Eigenstates of the quantum harmonic oscillator Professor M does Science 2021-04-07 | What do the energy eigenstates of the quantum harmonic oscillator look like? 📝 Problems+solutions:- Quantum harmonic oscillator I: professorm.learnworlds.com/course/quantum-harmonic-oscillator-i- Quantum harmonic oscillator II: professorm.learnworlds.com/course/quantum-harmonic-oscillator-ii- More learning material: professorm.learnworlds.com 📚 The eigenfunctions of the quantum harmonic oscillator are famously given by the product of a polynomial and a Gaussian. In this video, we derive their mathematical form and visualize a few representative states. 🐦 Follow me on Twitter: twitter.com/ProfMScience ⏮️ BACKGROUNDHarmonic oscillator: youtu.be/OdizRUe84bgLadder and number operators: youtu.be/Kb9twGd25P0Eigenvalues of the quantum harmonic oscillator: youtu.be/GkUXscdLQQ0Harmonic oscillator eigenvalues: youtu.be/GkUXscdLQQ0The parity operator: youtu.be/OsvXeTEQxygEven and odd operators: youtu.be/o8l6Yz9EHpsWave functions: youtu.be/2lr3aA4vaBsPosition and momentum: youtu.be/Yw2YrTLSq5U⏭️ WHAT NEXT? Hermite polynomials: youtu.be/p22UrUv9QdMCoherent states: youtu.be/x0wk98uMyys~Director and writer: BMProducer and designer: MC
The Hamiltonian in second quantization Professor M does Science 2024-04-30 | 📝 Problems+solutions:- Second quantization: professorm.learnworlds.com/course/second-quantization- Quantum field operators: [COMING SOON]💻 Book a 1:1 session: docs.google.com/forms/d/e/1FAIpQLScUL187erItvC7GPnNU2pelsueyVFr94nRq2A5Eq2aVRdGiIQ/viewform?pli=1📚 The Hamiltonian is the operator associated with the total energy of a quantum system. It plays a key role in quantum mechanics, as it is the operator that drives the time evolution of quantum systems. When we have a system of identical quantum particles, we work in the second quantization formulation of quantum mechanics. In this video, we write down the Hamiltonian operator of a general quantum system in second quantization. 0:00 Introduction0:43 Hamiltonian operator 3:54 Hamiltonian in second quantization9:16 Non-interacting Hamiltonian13:45 Hamiltonian in non-interacting basis15:22 Wrap-up⏮️ BACKGROUNDOccupation number representation: youtu.be/hTaqxOK8nGQFock space: youtu.be/jAw9WMkcCj0Creation and annihilation of bosons: youtu.be/BhK6u0bMqG0Creation and annihilation of fermions: youtu.be/HZ43XE89n8sOne body operators: youtu.be/8sXWslDlZU0Change of basis: youtu.be/6icXRi5lGWE⏭️ WHAT NEXT? Heisenberg picture in second quantization: [COMING SOON]Hamiltonian in terms of field operators: [COMING SOON]~Director and writer: BMProducer and designer: MC
Operators in terms of quantum field operators Professor M does Science 2024-04-17 | 📝 Problems+solutions:- Second quantization: professorm.learnworlds.com/course/second-quantization- Quantum field operators: [COMING SOON]💻 Book a 1:1 session: docs.google.com/forms/d/e/1FAIpQLScUL187erItvC7GPnNU2pelsueyVFr94nRq2A5Eq2aVRdGiIQ/viewform?pli=1📚 The basic building blocks of quantum field theory are the quantum field operators that describe the creation and annihilation of particles in space. As such, in quantum field theory we often need to write down general operators, such as the Hamiltonian operator, in terms of these quantum field operators. In this video, we discuss how to write general operators in terms of quantum field operators. 0:00 Introduction0:35 Quantum field operators3:28 One-body operators in terms of field operators10:06 Two-body operators in terms of field operators13:22 Operators interpreted as scattering of particles15:38 The case of particles with spin17:13 Example: local density operator20:43 Wrap-up⏮️ BACKGROUNDQuantum field operators: youtu.be/wAo0weNZVgQCommutators and anticommutators of field operators: youtu.be/cNDLHaVxIdASecond quantization: youtube.com/playlist?list=PL8W2boV7eVfnSqy1fs3CCNALSvnDDd-tb⏭️ WHAT NEXT? Hamiltonian in terms of field operators: [COMING SOON]Heisenberg picture for field operators: [COMING SOON]~Director and writer: BMProducer and designer: MC
Commutators and anticommutators of field operators Professor M does Science 2024-04-03 | 📝 Problems+solutions:- Second quantization: professorm.learnworlds.com/course/second-quantization- Quantum field operators: [COMING SOON]💻 Book a 1:1 session: docs.google.com/forms/d/e/1FAIpQLScUL187erItvC7GPnNU2pelsueyVFr94nRq2A5Eq2aVRdGiIQ/viewform?pli=1📚 Quantum field operators are the creation and annihilation operators associated with the position representation. As such, they have the usual properties of general creation and annihilation operators, and in this video we illustrate this by discussing the commutation relations of bosonic field operators and the anticommutation relations of fermionic field operators. 0:00 Introduction0:45 Quantum field operators2:57 Bosons: commutation relations7:40 Fermions: anticommutation relations8:46 Compact notation for commutators and anticommutators10:32 Commutators and anticommutators for particles with spin12:51 Wrap-up⏮️ BACKGROUNDQuantum field operators: youtu.be/wAo0weNZVgQSecond quantization: youtube.com/playlist?list=PL8W2boV7eVfnSqy1fs3CCNALSvnDDd-tb⏭️ WHAT NEXT? General operators in terms of field operators: youtu.be/pEIf8opLv08Hamiltonian in terms of field operators: [COMING SOON]Heisenberg picture for field operators: [COMING SOON]~Director and writer: BMProducer and designer: MC
Quantum field operators Professor M does Science 2024-03-20 | 📝 Problems+solutions:- Second quantization: professorm.learnworlds.com/course/second-quantization- Quantum field operators: [COMING SOON]💻 Book a 1:1 session: docs.google.com/forms/d/e/1FAIpQLScUL187erItvC7GPnNU2pelsueyVFr94nRq2A5Eq2aVRdGiIQ/viewform?pli=1📚 Quantum field operators are the creation and annihilation operators associated with the position representation. They describe the creation and annihilation of particles in space, and they form the basis of quantum field theory. In this video, we introduce the basics of quantum field operators.0:00 Introduction0:35 Changing basis in second quantization2:59 Position representation4:22 Quantum field operators12:10 Quantum field operators with spin15:29 Wrap-up⏮️ BACKGROUNDSecond quantization: youtube.com/playlist?list=PL8W2boV7eVfnSqy1fs3CCNALSvnDDd-tbChanging basis in second quantization: youtu.be/6icXRi5lGWEPosition representation: youtu.be/2lr3aA4vaBs⏭️ WHAT NEXT? Commutators and anticommutator of field operators: youtu.be/cNDLHaVxIdAGeneral operators in terms of field operators: youtu.be/pEIf8opLv08Hamiltonian in terms of field operators: [COMING SOON]Heisenberg picture for field operators: [COMING SOON]~Director and writer: BMProducer and designer: MC
Problems + Solutions! Professor M does Science 2024-03-14 | 📝 Problems+solutions: professorm.learnworlds.com💻 You can also book a 1:1 session: docs.google.com/forms/d/e/1FAIpQLScUL187erItvC7GPnNU2pelsueyVFr94nRq2A5Eq2aVRdGiIQ/viewform?pli=1
Homogeneous first order ordinary differential equations Professor M does Science 2024-02-28 | 💻 Book a 1:1 session: docs.google.com/forms/d/e/1FAIpQLScUL187erItvC7GPnNU2pelsueyVFr94nRq2A5Eq2aVRdGiIQ/viewform?pli=1📚 Homogeneous first order ordinary differential equations consist of a derivative equal to a homogeneous function of degree zero. In this video, we explain the general strategy to solve homogeneous differential equations, which involves a simple substitution that turns them into separable differential equations. We also solve, step by step, an example of a homogeneous equation, and we then plot the resulting solution. 0:00 Introduction0:24 Homogeneous first order differential equations2:43 Solution strategy6:45 Alternative forms for homogeneous equations9:24 Example17:15 Wrap-up⏭️ WHAT NEXT? Differential equation for tangent circles: [COMING SOON]Exact differential equations: [COMING SOON]Inexact differential equations: [COMING SOON]Linear differential equations: [COMING SOON]~Director and writer: BMProducer and designer: MCThumbnail image credits: Vecteezy.com
Differential equation for concentric circles Professor M does Science 2024-02-07 | 💻 Book a 1:1 session: docs.google.com/forms/d/e/1FAIpQLScUL187erItvC7GPnNU2pelsueyVFr94nRq2A5Eq2aVRdGiIQ/viewform?pli=1📚 In this video we cover the step by step solution of a separable differential equation. The equation we solve describes the family of all circles centered at the origin. 0:00 Intro0:31 Differential equation setup2:50 Solution to the differential equation4:49 Checking the solution7:01 Plotting the solution9:22 Wrap-up⏮️ BACKGROUNDSeparable differential equations: youtu.be/wU_jLOFHcQ4⏭️ WHAT NEXT? Homogeneous differential equations: [COMING SOON]Differential equation for tangent circles: [COMING SOON]Exact differential equations: [COMING SOON]Inexact differential equations: [COMING SOON]Linear differential equations: [COMING SOON]~Director and writer: BMProducer and designer: MCThumbnail image credits: Vecteezy.com
Separable first order ordinary differential equations Professor M does Science 2024-01-17 | 💻 Book a 1:1 session: docs.google.com/forms/d/e/1FAIpQLScUL187erItvC7GPnNU2pelsueyVFr94nRq2A5Eq2aVRdGiIQ/viewform?pli=1📚 Separable first order ordinary differential equations are the easiest differential equations of all. In this video, we explain the general strategy to solve separable differential equations, which involves a simple algebraic trick and evaluating some integrals. We also solve a very important example of a separable differential equation, that describing exponential growth, and we go over the solution step by step. 0:00 Introduction0:43 Separable first order differential equations4:57 Solution strategy7:41 Example: exponential growth14:13 Mathematical justification of solution strategy18:10 Wrap-up⏭️ WHAT NEXT? Differential equation for concentric circles: youtu.be/gPh_FUO6TGUHomogeneous differential equations: [COMING SOON]Differential equation for tangent circles: [COMING SOON]Exact differential equations: [COMING SOON]Inexact differential equations: [COMING SOON]Linear differential equations: [COMING SOON]~Director and writer: BMProducer and designer: MCThumbnail image credits: Vecteezy.com
Channel update: launching maths videos! Professor M does Science 2024-01-16 | We are going to start publishing videos on maths for science and engineering, with the first few on differential equations. You can already watch the first one here: youtu.be/wU_jLOFHcQ4
2-state quantum systems: energy eigenstates Professor M does Science 2023-10-16 | 💻 Book a 1:1 session: docs.google.com/forms/d/e/1FAIpQLScUL187erItvC7GPnNU2pelsueyVFr94nRq2A5Eq2aVRdGiIQ/viewform?pli=1📚 2-state quantum systems are key to understand phenomena ranging from the spin angular momentum of electrons and protons to the qubits of quantum computers. In this video, we calculate the energy eigenstates of a 2-state quantum system. 0:00 Introduction3:08 The "d dot sigma" operator8:21 Eigenvalues of the "d dot sigma" operator11:17 Eigenstates of the "d dot sigma" operator15:00 Energy eigenstates 20:15 Wrap-up⏮️ BACKGROUNDPauli matrices: youtu.be/2MsVD9ufguk2x2 matrix space: youtu.be/OdYNKLULvlUTwo-state quantum systems | Eigenvalues: youtu.be/kQnQpI2B72sOperators: youtu.be/pNFna7zZbgEMatrix formulation: youtu.be/wIwnb1ldYTIHermitian operators: youtu.be/XIgDUfyrLAYEigenvalues and eigenstates: youtu.be/p1zg-c1nvwQ⏭️ WHAT NEXT? Spin 1/2: [COMING SOON]Rabi oscillations: [COMING SOON]🧮 Double angle formulaeWikipedia: en.wikipedia.org/wiki/List_of_trigonometric_identities#Double-angle_formulaeWolfram MathWorld: mathworld.wolfram.com/Double-AngleFormulas.html~Director and writer: BMProducer and designer: MC
2-state quantum systems: energy eigenvalues Professor M does Science 2023-10-09 | 💻 Book a 1:1 session: docs.google.com/forms/d/e/1FAIpQLScUL187erItvC7GPnNU2pelsueyVFr94nRq2A5Eq2aVRdGiIQ/viewform?pli=1📚 2-state quantum systems are the simplest quantum systems, but they are also critically important in many areas, highlighting the spin angular momentum of spin 1/2 particles like the electron and the qubits that form the building blocks of quantum computers. In this video, we calculate the energy eigenvalues of a 2-state quantum system. 0:00 Introduction2:08 Eigenvalues of a 2-state quantum system7:59 Degenerate eigenvalues10:37 2-state quantum system in terms of Pauli matrices14:55 Re-deriving the eigenvalues of a 2-state quantum system18:05 Wrap-up⏮️ BACKGROUNDPauli matrices: youtu.be/2MsVD9ufguk2x2 matrix space: youtu.be/OdYNKLULvlUOperators: youtu.be/pNFna7zZbgEMatrix formulation: youtu.be/wIwnb1ldYTIHermitian operators: youtu.be/XIgDUfyrLAYEigenvalues and eigenstates: youtu.be/p1zg-c1nvwQ⏭️ WHAT NEXT? Two-state quantum systems | Eigenstates: youtu.be/lkVtvdJ3GbQSpin 1/2: [COMING SOON]Rabi oscillations: [COMING SOON]~Director and writer: BMProducer and designer: MC
2x2 matrices in terms of Pauli matrices Professor M does Science 2023-06-06 | 💻 Book a 1:1 session: docs.google.com/forms/d/e/1FAIpQLScUL187erItvC7GPnNU2pelsueyVFr94nRq2A5Eq2aVRdGiIQ/viewform?pli=1📚 2x2 matrices are used to represent operators acting on 2-dimensional state spaces such as that of the spin angular momentum of spin 1/2 particles like the electron. In this video, we show that we can write any 2x2 matrix in terms of the identity and Pauli matrices. This provides a convenient language for the description of 2-state quantum systems. 0:00 Introduction1:27 Pauli matrices2:09 2x2 matrices in terms of Pauli matrices10:29 Hermitian 2x2 matrices in terms of Pauli matrices14:58 Wrap-up⏮️ BACKGROUNDPauli matrices: youtu.be/2MsVD9ufgukOperators: youtu.be/pNFna7zZbgEMatrix formulation: youtu.be/wIwnb1ldYTIHermitian operators: youtu.be/XIgDUfyrLAY⏭️ WHAT NEXT? Two-state quantum systems | Eigenvalues: youtu.be/kQnQpI2B72sTwo-state quantum systems | Eigenstates: youtu.be/lkVtvdJ3GbQSpin 1/2: [COMING SOON]Rabi oscillations: [COMING SOON]~Director and writer: BMProducer and designer: MC
The Pauli matrices Professor M does Science 2023-02-28 | 💻 Book a 1:1 session: docs.google.com/forms/d/e/1FAIpQLScUL187erItvC7GPnNU2pelsueyVFr94nRq2A5Eq2aVRdGiIQ/viewform?pli=1📚 The Pauli matrices are a set of three matrices of dimension 2x2 that play a crucial role in many areas of quantum mechanics. Specifically, they provide a mathematical platform to explore any quantum system described by a two-dimensional state space. Examples include spin 1/2 particles like the electron, and qubits as the fundamental unit of quantum computing. In this video, we explore some the most important mathematical properties of the Pauli matrices.0:00 Introduction0:54 Pauli matrices1:54 Hermitian3:10 Involutory4:34 Unitary5:10 Determinant5:58 Trace6:31 Eigenvalues and eigenvectors9:27 Commutation relations13:22 Anticommuntation relations15:30 Wrap-up⏮️ BACKGROUNDHermitian operators: youtu.be/XIgDUfyrLAYUnitary operators: youtu.be/baIT6HaaYuQEigenvalues and eigenstates: youtu.be/p1zg-c1nvwQCommutator algebra: youtu.be/57xgSIV9PY0Angular momentum: youtu.be/Bo5qoaLsBOE⏭️ WHAT NEXT? 2x2 matrix space: youtu.be/OdYNKLULvlUTwo-state quantum systems | Eigenvalues: youtu.be/kQnQpI2B72sTwo-state quantum systems | Eigenstates: youtu.be/lkVtvdJ3GbQSpin 1/2: [COMING SOON]~Director and writer: BMProducer and designer: MC
The hydrogen spectral series Professor M does Science 2023-02-14 | 💻 Book a 1:1 session: docs.google.com/forms/d/e/1FAIpQLScUL187erItvC7GPnNU2pelsueyVFr94nRq2A5Eq2aVRdGiIQ/viewform?pli=1📚 The hydrogen atom is the simplest and most abundant of all elements in the Universe. In this video, we discuss the emission spectrum of hydrogen, which is mathematically given by the Rydberg formula. The emission spectrum of hydrogen was discovered before the development of atomic theory, and it played a key role in the discovery of quantum mechanics. We discuss the origin of the spectral lines, and classify them into groups such as the Lyman or Balmer series. We also discuss the importance that the hydrogen emission spectrum has for our understanding of large-scale structures in the Universe, such as star forming nebulas, and take a look at a beautiful image of the Orion nebula by the Hubble Space Telescope.0:00 Introduction2:22 Energy levels of hydrogen3:37 Photon emission5:23 Rydberg formula9:59 Emission spectrum of hydrogen10:50 Lyman series15:17 Balmer series19:25 Orion nebula19:49 Higher series23:28 Fine and hyperfine structure24:36 Wrap-up⏮️ BACKGROUNDHydrogen atom basics: youtu.be/lndTguV0u1gHydrogen atom | Maths: youtu.be/8NJm4Jkp0jYHydrogen atom | Eigenvalues and eigenstates: youtu.be/T78NkndaMb8Hydrogen atom | Ground state: youtu.be/r_C82Y6XNWYHydrogen atom | Energy spectrum: youtu.be/GV2CDZD2xNk⏭️ WHAT NEXT? Hydrogen fine structure: [COMING SOON]Hydrogen hyperfine structure: [COMING SOON]~Director and writer: BMProducer and designer: MC
The hydrogen atom energy spectrum Professor M does Science 2023-01-31 | 💻 Book a 1:1 session: docs.google.com/forms/d/e/1FAIpQLScUL187erItvC7GPnNU2pelsueyVFr94nRq2A5Eq2aVRdGiIQ/viewform?pli=1📚 The hydrogen atom is the simplest and most abundant of all elements in the Universe. In this video, we explore the energy eigenvalues of the hydrogen atom. We discuss the principal, azimuthal, and magnetic quantum numbers, as well as the associated eigenvalue degeneracies. We also introduce the concepts of atomic shell and atomic subshell, and describe the widely used spectroscopic notation. 0:00 Intro4:19 Energy eigenvalues degeneracies11:22 Hydrogen atom quantum numbers12:08 Atomic shells and atomic subshells18:46 Spectroscopic notation22:02 Wrap-up🐦 Follow me on Twitter: twitter.com/ProfMScience ⏮️ BACKGROUNDHydrogen atom basics: youtu.be/lndTguV0u1gHydrogen atom | Maths: youtu.be/8NJm4Jkp0jYHydrogen atom | Eigenvalues and eigenstates: youtu.be/T78NkndaMb8Hydrogen atom | Ground state: youtu.be/r_C82Y6XNWY⏭️ WHAT NEXT? Hydrogen atom | Spectral series: youtu.be/XawP16g7dJ8Hydrogen fine structure: [COMING SOON]Hydrogen hyperfine structure: [COMING SOON]Spin 1/2: [COMING SOON]~Director and writer: BMProducer and designer: MC
The hydrogen atom ground state Professor M does Science 2023-01-17 | 💻 Book a 1:1 session: docs.google.com/forms/d/e/1FAIpQLScUL187erItvC7GPnNU2pelsueyVFr94nRq2A5Eq2aVRdGiIQ/viewform?pli=1📚 The hydrogen atom is the simplest and most abundant of all elements. In this video, we study a range of properties of the ground state of the hydrogen atom. We explore the energy necessary to dissociate the proton and electron in hydrogen, called the ionization energy; the typical size of a hydrogen atom, called the Bohr radius; and the relative contribution of kinetic and potential energies in hydrogen, which we relate to the virial theorem.0:00 Intro1:12 Hydrogen atom basics4:04 Ground state energy: ionization energy8:50 Ground state wave function: Bohr radius22:55 Virial theorem29:20 Wrap-up⏮️ BACKGROUNDHydrogen atom basics: youtu.be/lndTguV0u1gHydrogen atom | Maths: youtu.be/8NJm4Jkp0jYHydrogen atom | Eigenvalues and eigenstates: youtu.be/T78NkndaMb8Central potentials: youtu.be/Y73ctxnP9gQSpherical harmonics: youtu.be/5PMqf3Hj-AwVirial theorem: youtu.be/QzUiVboHcKI⏭️ WHAT NEXT? Hydrogen atom | Energy spectrum: youtu.be/GV2CDZD2xNkHydrogen atom | Spectral series: youtu.be/XawP16g7dJ8~Director and writer: BMProducer and designer: MC
Hydrogen atom: eigenvalues and eigenfuctions Professor M does Science 2022-10-10 | 💻 Book a 1:1 session: docs.google.com/forms/d/e/1FAIpQLScUL187erItvC7GPnNU2pelsueyVFr94nRq2A5Eq2aVRdGiIQ/viewform?pli=1📚 The hydrogen atom is one of very few systems for which we can write down analytical expressions for its eigenvalues and eigenfunctions. In this video, we explicitly build the first few eigenvalues and eigenfunctions of the hydrogen atom. The eigenfunctions are written in terms of spherical harmonics because they are also eigenfunctions of orbital angular momentum. 0:00 Intro0:35 Hydrogen atom as a central potential4:53 Recap of the mathematical solution of the eigenvalue equation12:49 Ground state20:31 First excited state34:16 Second excited state37:34 Wrap-up⏮️ BACKGROUNDHydrogen atom basics: youtu.be/lndTguV0u1gHydrogen atom | Maths: youtu.be/8NJm4Jkp0jYCentral potentials: youtu.be/Y73ctxnP9gQRadial equation of central potentials: youtu.be/MsZP7yxpeFgOrbital angular momentum: youtu.be/EyGJ3JE9CgESpherical harmonics: youtu.be/5PMqf3Hj-Aw⏭️ WHAT NEXT? Hydrogen atom | Ground state: youtu.be/r_C82Y6XNWYHydrogen atom | Energy spectrum: youtu.be/GV2CDZD2xNkHydrogen atom | Spectral series: youtu.be/XawP16g7dJ8~Director and writer: BMProducer and designer: MC
3D isotropic quantum harmonic oscillator: eigenvalues and eigenstates Professor M does Science 2022-09-14 | 📝 Problems+solutions:- Quantum harmonic oscillator I: professorm.learnworlds.com/course/quantum-harmonic-oscillator-i- Quantum harmonic oscillator II: professorm.learnworlds.com/course/quantum-harmonic-oscillator-ii💻 Book a 1:1 session: docs.google.com/forms/d/e/1FAIpQLScUL187erItvC7GPnNU2pelsueyVFr94nRq2A5Eq2aVRdGiIQ/viewform?pli=1📚 The 3D isotropic quantum harmonic oscillator can be described using a Hamiltonian of a central potential. In this video, we explicitly build the first few eigenvalues and eigenstates of this system. The eigenstates are written in terms of spherical harmonics because they are also eigenstates of orbital angular momentum. 0:00 Intro1:34 3D isotropic quantum harmonic oscillator as a central potential5:35 Recap of the mathematical solution of the eigenvalue equation11:46 Ground state19:58 First excited state27:22 Second excited state30:21 Wrap-up⏮️ BACKGROUND1D quantum harmonic oscillator: youtu.be/OdizRUe84bg3D quantum harmonic oscillator in Cartesian coordinates: youtu.be/cZTnmlJfHdg3D quantum harmonic oscillator in spherical coordinates: youtu.be/jOQThICjLlwCentral potentials: youtu.be/Y73ctxnP9gQRadial equation of central potentials: youtu.be/MsZP7yxpeFgOrbital angular momentum: youtu.be/EyGJ3JE9CgESpherical harmonics: youtu.be/5PMqf3Hj-Aw⏭️ WHAT NEXT? Hydrogen atom: youtube.com/playlist?list=PL8W2boV7eVfnJbLf-p3-_7d51tskA0-Sa~Director and writer: BMProducer and designer: MC
The quantum virial theorem Professor M does Science 2022-06-30 | 💻 Book a 1:1 session: docs.google.com/forms/d/e/1FAIpQLScUL187erItvC7GPnNU2pelsueyVFr94nRq2A5Eq2aVRdGiIQ/viewform?pli=1📚 Evaluating expectation values in quantum mechanics typically requires lengthy maths. For example, if we work in the position representation in terms of wave functions, evaluating expectation values typically requires evaluating lengthy integrals. In this video we derive the virial theorem, which provides a very simple relation between the expectation value of the kinetic and potential energies of any quantum system. Such a relation allows us to bypass the lengthy evaluation of expectation values and learn about our system in a simple and transparent manner. We also discuss the special case of potentials described by homogeneous functions, in which the virial theorem takes an even simpler form. Examples we discuss include the quantum harmonic oscillator, the hydrogen atom, and also molecular systems and materials made of many atoms.0:00 Intro0:37 Hypervirial theorem2:57 Virial theorem for 1 particle in 1D9:13 Virial theorem for potentials described by homogeneous functions15:25 Virial theorem for multiple particles in 3D22:29 Wrap-up⏮️ BACKGROUNDEigenvalues and eigenstates: youtu.be/p1zg-c1nvwQHermitian operators: youtu.be/XIgDUfyrLAYExpectation values: youtu.be/rEm-Ejg5xekCommutator algebra: youtu.be/57xgSIV9PY0Functions of operators: youtu.be/nqLEbrzVsk4Quantum harmonic oscillator playlist: youtube.com/playlist?list=PL8W2boV7eVfmdWs3CsaGfoITHURXvHOGmHydrogen atom playlist: youtube.com/playlist?list=PL8W2boV7eVfnJbLf-p3-_7d51tskA0-Sa⏭️ WHAT NEXT? Hydrogen atom ground state: [COMING SOON]~Director and writer: BMProducer and designer: MC
Schrödinger vs. Heisenberg pictures of quantum mechanics Professor M does Science 2022-04-20 | 💻 Book a 1:1 session: docs.google.com/forms/d/e/1FAIpQLScUL187erItvC7GPnNU2pelsueyVFr94nRq2A5Eq2aVRdGiIQ/viewform?pli=1📚 Time evolution in quantum mechanics can be described in various alternative but equivalent ways, each of which referred to as a "picture". We can relate the different pictures by unitary transformations driven by time evolution operators. In this video, we compare two of the most common pictures, the Schrödinger picture and the Heisenberg picture. In the Schrödinger picture, time evolution is encoded by states, while in the Heisenberg picture states are time independent and the time evolution is encoded by the observables. 0:00 Intro0:52 Schrodinger picture4:52 Time evolution operator6:54 Heisenberg picture - simple motivation11:08 Unitary transformations13:32 Pictures of time evolution in quantum mechanics16:28 Heisenberg picture22:52 Dynamics27:31 Conservative systems30:14 Why use the Heisenberg picture?32:42 Wrap-up⏮️ BACKGROUNDSchrödinger equation: youtu.be/CKpx9hkQ3HMTime evolution operator: youtu.be/zqmU4dW03aMUnitary operators: youtu.be/baIT6HaaYuQExpectation values: youtu.be/rEm-Ejg5xekMeasurements: youtu.be/u1R3kRWh1ekThe Ehrenfest theorem: youtu.be/55RQCCtdlko⏭️ WHAT NEXT? Interaction picture: [COMING SOON]~Director and writer: BMProducer and designer: MC
Hydrogen atom: power series solution Professor M does Science 2022-03-17 | 💻 Book a 1:1 session: docs.google.com/forms/d/e/1FAIpQLScUL187erItvC7GPnNU2pelsueyVFr94nRq2A5Eq2aVRdGiIQ/viewform?pli=1📚 The hydrogen atom can be described using a Hamiltonian of a central potential. In this video, we go over the mathematical solution of the eigenvalue equation of the Hamiltonian. This requires the solution of a complex differential equation, and involves interesting concepts such as limiting behaviors and power series expansions. 0:00 Intro1:00 Hydrogen as a central potential7:27 Radial equation 9:46 Bound vs unbound states12:52 Simplifying notation16:38 Radial equation solution33:07 Quantized energy eigenvalues39:56 Energy eigenfunctions45:47 Wrap-up⏮️ BACKGROUNDHydrogen atom basics: youtu.be/lndTguV0u1g3D quantum harmonic oscillator: youtu.be/jOQThICjLlwCentral potentials: youtu.be/Y73ctxnP9gQRadial equation of central potentials: youtu.be/MsZP7yxpeFgOrbital angular momentum: youtu.be/EyGJ3JE9CgESpherical harmonics: youtu.be/5PMqf3Hj-AwBound vs unbound states: [COMING SOON]⏭️ WHAT NEXT? Hydrogen atom | Eigenvalues and eigenfunctions: youtu.be/T78NkndaMb8Hydrogen atom | Ground state: youtu.be/r_C82Y6XNWYHydrogen atom | Energy spectrum: youtu.be/GV2CDZD2xNkHydrogen atom | Spectral series: youtu.be/XawP16g7dJ8~Director and writer: BMProducer and designer: MC
3D isotropic quantum harmonic oscillator: power series solution Professor M does Science 2022-02-24 | 📝 Problems+solutions:- Quantum harmonic oscillator I: professorm.learnworlds.com/course/quantum-harmonic-oscillator-i- Quantum harmonic oscillator II: professorm.learnworlds.com/course/quantum-harmonic-oscillator-ii💻 Book a 1:1 session: docs.google.com/forms/d/e/1FAIpQLScUL187erItvC7GPnNU2pelsueyVFr94nRq2A5Eq2aVRdGiIQ/viewform?pli=1📚 The 3D isotropic quantum harmonic oscillator can be described using a Hamiltonian of a central potential. In this video, we go over the mathematical solution of the eigenvalue equation of the Hamiltonian. This requires the solution of a complex differential equation, and involves interesting concepts such as limiting behaviors and power series expansions. 0:00 Intro1:47 3D isotropic quantum harmonic oscillator as a central potential8:18 Radial equation solution36:22 Wrap-up⏮️ BACKGROUND1D quantum harmonic oscillator: youtu.be/OdizRUe84bg3D quantum harmonic oscillator in Cartesian coordinates: youtu.be/cZTnmlJfHdgCentral potentials: youtu.be/Y73ctxnP9gQRadial equation of central potentials: youtu.be/MsZP7yxpeFgOrbital angular momentum: youtu.be/EyGJ3JE9CgESpherical harmonics: youtu.be/5PMqf3Hj-Aw⏭️ WHAT NEXT? 3D isotropic quantum harmonic oscillator | eigenvalues and eigenstates: youtu.be/3Ipcr9tMlxU~Director and writer: BMProducer and designer: MC
The hydrogen atom Professor M does Science 2022-02-02 | 💻 Book a 1:1 session: docs.google.com/forms/d/e/1FAIpQLScUL187erItvC7GPnNU2pelsueyVFr94nRq2A5Eq2aVRdGiIQ/viewform?pli=1📚 The hydrogen atom is an iconic system in both physics and chemistry. Hydrogen, formed of a single proton and a single electron, is the simplest and most abundant of all elements. Furthermore, the eigenvalue equation of the Hamiltonian of the hydrogen atom is one of a few select systems that we can fully solve analytically in quantum mechanics. As such, it informs our understanding of more complex systems such as other heavier elements. In this video, we introduce the hydrogen atom and write down the Hamiltonian describing the relative motion of the proton and electron, the starting point for the quantum mechanical study of hydrogen.0:00 Intro1:07 A proton and an electron6:34 Hamiltonian of the hydrogen atom8:28 Relative motion of proton and electron17:54 Wrap-up🐦 Follow me on Twitter: twitter.com/ProfMScience ⏮️ BACKGROUNDTwo interacting quantum particles: youtu.be/kOupIEhYdY8⏭️ WHAT NEXT? Hydrogen atom | Maths: youtu.be/8NJm4Jkp0jYHydrogen atom | Eigenvalues and eigenstates: youtu.be/T78NkndaMb8Hydrogen atom | Ground state: youtu.be/r_C82Y6XNWYHydrogen atom | Energy spectrum: youtu.be/GV2CDZD2xNkHydrogen atom | Spectral series: youtu.be/XawP16g7dJ8~Director and writer: BMProducer and designer: MC
Two interacting quantum particles: relative motion Professor M does Science 2021-11-24 | 💻 Book a 1:1 session: docs.google.com/forms/d/e/1FAIpQLScUL187erItvC7GPnNU2pelsueyVFr94nRq2A5Eq2aVRdGiIQ/viewform?pli=1📚 Systems of two interacting quantum particles play a major role in physics, perhaps the most important one being the hydrogen atom, made up of a proton and an electron interacting via the Coulomb potential. In this video, we learn that we can solve the mathematical problem of two quantum particles interacting via a potential that depends on their relative position by solving two simpler problems of two fictitious non-interacting particles. As such, it sets the foundation to study the hydrogen atom in quantum mechanics.0:00 Intro1:13 Two-particle systems4:24 Change of variables13:04 Hamiltonian of two interacting particles20:58 Separable Hamiltonians24:35 Center of mass Hamiltonian26:02 Relative Hamiltonian28:55 Wrap-up⏮️ BACKGROUNDTensor products: youtu.be/kz3206S2B6QEigenvalues and eigenstates: youtu.be/p1zg-c1nvwQPosition representation: youtu.be/Yw2YrTLSq5UFree particle: [COMING SOON]Central potentials: youtu.be/Y73ctxnP9gQ⏭️ WHAT NEXT? Hydrogen atom: youtu.be/lndTguV0u1g~Director and writer: BMProducer and designer: MC
The spherical harmonics Professor M does Science 2021-10-28 | 💻 Book a 1:1 session: docs.google.com/forms/d/e/1FAIpQLScUL187erItvC7GPnNU2pelsueyVFr94nRq2A5Eq2aVRdGiIQ/viewform?pli=1📚 The spherical harmonics are the eigenstates of orbital angular momentum in quantum mechanics. As such, they feature in the wave functions of many quantum problems, including the 3D quantum harmonic oscillator and the hydrogen atom. In this video, we write down the mathematical form of the first few spherical harmonics, and we also visualize them with animated 3D graphics. 0:00 Intro0:45 Definition of the spherical harmonics3:09 How do we visualize spherical harmonics?6:37 l=0 spherical harmonic8:14 l=1 spherical harmonics17:16 l=2 spherical harmonics18:53 l=3 spherical harmonics19:50 An alternative view of the spherical harmonics23:01 Wrap-up🐦 Follow me on Twitter: twitter.com/ProfMScience ⏮️ BACKGROUNDOrbital angular momentum: youtu.be/EyGJ3JE9CgEOrbital angular momentum eigenvalues: youtu.be/5h5mbL5rco0Orbital angular momentum eigenfunctions: youtu.be/Gk2XNmIHVwo⏭️ WHAT NEXT? The 3D quantum harmonic oscillator eigenfunctions: [COMING SOON]The hydrogen atom eigenfunctions: [COMING SOON]👩💻 GITHUBPlot the spherical harmonics yourself: nbviewer.org/github/profmscience/spherical-harmonics/blob/main/spherical-harmonics-for-github.ipynb~Director and writer: BMProducer and designer: MC
Degeneracies of the 3D quantum harmonic oscillator Professor M does Science 2021-10-06 | 📝 Problems+solutions:- Quantum harmonic oscillator I: professorm.learnworlds.com/course/quantum-harmonic-oscillator-i- Quantum harmonic oscillator II: professorm.learnworlds.com/course/quantum-harmonic-oscillator-ii💻 Book a 1:1 session: docs.google.com/forms/d/e/1FAIpQLScUL187erItvC7GPnNU2pelsueyVFr94nRq2A5Eq2aVRdGiIQ/viewform?pli=1📚 The 3D quantum harmonic oscillator can be described as a simple combination of three 1D quantum harmonic oscillators along the three Cartesian directions. Despite this apparent simplicity, when the 3D oscillator is isotropic, the energy spectrum becomes highly degenerate. In this video, we explore these degeneracies by considering the different states that share the same energy, visualizing the associated eigenfunctions. 0:00 Intro2:41 3D isotropic quantum harmonic oscillator 9:31 Ground state11:56 First excited state15:50 Second excited state19:58 General case22:53 Wrap-up⏮️ BACKGROUND1D quantum harmonic oscillator: youtu.be/OdizRUe84bgEigenvalues of the quantum harmonic oscillator: youtu.be/GkUXscdLQQ0Eigenstates of the quantum harmonic oscillator: youtu.be/0o-LoJRtxDc3D quantum harmonic oscillator: youtu.be/cZTnmlJfHdg⏭️ WHAT NEXT? The 3D quantum harmonic oscillator as a central potential: [COMING SOON]~Director and writer: BMProducer and designer: MC
The 3D quantum harmonic oscillator Professor M does Science 2021-09-22 | 📝 Problems+solutions:- Quantum harmonic oscillator I: professorm.learnworlds.com/course/quantum-harmonic-oscillator-i- Quantum harmonic oscillator II: professorm.learnworlds.com/course/quantum-harmonic-oscillator-ii💻 Book a 1:1 session: docs.google.com/forms/d/e/1FAIpQLScUL187erItvC7GPnNU2pelsueyVFr94nRq2A5Eq2aVRdGiIQ/viewform?pli=1📚 The harmonic oscillator is one of the most important systems in quantum mechanics, used to describe a plethora of phenomena, from atomic vibrations (phonons) to light. In this video, we use the properties of tensor product state spaces to solve the 3-dimensional quantum harmonic oscillator from our knowledge of the solutions of the 1-dimensional counterpart.0:00 Intro1:07 3D quantum harmonic oscillator Hamiltonian6:39 Solving the eigenvalue equation9:02 Eigenvalues9:58 Eigenstates15:43 Eigenfunctions19:22 Wrap-up⏮️ BACKGROUND1D quantum harmonic oscillator: youtu.be/OdizRUe84bgLadder operators: youtu.be/Kb9twGd25P0Eigenvalues of the quantum harmonic oscillator: youtu.be/GkUXscdLQQ0Eigenstates of the quantum harmonic oscillator: youtu.be/0o-LoJRtxDcHermite polynomials: youtu.be/p22UrUv9QdMCoherent quantum states: youtu.be/x0wk98uMyys Tensor product state spaces: youtu.be/kz3206S2B6QEigenvalues and eigenstates in tensor product spaces: youtu.be/T3ynwXrE0Xw⏭️ WHAT NEXT? Degeneracies of the isotropic quantum harmonic oscillator: youtu.be/K5YsxsHXGHcThe 3D quantum harmonic oscillator as a central potential: [COMING SOON]~Director and writer: BMProducer and designer: MC
Eigenvalues and eigenstates in tensor product state spaces Professor M does Science 2021-09-08 | 💻 Book a 1:1 session: docs.google.com/forms/d/e/1FAIpQLScUL187erItvC7GPnNU2pelsueyVFr94nRq2A5Eq2aVRdGiIQ/viewform?pli=1📚 Tensor product state spaces allow us to mathematically describe quantum systems of multiple degrees of freedom (e.g. particles moving in 3D, orbital and spin degrees of freedom, or multi-particle systems). In this video, we learn how we can build the eigenvalues and eigenstates of operators acting in tensor product state spaces without having to solve the full problem. Instead, these eigenvalues and eigenstates can be constructed by combining the eigenvalues and eigenstates associated with the individual state spaces making up the tensor product space.⏮️ BACKGROUNDEigenvalues and eigenstates: youtu.be/p1zg-c1nvwQTensor products: youtu.be/kz3206S2B6Q⏭️ WHAT NEXT?3D quantum harmonic oscillator: youtu.be/cZTnmlJfHdgTwo interacting quantum particles: [COMING SOON]~ Director and writer: BMProducer and designer: MC
Coherent state wave function || Concepts Professor M does Science 2021-08-25 | 📝 Problems+solutions:- Quantum harmonic oscillator I: professorm.learnworlds.com/course/quantum-harmonic-oscillator-i- Quantum harmonic oscillator II: professorm.learnworlds.com/course/quantum-harmonic-oscillator-ii💻 Book a 1:1 session: docs.google.com/forms/d/e/1FAIpQLScUL187erItvC7GPnNU2pelsueyVFr94nRq2A5Eq2aVRdGiIQ/viewform?pli=1📚 Coherent states of the quantum harmonic oscillator are those states that most closely resemble the classical oscillator. In this video we explore this from the perspective of the wave function of coherent states. We find that it has a Gaussian shape, which is consistent with the fact that coherent states are minimum uncertainty states. We also explore the time evolution of the coherent state wave function, revealing the back-and-forth motion that we traditionally associated with harmonic oscillators. 0:00 Intro4:16 Coherent state wave function8:06 Minimum uncertainty state9:40 Displacement operator11:02 Time evolution14:31 Wrap-up⏮️ BACKGROUNDQuantum harmonic oscillator: youtu.be/OdizRUe84bgLadder operators: youtu.be/Kb9twGd25P0Quantum harmonic oscillator eigenstates: youtu.be/0o-LoJRtxDcCoherent states: youtu.be/x0wk98uMyysQuasi-classical states: youtu.be/0ef1rLO6DTUDisplacement operator: youtu.be/ypRTLIo-IIcMinimum uncertainty states: youtu.be/cX9JiQ8nEL4⏭️ WHAT NEXT? Coherent state wave function || Maths: youtu.be/0p9pH85SLIU~Director and writer: BMProducer and designer: MC
Coherent state wave function || Maths Professor M does Science 2021-08-11 | 📝 Problems+solutions:- Quantum harmonic oscillator I: professorm.learnworlds.com/course/quantum-harmonic-oscillator-i- Quantum harmonic oscillator II: professorm.learnworlds.com/course/quantum-harmonic-oscillator-ii💻 Book a 1:1 session: docs.google.com/forms/d/e/1FAIpQLScUL187erItvC7GPnNU2pelsueyVFr94nRq2A5Eq2aVRdGiIQ/viewform?pli=1📚 Coherent states of the quantum harmonic oscillator are those states that most closely resemble the classical oscillator. In this video we derive the mathematical form that coherent states take when we write them as wave functions. Check out the companion video for the conceptual interpretation of the wave function we derive in this video.0:00 Intro3:16 Wave functions4:44 Mathematical derivation14:39 Coherent state wave function16:03 Wrap-up⏮️ BACKGROUNDQuantum harmonic oscillator: youtu.be/OdizRUe84bgLadder operators: youtu.be/Kb9twGd25P0Quantum harmonic oscillator eigenstates: youtu.be/0o-LoJRtxDcCoherent states: youtu.be/x0wk98uMyysQuasi-classical states: youtu.be/0ef1rLO6DTUDisplacement operator: youtu.be/ypRTLIo-IIcFunctions of operators: youtu.be/nqLEbrzVsk4Wave functions: youtu.be/2lr3aA4vaBsTranslation operator: youtu.be/978mMgGYs1M⏭️ WHAT NEXT? Coherent state wave function || Concepts: youtu.be/-MaF_TzD4Q8~Director and writer: BMProducer and designer: MC
The displacement operator in quantum mechanics Professor M does Science 2021-07-28 | 📝 Problems+solutions:- Quantum harmonic oscillator I: professorm.learnworlds.com/course/quantum-harmonic-oscillator-i- Quantum harmonic oscillator II: professorm.learnworlds.com/course/quantum-harmonic-oscillator-ii💻 Book a 1:1 session: docs.google.com/forms/d/e/1FAIpQLScUL187erItvC7GPnNU2pelsueyVFr94nRq2A5Eq2aVRdGiIQ/viewform?pli=1📚 In this video we learn about the mathematical properties of the displacement operator in the context of the quantum harmonic oscillator. We also learn that the application of the displacement operator on the ground state of the harmonic oscillator generates a coherent state.0:00 Intro2:28 Definition5:54 Mathematical properties15:36 Coherent states19:45 Wrap-up⏮️ BACKGROUNDQuantum harmonic oscillator: youtu.be/OdizRUe84bgLadder operators: youtu.be/Kb9twGd25P0Coherent states: youtu.be/x0wk98uMyys⏭️ WHAT NEXT? Coherent state wave function || Maths: youtu.be/0p9pH85SLIUCoherent state wave function || Concepts: youtu.be/-MaF_TzD4Q8~Director and writer: BMProducer and designer: MC
Minimum uncertainty states in quantum mechanics Professor M does Science 2021-07-14 | 💻 Book a 1:1 session: docs.google.com/forms/d/e/1FAIpQLScUL187erItvC7GPnNU2pelsueyVFr94nRq2A5Eq2aVRdGiIQ/viewform?pli=1📚 The Heisenberg uncertainty principle between position and momentum is one of the most famous results in quantum mechanics. It states that the product of the uncertainty in position and the uncertainty in momentum has a lower bound. In this video, we ask under which conditions the product of uncertainties takes its minimum possible value. We find that the resulting minimum uncertainty states must be Gaussian, and examples include the ground state of the quantum harmonic oscillator and coherent states.0:00 Intro3:09 Uncertainty principle between position and momentum13:03 Minimum uncertainty states24:30 Wrap-up⏮️ BACKGROUNDExpectation values: youtu.be/rEm-Ejg5xekThe Heisenberg uncertainty principle || Concepts: youtu.be/pfjQtyLBBHwThe Heisenberg uncertainty principle || Proof: youtu.be/fsC5Mhd7YUcPosition representation || Wave functions: youtu.be/2lr3aA4vaBsPosition representation || Operators: youtu.be/Yw2YrTLSq5U⏭️ WHAT NEXT?Quantum harmonic oscillator wave functions: youtu.be/0o-LoJRtxDcCoherent states wave functions: youtu.be/-MaF_TzD4Q8~Director and writer: BMProducer and designer: MC
The time-energy uncertainty principle Professor M does Science 2021-06-30 | 💻 Book a 1:1 session: docs.google.com/forms/d/e/1FAIpQLScUL187erItvC7GPnNU2pelsueyVFr94nRq2A5Eq2aVRdGiIQ/viewform?pli=1📚 The Heisenberg uncertainty principle is one of the most famous results in quantum mechanics. The most quoted form of this principle involves the uncertainty relation between position and momentum. Another rather famous form of the uncertainty principle is that between time and energy. However, the time-energy uncertainty principle is fundamentally different from any other uncertainty principle. This is because time is not an observable in quantum mechanics. In this video, we'll explore the real meaning of the time-energy uncertainty principle.0:00 Intro1:03 General uncertainty principles7:18 Time-energy uncertainty principle15:52 Examples17:51 Wrap-up⏮️ BACKGROUNDSchrödinger equation: youtu.be/CKpx9hkQ3HMExpectation values and root mean square deviations: youtu.be/rEm-Ejg5xekThe Heisenberg uncertainty principle || Concepts: youtu.be/pfjQtyLBBHwThe Heisenberg uncertainty principle || Proof: youtu.be/fsC5Mhd7YUcEigenvalues and eigenstates: youtu.be/p1zg-c1nvwQEhrenfest's theorem: youtu.be/55RQCCtdlko~Director and writer: BMProducer and designer: MC
Quasi-classical states: does the quantum harmonic oscillator ACTUALLY oscillate?? Professor M does Science 2021-06-23 | 📝 Problems+solutions:- Quantum harmonic oscillator I: professorm.learnworlds.com/course/quantum-harmonic-oscillator-i- Quantum harmonic oscillator II: professorm.learnworlds.com/course/quantum-harmonic-oscillator-ii💻 Book a 1:1 session: docs.google.com/forms/d/e/1FAIpQLScUL187erItvC7GPnNU2pelsueyVFr94nRq2A5Eq2aVRdGiIQ/viewform?pli=1📚 When we think of a harmonic oscillator what typically comes to mind is the back-and-forth motion of, for example, a mass attached to a spring. For the quantum harmonic oscillator, we mostly discuss the properties of energy eigenstates, but these energy eigenstates do not oscillate! In this video, we ask the question of whether there are any states of the quantum harmonic oscillator that do oscillate. The answer is yes: the coherent states are those that most closely resemble the classical motion, and for this reason they are also called "quasi-classical states".0:00 Intro1:06 Classical harmonic oscillator2:50 Quantum harmonic oscillator5:40 Classical oscillator vs energy eigenstates7:39 Coherent states18:53 Classical oscillator vs coherent states22:18 Wrap-up⏮️ BACKGROUNDQuantum harmonic oscillator: youtu.be/OdizRUe84bgLadder operators: youtu.be/Kb9twGd25P0Quantum harmonic oscillator eigenvalues: youtu.be/GkUXscdLQQ0Ehrenfest's theorem: youtu.be/55RQCCtdlkoCoherent states: youtu.be/x0wk98uMyys⏭️ WHAT NEXT? Displacement operator: youtu.be/ypRTLIo-IIcCoherent state wave function || Maths: youtu.be/0p9pH85SLIUCoherent state wave function || Concepts: youtu.be/-MaF_TzD4Q8~Director and writer: BMProducer and designer: MC
Coherent states in quantum mechanics Professor M does Science 2021-06-16 | 📝 Problems+solutions:- Quantum harmonic oscillator I: professorm.learnworlds.com/course/quantum-harmonic-oscillator-i- Quantum harmonic oscillator II: professorm.learnworlds.com/course/quantum-harmonic-oscillator-ii💻 Book a 1:1 session: docs.google.com/forms/d/e/1FAIpQLScUL187erItvC7GPnNU2pelsueyVFr94nRq2A5Eq2aVRdGiIQ/viewform?pli=1📚 The coherent states of the quantum harmonic oscillator are defined as the eigenstates of the lowering operator. This seemingly simple definition leads to a plethora of properties that make coherent all-important. For example, they are the states that most closely resemble the motion of a classical harmonic oscillator, and in this context they are called quasi-classical states. As another example, they play a key role in the study of quantum optics. In this video, we define coherent states and investigate some of their most basic properties.0:00 Intro2:41 Definition of coherent states4:16 Coherent states in the energy basis12:04 Time evolution of coherent states16:19 The raising operator has no eigenstates21:04 Wrap-up🐦 Follow me on Twitter: twitter.com/ProfMScience ⏮️ BACKGROUNDQuantum harmonic oscillator: youtu.be/OdizRUe84bgLadder operators: youtu.be/Kb9twGd25P0Quantum harmonic oscillator eigenvalues: youtu.be/GkUXscdLQQ0Eigenvalues and eigenstates: youtu.be/p1zg-c1nvwQRepresentations: youtu.be/rp2k2oR5ZQ8Schrödinger equation: youtu.be/CKpx9hkQ3HM⏭️ WHAT NEXT? Quasi-classical states: youtu.be/0ef1rLO6DTUDisplacement operator: youtu.be/ypRTLIo-IIcCoherent state wave function || Maths: youtu.be/0p9pH85SLIUCoherent state wave function || Concepts: youtu.be/-MaF_TzD4Q8~Director and writer: BMProducer and designer: MC
Constants of motion in quantum mechanics Professor M does Science 2021-06-09 | 💻 Book a 1:1 session: docs.google.com/forms/d/e/1FAIpQLScUL187erItvC7GPnNU2pelsueyVFr94nRq2A5Eq2aVRdGiIQ/viewform?pli=1📚 In classical mechanics, a constant of motion is a physical quantity that does not change in time. In this video we explore constants of motion in quantum mechanics. We find that for an observable that is a constant of motion, the following quantities are time independent: the expectation value, the eigenstates, and the probabilities of a measurement outcome. 0:00 Intro0:43 Definition2:00 Time-independent expectation values3:14 Common set of eigenstates with H5:25 Time-independent outcome probabilities8:21 Wrap-up⏮️ BACKGROUNDSchrödinger equation: youtu.be/CKpx9hkQ3HMEhrenfest theorem: youtu.be/55RQCCtdlkoCompatible observables: youtu.be/IhJvX4H7xkAMeasurements | Concepts: youtu.be/u1R3kRWh1ek⏭️ WHAT NEXT? Central potentials: youtu.be/Y73ctxnP9gQ~Director and writer: BMProducer and designer: MC
The Ehrenfest theorem Professor M does Science 2021-06-02 | What is the relation between quantum and classical mechanics? 📚 In this video we explore the time dependence of expectation values of observables in quantum mechanics. When these ideas are applied to the position and momentum of a quantum particle, we end up with Ehrenfest's theorem. This theorem shows that the equations obeyed by the expectation values of position and momentum operators are very similar to the equations of motion of a classical particle. As such, Ehrenfest's theorem allows us to make a connection between the quantum microscopic world and the classical macroscopic world. 🐦 Follow me on Twitter: twitter.com/ProfMScience ⏮️ BACKGROUNDSchrödinger equation: youtu.be/CKpx9hkQ3HMExpectation values and root mean square deviations: youtu.be/rEm-Ejg5xekState space and dual space: youtu.be/hJoWM9jf0gUHermitian operators: youtu.be/XIgDUfyrLAYCommutator algebra: youtu.be/57xgSIV9PY0Functions of operators: youtu.be/nqLEbrzVsk4Wave packets: [COMING SOON]⏭️ WHAT NEXT? Constants of motion: youtu.be/r38VW3JwBAk~Director and writer: BMProducer and designer: MC
Functions of operators in quantum mechanics Professor M does Science 2021-05-26 | 💻 Book a 1:1 session: docs.google.com/forms/d/e/1FAIpQLScUL187erItvC7GPnNU2pelsueyVFr94nRq2A5Eq2aVRdGiIQ/viewform?pli=1📚 Many important phenomena in quantum mechanics are described by functions of operators. For example, spatial translations are given by an exponential function of the momentum operator, and time evolution is given by an exponential function of the Hamiltonian. In this video we explain how to construct the function of an operator, and explore what their properties are.⏮️ BACKGROUNDState space and dual space: youtu.be/hJoWM9jf0gUOperators: youtu.be/pNFna7zZbgEAdjoint operator: youtu.be/b_DcsVCtP5ICommutator algebra: youtu.be/57xgSIV9PY0Eigenvalues and eigenstates: youtu.be/p1zg-c1nvwQ⏭️ WHAT NEXT?Hermitian operators: youtu.be/XIgDUfyrLAYUnitary operators: youtu.be/baIT6HaaYuQTranslation operators: youtu.be/978mMgGYs1MTime evolution operator: youtu.be/zqmU4dW03aM📖 READ MOREIf you are interested in the mathematics behind the Baker-Campbell-Hausdorff formula, further details can be found in the following links: * http://webhome.phy.duke.edu/~mehen/760/ProblemSets/BCH.pdf* http://math.columbia.edu/~rzhang/files/BCHFormula.pdf~ Director and writer: BMProducer and designer: MC
The radial equation of central potentials Professor M does Science 2021-05-19 | How can we describe the radial motion of a quantum particle moving in a central potential?📚 A central potential is a potential that only depends on the distance from the origin, but does not depend on the orientation about the origin. This means that the motion of a particle moving in a central potential can be separated into its radial and angular parts. The angular part is fully determined by the equations of orbital angular momentum in quantum mechanics. In this video we explore the equation obeyed by the radial part. We also discuss the related and important concept of an effective potential.🐦 Follow me on Twitter: twitter.com/ProfMScience ⏮️ BACKGROUNDCentral potentials: youtu.be/Y73ctxnP9gQOrbital angular momentum: youtu.be/EyGJ3JE9CgEOrbital angular momentum eigenvalues: youtu.be/5h5mbL5rco0Orbital angular momentum eigenfunctions: youtu.be/Gk2XNmIHVwoPosition representation: youtu.be/Yw2YrTLSq5U⏭️ WHAT NEXT? 3D harmonic oscillator: [COMING SOON]Hydrogen atom: [COMING SOON]~Director and writer: BMProducer and designer: MC
Central potentials in quantum mechanics Professor M does Science 2021-05-12 | How can we study a particle moving in a central potential in quantum mechanics?📚 A central potential is a potential that only depends on the distance from the origin, but does not depend on the orientation about the origin. Central potentials feature in a range of physical system, from the 3-dimensional isotropic harmonic oscillator, to the relative motion of the electron and proton in a hydrogen atom. In this video we introduce central potentials and show how they are related to orbital angular momentum. 🐦 Follow me on Twitter: twitter.com/ProfMScience ⏮️ BACKGROUNDOrbital angular momentum: youtu.be/EyGJ3JE9CgEOrbital angular momentum eigenvalues: youtu.be/5h5mbL5rco0Orbital angular momentum eigenfunctions: youtu.be/Gk2XNmIHVwoPosition representation: youtu.be/Yw2YrTLSq5UConstants of motion: youtu.be/r38VW3JwBAk⏭️ WHAT NEXT? Central potentials | Radial equation: youtu.be/MsZP7yxpeFg3D harmonic oscillator: [COMING SOON]Hydrogen atom: [COMING SOON]~Director and writer: BMProducer and designer: MC
Commutator algebra in quantum mechanics Professor M does Science 2021-05-05 | In this video we go over a series of exercises to understand the mathematical properties of commutators. 📚 Operators in quantum mechanics do not always commute, which makes the commutator a key quantity. The best known example is the commutator between position and momentum, which is equal to the imaginary unit multiplied by Planck's constant, but commutators feature in many other problems. In this video we explore some of the mathematical properties of the commutator, which will prove extremely useful in our study of quantum mechanics.🐦 Follow me on Twitter: twitter.com/ProfMScience ⏮️ BACKGROUNDState space and dual space: youtu.be/hJoWM9jf0gUOperators: youtu.be/pNFna7zZbgEAdjoint operator: youtu.be/b_DcsVCtP5I⏭️ WHAT NEXT?Functions of operators: youtu.be/nqLEbrzVsk4Position and momentum: youtu.be/Yw2YrTLSq5U~ Director and writer: BMProducer and designer: MC
The time evolution operator in quantum mechanics Professor M does Science 2021-04-28 | 💻 Book a 1:1 session: docs.google.com/forms/d/e/1FAIpQLScUL187erItvC7GPnNU2pelsueyVFr94nRq2A5Eq2aVRdGiIQ/viewform?pli=1📚 In this video we learn about the properties of the time evolution operator in quantum mechanics. This operators provides an alternative but equivalent way to the Schrödinger equation for the study of time evolution of quantum systems. ⏮️ BACKGROUNDOperators in Quantum Mechanics: youtu.be/pNFna7zZbgE Unitary operators: youtu.be/baIT6HaaYuQSchrödinger equation: youtu.be/CKpx9hkQ3HMFunctions of operators: youtu.be/nqLEbrzVsk4⏭️ WHAT NEXT? Schrödinger vs. Heisenberg picture: youtu.be/2DM5DSDmP4cInteraction picture: [COMING SOON]~Director and writer: BMProducer and designer: MC
Unitary operators in quantum mechanics Professor M does Science 2021-04-21 | 💻 Book a 1:1 session: docs.google.com/forms/d/e/1FAIpQLScUL187erItvC7GPnNU2pelsueyVFr94nRq2A5Eq2aVRdGiIQ/viewform?pli=1📚 Unitary operators in quantum mechanics are used to describe physical processes such as spatial translations and time evolution. In this video, we discuss the basic properties of unitary operators and how we can transform quantum states and observables under the action of unitary operators. ⏮️ BACKGROUNDDirac notation in state space: youtu.be/hJoWM9jf0gUOperators: youtu.be/pNFna7zZbgERepresentations: youtu.be/rp2k2oR5ZQ8Matrix formulation: youtu.be/wIwnb1ldYTIEigenvalues and eigenstates: youtu.be/p1zg-c1nvwQ⏭️ WHAT NEXT?Translation operators: youtu.be/978mMgGYs1MTime evolution operator: youtu.be/zqmU4dW03aM~ Director and writer: BMProducer and designer: MC
Hermite polynomials and the quantum harmonic oscillator Professor M does Science 2021-04-14 | 📝 Problems+solutions:- Quantum harmonic oscillator I: professorm.learnworlds.com/course/quantum-harmonic-oscillator-i- Quantum harmonic oscillator II: professorm.learnworlds.com/course/quantum-harmonic-oscillator-ii💻 Book a 1:1 session: docs.google.com/forms/d/e/1FAIpQLScUL187erItvC7GPnNU2pelsueyVFr94nRq2A5Eq2aVRdGiIQ/viewform?pli=1📚 The eigenfunctions of the quantum harmonic oscillator can be written as a product of a Hermite polynomial and a Gaussian. In this video, we first introduce some of the mathematical properties of Hermite polynomials, and we then prove their relation to the eigenstates of the quantum harmonic oscillator.⏮️ BACKGROUNDHarmonic oscillator: youtu.be/OdizRUe84bgLadder operators: youtu.be/Kb9twGd25P0Eigenvalues of the quantum harmonic oscillator: youtu.be/GkUXscdLQQ0Eigenstates of the quantum harmonic oscillator: youtu.be/0o-LoJRtxDcTranslation operator: youtu.be/978mMgGYs1M~Director and writer: BMProducer and designer: MC
Eigenvalues of the quantum harmonic oscillator Professor M does Science 2021-03-31 | What are the possible energy eigenvalues of the quantum harmonic oscillator? 📝 Problems+solutions:- Quantum harmonic oscillator I: professorm.learnworlds.com/course/quantum-harmonic-oscillator-i- Quantum harmonic oscillator II: professorm.learnworlds.com/course/quantum-harmonic-oscillator-ii- More learning material: professorm.learnworlds.com 📚 The harmonic oscillator is an iconic potential in both classical and quantum mechanics. In quantum mechanics, the energy is quantized: the eigenvalues can only take some special values. In this video, we derive the possible energy eigenvalues using an elegant approach first proposed by Paul Dirac that exploits the properties of the ladder operators. 🐦 Follow me on Twitter: twitter.com/ProfMScience ⏮️ BACKGROUNDHarmonic oscillator: youtu.be/OdizRUe84bgLadder operators: youtu.be/Kb9twGd25P0⏭️ WHAT NEXT? Eigenstates of the quantum harmonic oscillator: youtu.be/0o-LoJRtxDc~Director and writer: BMProducer and designer: MC
Even and odd operators in quantum mechanics Professor M does Science 2021-03-24 | What are even and odd operators? 📚 Simply put, even operators commute with the parity operator, and odd operators anticommute with the parity operator. Of these, even operators are the most interesting and useful because they share a common set of eigenstates with the parity operator. This implies that the eigenstates of even operators are either even or odd states. There are many systems described by even Hamiltonians (from the infinite square well to the hydrogen atom), and in all these systems we can, before doing any calculations, say that the energy eigenstates must be even or odd states.🐦 Follow me on Twitter: twitter.com/ProfMScience ⏮️ BACKGROUNDParity operator: youtu.be/OsvXeTEQxygCompatible observables: youtu.be/IhJvX4H7xkAWave functions: youtu.be/2lr3aA4vaBsProjection operators: youtu.be/M9V4hhqyrKQEigenvalues and eigenstates: youtu.be/p1zg-c1nvwQ⏭️ WHAT NEXT? Infinite square well: youtu.be/pbZN8Pd8kacQuantum harmonic oscillator: [COMING SOON]Hydrogen atom: [COMING SOON]~Director and writer: BMProducer and designer: MC
Ladder and number operators of the quantum harmonic oscillator Professor M does Science 2021-03-17 | 📝 Problems+solutions:- Quantum harmonic oscillator I: professorm.learnworlds.com/course/quantum-harmonic-oscillator-i- Quantum harmonic oscillator II: professorm.learnworlds.com/course/quantum-harmonic-oscillator-ii💻 Book a 1:1 session: docs.google.com/forms/d/e/1FAIpQLScUL187erItvC7GPnNU2pelsueyVFr94nRq2A5Eq2aVRdGiIQ/viewform?pli=1📚 Ladder operators allow us to increase or decrease the energy of a quantum harmonic oscillator by a discrete amount, called a quantum of energy. The number operator allows us to count the number of energy quanta in the system. In this video, we derive a long list of properties of ladder and number operators that will prove extremely useful in our study of the quantum harmonic oscillator.🐦 Follow me on Twitter: twitter.com/ProfMScience ⏮️ BACKGROUNDHermitian operators: youtu.be/XIgDUfyrLAYHarmonic oscillator: youtu.be/OdizRUe84bg⏭️ WHAT NEXT? Eigenvalues of the quantum harmonic oscillator: youtu.be/GkUXscdLQQ0Eigenstates of the quantum harmonic oscillator: youtu.be/0o-LoJRtxDcLadder operators in angular momentum: youtu.be/yGvfqRfw1BE~Director and writer: BMProducer and designer: MC
The parity operator in quantum mechanics Professor M does Science 2021-03-10 | Why is the parity operator important? 📚 Considering a Cartesian coordinate system, the parity operator reflects a quantum state about the origin of coordinates, and is therefore also called the "space inversion operator". In this video, we explore the fundamental properties of the parity operator, for example we learn that its eigenstates are described by either even or odd wave functions. The parity operator is particularly important in systems with inversion symmetry, which include a wide range of examples, from the quantum harmonic oscillator to the hydrogen atom. 🐦 Follow me on Twitter: twitter.com/ProfMScience ⏮️ BACKGROUNDHermitian operators: youtu.be/XIgDUfyrLAYUnitary operators: youtu.be/baIT6HaaYuQProjection operators: youtu.be/M9V4hhqyrKQEigenvalues and eigenstates: youtu.be/p1zg-c1nvwQWave functions: youtu.be/2lr3aA4vaBs⏭️ WHAT NEXT? Even and odd operators: youtu.be/o8l6Yz9EHpsSelection rules: [COMING SOON]~Director and writer: BMProducer and designer: MC
The quantum harmonic oscillator Professor M does Science 2021-03-03 | 📝 Problems+solutions:- Quantum harmonic oscillator I: professorm.learnworlds.com/course/quantum-harmonic-oscillator-i- Quantum harmonic oscillator II: professorm.learnworlds.com/course/quantum-harmonic-oscillator-ii💻 Book a 1:1 session: docs.google.com/forms/d/e/1FAIpQLScUL187erItvC7GPnNU2pelsueyVFr94nRq2A5Eq2aVRdGiIQ/viewform?pli=1📚 The harmonic potential is key in understanding many classical physics problems, from the vibrations of strings to the behavior of electronic circuits. In quantum mechanics, the harmonic oscillator plays a similarly important role, and in this video we explore the reasons why. ⏮️ BACKGROUNDOperators: youtu.be/pNFna7zZbgE Position and momentum: youtu.be/Yw2YrTLSq5U⏭️ WHAT NEXT? Ladder and number operators: youtu.be/Kb9twGd25P0Eigenvalues of the quantum harmonic oscillator: youtu.be/GkUXscdLQQ0Eigenstates of the quantum harmonic oscillator: youtu.be/0o-LoJRtxDc~Director and writer: BMProducer and designer: MC
