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robphippen

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- TL;DR Summary
- Does this quote from 'Quantum Mechanics: The Theoretical Minimum' make it self-evident that a spin state can be represented by a 2d vector space?

There is a passage in this book where I don't follow the logic;

In this short quotation from 'Quantum Mechanics: The Theoretical Minimum' by Leonard Susskind and Art Friedman

The book goes on to make it crystal clear that this final statement, that a spin can be represented via a 2d vector space, is true and I totally follow those arguments. So I am

However, the wording of the quote above seems to imply that the argument above

In this short quotation from 'Quantum Mechanics: The Theoretical Minimum' by Leonard Susskind and Art Friedman

- [itex]\mathcal{A}[/itex] represents the apparatus that is performing the measurement
- the apparatus can be oriented (in principle) in any direction
- the item being measured is a single spin

Let’s begin by labeling the possible spin states along the three coordinate axes.

If [itex]\mathcal{A}[/itex] is oriented along the z axis, the two possible states that can be prepared correspond to [itex]\sigma_{z}[/itex] = ± 1. Let’s call them up and down and denote them by ket-vectors | u > and | d >. Thus, when the apparatus is oriented along the z axis and registers + 1, the state | u > has been prepared.

On the other hand, if the apparatus is oriented along the x-axis and registers − 1, the state | l > has been prepared. We’ll call it left.

If [itex]\mathcal{A}[/itex] is along the y axis, it can prepare the states | i > and | o > (in and out).

You get the idea. The idea that there are no hidden variables has a very simple mathematical representation: the space of states for a single spin has only two dimensions. This point deserves emphasis:

All possible spin states can be represented in a two-dimensional vector space.”

The book goes on to make it crystal clear that this final statement, that a spin can be represented via a 2d vector space, is true and I totally follow those arguments. So I am

**not**asking if the statement is true.However, the wording of the quote above seems to imply that the argument above

**in itself**is enough to establish that a spin can be represented via a two-dimensional vector space. That is not obvious to me**from this section alone.**Can anyone enlighten me?