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Alright, let’s do a quick Gram–Schmidt self-check run. You’re driving. I’m the friendly voice in the passenger seat telling you when to blink. We start with vectors v1, v2, v3… maybe in a real vector space, maybe complex. Either way, same vibe: we’re building an orthonormal set one step at a time. Step 1: First vector. Pause and predict: what do we do with v1? …Pause. Yep. We take u1 = v1. No projections yet, because there’s nothing to be orthogonal to. Then we normalize: e1 = u1 / ||u1||. Reason: we want length 1. Quick reminder: in complex spaces, ||u1|| comes from ⟨u1|u1⟩, and the inner product uses conjugation on the first slot. So ⟨u|v⟩ means “conjugate u, then dot with v.” Don’t let your i’s run wild. Step 2: Now v2. Pause: what’s the next move? …Pause. We subtract off the piece of v2 that points in the e1 direction. The projection. So: u2 = v2 − proj_{e1}(v2). And that projection is ⟨e1|v2⟩ e1. Why that exact form? Because e1 is unit length, so ⟨e1|v2⟩ is literally “how much v2 leans toward e1.” Tiny but crucial reminder: it’s ⟨e1|v2⟩, not ⟨v2|e1⟩. Swap them and you might accidentally conjugate the wrong thing. That’s like putting your shoes on the wrong feet. You can walk, but you’ll regret it. Now pause again: after subtracting, what do we do? …Pause. Normalize! e2 = u2 / ||u2||. Reason: we want orthonormal, not just orthogonal. Step 3: Now v3. This is where people forget something. Pause and predict: what projections do we subtract? …Pause. Both. You subtract the parts along e1 and e2. u3 = v3 − ⟨e1|v3⟩ e1 − ⟨e2|v3⟩ e2. Quick reason: u3 must be orthogonal to everything we already accepted into the team. Pause: after you simplify u3, what’s next? …Pause. Normalize again: e3 = u3 / ||u3||. And that’s the whole rhythm: project, subtract, simplify, normalize. Repeat until you run out of vectors… or patience. Now, the final self-check. This is how you know you didn’t make a sneaky inner-product mistake. First check: norms. Each ei should satisfy ||ei|| = 1. That means ⟨ei|ei⟩ = 1. Second check: pairwise orthogonality. For i ≠ j, you want ⟨ei|ej⟩ = 0. If you get a tiny mess, re-check your projection scalars and—say it with me—conjugation in ⟨u|v⟩. If all norms are 1 and all cross inner products are 0, congrats: you built an orthonormal set. Recap: take v1, normalize. Then for each next vk: subtract projections onto earlier e’s using ⟨ei|vk⟩, then normalize. You’ve got this. Gram–Schmidt isn’t scary—just very picky, like a cat with math homework.

Course

Introductory Non‑Relativistic Quantum Mechanics: Postulates, Ope

12 units57 lessons

Topics

Quantum Physics (Non-relativistic Quantum Mechanics)Mathematical PhysicsLinear AlgebraDifferential Equations / Boundary-Value ProblemsComplex Analysis (foundational tools)

About this course

Develop an intuition-first, problem-solving mastery of non-relativistic quantum mechanics using the postulates and operator formalism. Core topics include states and representations (kets/bras and wavefunctions), normalization and phase, the Born rule and expectation values, measurement as projectors and spectral decomposition, and unitary time evolution via the Schrödinger equation and U(t)=e^{-iHt/ħ}. Apply commutators and uncertainty relations, switch between position and momentum pictures, and solve standard Hamiltonians: 1D wells and barriers (including tunneling and probability current), the harmonic oscillator (ladder operators), angular momentum and spin-1/2, and the hydrogen atom. Gain moderate facility with approximation methods such as time-independent perturbation theory, the variational principle, and optional WKB.