Learn Quantum Mechanics with Haskell

To learn quantum mechanics, one must become adept in the use of various mathematical structures that make up the theory; one must also become familiar with some basic laboratory experiments that the theory is designed to explain. The laboratory ideas are naturally expressed in one language, and the theoretical ideas in another. We present a method for learning quantum mechanics that begins with a laboratory language for the description and simulation of simple but essential laboratory experiments, so that students can gain some intuition about the phenomena that a theory of quantum mechanics needs to explain. Then, in parallel with the introduction of the mathematical framework on which quantum mechanics is based, we introduce a calculational language for describing important mathematical objects and operations, allowing students to do calculations in quantum mechanics, including calculations that cannot be done by hand. Finally, we ask students to use the calculational language to implement a simplified version of the laboratory language, bringing together the theoretical and laboratory ideas.Comment: In Proceedings TFPIE 2015/6, arXiv:1611.0865

Learn Physics by Programming in Haskell

We describe a method for deepening a student's understanding of basic physics by asking the student to express physical ideas in a functional programming language. The method is implemented in a second-year course in computational physics at Lebanon Valley College. We argue that the structure of Newtonian mechanics is clarified by its expression in a language (Haskell) that supports higher-order functions, types, and type classes. In electromagnetic theory, the type signatures of functions that calculate electric and magnetic fields clearly express the functional dependency on the charge and current distributions that produce the fields. Many of the ideas in basic physics are well-captured by a type or a function.Comment: In Proceedings TFPIE 2014, arXiv:1412.473

P wave velocity variations in the Coso Region, California, derived from local earthquake travel times

Inversion of 4036 P wave travel time residuals from 429 local earthquakes using a tomographic scheme provides information about three-dimensional upper crustal velocity variations in the Indian Wells Valley-Coso region of southeastern California. The residuals are calculated relative to a Coso-specific velocity model, corrected for station elevation, weighted, and back-projected along their ray paths through models defined with layers of blocks. Slowness variations in the surface layer reflect local geology, including slow velocities for the sedimentary basins of Indian Wells and Rose valleys and relatively fast velocities for the Sierra Nevada and Argus Mountains. In the depth range of 3–5 km the inversion images an area of reduced compressional velocity in western and northern Indian Wells Valley but finds no major velocity variations beneath the Coso volcanic field to the north. These results are consistent with a recent study of anomalous shear wave attenuation in the Coso region. Between 5 and 10 km depth, low-velocity areas (7% slow) appear at the southern end of the Coso volcanics, reaching east to the Coso Basin. Numerical tests of the inversion's resolution and sensitivity to noise indicate that these major anomalies are significant and well-resolved, while other apparent velocity variations in poorly sampled areas are probably artifacts. The seismic data alone are not sufficient to uniquely characterize the physical state of these low-velocity regions. Because of the Coso region's history of Pleistocene bimodal volcanism, high heat flow, geothermal activity, geodetic deformation, and seismic activity, one possibility is to link the zones of decreased P velocity to contemporary magmatic activity

Maximum stabilizer dimension for nonproduct states

Composite quantum states can be classified by how they behave under local unitary transformations. Each quantum state has a stabilizer subgroup and a corresponding Lie algebra, the structure of which is a local unitary invariant. In this paper, we study the structure of the stabilizer subalgebra for n-qubit pure states, and find its maximum dimension to be n-1 for nonproduct states of three qubits and higher. The n-qubit Greenberger-Horne-Zeilinger state has a stabilizer subalgebra that achieves the maximum possible dimension for pure nonproduct states. The converse, however, is not true: we show examples of pure 4-qubit states that achieve the maximum nonproduct stabilizer dimension, but have stabilizer subalgebra structures different from that of the n-qubit GHZ state.Comment: 6 page

Werner state structure and entanglement classification

We present applications of the representation theory of Lie groups to the analysis of structure and local unitary classification of Werner states, sometimes called the {\em decoherence-free} states, which are states of $n$ quantum bits left unchanged by local transformations that are the same on each particle. We introduce a multiqubit generalization of the singlet state, and a construction that assembles these into Werner states.Comment: 9 pages, 2 figures, minor changes and corrections for version