Understanding spin states along different axes

I am reading an introductory book about Quantum Mechanics, and I have a problem understanding one concept that is probably very simple. In particluar, I have two questions.

My book says that a spin state can be expressed as a linear combination of two orthogonal ket vectors (in Hilbert space). Let's imagine a 3D space with the three axes $x,y,z$. If we measure spin along $z$ and we get $+1$, then we call that state $|u\rangle$ (as in "up"). Vice versa, if we get -1, we'll call that state |$d\rangle$. We do this with the other axes and we get the states $|r\rangle$ ("right") and $|l\rangle$ ("left") for the x axis and the states $|i\rangle$ ("in") and $|o\rangle$ ("out") for the y axis.

We then pick two of these vectors arbitrarily (although they need to be orthogonal), so my book takes $|u\rangle$ and $|d\rangle$. Any state $|A\rangle$ can be expressed like this: $$|A\rangle=\alpha_1|u\rangle+\alpha_2 |d\rangle,$$ where $\alpha_1$ and $\alpha_2$ are complex numbers and are numerically equal to the square root of the probability that, measuring the spin along the z axis and having observed $|A\rangle$, one will respectively get the state $|u\rangle$ or $|d\rangle$.

My first question is why is it so? I mean, why are they square roots and not simply the probabilities? Anyway, it is pretty obvious if that's the case that $$|r\rangle=\frac{1}{\sqrt2}|u\rangle+\frac{1}{\sqrt2} |d\rangle,$$ since the direction "right" is orthogonal (spatially) to the $z$ axis, so the spin $+1$ and $-1$ are equally probable.

My second question is in understanding why my book then says that $$|l\rangle=\frac{1}{\sqrt2}|u\rangle-\frac{1}{\sqrt2} |d\rangle.$$ Why is that? Why is there a minus? In writing the "in" and "out" states the thing gets even weirder (with the appearance of the imaginary unit), but I don't wanna go into that before clearly understanding this. Thank you for your time!

Are you reading Susskinds Q.M. Theoretical Minimum ?

If you are, it is the same principle anyway in all books, but remember Susskind has two basis vectors, up and down, and he wants to express (left and right), and (in and out), in terms of combinations of those up and down vectors.

The only way he can do that is by using linear combinations of up and down, and by using the dreaded $i$. The way to test it is see what you get when you multiply left by right, etc, using Dirac notation and remembering the rules of orthogonal kets and bras. It takes a bit of practice, because you are learning 3 different things at the same time.

My first question is why is it so? I mean, why are they square roots and not simply the probabilities?

What happens when you get the square of a square root?

The superpositions contain amplitudes, not probabilities. This allows us to have more freedom, as long as we can get back to real numbers.

The coefficients $\alpha$ are amplitudes, not probabilities, so they may be complex numbers which, when squared, produce real probabilities. Whether the amplitudes are complex or real, they must produce real numbers as probabilities, a probability of 1/4$i$ is meaningless.

My second question is in understanding why my book then says that $$|l\rangle=\frac{1}{\sqrt2}|u\rangle-\frac{1}{\sqrt2} |d\rangle.$$ Why is that? Why is there a minus?

You are limited in your choice of combinations of up and down, to express the other orthogonal axis in, and linear superpositions are allowed, so a minus is as good as a plus.

It is vital that you understand the difference between amplitudes, which do allow complex numbers, and probabilities, which don't. You are going from amplitudes to probabilities, under the constraint that probabilities must total 1.

• > "so a minus is as good as a plus." Does this mean that the values obtained for |r> are independent from the values that are obtained for |l>? (sorry I'm writing from my phone) Apr 17 '19 at 11:07