Name: Expectation values in quantum mechanics @ProfessorMdoesScience
Uploaded: 2024-10-22T08:17:00-05:00
Duration: 8 min 17 s
Description: Expectation values in quantum mechanics

Expectation values in quantum mechanics

Professor M does Science 2024-10-22 | Expectation values in quantum mechanics

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 Introduction
0:43 Hamiltonian operator
3:54 Hamiltonian in second quantization
9:16 Non-interacting Hamiltonian
13:45 Hamiltonian in non-interacting basis
15:22 Wrap-up

⏮️ BACKGROUND
Occupation number representation: youtu.be/hTaqxOK8nGQ
Fock space: youtu.be/jAw9WMkcCj0
Creation and annihilation of bosons: youtu.be/BhK6u0bMqG0
Creation and annihilation of fermions: youtu.be/HZ43XE89n8s
One body operators: youtu.be/8sXWslDlZU0
Change of basis: youtu.be/6icXRi5lGWE

⏭️ WHAT NEXT?
Heisenberg picture in second quantization: [COMING SOON]
Hamiltonian in terms of field operators: [COMING SOON]

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Director and writer: BM
Producer 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 Introduction
0:35 Quantum field operators
3:28 One-body operators in terms of field operators
10:06 Two-body operators in terms of field operators
13:22 Operators interpreted as scattering of particles
15:38 The case of particles with spin
17:13 Example: local density operator
20:43 Wrap-up

⏮️ BACKGROUND
Quantum field operators: youtu.be/wAo0weNZVgQ
Commutators and anticommutators of field operators: youtu.be/cNDLHaVxIdA
Second 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: BM
Producer 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 Introduction
0:45 Quantum field operators
2:57 Bosons: commutation relations
7:40 Fermions: anticommutation relations
8:46 Compact notation for commutators and anticommutators
10:32 Commutators and anticommutators for particles with spin
12:51 Wrap-up

⏮️ BACKGROUND
Quantum field operators: youtu.be/wAo0weNZVgQ
Second quantization: youtube.com/playlist?list=PL8W2boV7eVfnSqy1fs3CCNALSvnDDd-tb

⏭️ WHAT NEXT?
General operators in terms of field operators: youtu.be/pEIf8opLv08
Hamiltonian in terms of field operators: [COMING SOON]
Heisenberg picture for field operators: [COMING SOON]

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Director and writer: BM
Producer 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 Introduction
0:35 Changing basis in second quantization
2:59 Position representation
4:22 Quantum field operators
12:10 Quantum field operators with spin
15:29 Wrap-up

⏮️ BACKGROUND
Second quantization: youtube.com/playlist?list=PL8W2boV7eVfnSqy1fs3CCNALSvnDDd-tb
Changing basis in second quantization: youtu.be/6icXRi5lGWE
Position representation: youtu.be/2lr3aA4vaBs

⏭️ WHAT NEXT?
Commutators and anticommutator of field operators: youtu.be/cNDLHaVxIdA
General operators in terms of field operators: youtu.be/pEIf8opLv08
Hamiltonian in terms of field operators: [COMING SOON]
Heisenberg picture for field operators: [COMING SOON]

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Director and writer: BM
Producer 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 Introduction
0:24 Homogeneous first order differential equations
2:43 Solution strategy
6:45 Alternative forms for homogeneous equations
9:24 Example
17:15 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]

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Director and writer: BM
Producer and designer: MC
Thumbnail 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 Intro
0:31 Differential equation setup
2:50 Solution to the differential equation
4:49 Checking the solution
7:01 Plotting the solution
9:22 Wrap-up

⏮️ BACKGROUND
Separable 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]

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Director and writer: BM
Producer and designer: MC
Thumbnail 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 Introduction
0:43 Separable first order differential equations
4:57 Solution strategy
7:41 Example: exponential growth
14:13 Mathematical justification of solution strategy
18:10 Wrap-up

⏭️ WHAT NEXT?
Differential equation for concentric circles: youtu.be/gPh_FUO6TGU
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: BM
Producer and designer: MC
Thumbnail 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 Introduction
3:08 The "d dot sigma" operator
8:21 Eigenvalues of the "d dot sigma" operator
11:17 Eigenstates of the "d dot sigma" operator
15:00 Energy eigenstates
20:15 Wrap-up

⏮️ BACKGROUND
Pauli matrices: youtu.be/2MsVD9ufguk
2x2 matrix space: youtu.be/OdYNKLULvlU
Two-state quantum systems | Eigenvalues: youtu.be/kQnQpI2B72s
Operators: youtu.be/pNFna7zZbgE
Matrix formulation: youtu.be/wIwnb1ldYTI
Hermitian operators: youtu.be/XIgDUfyrLAY
Eigenvalues and eigenstates: youtu.be/p1zg-c1nvwQ

⏭️ WHAT NEXT?
Spin 1/2: [COMING SOON]
Rabi oscillations: [COMING SOON]

🧮 Double angle formulae
Wikipedia: en.wikipedia.org/wiki/List_of_trigonometric_identities#Double-angle_formulae
Wolfram MathWorld: mathworld.wolfram.com/Double-AngleFormulas.html

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Director and writer: BM
Producer 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 Introduction
2:08 Eigenvalues of a 2-state quantum system
7:59 Degenerate eigenvalues
10:37 2-state quantum system in terms of Pauli matrices
14:55 Re-deriving the eigenvalues of a 2-state quantum system
18:05 Wrap-up

⏮️ BACKGROUND
Pauli matrices: youtu.be/2MsVD9ufguk
2x2 matrix space: youtu.be/OdYNKLULvlU
Operators: youtu.be/pNFna7zZbgE
Matrix formulation: youtu.be/wIwnb1ldYTI
Hermitian operators: youtu.be/XIgDUfyrLAY
Eigenvalues and eigenstates: youtu.be/p1zg-c1nvwQ

⏭️ WHAT NEXT?
Two-state quantum systems | Eigenstates: youtu.be/lkVtvdJ3GbQ
Spin 1/2: [COMING SOON]
Rabi oscillations: [COMING SOON]

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Director and writer: BM
Producer 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 Introduction
1:27 Pauli matrices
2:09 2x2 matrices in terms of Pauli matrices
10:29 Hermitian 2x2 matrices in terms of Pauli matrices
14:58 Wrap-up

⏮️ BACKGROUND
Pauli matrices: youtu.be/2MsVD9ufguk
Operators: youtu.be/pNFna7zZbgE
Matrix formulation: youtu.be/wIwnb1ldYTI
Hermitian operators: youtu.be/XIgDUfyrLAY

⏭️ WHAT NEXT?
Two-state quantum systems | Eigenvalues: youtu.be/kQnQpI2B72s
Two-state quantum systems | Eigenstates: youtu.be/lkVtvdJ3GbQ
Spin 1/2: [COMING SOON]
Rabi oscillations: [COMING SOON]

~
Director and writer: BM
Producer 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 Introduction
0:54 Pauli matrices
1:54 Hermitian
3:10 Involutory
4:34 Unitary
5:10 Determinant
5:58 Trace
6:31 Eigenvalues and eigenvectors
9:27 Commutation relations
13:22 Anticommuntation relations
15:30 Wrap-up

⏮️ BACKGROUND
Hermitian operators: youtu.be/XIgDUfyrLAY
Unitary operators: youtu.be/baIT6HaaYuQ
Eigenvalues and eigenstates: youtu.be/p1zg-c1nvwQ
Commutator algebra: youtu.be/57xgSIV9PY0
Angular momentum: youtu.be/Bo5qoaLsBOE

⏭️ WHAT NEXT?
2x2 matrix space: youtu.be/OdYNKLULvlU
Two-state quantum systems | Eigenvalues: youtu.be/kQnQpI2B72s
Two-state quantum systems | Eigenstates: youtu.be/lkVtvdJ3GbQ
Spin 1/2: [COMING SOON]

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Director and writer: BM
Producer 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 Introduction
2:22 Energy levels of hydrogen
3:37 Photon emission
5:23 Rydberg formula
9:59 Emission spectrum of hydrogen
10:50 Lyman series
15:17 Balmer series
19:25 Orion nebula
19:49 Higher series
23:28 Fine and hyperfine structure
24:36 Wrap-up

⏮️ BACKGROUND
Hydrogen atom basics: youtu.be/lndTguV0u1g
Hydrogen atom | Maths: youtu.be/8NJm4Jkp0jY
Hydrogen atom | Eigenvalues and eigenstates: youtu.be/T78NkndaMb8
Hydrogen atom | Ground state: youtu.be/r_C82Y6XNWY
Hydrogen atom | Energy spectrum: youtu.be/GV2CDZD2xNk

⏭️ WHAT NEXT?
Hydrogen fine structure: [COMING SOON]
Hydrogen hyperfine structure: [COMING SOON]

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Director and writer: BM
Producer 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 Intro
4:19 Energy eigenvalues degeneracies
11:22 Hydrogen atom quantum numbers
12:08 Atomic shells and atomic subshells
18:46 Spectroscopic notation
22:02 Wrap-up

🐦 Follow me on Twitter: twitter.com/ProfMScience

⏮️ BACKGROUND
Hydrogen atom basics: youtu.be/lndTguV0u1g
Hydrogen atom | Maths: youtu.be/8NJm4Jkp0jY
Hydrogen atom | Eigenvalues and eigenstates: youtu.be/T78NkndaMb8
Hydrogen atom | Ground state: youtu.be/r_C82Y6XNWY

⏭️ WHAT NEXT?
Hydrogen atom | Spectral series: youtu.be/XawP16g7dJ8
Hydrogen fine structure: [COMING SOON]
Hydrogen hyperfine structure: [COMING SOON]
Spin 1/2: [COMING SOON]

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Director and writer: BM
Producer 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 Intro
1:12 Hydrogen atom basics
4:04 Ground state energy: ionization energy
8:50 Ground state wave function: Bohr radius
22:55 Virial theorem
29:20 Wrap-up

⏮️ BACKGROUND
Hydrogen atom basics: youtu.be/lndTguV0u1g
Hydrogen atom | Maths: youtu.be/8NJm4Jkp0jY
Hydrogen atom | Eigenvalues and eigenstates: youtu.be/T78NkndaMb8
Central potentials: youtu.be/Y73ctxnP9gQ
Spherical harmonics: youtu.be/5PMqf3Hj-Aw
Virial theorem: youtu.be/QzUiVboHcKI

⏭️ WHAT NEXT?
Hydrogen atom | Energy spectrum: youtu.be/GV2CDZD2xNk
Hydrogen atom | Spectral series: youtu.be/XawP16g7dJ8

~
Director and writer: BM
Producer 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 Intro
0:35 Hydrogen atom as a central potential
4:53 Recap of the mathematical solution of the eigenvalue equation
12:49 Ground state
20:31 First excited state
34:16 Second excited state
37:34 Wrap-up

⏮️ BACKGROUND
Hydrogen atom basics: youtu.be/lndTguV0u1g
Hydrogen atom | Maths: youtu.be/8NJm4Jkp0jY
Central potentials: youtu.be/Y73ctxnP9gQ
Radial equation of central potentials: youtu.be/MsZP7yxpeFg
Orbital angular momentum: youtu.be/EyGJ3JE9CgE
Spherical harmonics: youtu.be/5PMqf3Hj-Aw

⏭️ WHAT NEXT?
Hydrogen atom | Ground state: youtu.be/r_C82Y6XNWY
Hydrogen atom | Energy spectrum: youtu.be/GV2CDZD2xNk
Hydrogen atom | Spectral series: youtu.be/XawP16g7dJ8

~
Director and writer: BM
Producer 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 Intro
1:34 3D isotropic quantum harmonic oscillator as a central potential
5:35 Recap of the mathematical solution of the eigenvalue equation
11:46 Ground state
19:58 First excited state
27:22 Second excited state
30:21 Wrap-up

⏮️ BACKGROUND
1D quantum harmonic oscillator: youtu.be/OdizRUe84bg
3D quantum harmonic oscillator in Cartesian coordinates: youtu.be/cZTnmlJfHdg
3D quantum harmonic oscillator in spherical coordinates: youtu.be/jOQThICjLlw
Central potentials: youtu.be/Y73ctxnP9gQ
Radial equation of central potentials: youtu.be/MsZP7yxpeFg
Orbital angular momentum: youtu.be/EyGJ3JE9CgE
Spherical harmonics: youtu.be/5PMqf3Hj-Aw

⏭️ WHAT NEXT?
Hydrogen atom: youtube.com/playlist?list=PL8W2boV7eVfnJbLf-p3-_7d51tskA0-Sa

~
Director and writer: BM
Producer 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 Intro
0:37 Hypervirial theorem
2:57 Virial theorem for 1 particle in 1D
9:13 Virial theorem for potentials described by homogeneous functions
15:25 Virial theorem for multiple particles in 3D
22:29 Wrap-up

⏮️ BACKGROUND
Eigenvalues and eigenstates: youtu.be/p1zg-c1nvwQ
Hermitian operators: youtu.be/XIgDUfyrLAY
Expectation values: youtu.be/rEm-Ejg5xek
Commutator algebra: youtu.be/57xgSIV9PY0
Functions of operators: youtu.be/nqLEbrzVsk4
Quantum harmonic oscillator playlist: youtube.com/playlist?list=PL8W2boV7eVfmdWs3CsaGfoITHURXvHOGm
Hydrogen atom playlist: youtube.com/playlist?list=PL8W2boV7eVfnJbLf-p3-_7d51tskA0-Sa

⏭️ WHAT NEXT?
Hydrogen atom ground state: [COMING SOON]

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Director and writer: BM
Producer 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 Intro
0:52 Schrodinger picture
4:52 Time evolution operator
6:54 Heisenberg picture - simple motivation
11:08 Unitary transformations
13:32 Pictures of time evolution in quantum mechanics
16:28 Heisenberg picture
22:52 Dynamics
27:31 Conservative systems
30:14 Why use the Heisenberg picture?
32:42 Wrap-up

⏮️ BACKGROUND
Schrödinger equation: youtu.be/CKpx9hkQ3HM
Time evolution operator: youtu.be/zqmU4dW03aM
Unitary operators: youtu.be/baIT6HaaYuQ
Expectation values: youtu.be/rEm-Ejg5xek
Measurements: youtu.be/u1R3kRWh1ek
The Ehrenfest theorem: youtu.be/55RQCCtdlko

⏭️ WHAT NEXT?
Interaction picture: [COMING SOON]

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Director and writer: BM
Producer 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 Intro
1:00 Hydrogen as a central potential
7:27 Radial equation
9:46 Bound vs unbound states
12:52 Simplifying notation
16:38 Radial equation solution
33:07 Quantized energy eigenvalues
39:56 Energy eigenfunctions
45:47 Wrap-up

⏮️ BACKGROUND
Hydrogen atom basics: youtu.be/lndTguV0u1g
3D quantum harmonic oscillator: youtu.be/jOQThICjLlw
Central potentials: youtu.be/Y73ctxnP9gQ
Radial equation of central potentials: youtu.be/MsZP7yxpeFg
Orbital angular momentum: youtu.be/EyGJ3JE9CgE
Spherical harmonics: youtu.be/5PMqf3Hj-Aw
Bound vs unbound states: [COMING SOON]

⏭️ WHAT NEXT?
Hydrogen atom | Eigenvalues and eigenfunctions: youtu.be/T78NkndaMb8
Hydrogen atom | Ground state: youtu.be/r_C82Y6XNWY
Hydrogen atom | Energy spectrum: youtu.be/GV2CDZD2xNk
Hydrogen atom | Spectral series: youtu.be/XawP16g7dJ8

~
Director and writer: BM
Producer 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 Intro
1:47 3D isotropic quantum harmonic oscillator as a central potential
8:18 Radial equation solution
36:22 Wrap-up

⏮️ BACKGROUND
1D quantum harmonic oscillator: youtu.be/OdizRUe84bg
3D quantum harmonic oscillator in Cartesian coordinates: youtu.be/cZTnmlJfHdg
Central potentials: youtu.be/Y73ctxnP9gQ
Radial equation of central potentials: youtu.be/MsZP7yxpeFg
Orbital angular momentum: youtu.be/EyGJ3JE9CgE
Spherical harmonics: youtu.be/5PMqf3Hj-Aw

⏭️ WHAT NEXT?
3D isotropic quantum harmonic oscillator | eigenvalues and eigenstates: youtu.be/3Ipcr9tMlxU

~
Director and writer: BM
Producer 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 Intro
1:07 A proton and an electron
6:34 Hamiltonian of the hydrogen atom
8:28 Relative motion of proton and electron
17:54 Wrap-up

🐦 Follow me on Twitter: twitter.com/ProfMScience

⏮️ BACKGROUND
Two interacting quantum particles: youtu.be/kOupIEhYdY8

⏭️ WHAT NEXT?
Hydrogen atom | Maths: youtu.be/8NJm4Jkp0jY
Hydrogen atom | Eigenvalues and eigenstates: youtu.be/T78NkndaMb8
Hydrogen atom | Ground state: youtu.be/r_C82Y6XNWY
Hydrogen atom | Energy spectrum: youtu.be/GV2CDZD2xNk
Hydrogen atom | Spectral series: youtu.be/XawP16g7dJ8

~
Director and writer: BM
Producer 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 Intro
1:13 Two-particle systems
4:24 Change of variables
13:04 Hamiltonian of two interacting particles
20:58 Separable Hamiltonians
24:35 Center of mass Hamiltonian
26:02 Relative Hamiltonian
28:55 Wrap-up

⏮️ BACKGROUND
Tensor products: youtu.be/kz3206S2B6Q
Eigenvalues and eigenstates: youtu.be/p1zg-c1nvwQ
Position representation: youtu.be/Yw2YrTLSq5U
Free particle: [COMING SOON]
Central potentials: youtu.be/Y73ctxnP9gQ

⏭️ WHAT NEXT?
Hydrogen atom: youtu.be/lndTguV0u1g

~
Director and writer: BM
Producer 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 Intro
0:45 Definition of the spherical harmonics
3:09 How do we visualize spherical harmonics?
6:37 l=0 spherical harmonic
8:14 l=1 spherical harmonics
17:16 l=2 spherical harmonics
18:53 l=3 spherical harmonics
19:50 An alternative view of the spherical harmonics
23:01 Wrap-up

🐦 Follow me on Twitter: twitter.com/ProfMScience

⏮️ BACKGROUND
Orbital angular momentum: youtu.be/EyGJ3JE9CgE
Orbital angular momentum eigenvalues: youtu.be/5h5mbL5rco0
Orbital angular momentum eigenfunctions: youtu.be/Gk2XNmIHVwo

⏭️ WHAT NEXT?
The 3D quantum harmonic oscillator eigenfunctions: [COMING SOON]
The hydrogen atom eigenfunctions: [COMING SOON]

👩‍💻 GITHUB
Plot the spherical harmonics yourself: nbviewer.org/github/profmscience/spherical-harmonics/blob/main/spherical-harmonics-for-github.ipynb

~
Director and writer: BM
Producer 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 Intro
2:41 3D isotropic quantum harmonic oscillator
9:31 Ground state
11:56 First excited state
15:50 Second excited state
19:58 General case
22:53 Wrap-up

⏮️ BACKGROUND
1D quantum harmonic oscillator: youtu.be/OdizRUe84bg
Eigenvalues of the quantum harmonic oscillator: youtu.be/GkUXscdLQQ0
Eigenstates of the quantum harmonic oscillator: youtu.be/0o-LoJRtxDc
3D quantum harmonic oscillator: youtu.be/cZTnmlJfHdg

⏭️ WHAT NEXT?
The 3D quantum harmonic oscillator as a central potential: [COMING SOON]

~
Director and writer: BM
Producer 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 Intro
1:07 3D quantum harmonic oscillator Hamiltonian
6:39 Solving the eigenvalue equation
9:02 Eigenvalues
9:58 Eigenstates
15:43 Eigenfunctions
19:22 Wrap-up

⏮️ BACKGROUND
1D quantum harmonic oscillator: youtu.be/OdizRUe84bg
Ladder operators: youtu.be/Kb9twGd25P0
Eigenvalues of the quantum harmonic oscillator: youtu.be/GkUXscdLQQ0
Eigenstates of the quantum harmonic oscillator: youtu.be/0o-LoJRtxDc
Hermite polynomials: youtu.be/p22UrUv9QdM
Coherent quantum states: youtu.be/x0wk98uMyys
Tensor product state spaces: youtu.be/kz3206S2B6Q
Eigenvalues and eigenstates in tensor product spaces: youtu.be/T3ynwXrE0Xw

⏭️ WHAT NEXT?
Degeneracies of the isotropic quantum harmonic oscillator: youtu.be/K5YsxsHXGHc
The 3D quantum harmonic oscillator as a central potential: [COMING SOON]

~
Director and writer: BM
Producer 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.

⏮️ BACKGROUND
Eigenvalues and eigenstates: youtu.be/p1zg-c1nvwQ
Tensor products: youtu.be/kz3206S2B6Q

⏭️ WHAT NEXT?
3D quantum harmonic oscillator: youtu.be/cZTnmlJfHdg
Two interacting quantum particles: [COMING SOON]

~
Director and writer: BM
Producer 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 Intro
4:16 Coherent state wave function
8:06 Minimum uncertainty state
9:40 Displacement operator
11:02 Time evolution
14:31 Wrap-up

⏮️ BACKGROUND
Quantum harmonic oscillator: youtu.be/OdizRUe84bg
Ladder operators: youtu.be/Kb9twGd25P0
Quantum harmonic oscillator eigenstates: youtu.be/0o-LoJRtxDc
Coherent states: youtu.be/x0wk98uMyys
Quasi-classical states: youtu.be/0ef1rLO6DTU
Displacement operator: youtu.be/ypRTLIo-IIc
Minimum uncertainty states: youtu.be/cX9JiQ8nEL4

⏭️ WHAT NEXT?
Coherent state wave function || Maths: youtu.be/0p9pH85SLIU

~
Director and writer: BM
Producer 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 Intro
3:16 Wave functions
4:44 Mathematical derivation
14:39 Coherent state wave function
16:03 Wrap-up

⏮️ BACKGROUND
Quantum harmonic oscillator: youtu.be/OdizRUe84bg
Ladder operators: youtu.be/Kb9twGd25P0
Quantum harmonic oscillator eigenstates: youtu.be/0o-LoJRtxDc
Coherent states: youtu.be/x0wk98uMyys
Quasi-classical states: youtu.be/0ef1rLO6DTU
Displacement operator: youtu.be/ypRTLIo-IIc
Functions of operators: youtu.be/nqLEbrzVsk4
Wave functions: youtu.be/2lr3aA4vaBs
Translation operator: youtu.be/978mMgGYs1M

⏭️ WHAT NEXT?
Coherent state wave function || Concepts: youtu.be/-MaF_TzD4Q8

~
Director and writer: BM
Producer 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 Intro
2:28 Definition
5:54 Mathematical properties
15:36 Coherent states
19:45 Wrap-up

⏮️ BACKGROUND
Quantum harmonic oscillator: youtu.be/OdizRUe84bg
Ladder operators: youtu.be/Kb9twGd25P0
Coherent states: youtu.be/x0wk98uMyys

⏭️ WHAT NEXT?
Coherent state wave function || Maths: youtu.be/0p9pH85SLIU
Coherent state wave function || Concepts: youtu.be/-MaF_TzD4Q8

~
Director and writer: BM
Producer 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 Intro
3:09 Uncertainty principle between position and momentum
13:03 Minimum uncertainty states
24:30 Wrap-up

⏮️ BACKGROUND
Expectation values: youtu.be/rEm-Ejg5xek
The Heisenberg uncertainty principle || Concepts: youtu.be/pfjQtyLBBHw
The Heisenberg uncertainty principle || Proof: youtu.be/fsC5Mhd7YUc
Position representation || Wave functions: youtu.be/2lr3aA4vaBs
Position representation || Operators: youtu.be/Yw2YrTLSq5U

⏭️ WHAT NEXT?
Quantum harmonic oscillator wave functions: youtu.be/0o-LoJRtxDc
Coherent states wave functions: youtu.be/-MaF_TzD4Q8

~
Director and writer: BM
Producer 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 Intro
1:03 General uncertainty principles
7:18 Time-energy uncertainty principle
15:52 Examples
17:51 Wrap-up

⏮️ BACKGROUND
Schrödinger equation: youtu.be/CKpx9hkQ3HM
Expectation values and root mean square deviations: youtu.be/rEm-Ejg5xek
The Heisenberg uncertainty principle || Concepts: youtu.be/pfjQtyLBBHw
The Heisenberg uncertainty principle || Proof: youtu.be/fsC5Mhd7YUc
Eigenvalues and eigenstates: youtu.be/p1zg-c1nvwQ
Ehrenfest's theorem: youtu.be/55RQCCtdlko

~
Director and writer: BM
Producer 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 Intro
1:06 Classical harmonic oscillator
2:50 Quantum harmonic oscillator
5:40 Classical oscillator vs energy eigenstates
7:39 Coherent states
18:53 Classical oscillator vs coherent states
22:18 Wrap-up

⏮️ BACKGROUND
Quantum harmonic oscillator: youtu.be/OdizRUe84bg
Ladder operators: youtu.be/Kb9twGd25P0
Quantum harmonic oscillator eigenvalues: youtu.be/GkUXscdLQQ0
Ehrenfest's theorem: youtu.be/55RQCCtdlko
Coherent states: youtu.be/x0wk98uMyys

⏭️ WHAT NEXT?
Displacement operator: youtu.be/ypRTLIo-IIc
Coherent state wave function || Maths: youtu.be/0p9pH85SLIU
Coherent state wave function || Concepts: youtu.be/-MaF_TzD4Q8

~
Director and writer: BM
Producer 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 Intro
2:41 Definition of coherent states
4:16 Coherent states in the energy basis
12:04 Time evolution of coherent states
16:19 The raising operator has no eigenstates
21:04 Wrap-up

🐦 Follow me on Twitter: twitter.com/ProfMScience

⏮️ BACKGROUND
Quantum harmonic oscillator: youtu.be/OdizRUe84bg
Ladder operators: youtu.be/Kb9twGd25P0
Quantum harmonic oscillator eigenvalues: youtu.be/GkUXscdLQQ0
Eigenvalues and eigenstates: youtu.be/p1zg-c1nvwQ
Representations: youtu.be/rp2k2oR5ZQ8
Schrödinger equation: youtu.be/CKpx9hkQ3HM

⏭️ WHAT NEXT?
Quasi-classical states: youtu.be/0ef1rLO6DTU
Displacement operator: youtu.be/ypRTLIo-IIc
Coherent state wave function || Maths: youtu.be/0p9pH85SLIU
Coherent state wave function || Concepts: youtu.be/-MaF_TzD4Q8

~
Director and writer: BM
Producer 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 Intro
0:43 Definition
2:00 Time-independent expectation values
3:14 Common set of eigenstates with H
5:25 Time-independent outcome probabilities
8:21 Wrap-up

⏮️ BACKGROUND
Schrödinger equation: youtu.be/CKpx9hkQ3HM
Ehrenfest theorem: youtu.be/55RQCCtdlko
Compatible observables: youtu.be/IhJvX4H7xkA
Measurements | Concepts: youtu.be/u1R3kRWh1ek

⏭️ WHAT NEXT?
Central potentials: youtu.be/Y73ctxnP9gQ

~
Director and writer: BM
Producer 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

⏮️ BACKGROUND
Schrödinger equation: youtu.be/CKpx9hkQ3HM
Expectation values and root mean square deviations: youtu.be/rEm-Ejg5xek
State space and dual space: youtu.be/hJoWM9jf0gU
Hermitian operators: youtu.be/XIgDUfyrLAY
Commutator algebra: youtu.be/57xgSIV9PY0
Functions of operators: youtu.be/nqLEbrzVsk4
Wave packets: [COMING SOON]

⏭️ WHAT NEXT?
Constants of motion: youtu.be/r38VW3JwBAk

~
Director and writer: BM
Producer 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.

⏮️ BACKGROUND
State space and dual space: youtu.be/hJoWM9jf0gU
Operators: youtu.be/pNFna7zZbgE
Adjoint operator: youtu.be/b_DcsVCtP5I
Commutator algebra: youtu.be/57xgSIV9PY0
Eigenvalues and eigenstates: youtu.be/p1zg-c1nvwQ

⏭️ WHAT NEXT?
Hermitian operators: youtu.be/XIgDUfyrLAY
Unitary operators: youtu.be/baIT6HaaYuQ
Translation operators: youtu.be/978mMgGYs1M
Time evolution operator: youtu.be/zqmU4dW03aM

📖 READ MORE
If 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: BM
Producer 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

⏮️ BACKGROUND
Central potentials: youtu.be/Y73ctxnP9gQ
Orbital angular momentum: youtu.be/EyGJ3JE9CgE
Orbital angular momentum eigenvalues: youtu.be/5h5mbL5rco0
Orbital angular momentum eigenfunctions: youtu.be/Gk2XNmIHVwo
Position representation: youtu.be/Yw2YrTLSq5U

⏭️ WHAT NEXT?
3D harmonic oscillator: [COMING SOON]
Hydrogen atom: [COMING SOON]

~
Director and writer: BM
Producer 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.

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⏮️ BACKGROUND
Orbital angular momentum: youtu.be/EyGJ3JE9CgE
Orbital angular momentum eigenvalues: youtu.be/5h5mbL5rco0
Orbital angular momentum eigenfunctions: youtu.be/Gk2XNmIHVwo
Position representation: youtu.be/Yw2YrTLSq5U
Constants of motion: youtu.be/r38VW3JwBAk

⏭️ WHAT NEXT?
Central potentials | Radial equation: youtu.be/MsZP7yxpeFg
3D harmonic oscillator: [COMING SOON]
Hydrogen atom: [COMING SOON]

~
Director and writer: BM
Producer 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

⏮️ BACKGROUND
State space and dual space: youtu.be/hJoWM9jf0gU
Operators: youtu.be/pNFna7zZbgE
Adjoint operator: youtu.be/b_DcsVCtP5I

⏭️ WHAT NEXT?
Functions of operators: youtu.be/nqLEbrzVsk4
Position and momentum: youtu.be/Yw2YrTLSq5U

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Director and writer: BM
Producer 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.

⏮️ BACKGROUND
Operators in Quantum Mechanics: youtu.be/pNFna7zZbgE
Unitary operators: youtu.be/baIT6HaaYuQ
Schrödinger equation: youtu.be/CKpx9hkQ3HM
Functions of operators: youtu.be/nqLEbrzVsk4

⏭️ WHAT NEXT?
Schrödinger vs. Heisenberg picture: youtu.be/2DM5DSDmP4c
Interaction picture: [COMING SOON]

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Director and writer: BM
Producer 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.

⏮️ BACKGROUND
Dirac notation in state space: youtu.be/hJoWM9jf0gU
Operators: youtu.be/pNFna7zZbgE
Representations: youtu.be/rp2k2oR5ZQ8
Matrix formulation: youtu.be/wIwnb1ldYTI
Eigenvalues and eigenstates: youtu.be/p1zg-c1nvwQ

⏭️ WHAT NEXT?
Translation operators: youtu.be/978mMgGYs1M
Time evolution operator: youtu.be/zqmU4dW03aM

~
Director and writer: BM
Producer 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.

⏮️ BACKGROUND
Harmonic oscillator: youtu.be/OdizRUe84bg
Ladder operators: youtu.be/Kb9twGd25P0
Eigenvalues of the quantum harmonic oscillator: youtu.be/GkUXscdLQQ0
Eigenstates of the quantum harmonic oscillator: youtu.be/0o-LoJRtxDc
Translation operator: youtu.be/978mMgGYs1M

~
Director and writer: BM
Producer and designer: MC

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.

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⏮️ BACKGROUND
Harmonic oscillator: youtu.be/OdizRUe84bg
Ladder and number operators: youtu.be/Kb9twGd25P0
Eigenvalues of the quantum harmonic oscillator: youtu.be/GkUXscdLQQ0
Harmonic oscillator eigenvalues: youtu.be/GkUXscdLQQ0
The parity operator: youtu.be/OsvXeTEQxyg
Even and odd operators: youtu.be/o8l6Yz9EHps
Wave functions: youtu.be/2lr3aA4vaBs
Position and momentum: youtu.be/Yw2YrTLSq5U

⏭️ WHAT NEXT?
Hermite polynomials: youtu.be/p22UrUv9QdM
Coherent states: youtu.be/x0wk98uMyys

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Director and writer: BM
Producer 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.

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⏮️ BACKGROUND
Harmonic oscillator: youtu.be/OdizRUe84bg
Ladder operators: youtu.be/Kb9twGd25P0

⏭️ WHAT NEXT?
Eigenstates of the quantum harmonic oscillator: youtu.be/0o-LoJRtxDc

~
Director and writer: BM
Producer 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.

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⏮️ BACKGROUND
Parity operator: youtu.be/OsvXeTEQxyg
Compatible observables: youtu.be/IhJvX4H7xkA
Wave functions: youtu.be/2lr3aA4vaBs
Projection operators: youtu.be/M9V4hhqyrKQ
Eigenvalues and eigenstates: youtu.be/p1zg-c1nvwQ

⏭️ WHAT NEXT?
Infinite square well: youtu.be/pbZN8Pd8kac
Quantum harmonic oscillator: [COMING SOON]
Hydrogen atom: [COMING SOON]

~
Director and writer: BM
Producer 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

⏮️ BACKGROUND
Hermitian operators: youtu.be/XIgDUfyrLAY
Harmonic oscillator: youtu.be/OdizRUe84bg

⏭️ WHAT NEXT?
Eigenvalues of the quantum harmonic oscillator: youtu.be/GkUXscdLQQ0
Eigenstates of the quantum harmonic oscillator: youtu.be/0o-LoJRtxDc
Ladder operators in angular momentum: youtu.be/yGvfqRfw1BE

~
Director and writer: BM
Producer 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

⏮️ BACKGROUND
Hermitian operators: youtu.be/XIgDUfyrLAY
Unitary operators: youtu.be/baIT6HaaYuQ
Projection operators: youtu.be/M9V4hhqyrKQ
Eigenvalues and eigenstates: youtu.be/p1zg-c1nvwQ
Wave functions: youtu.be/2lr3aA4vaBs

⏭️ WHAT NEXT?
Even and odd operators: youtu.be/o8l6Yz9EHps
Selection rules: [COMING SOON]
~
Director and writer: BM
Producer 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.

⏮️ BACKGROUND
Operators: youtu.be/pNFna7zZbgE
Position and momentum: youtu.be/Yw2YrTLSq5U

⏭️ WHAT NEXT?
Ladder and number operators: youtu.be/Kb9twGd25P0
Eigenvalues of the quantum harmonic oscillator: youtu.be/GkUXscdLQQ0
Eigenstates of the quantum harmonic oscillator: youtu.be/0o-LoJRtxDc

~
Director and writer: BM
Producer and designer: MC