{{page>wiki:headers:hheader}} ===== Lecture: Introduction to Bra-ket Notation (?? minutes) ===== - To stimulate discussion of several different vector representations, students each write down a representation of a 2D vector on their small whiteboards. Then all representations are written on the board and discussed. The instructor should wrap-up the discussion by presenting any common representations that the students do not mention, e.g., $v_x \hat{i} + v_y \hat{j}$, $\Vec{v}$, $(v_{x}, v_{y})$, $\pmatrix{v_x \\ v_y}$. - Students are introduced to the basics of bra-ket notation and given a brief handout to introduce them to how to use it and how it relates to other vector representations: * {{courses:preface:order:brakettablecorrected.pdf|Vector Calculations in Bra-ket notation (.pdf)}} * {{courses:preface:order:brakettablecorrected.docx|Vector Calculations in Bra-ket notation (.docx)}}. - In this lecture, we introduce the representation operator $\doteq$ which indicates that two vectors are not equal, but that one is a representation of the other in a particular basis. For example: $$\Vec {v~} \doteq \pmatrix{v_x \\ v_y}$$ - To get students to begin thinking about the use of complex vectors, we discuss how to draw a 2-dimensional complex vector. Students struggle with this idea in QM since representing a 2D complex vector requires a 4D space. We reinforce the idea that complex vectors work just like real vectors as long as you remember to take the complex conjugate of elements of Bra's. {{page>wiki:footers:courses:prfooter}}