Bra Ket Notation

Just a few notes here about shiny things that caught my eye regarding bra and ket notation in the quantum mechanics II lecture last night.

Inner Products
Inner products are the Hilbert space, quantum mechanical, state vector equivalent of the dot product for more standard vectors like position or velocity.  The unit basis ket , at least in our class, is written as

where j is the index of the component.  Associating back to Cartesian coordiantes, 1 would denote x, 2 would denote y, and 3 would denote z.  The ket vector is the same symbol in a ket and when the two are applied to each other we get the inner product

.

In other words, the inner product only produces contributions from like basis vectors, just like the dot product.

So here's the cool bit, the following all accomplish about the same thing, they find a number proportional to the magnitude of the component of one vector, lying along another vector whether those  vectors are what we most typically call vectors, or what we call bras and kets, or what we call wavefunctions:

The Projection Operator

Applying the following notation, (the projection operator), had continued to confuse me until last week.

To pull out the portion of a bra vector that points along the basis vector u sub j, the above can be applied as follows:

Applying the projection operator always seemed a bit daunting to me until I saw the entire operation written in several steps utilizing the associative property and the fact that bras and kets commute with scalars as

.

Cool note:

The projection operator is an example of an outer product.  An outer product of two vectors produces a square matrix.  The trace of the matrix is equal to the dot, (or inner), product of the two vectors.

Picture of the Day:

From 1/21/13