Quantum mechanics is for many students also one of the more challenging of the core courses because, for the first time, their day-to-day experiences about the world provides little intuition about the quantum world, and they are forced to rely on an abstract mathematical theory to understand electrons, atoms and molecules. For example, the familiar concepts of the path of a particle, the velocity of a particle, and a force acting on a particle do not show up in quantum mechanics. One instead has to gain intuition about atoms and molecules by working with more abstract concepts such as kets (vectors in a complex-valued Hilbert space) and Hermitian operators.
Quantum mechanics is one of the most important core courses to take and understand because of the prominent role of quantum mechanics in nearly all frontiers of 21st-century science and engineering. Quantum mechanics lies at the heart of chemistry, material science, and nanoscience. Quantum mechanics is crucial for the design of ever smaller computer chips that are needed to produce ever more powerful and energy efficient computers, and quantum mechanics may lead to radical quantum computers that are more powerful than any existing digital computer for solving certain classes of problems. (An example would be factoring a large integer, a difficult mathematical problem for which an efficient algorithm would disrupt the world economy by breaking the most widely used encryption methods). On the science side, some of the most profound questions about the universe such as where did it come from (the Big Bang), what is it made of and why (the zoo of subatomic particles), what will happen to the universe (the accelerated expansion of the universe, dark matter, and dark energy), and even what is space and time---all these questions assume knowledge of quantum mechanics as a starting point.
Duke students first learn about quantum mechanics and some of its applications in Physics 264. Physics 464 continues from 264 by discussing a mathematically abstract but powerful formalism invented by Dirac and others that generalizes the Schrodinger equation (which describes the evolution of a wave function that depends on space and time) to include systems that have spin (intrinsic angular momentum states), which have no spatial structure but do evolve in time. The Dirac formalism greatly clarifies the logical structure of quantum mechanics and also relates that structure to the Hamiltonian and Lagrangian formulations of classical mechanics that were discussed in Physics 361.
The Dirac formalism, in its essence, says that to understand nature at the atomic level, one needs to calculate eigenvalues and eigenvectors of certain matrices that may be of infinite size. The eigenvalues predict precisely what values of a certain quantity like energy or momentum will be observed in a particular experiment, while the corresponding eigenvectors are used to predict the probability of observing a particular value at a particular time, based on the initial state of the experiment. There is therefore a remarkable relation between linear algebra (eigenvalues and eigenvectors of Hermitian matrices) and nature at the quantum level. Why this is the case remains a puzzle (no one knows why the world obeys quantum mechanics) but tens of thousands of experiments have so far confirmed quantum mechanics to be correct, and no experiment has so far shown quantum mechanics to be wrong.