Topics in Quantum Mechanics

Introduction

In this section, some topics in quantum mechanics will be presented. These are outside the main topic of the book, which is focused on quantum computing and quantum error prevention. Here the topics are of quantum mechanics, but not directly having to do with those main topics of the book.

Time Independent Perturbation Theory

A fairly good introductory treatment of this topic can be found in Griffiths' book [1]

Unitary transformations are represented in a circuit diagram with a box around the unitary transformation. Consider a unitary transformation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} on a single qubit state Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\vert{\psi}\right\rangle} . If the result of the transformation is $\left\vert {\psi ^{\prime }}\right\rangle$, we can then write

	Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\vert{\psi^\prime}\right\rangle = V\left\vert{\psi}\right\rangle.}	(2.5)

The corresponding circuit diagram is shown in Fig. 2.1.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\vert{\psi}\right\rangle}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\vert{\psi^\prime}\right\rangle}

Figure 2.1: Circuit diagram for a one-qubit gate that implements the unitary transformation $V\,\!$. The input state Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\vert{\psi}\right\rangle} is on the left and the output, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\vert{\psi^\prime}\right\rangle} , is on the right.