The Road to Reality, Roger Penrose (2007). Great companion to other more in-depth books, explains how all the parts of physics ‘fit together’ in the larger theory. Requires simple understanding of differential geometry (and is in fact a great companion to any rigorous book on differential geometry). I’ve started with the chapter on Lagrangian and Hamiltonian mechanics, and paired it with the Taylor textbook.
Classical Mechanics, Charles Taylor (2005). Easy to read, well explained, at the level of 2nd undergrad course in mechanics. Reputed to have the best explanation of the Lagrangian. Replete with well picked exercises. Requires a bit of calculus - derivatives, integrals, very basic PDEs. No differential geometry required.
Gravitation, Misner, Thorne, Wheeler (1973). I have not read it, but it’s supposed to be ‘the’ reference on gravity as of the 1970s. And it’s supposed to have really clear physical interpretation of differential geometric concepts.
Quantum Mechanics in Simple Matrix Form, Thomas F. Jordan (1986). An exposition in the simplest operator form, starting from 2x2 matrices, and avoiding any discussion of eigenvalues, hermitian matrices, etc.
101 Quantum Questions, Kenneth W. Ford (2011). For some reason, this is an easier read, perhaps because it speaks, at least in the earlier sections, of the experiments that resulted in the discoveries of various elementary particles and quantum effects. This makes it an ideal companion to the more mathematically inclined books.
Lecture 3. Proof of the principle of least action. Equivalence with Newton’s laws. Examples.
Lecture 4. Lagrangian in polar coordinates. Continuous symmetries and the Noether Theorem. Conservation of energy. Hamiltonian.
Lecture 5 Examples. Conservation of energy. Noether’s theorem.
Lecture 6 Using Lagrangians to solve simple problems: Object sliding on inclined plane which slides on horizontal plane. Double pendulum. The Hamiltonian. Hamilton’s equations. Example: the harmonic oscillator.
Lecture 7 The Hamiltonian as energy. Divergence. The Liouville theorem. Definition of Poisson bracket.
Lecture 8 Poisson brackets \(\{F, G\}\). Poisson brackets in coordinates $q, p$. Derivation of \(\dot{F} = \{F, H\}\) for the Hamiltonian \(H\). Symmetries and conservation seen through the Poisson bracket - the Hamiltonian version of the Noether theorem. Angular momentum \(L_x = yp_z - zp_y\), \(L_y = zp_x - xp_z\), \(L_z = xp_y - yp_x\), its Poisson brackets with \(x, y, z\) and other angular momentum components \(L_x, L_y, L_z\). The gyroscope. Not mentioned: Poisson brackets also satisfy a Jacoby identity.
Lecture 9 Gradient, curl, divergence. Divergence of curl is zero. Fields with zero divergence are curls. Curl of gradient is zero. Fields with zero curl are gradients. Electrostatic field \(\vec{E}\). Magnetic field \(\vec{B}\). Comparison of magnetic force with Coriolis force. Zero divergence of magnetic field. Vector potential \(\vec{A}\) defined so its curl is \(\vec{B}\). Non-uniqueness of \(\vec{A}\), up to gauge transformation \(\vec{A} + \vec{\nabla}f\). Origin of term gauge as historical glitch. Lorenz force \(\vec{F} = e (-\vec{\nabla}V + \frac{\vec{v}}{c} \times \vec{B})\). Lagrangian for Lorenz force \(\mathcal{L} = \frac{m}{2} \dot{x}^2 - eV + \frac{e}{c} \vec{A} \cdot \dot{x}\). Gauge invariance of the Lagrangian least action.
Lecture 10 Lagrangian of a particle in a magnetic field. Gauge invariance. Mechanical vs. canonical momentum. The Hamiltonian. Motion of a charged particle in a uniform magnetic field. Non-existence in practice of magnetic monopoles. Theoretical construction of a magnetic monopole.
Lecture 1 Classical bits. States in classical Physics. Transformations. Reversibility. Computability of states and transformations. Vectors, matrices.
Lecture 2: Electron spin. Complex vector spaces. Bra-kets. Basis \(\ket{u}, \ket{d}\). Quantum bit. Invariance to multiplication by complex scalar. Hermitian matrices as observables.
Lecture 3 Pauli matrices \(\sigma_1, \sigma_2, \sigma_3\). Spin. Pauli matrices action on basis: \(\sigma_1 \ket{u} = \ket{d}\), \(\sigma_1 \ket{d} = \ket{u}\), \(\sigma_2 \ket{u} = i \ket{d}\), \(\sigma_2 \ket{d} = - i \ket{u}\), \(\sigma_3 \ket{u} = \ket{u}\), \(\sigma_3 \ket{d} = - \ket{d}\). Probabilistic interpretation. Each spin corresponds to a spin direction in space.
Lecture 4 Probabilities between any two spin directions. There always exist a direction for which a vector is an eigenvector of measuring spin in that direction. Spin state of two electrons. They have six not four degrees of freedom. Product spins. Quantum entagled pair of electrons. How to create an entangled pair - the singlet state.
Lecture 5, slides. Singlet state. Bell inequality. Violated by singlet first spin measured at 0 degrees, 45 degrees, 90 degrees in xz axis. Conclusion - no deterministic hidden variable model explains quantum mechanics.
Lecture 6 Projections operators, Bell inequality again. Hermitian operator \(K\) as observable, eigenvalue \(\lambda\) as observed value. Probability postulate: For state $\ket{\psi}$ (normalized), $\text{Prob}(K=\lambda) = \bra{\psi} P_\lambda \ket{\psi} = \sum_a \lvert \langle a \lvert \psi \rangle \rvert^2$ (sum of probabilities over orthogonal ways to realize the property). Two-slit experiment, interference, how measurement creates entanglement and destroys interference.