- #1

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

What it means ?

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- Thread starter Quarlep
- Start date

- #1

- 257

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

What it means ?

- #2

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- #3

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If you have The Theoritical Minimum book you can look it Lecture 2.2

- #4

- 470

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If you read a few line below the vector space sentece the author says exactly what I wrote before.

- #5

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I just dind understand how can we show left right spin states just use up and down states.

- #6

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\begin{align}

|l\rangle=&\frac{1}{\sqrt{2}}\left(|1/2\rangle+|-1/2\rangle\right) \\

|r\rangle=&\frac{1}{\sqrt{2}}\left(|1/2\rangle-|-1/2\rangle\right).

\end{align}

So, you can express the left/right states in terms of the up and down ones.

- #7

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- #8

Nugatory

Mentor

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I just don't understand how can we show left right spin states just use up and down states.

It works the same way that we can describe any compass direction using just two directions. Say we're allowed to use just north and east and we represent them as the vectors ##\vec{N}## and ##\vec{E}##.... Then south is ##-\vec{N}##, west is ##-\vec{E}##, northwest is ##\frac{\sqrt{2}}{2}(\vec{N}-\vec{E})## and so forth.

The confusing thing about doing this with spin is that the associated magnetic moments point in opposite spatial directions, so you are tempted to think that spin-down, ##|D\rangle##, is equal to ##-|U\rangle##, the negative of spin-up. But it's not; spin-up and spin-down are orthogonal vectors in the abstract vector space. The easiest way to see this is to look at the representation of these vectors as 1x2 matrices. You'll see that not only is ##|U\rangle=|\psi_{z+}\rangle## not the negative of ##|D\rangle=|\psi_{z-}\rangle## but also that their product is zero and their sum is equal to ##|\psi_{x+}\rangle## which is a left-right spin state.

- #9

Nugatory

Mentor

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A fairly common convention is that if you are standing at the source looking at the detector, the z-axis is up-down, the x-axis is left-right, and you need a slightly more ingenious setup if you're doing three-axis measurements.

- #10

blue_leaf77

Science Advisor

Homework Helper

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A fairly common convention is that if you are standing at the source looking at the detector, the z-axis is up-down, the x-axis is left-right, and you need a slightly more ingenious setup if you're doing three-axis measurements.

Probably using a magnet coil oriented along the beam propagation direction will help.

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