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Alright, let’s make friends with Euler’s formula: e to the i-theta equals cos theta plus i sin theta. At first, that looks like math wearing a costume. But it’s really just… rotation. Picture a clock hand on the complex plane. The horizontal axis is “real,” the vertical axis is “imaginary.” One step to the right is 1. One step up is i. Now imagine a hand of length 1, starting at 1 on the real axis. If you rotate it by an angle theta, where does the tip land? Its x-coordinate is cos theta. Its y-coordinate is sin theta. So the point is: cos theta plus i sin theta. That’s Euler’s formula: e^{iθ} is literally the unit-length arrow pointing at angle θ. Now here’s the key idea: why does multiplying by e^{iθ} rotate? Because complex multiplication combines two things at once: - it multiplies lengths, and - it adds angles. So if you have some complex number z, it’s like an arrow with some length and some direction. When you multiply by e^{iθ}, you’re multiplying by a unit arrow that points at angle θ. Length stays the same… but the direction gets shifted by θ. So: z times e^{iθ} is z rotated by θ. Clean. Powerful. Slightly magical. Okay—oscillations time. A real cosine like A cos(ωt + φ) is a wave in time. Here’s the important split: - ωt is the time dependence. It keeps changing. - φ is the fixed phase. It’s a constant offset. Euler lets us package that cosine into a spinning arrow: A cos(ωt + φ) equals the real part of A e^{i(ωt + φ)}. Why “real part”? Because e^{i(ωt+φ)} equals cos(ωt+φ) plus i sin(ωt+φ). Multiply by A, then the real part is exactly A cos(ωt+φ). So you can think of A e^{i(ωt + φ)} as a phasor: an arrow of length A, spinning at angular speed ω. The spin—ωt—is the time part. The initial angle—φ—is the fixed phase. Now the best part: adding oscillations. If two signals have the same ω, their phasors spin together. Same rotation speed. So their sum is just… vector addition in the complex plane. Add the arrows, get a new arrow. That new arrow has: - a new amplitude (its length), and - a new fixed phase (its angle). The time dependence ωt is still there, same as before. Let’s do a concrete example. Say we have: x1(t) = cos(ωt) and x2(t) = cos(ωt + 60°). Write them as phasors with amplitude 1: Phasor 1: 1·e^{i(ωt + 0°)} Phasor 2: 1·e^{i(ωt + 60°)} Factor out the shared time dependence e^{iωt}: Sum = e^{iωt} [ e^{i0°} + e^{i60°} ]. Now the bracket is time-independent. It’s just two fixed arrows: One at 0°, one at 60°. Let’s add them. In components: e^{i0°} = 1 + i0. e^{i60°} = cos60° + i sin60° = 0.5 + i(√3/2). Add: (1 + 0.5) + i(0 + √3/2) = 1.5 + i(√3/2). Now convert that to “length and angle.” Length is: R = sqrt(1.5^2 + (√3/2)^2) = sqrt(2.25 + 0.75) = sqrt(3) ≈ 1.732. Angle is: α = arctan( (√3/2) / 1.5 ) = arctan( √3 / 3 ) = 30°. So the whole sum is: Re{ e^{iωt} · R e^{iα} } = Re{ R e^{i(ωt + α)} } which means: x1(t) + x2(t) = R cos(ωt + α) = √3 · cos(ωt + 30°). Interpretation: same frequency ω. The time dependence is still ωt. But the amplitude grew to √3, and the fixed phase shifted to 30°. And that’s the phasor superpower: messy trig becomes simple arrow math. Quick recap: e^{iθ} is a rotation. Multiplying by it rotates an arrow. A cosine is the real part of a spinning phasor. And adding same-frequency waves is just adding vectors. If that clicked even a little, you’re already doing complex numbers the cool way.

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.