Andrew J. Larkoski - Quantum Mechanics: A Mathematical Introduction
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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.
Lets first consider a function of one variable, 
. Here, 
is a position, for example, and 
might be the amplitude of a jiggled rope at position 
. Precisely what function 
is, is not relevant for our current discussion. We can draw an example function as illustrated in . On this plot, I have identified the point 
at which the function takes a value 
. Now, from 
, how do I move along in 
to get to a new point a distance 
to the right? The function value at this new point is, of course, 
. However, to get there from the point 
, that is, to establish the value 
exclusively from data at 
, we can use the Taylor expansion
Representation of the action of displacement from point 
to point 
of a function 
.
(2.1) To move a distance 
away from 
, I need to know all of the derivatives of 
, evaluated at 
. This seems very complicated and like we would actually need an infinite amount of data to proceed.
However, lets re-write this Taylor expansion in the compact form
(2.2) Further, because the derivative is a linear operator, we can write
(2.3) Now, the sum in parentheses looks very familiar. If we just think of the multiple derivative 
as the multiplication of 
with itself 
times, the sum has the form
(2.4) which is just the exponential function! Using this generous interpretation for now, the sum over derivatives is then
(2.5) So, the Taylor expansion of the function 
can be represented as
(2.6) This awkward exponential of a derivative is defined by its Taylor expansion. In this construction, we exploited the linearity of the derivative as an operator that acts on functions. In this chapter, we will remind about the properties of linear operators and then use that to provide a profoundly new interpretation of the derivative that we will exploit.
You are most likely familiar with the property of linearity from a course on linear algebra (the similarity of names is not a coincidence). For a matrix 
and vectors 
and 
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