Starting in this chapter, and in the following two chapters, we will take all of the abstractness of quantum mechanics that we have developed and make it concrete in actual examples. Our goal for these examples will be to solve the time-independent eigenvalue problem for the Hamiltonian $\hat{H}$

Welcome to the world of quantum mechanics! I’ll be your guide through this fascinating, counterintuitive, and extremely rich subject. Quantum mechanics underlies nearly all of contemporary physics research; from the inner workings of atoms, to properties of materials, to the physics of neutron stars, to what happens at a black hole. Additionally, quantum mechanics and its consequences are exploited in modern technology and are beginning to produce major breakthroughs in computing. In this book, we will introduce the formalism, axioms, and a new way of thinking quantum mechanically that will require a re-interpretation of the classical physics you have learned.

In this chapter we will discuss the , for which the potential is simply 0 for all $x\in (-\mathrm{\infty },\mathrm{\infty })$. That is, the Hamiltonian is just the kinetic energy

In this chapter, we will study the quantumwhich, just as in classical mechanics, will be our “canonical” quantum system and one which we can apply to a huge variety of systems. Unlike the infinite square well, we will find that the harmonic oscillator is not most naturally expressed in position space. This new formulation of the harmonic oscillator’s Hamiltonian will demonstrate the power of exploiting commutation relations in quantum mechanics and will be a foundation for analyses in future chapters. Eigenstates of the harmonic oscillator, especially its lowest-energy state, have a deep connection to the Heisenberg uncertainty principle and open the door to studying a whole class of states that bridge the chasm between quantum wavefunctions and classical particles.

Our study of linear algebra in the previous chapter suggested some interesting features of a deeper structure. First, as a study of the derivative operator suggested, we will consider a complex vector space, spanned by some set of orthonormal vectors $\{{\vec{v}}_i\}$, where orthonormality is defined by

In this chapter, we are going to step back from the exact solutions of the Schrödinger equation (i.e., diagonalization of the Hamiltonian) and introduce methods for approximation. We have nearly exhausted those problems for which there exists a known, exact, analytic solution to the Schrödinger equation, and much of modern research on quantum mechanics and its generalizations focuses around approaches to approximating the eigenvalues of the Hamiltonian. We will just barely scratch the surface of techniques for approximation in this chapter, focusing on four different methods. The techniques we introduce here are general techniques for determining approximations to eigenvalues of linear operators or matrices, but applied to the problem of solving a quantum system, namely, diagonalization of its Hamiltonian.

Up until this point, we have essentially exclusively studied the quantum mechanics of individual particles. In this chapter, we will consider ensembles of quantum mechanical particles and attempt to understand their collective dynamics (or at least provide a brief introduction). By “ensemble,” I mean that we are considering a quantum system with more than one particle in it. So, instead of, say, one spin-1/2 particle, we will consider a collection of N spin-1/2 particles, or N particles in the infinite square well, or something like that. With this set-up, we can immediately see that the quantum description of states on a Hilbert space is insufficient and incomplete for describing this collection of particles.

With scattering and its formalism in the previous chapter, we pivot to studying quantum mechanics in more than one dimension. In this and the next chapter, we will work to generalize our formulation of quantum mechanics to be more realistic (i.e., actually account for the multiple spatial dimensions that we experience in our universe). In this chapter, we will introduce a profound consequence of living in multiple spatial dimensions and a study of angular momentum in quantum mechanics. We will start by setting the stage for describing rotations and later, construct the complete theory of angular momentum, at least as much as we need here.

Quantum mechanics is ultimately and fundamentally a framework for understanding Nature through the formalism of linear algebra, vector spaces, and the like. In principle, your study of linear algebra from an introductory mathematics course would be necessary and sufficient background for studying quantum mechanics, but such a course is typically firmly rooted in the study of matrices and their properties. This is definitely relevant for quantum mechanics, but we will need a more general approach to linear algebra to be able to describe the dynamics of interesting physical systems and make predictions. Perhaps the most familiar and important linear operator, the derivative, was not even discussed within that context in a course on linear algebra. Because of this familiarity, studying properties of the derivative is an excellent place to begin to dip our toes into the shallow waters of the formalism of quantum mechanics.

In this chapter, we will introduce the , which is a formulation of quantum mechanics equivalent to the Schrödinger equation, but has a profoundly distinct interpretation. Further, the path integral is very easily extended to incorporate special relativity, which is very challenging and inconvenient within the context of the Schrödinger equation. So, what is the idea of this path integral? Our goal will be to calculate the amplitude for a quantum mechanical particle that starts at position xi at time t = 0 and ends at position xf at some later time $t=T>0$. In some sense, this question is analogous to what you ask in an introduction to kinematics in introductory physics; however, its analysis in quantum mechanics will prove to be a bit more complicated than that in the first week of your first course in physics.