https://gdpark.blog/tags/quantum-mechanics/Recent content in Quantum Mechanics on gdpark.blogHugoenWed, 19 Aug 2015 00:00:00 +0000https://gdpark.blog/posts/quantum-mechanics-01-introduction-to-quantum-mechanics-on-probability/Mon, 10 Aug 2015 00:00:00 +0000https://gdpark.blog/posts/quantum-mechanics-01-introduction-to-quantum-mechanics-on-probability/Two weeks into quantum mechanics and already in full hair-loss mode — here&rsquo;s my honest attempt to make sense of the Schrödinger equation and what a wave function even is.https://gdpark.blog/posts/quantum-mechanics-02-normalization/Mon, 10 Aug 2015 00:00:00 +0000https://gdpark.blog/posts/quantum-mechanics-02-normalization/Why normalizing your wavefunction once is enough — turns out the Schrödinger equation secretly guarantees it stays normalized for all time.https://gdpark.blog/posts/quantum-mechanics-03-the-momentum-operator/Tue, 11 Aug 2015 00:00:00 +0000https://gdpark.blog/posts/quantum-mechanics-03-the-momentum-operator/Cranking through the math to differentiate ⟨x⟩ with respect to time — and stumbling our way into the momentum operator.https://gdpark.blog/posts/quantum-mechanics-04-the-time-independent-schr-dinger-equation/Tue, 11 Aug 2015 00:00:00 +0000https://gdpark.blog/posts/quantum-mechanics-04-the-time-independent-schr-dinger-equation/A breezy play-by-play of how Planck, Einstein, de Broglie, and Schrödinger loaded the bases and derived the time-independent Schrödinger equation — no mysticism required!https://gdpark.blog/posts/quantum-mechanics-05-the-time-independent-schr-dinger-equation-part-2/Tue, 11 Aug 2015 00:00:00 +0000https://gdpark.blog/posts/quantum-mechanics-05-the-time-independent-schr-dinger-equation-part-2/We assume the wave function splits into space and time parts, plug it into the Schrödinger equation, and crank out two way simpler ODEs — same trick as E&amp;M!https://gdpark.blog/posts/quantum-mechanics-06-the-infinite-square-well/Tue, 11 Aug 2015 00:00:00 +0000https://gdpark.blog/posts/quantum-mechanics-06-the-infinite-square-well/We dive into the infinite square well, crank through boundary conditions on the Schrödinger equation, and land on quantized wave functions — same moves as E&amp;M, surprisingly!https://gdpark.blog/posts/quantum-mechanics-07-the-harmonic-oscillator-and-ladder-operators/Thu, 13 Aug 2015 00:00:00 +0000https://gdpark.blog/posts/quantum-mechanics-07-the-harmonic-oscillator-and-ladder-operators/We tackle case 2 of V(x) — the harmonic oscillator — unpack why springs are <em>everywhere</em> in physics, then get into ladder operators to solve it.https://gdpark.blog/posts/quantum-mechanics-08-ladder-operators/Thu, 13 Aug 2015 00:00:00 +0000https://gdpark.blog/posts/quantum-mechanics-08-ladder-operators/We pick up the ladder-operator trick to climb from the ground state through every ψ_n, then chase down the normalization constants c_n and d_n purely in terms of n.https://gdpark.blog/posts/quantum-mechanics-09-algebraic-solution-to-the-harmonic-oscillator/Fri, 14 Aug 2015 00:00:00 +0000https://gdpark.blog/posts/quantum-mechanics-09-algebraic-solution-to-the-harmonic-oscillator/Picking up right where we left off — muscling through the harmonic oscillator&rsquo;s Schrödinger equation with a slick substitution and some algebraic tricks!https://gdpark.blog/posts/quantum-mechanics-10-the-free-particle/Mon, 17 Aug 2015 00:00:00 +0000https://gdpark.blog/posts/quantum-mechanics-10-the-free-particle/We crack open the free particle case (V(x) = 0 everywhere) — it looks deceptively easy until two existential problems crash the party almost immediately.https://gdpark.blog/posts/quantum-mechanics-11-the-delta-function-potential/Mon, 17 Aug 2015 00:00:00 +0000https://gdpark.blog/posts/quantum-mechanics-11-the-delta-function-potential/Time for the delta-function potential — we sort out bound vs. scattering states, meet the Dirac delta, and motivate why V(x) = −αδ(x) is a solid model for electrons in a material.https://gdpark.blog/posts/quantum-mechanics-12-delta-function-potential-scattering-states-and-quantum-tunne/Mon, 17 Aug 2015 00:00:00 +0000https://gdpark.blog/posts/quantum-mechanics-12-delta-function-potential-scattering-states-and-quantum-tunne/Tackling the E&gt;0 case of the delta-function potential — scattering states, complex exponentials that refuse to die, and how tunneling actually works.https://gdpark.blog/posts/quantum-mechanics-13-the-finite-square-well/Mon, 17 Aug 2015 00:00:00 +0000https://gdpark.blog/posts/quantum-mechanics-13-the-finite-square-well/Tackling the finite square well — the sneaky middle ground between the infinite well and the delta function — through bound states, boundary conditions, and all that fun stuff.https://gdpark.blog/posts/quantum-mechanics-14-formalism-hilbert-space/Mon, 17 Aug 2015 00:00:00 +0000https://gdpark.blog/posts/quantum-mechanics-14-formalism-hilbert-space/We finally crack open the math behind quantum mechanics — why it works, where it came from, and how Heisenberg&rsquo;s beach epiphany kicked the whole thing off.https://gdpark.blog/posts/quantum-mechanics-15-formalism-observables-and-hermitian-operators/Mon, 17 Aug 2015 00:00:00 +0000https://gdpark.blog/posts/quantum-mechanics-15-formalism-observables-and-hermitian-operators/Why measurement in QM always spits out real numbers — and how that forces every observable to be a Hermitian operator. (Daggers are literally knives, lol.)https://gdpark.blog/posts/quantum-mechanics-16-formalism-determinate-states/Mon, 17 Aug 2015 00:00:00 +0000https://gdpark.blog/posts/quantum-mechanics-16-formalism-determinate-states/Transcribing Griffiths on determinate states — turns out demanding zero standard deviation just pops out the eigenvalue equation, and it all chains together beautifully.https://gdpark.blog/posts/quantum-mechanics-17-the-uncertainty-principle/Tue, 18 Aug 2015 00:00:00 +0000https://gdpark.blog/posts/quantum-mechanics-17-the-uncertainty-principle/We derive the generalized uncertainty principle and chase down exactly when it hits its minimum — turns out the answer is when ψ is Gaussian, and here&rsquo;s why!https://gdpark.blog/posts/quantum-mechanics-18-two-level-systems/Tue, 18 Aug 2015 00:00:00 +0000https://gdpark.blog/posts/quantum-mechanics-18-two-level-systems/We set up a two-state quantum system with a 2×2 Hermitian Hamiltonian and crank out the eigenvalues and eigenvectors using the Schrödinger equation.https://gdpark.blog/posts/quantum-mechanics-19-the-schr-dinger-equation-in-spherical-coordinates-3d/Wed, 19 Aug 2015 00:00:00 +0000https://gdpark.blog/posts/quantum-mechanics-19-the-schr-dinger-equation-in-spherical-coordinates-3d/We bump quantum mechanics up to 3D for the hydrogen atom, watch Cartesian coordinates immediately implode, and brace ourselves for the spherical coordinate nightmare.https://gdpark.blog/posts/quantum-mechanics-20-the-hydrogen-atom/Wed, 19 Aug 2015 00:00:00 +0000https://gdpark.blog/posts/quantum-mechanics-20-the-hydrogen-atom/We finally plug the real hydrogen electric potential into the radial equation, grind through the algebra, and find R(r) — normalization and all.