Hands-on exercises are designed for undergraduate physical chemistry students to derive two-dimensional quantum chemistry from scratch for the H atom and H[subscript 2] molecule, both in the ground state and excited states. By reducing the mathematical complexity of the traditional quantum chemistry teaching, these exercises can be completed independently by students who have basic calculus skills. Students are expected to convert the Laplacian into the polar coordinates; solve the Schrödinger equations for the two-dimensional H atom; write out the electron configurations and atomic terms for many-electron atoms; solve the Schrödinger equations for the H[subscript 2] molecule; obtain its potential energy surfaces in the ground state and excited states; examine the individual kinetic, attractive, and repulsive energy components; and plot and visualize the bonding and antibonding molecular orbitals of the two-dimensional H[subscript 2]. In summary, various exercises are designed for students to derive quantum chemistry for atoms and molecules in two dimensions, from scratch, by themselves. These exercises help reinforce students' understanding of quantum chemistry of real atoms and molecules in three dimensions; they also challenge students to derive quantum chemistry for a fictitious two-dimensional world and to critically evaluate the validity of their derivations. These challenges may help students climb up Bloom's Taxonomy Pyramid of educational objectives toward its highest levels: synthesis of new knowledge and critical evaluation.