Quantum Mechanics

Fall 2025 Review in Tweets

Happy first day of school! This semester, I'm teaching:

1. #PHY131 Quantum Physics and Technology for Everyone #QuantumForEveryone, a new 2-credit general education course

2. #PHY201 General Physics for the Life Sciences 1 #GenPhys1

3. #PHY531 #QuantumMechanics

— Tom Wong (@thomasgwong) August 19, 2025

Today in #PHY531 #QuantumMechanics: One at a time, electrons are shot at two slits. The resulting distribution resembles a wave's interference pattern. A wave determines the likelihood of the particle's position. It's particle-wave duality! Picture from https://t.co/GmDrhSJFWH. pic.twitter.com/67f4Frw85p

— Tom Wong (@thomasgwong) August 20, 2025

Today in #PHY531 #QuantumMechanics: The wave function for a wave on a string describes the height of the wave, but a quantum wave function is statistical. Its norm-square gives the probability density of where the particle will be found, and it collapses upon measurement.

— Tom Wong (@thomasgwong) August 22, 2025

Today in #PHY531 #QuantumMechanics: Quantum waves are statistical, so let's review probability. Averages include mode (most probable value), median (middle value), & mean (expected value). For spread, the average deviations from the mean is zero, so average the square deviation.

— Tom Wong (@thomasgwong) August 25, 2025

Today in #PHY531 #QuantumMechanics: Review of probability with continuous variables, which mirrors the equations for discrete variables. pic.twitter.com/zuM7CY780b

— Tom Wong (@thomasgwong) August 27, 2025

Today in #PHY531 #QuantumMechanics: For a plane wave to obey the de Broglie relations, energy and momentum are differential operators. The Hamiltonian is kinetic plus potential energy, yielding Schrödinger's equation. Experiments show the equation holds for other wave functions. pic.twitter.com/4DuxXKjwVb

— Tom Wong (@thomasgwong) August 29, 2025

Today in #PHY531 #QuantumMechanics: The expectation value of a quantity is obtained replacing momentum with an operator, sandwiching, and integrating over space. Heisenberg's uncertainty principle quantifies the tradeoff of the standard deviation of position and momentum. pic.twitter.com/mVTNi0sMxG

— Tom Wong (@thomasgwong) September 3, 2025

Today in #PHY531 #QuantumMechanics: For a time-independent potential, the particular solutions to Schrödinger's equation are separable, obtained from the eigenfunctions/values of the Hamiltonian, have constant expectation values, and have definite energy equal to the eigenvalue. pic.twitter.com/yRMb2T8Qni

— Tom Wong (@thomasgwong) September 8, 2025

Today in #PHY531 #QuantumMechanics: For a particle trapped in a infinitely tall box, the states of definite energy are sine waves with nodes at the ends, and the corresponding energies En are quadratic in n. pic.twitter.com/ddr4DD3Wvd

— Tom Wong (@thomasgwong) September 11, 2025

Today in #PHY531 #QuantumMechanics: A particle in a box whose initial wave function is an upside-down parabola can be written as a linear combination of energy eigenfunctions, which is its Fourier series. Its probability density function evolves as shown. pic.twitter.com/33punZsoue

— Tom Wong (@thomasgwong) September 13, 2025

Today in #PHY531 #QuantumMechanics: Review for the first midterm. I've collected my tweets so far at https://t.co/MgumaBuzdr.

— Tom Wong (@thomasgwong) September 15, 2025

Today in #PHY531 #QuantumMechanics: The first midterm! Here's the first problem: pic.twitter.com/h57ebwcbcI

— Tom Wong (@thomasgwong) September 17, 2025

Today in #PHY531 #QuantumMechanics: Classically, we can trap a mass by attaching it to a spring, resulting in simple harmonic motion. Quantumly, we can similarly trap a particle with a harmonic potential. We started finding the eigenfunctions and eigenvalues of the Hamiltonian.

— Tom Wong (@thomasgwong) September 19, 2025

Today in #PHY531 #QuantumMechanics: For the quantum harmonic oscillator, the eigenfunctions of the Hamiltonian is a power series times a Gaussian. If the series is infinite, the eigenfunction is unnormalizable. So it must be a finite polynomial, yielding quantized eigenenergies.

— Tom Wong (@thomasgwong) September 22, 2025

Today in #PHY531 #QuantumMechanics: For a particle trapped in a harmonic potential, the energy eigenfunctions are Hermite polynomials times a Gaussian. Here is how an initial Gaussian wave function, of a different width, evolves in the potential. pic.twitter.com/hyn8jP2Rnd

— Tom Wong (@thomasgwong) September 24, 2025

Today in #PHY531 #QuantumMechanics: For the quantum harmonic oscillator, the Hamiltonian can be written in terms of raising and lower operators, which don't commute. Given an eigenfunction, these operators give new eigenfunctions with energies offset by ħω, creating a "ladder." pic.twitter.com/LhagYV1bln

— Tom Wong (@thomasgwong) September 27, 2025

Today in #PHY531 #QuantumMechanics: The raising and lowering operators of the quantum Harmonic oscillator are Hermitian conjugates of each other. Using this, we derived the normalization constant of the eigenfunctions of the Hamiltonian, yielding our final "ladder." pic.twitter.com/f7QiAsJUct

— Tom Wong (@thomasgwong) September 29, 2025

Grading #PHY531 #QuantumMechanics homework. A student's assignment abruptly ended with a statement, "and that's where I slept." I responded. pic.twitter.com/dxUr1G1uI9

— Tom Wong (@thomasgwong) October 2, 2025

Today in #PHY531 #QuantumMechanics: A free particle is a superposition of plane waves, whose contributions are given by the Fourier transform of the initial wave function. A square-shaped wave function spreads out, as pictured. pic.twitter.com/tCg5KgWyzY

— Tom Wong (@thomasgwong) October 3, 2025

Today in #PHY531 #QuantumMechanics: The eigenfunctions of the Hamiltonian depend on the potential. Potentials that trap particles yield bound states (normalizable and discrete energies). Potentials that don't trap yield scattering states (unnormalizable and continuous energies).

— Tom Wong (@thomasgwong) October 7, 2025

Yesterday in #PHY531 #QuantumMechanics: A classical particle cannot transmit through a delta function barrier, but a quantum particle may tunnel through with probability that increases with the energy of the particle.

— Tom Wong (@thomasgwong) October 10, 2025

Today in #PHY531 #QuantumMechanics: For a delta-function well, a classical particle with E > 0 always transmits, but a quantum particle can reflect. It's like the opposite of tunneling. With E < 0, there is one bound state that decays exponentially from the well.

— Tom Wong (@thomasgwong) October 10, 2025

Today in #PHY531 #QuantumMechanics: For a finite square well, a classical particle with positive energy transmits with probability 1. But a quantum particle only transmits with certainty for particular energies, and more energy does not necessarily mean more transmission.

— Tom Wong (@thomasgwong) October 21, 2025

Yesterday in #PHY531 #QuantumMechanics: For a quantum particle in a box, the number of allowed energies increases as the box gets wider or deeper. In Dirac notation, we express column vectors as kets and their conjugate transposes as bras, and their inner products brackets.

— Tom Wong (@thomasgwong) October 23, 2025

Today in #PHY531 #QuantumMechanics: Generalizing of linear algebra to complex vector spaces and Hilbert space, including the norm, normalization, orthogonality, completeness, and Cauchy-Schwarz inequality.

— Tom Wong (@thomasgwong) October 24, 2025

Yesterday in #PHY531 #QuantumMechanics: A Hermitian (self-adjoint) operator equals its Hermitian conjugate (adjoint). It is an observable. Its eigenvectors are determinate states of the observable, whose measurement outcomes are the eigenvalues.

— Tom Wong (@thomasgwong) October 28, 2025

Today in #PHY531 #QuantumMechanics: We derived the generalized uncertainty principle (pictured). If two observables commute, it is possible to find simultaneous determinate states (eigenfunctions) of both, which have guaranteed outcomes (eigenvalues) with zero spread. pic.twitter.com/AcgdJf4J9J

— Tom Wong (@thomasgwong) October 30, 2025

Yesterday in #PHY531 #QuantumMechanics: In spherical coords, the time-indep Schrödinger eq is a PDE for (r,θ,φ). When V = V(r), separate into an ODE for r and a PDE for (θ,φ). Separate the latter into ODEs for θ and φ, yielding associated Legendre ftns and complex exponentials.

— Tom Wong (@thomasgwong) November 4, 2025

Yesterday in #PHY531 #QuantumMechanics: For a spherically symmetric potential, the angular solution is spherical harmonics. The radial solution evolves with an effective potential, where angular momentum keeps an electron from falling into a proton.

— Tom Wong (@thomasgwong) November 7, 2025

Lately in #PHY531 #QuantumMechanics: For the hydrogen atom, the radial wave function is a decaying exponential times a polynomial, and this yields the discrete energies. Multiplying these by spherical harmonics, we get the energy eigenfunctions. pic.twitter.com/ZVwv98b1aG

— Tom Wong (@thomasgwong) November 11, 2025

Yesterday in #PHY531 #QuantumMechanics: Angular momentum is a vector. Its components don't commute, but they do individually commute with the magnitude of the vector. So, we can simultaneously know the magnitude of angular momentum and one component, but not the other components.

— Tom Wong (@thomasgwong) November 13, 2025

Yesterday in #PHY531 #QuantumMechanics: The possible z-components of angular momentum are equally spaced by hbar, forming a ladder obtained by raising and lowering operators. The top and bottom values are equal and opposite, equalling an integer or half integer of hbar.

— Tom Wong (@thomasgwong) November 16, 2025

Last time in #PHY531 #QuantumMechanics: L² and Lz have quantum numbers ℓ and 𝑚 w/ ℓ integer or half-integer. Their eigenvalue relations are solved by spherical harmonics w/ integer ℓ. Intrinsic angular momentum (spin) has integer or half-integer ℓ, and extrinsic integer ℓ.

— Tom Wong (@thomasgwong) November 19, 2025

Today in #PHY531 #QuantumMechanics: A spin-1/2 particle has spin quantum number 𝘴 = 1/2 and spin magnetic quantum number 𝘮ₛ = 1/2 or -1/2. Call these spin up and spin down and express them as spinors/vectors. The components of spin are ℏ/2 times the Pauli matrices. pic.twitter.com/cJCfgKcOpd

— Tom Wong (@thomasgwong) November 19, 2025

Today in #PHY531 #QuantumMechanics: For a spin-1/2 particle, say a measurement of the z-component of spin yields ℏ/2. Measuring the x-component and then the z-component again, we now can get ℏ/2 or -ℏ/2. The z and x-components are incompatible/noncommuting observables.

— Tom Wong (@thomasgwong) November 22, 2025

Today in #PHY531 #QuantumMechanics: In a uniform magnetic field, the expected value of a spin-1/2 system precesses around the magnetic field. In a Stern-Gerlach apparatus, the spin is deflected along or against a non-uniform field, showing that it's quantized.

— Tom Wong (@thomasgwong) November 24, 2025

Today in #PHY531 #QuantumMechanics: Just as spin-1/2 operators can be written as 2x2 matrices, spin-1 operators are 3x3 matrices. We worked out the magnitude-squared of spin; its x, y, and z components; and the raising and lowering operators.

— Tom Wong (@thomasgwong) December 2, 2025

Yesterday in #PHY531 #QuantumMechanics: A spin-s1 particle and a spin-s2 particle can have total spin ranging from s1+s2 to |s1 - s2|. The eigenstates of the total spin are linear combinations of the eigenstates of the individual spins, with Clebsch-Gordon coefficients.

— Tom Wong (@thomasgwong) December 4, 2025