How Does the Preferred Basis Problem Impact Quantum State Representation?

[bon]

Homework Statement

I'm trying to understand the preferred basis problem in the foundations of QM

Ok so I read somewhere that in general any state can be decomposed in different ways.

I don't quite see how this is meant to work

Suppose 'up' / 'down' represent z component of ang mom, then one spin state is

psi = 1/root 2 |up> + 1/root 2 |down>

What's another way to represent this same state in a different basis? How does it relate to the preferred basis problem?

Thanks

Homework Equations

The Attempt at a Solution

 

[Chopin]

Are you familiar with change-of-basis problems for regular finite-dimensional vector spaces? For instance, if you take a plain old Cartesian graph, you can define a set of basis vectors (up-down, side-to-side). Then a vector pointing to the right is (1,0), and a vector pointing up is (0,1). A vector with unit norm pointing at a 45 degree angle is 1/sqrt(2) (1, 1). However, you could also define the basis vectors as (45 degrees right, 45 degrees left), in which case the 45 degree vector becomes (1,0), and the up-down and side-to-side vectors are mixed.

In QM it works exactly the same way. In the spin system, you have Spin Z+ and Spin Z-, so you can define those as the basis vectors and have (1,0) and (0,1) as the two basis states. However, you can also define spin in a different direction, in which case the Z+ and Z- states will be represented as a mixed vector, just like the 45-degree vector above.