# sets

4

Lego Kwik-E-Mart - Springfield’s Favorite Convenience Store

Images from Brickset

6

Native Spring Wildflowers Of Marin California

California Poppy, Wild Rose, Indian Paintbrush, Douglas Iris, Canyon Larkspur and Mission Bells. Get to know your natives!

(Marin, California - 4/2015)

2

## ♡  ♥  Jamie, Dakota & Hearts  ♥  ♡

◕‿◕         ◕‿◕

10

Vadim Voitekhovitch | Oil on Canvas
steel moon
old harbor
leviathan
heavenly guardian
stolen sky
chasing time
tide
the last stagecoach
curfew
arrival

Quantum Physics

## Visualisation of Hilbert space

Cont’d from “Hilbert space”, see “Dirac notation: Analogy with Cartesian vectors

Now, in classical mechanics we deal with a vector space called Euclidean space (or Cartesian space), which is defined by the unitary vector components ex, ey and ez extending in a particular direction to infinity. Therefore this space is entirely filled by the vector e.

Hilbert space is, by contrast, a much more generalised and flexible version of this space. In Hilbert space, an “axis” can be any function or vector (not only e), which can extend for any distance or to infinity.

Imagine a space containing a number of representations completely orthogonal (i.e. at π∕2 radians, or 90°) to one another which are each represented by  a different basis function or vector. Each can have vectors projected on to them to reveal a different property of a corresponding vector or function. Suppose we have the basis functions ψi for i = 1 to 7 in this Hilbert space.

Notice how each basis vector is linearly dependent of every other basis vector within its set, meaning they are orthogonal and have an inner product of 0 with respect to one another.

Let’s now introduce into it a vector ψ, which has projections onto (or “components in”) each of the basis vectors ψi given by

As we can see, we’ve introduced this new vector in red and ‘visualised’ projecting it onto each of its basis vectors. We’ve previously discussed the similarities between Euclidean space and Hilbert space and can here draw a parallel between the process of resolving vectors on to their Cartesian axes and the projection of a state onto its basis.