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10 videos · 3 h 30 min
Explainer · YouTube
Parth G
Parth G explains what a quantum operator is and why commutators, which measure whether the order of two operations matters, are so useful, including their role in Ehrenfest's theorem. Watch it first, then come back to see the same ideas written in Dirac's notation and worked through with real two-by-two matrices.
Explainer · YouTube
3Blue1Brown
Not a quantum video, but the clearest picture anywhere of what an eigenvector is: a direction that a transformation only stretches, squashes or flips, never turns. Hold that image through the session, because a quantum eigenstate is exactly such a direction, and its eigenvalue is a possible measurement result.
Explainer · YouTube
Sabine Hossenfelder
Sabine Hossenfelder argues that the mathematics of quantum mechanics is mostly linear algebra, and only looks strange because of Dirac's bra-ket notation. She covers kets as vectors with complex components, bras, Born's rule and the measurement update, so she previews section 2 too: watch how a bra carries the complex conjugates of its ket's coefficients, and why every state must have length one (her description notes that the left side of the equation at 4:44 should not be squared).
Explainer · YouTube
3Blue1Brown
Grant Sanderson and Henry Reich build the maths of superposition from polarized light, a state that is literally an arrow with two components, and connect it to classical waves. The chapters 'Photons and measurement' (12:51) and 'Polarization experiments' (15:40) are section 2 in action: one photon at a time, the chance of passing a polarizing filter is the squared component along the filter's direction, which is Malus's law.
Explainer · YouTube
Quantum Sense
An animated chapter of Quantum Sense's 'Maths of Quantum Mechanics' series. It explains what a Hermitian operator is and some of its properties, then uses physical intuition to show that observables must be Hermitian operators. It sits between section 3's definition of the adjoint and Zwiebach's more formal lecture below, and its description links a proof that a bra is the adjoint of its ket: section 2's row of conjugated components, in operator language.
Explainer · YouTube
Parth G
A beginner-level tour of the eigenvalue equation, which Parth G calls, in his opinion, the second most important equation in quantum mechanics. He moves from vectors a matrix only stretches, to measurement operators and electron spin (2:47), to measuring a state that is not an eigenstate (6:25). That is section 4's point: each eigenvalue is a possible result, and in the matching eigenstate that result comes back every time.
Explainer · YouTube
Professor M does Science
A property-by-property tour of the three Pauli matrices, with chapters on being Hermitian (1:54), squaring to the identity (3:10, 'involutory'), unitarity, determinant and trace, and eigenvalues and eigenvectors (6:31). Check each property against section 5, then keep going: the chapter on commutation relations (9:27) previews section 6.
Lecture · MIT
MIT OpenCourseWare · Barton Zwiebach · 2013
Barton Zwiebach's lecture from MIT's 8.05 Quantum Physics II (Fall 2013) builds Dirac's notation from the inner product upward: bras and kets, projection operators, the adjoint of an operator, Hermitian and unitary operators, and the uncertainty of a Hermitian operator. It runs well ahead of this course, so treat it as a view of where today's grammar leads.
Lecture · MIT
MIT OpenCourseWare · Barton Zwiebach
Zwiebach on what makes Hermitian operators special. They always have enough eigenfunctions to span the whole space of states (the spectral theorem), and eigenfunctions with different eigenvalues are automatically orthogonal: two of section 3's three guarantees. Watch how he handles degenerate eigenvalues (from 10:52), where an orthonormal set has to be chosen rather than coming for free.
Lecture · MIT
MIT OpenCourseWare · Barton Zwiebach
Zwiebach writes down the Pauli matrices, multiplies σ1 and σ2 in both orders and finds the commutator 2iσ3, section 6's cycle. He then states, without proof, that no two matrices can have a plain number as their commutator, so matrices for x and p must be infinite-dimensional; section 7's callout gives the trace argument he leaves to you. From 6:07 he moves to the Schrödinger equation in three dimensions, ending with the commutators of the x, y and z positions and momenta.