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Alright—let’s meet one of the most famous translations in quantum mechanics. We start with an abstract object: the ket, |ψ⟩. It’s the full description of a quantum state… but it’s kind of like saying, “I have a song,” without telling me the notes. So we pick a “coordinate system” for states. If we choose position, we ask: what does this state look like at each position x? That’s where this comes in: ψ(x) = ⟨x|ψ⟩. Read it like: “the amplitude for finding the state |ψ⟩ at position x.” Not the probability. The amplitude. And this difference matters. Amplitudes can be negative, or even complex—like having a direction and a phase. Probabilities can’t do that. Probabilities are boring in a good way: they’re always real and nonnegative. So how do we get probability from an amplitude? We take the absolute value squared: |ψ(x)|². That is the probability density for position. Notice I said “density,” not “probability at exactly x.” Because in continuous space, the probability of landing on one exact point is basically zero—like trying to hit exactly 3.0000000000 meters with infinite precision. Not happening. Instead, |ψ(x)|² tells you probability per unit length. So if you want an actual probability, you take a little interval—say from x to x plus a tiny dx—and then the probability is roughly |ψ(x)|² times dx. Density times width equals probability. Nice and physical. Let’s do a story. Imagine two quantum “characters.” Character one is a localized packet: ψ(x) is big around, say, x = 2 meters, and tiny everywhere else. So |ψ(x)|² has a tall bump near 2 meters. If you measure position, you’ll very likely find it near 2. Character two is spread out: ψ(x) is sort of gently nonzero over a wide region. Now |ψ(x)|² is more like a low, wide hill. No single spot is especially likely—your measurement could land in lots of places. Same kind of particle, same rules—just different states. And importantly: ψ(x) isn’t “where the particle is.” It’s the state written in the position language. Now here’s the cool part: position is not the only language. You can represent the same exact abstract state |ψ⟩ in momentum space too, like ϕ(p) = ⟨p|ψ⟩. Different representation, same underlying |ψ⟩—just like the same song written in sheet music or played by ear. Quick recap: ψ(x) = ⟨x|ψ⟩ is the position-space amplitude. |ψ(x)|² is the probability density—probability per unit length. And different representations are just different ways to describe the same state. You’ve got this. Quantum mechanics is weird, yes—but you’re learning to speak its language.

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.