course. A financial analyst may benefit from a review of probability theory.
See salaries and explore the career path for Financial Analyst
Biostatistician
Biostatisticians apply statistical methods to solve problems in biology and medicine. While this course, *Math Intuition for Quantum Mechanics & Quantum Field Theory* is not directly relevant to biostatistics, it may help develop strong mathematical and analytical skills that may be transferrable. The course's coverage of multivariable calculus, probability theory, and random variables helps to develop logical reasoning abilities. The course may be useful to learn how to approach complex problems and develop mathematical models, which may be beneficial in any field of quantitative analysis. Biostatisticians may find the section on theory of probability useful.
See salaries and explore the career path for Biostatistician

Reading list

We've selected two books that we think will supplement your learning. Use these to develop background knowledge, enrich your coursework, and gain a deeper understanding of the topics covered in Math Intuition for Quantum Mechanics & Quantum Field Theory.
Cover image
Introduction to Electrodynamics
Save
Provides a clear and accessible introduction to the principles of quantum mechanics. It is widely used as a textbook in undergraduate physics courses. Griffiths' book offers a balanced approach, covering both the conceptual foundations and the mathematical formalism of quantum mechanics. It is particularly helpful for solidifying the concepts introduced in the first part of the course.
Introduction to Electrodynamics
Hardcover
$$$
Introduction to Electrodynamics
Paperback
$$$
INTRODUCTION TO ELECTRODYNAMICS, 4TH EDITION
Paperback
$$
Introduction to Electrodynamics
Hardcover
$$$
Introduction to Electrodynamics
Kindle Edition
$$$
Introduction to Electrodynamics
Kindle Edition
$$$
Cover image
Einstein Gravity in a Nutshell
Save
Offers a relatively accessible introduction to Quantum Field Theory, focusing on the key concepts and techniques. It is known for its engaging writing style and its emphasis on physical intuition. Zee's book is particularly useful for gaining a broader perspective on QFT and for understanding the connections between different topics. It serves as excellent additional reading to complement the course material.
Einstein Gravity in a Nutshell
Paperback
Check
Einstein Gravity in a Nutshell
Kindle Edition
Check

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Effort
43 total hours
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Udemy
Instructor
Emanuele Pesaresi
Language
English
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