## Lecture: Introduction to Bra-ket Notation (?? minutes)

1. 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., $v_x \hat{i} + v_y \hat{j}$, $\Vec{v}$, $(v_{x}, v_{y})$, $\pmatrix{v_x \\ v_y}$.
2. 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:
3. 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: $$\Vec {v } \doteq \pmatrix{v_x \\ v_y}$$
4. 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.

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