Physics · 19
Attempt these before you open a book or a video. Write the argument in the work box and the conclusion in the answer box. Check counts the problem only when both pass, and it does not open the solution. The solution stays locked for twenty minutes after you start the timer.
Attack 1
In the photoelectric effect, why does a brighter light not eject electrons if the frequency is below a threshold?
Timer not started. Twenty minutes of real work before a solution.
Each electron absorbs one photon of energy \(hf\). If \(hf\) is less than the work function, no electron leaves, no matter how many photons arrive. Intensity changes the number of photons, not the energy of each one. \(K_{\max}=hf-\phi\).
Attack 2
State the de Broglie relation and the wavelength of an electron with momentum \(p\).
Timer not started. Twenty minutes of real work before a solution.
\(\lambda=h/p\). It is a statement about the wavelength of the wavefunction associated with a free particle of momentum \(p\), not a claim that the electron is a little wave on a string.
Attack 3
For a particle in a one-dimensional infinite well from 0 to \(a\), why are the energies proportional to \(n^2\)?
Timer not started. Twenty minutes of real work before a solution.
The stationary states vanish at the walls, so the wavefunction is a sine with \(k a=n\pi\). Then \(E=\hbar^2 k^2/2m\), so \(E_n=n^2\pi^2\hbar^2/(2ma^2)\).
Modern physics on this page is the break with classical mechanics that happens in the first quantum course and the special-relativity chapter that usually sits beside it: photons, de Broglie waves, the Schrödinger equation in one dimension, the infinite well, the harmonic oscillator, tunneling, and the hydrogen atom at the level of its spectrum. Special relativity is the statements you need in order not to lie about simultaneity, time dilation, length contraction, and \(E^2=p^2c^2+m^2c^4\).
This is an entrance, not a graduate quantum course. OpenStax Volume 3 contains the standard modern-physics chapters. MIT 8.04 is the real quantum course and it is the ceiling’s doorway: if you can do its first problem sets, you are past “modern physics” and into quantum mechanics. Griffiths’ Introduction to Quantum Mechanics is the paid book people use next; no edition was verified, so no ISBN is printed. Feynman Volume III is the preview and will not replace 8.04.
A wavefunction is not a smear of the particle. \(|\psi|^2\) is a probability density. Say that sentence correctly or the rest of the page will be mush.
Statements only. Write the argument before you reveal it. A revealed proof you did not attempt is a paragraph you will not be able to rebuild at 2am.
Photons and relativity, briefly. \(E=hf\), \(p=h/\lambda\) for a photon, and \(E^2=p^2c^2+m^2c^4\) for a particle. Time dilation and length contraction follow from the invariance of the interval.
The photon relations are the empirical start of the subject, tied to the photoelectric effect and Compton scattering. Time dilation is a light-clock argument in one frame versus another. Write the light clock once.
Schrödinger. \(i\hbar\partial_t\Psi=\hat H\Psi\). For a stationary state, \(\Psi(x,t)=\psi(x)e^{-iEt/\hbar}\) and \(\hat H\psi=E\psi\). \(\int|\Psi|^2=1\). Observables are expectation values.
The separation of the time-dependent equation when \(H\) is time-independent is ordinary separation of variables. Normalization is the requirement that the probability density integrates to 1. The expectation-value formula is the definition you will use; derive \(d\langle x\rangle/dt=\langle p\rangle/m\) for a standard Hamiltonian once.
Infinite well and oscillator. Infinite well energies are \(n^2\pi^2\hbar^2/(2ma^2)\), wavefunctions sines. The harmonic oscillator has \(E_n=\hbar\omega(n+\tfrac12)\), and the ground state is a Gaussian.
The well is the attack. The oscillator can be solved with Hermite polynomials or with ladder operators. Learn one method well enough to get the zero-point energy without looking.
Tunneling and hydrogen. A barrier of finite height gives exponential decay inside the barrier and a nonzero transmission probability. The hydrogen spectrum is \(E_n=-13.6\,\mathrm{eV}/n^2\), from the Coulomb Schrödinger equation, with degeneracy you should be able to state.
Tunneling is matching oscillatory and exponential solutions and their derivatives at the boundaries. The hydrogen energies you may derive for the radial ground-state ansatz and otherwise quote the separation in spherical coordinates after you have seen it once.
Spin, stated. Electron spin is a two-state system. The Pauli matrices represent the spin operators in the \(S_z\) basis. A spin measurement along an axis yields \(\pm\hbar/2\).
This is linear algebra: a Hermitian operator on \(\mathbb{C}^2\) with the right commutation relations. Compute one expectation value, \(\langle +x|S_z|+x\rangle\), so the matrix language is not decorative.
Four rungs, in order. Do not skip a rung because the next one looks more serious.
Watch or read once. Do not study from this.
Feynman Lectures, Volume III, from the official site
Read once, do not study from this. Volume III starts from amplitudes. It is a preview of the taste of the subject. 8.04 is the course.
The thing you work through.
OpenStax Volume 3, modern physics chapters, then MIT 8.04
OpenStax is the gentle entrance. 8.04 is the real course and becomes the main text as soon as the infinite well feels easy: https://ocw.mit.edu/courses/8-04-quantum-physics-i-spring-2016/
The problem book.
8.04 problem sets
This is the hard layer. Griffiths is the paid book to buy when you want a second source; no edition was verified.
The hardest questions that exist at this level.
8.04 exams
The ceiling of this page. A graduate quals in quantum is the next ceiling and is not linked because no specific public exam was verified.
Names and links. The problem text stays on the source site. Copyrighted contest problems are not copied here. What follows is where they are and why they hurt.
Pass: derive the infinite-well spectrum, state the photoelectric equation and the reason for the threshold, and complete one early 8.04 problem set. Fail if \(|\psi|^2\) is described as the particle being smeared out, without the word probability.
Five sets. Finish a set before you touch the next. Full sentences, not a circled answer.
Ten problems to meet again. Logging a writeup schedules them at 1 day, 7 days, and 30 days in this browser.
Explain the photoelectric threshold, derive the infinite-well energies, and say what \(|\psi|^2\) is. No notes.