This is my first question on this site so please improve the question if needed. I am a beginner in Quantum Mechanics and so to study about it, I purchased the book "The Theoretical Minimum : Quantum Mechanics by Leonard Susskind" as an introduction to Quantum Mechanics. I am currently on lecture-2 which is about Quantum States and the author is talking about "spin states" of a system. I have completed "spin along z-axis" and "spin along x-axis" but I am having a doubt in "spin along y-axis". In the spin along y-axis section, the author uses "in" and "out" states labelled as $|i\rangle$ (in) and $|o\rangle$ (out). Then the author mentions that they are orthogonal states i.e. $\langle i|o\rangle = 0$ and vice-versa which mean that if the spin is in then it has $0$ probability to be out which I am clear with. Then the author uses statistical results of our experiments and writes :
$$\begin{align*}
\langle o|u\rangle \langle u|o\rangle &= \frac12 \tag{1}\\
\langle o|d\rangle \langle d|o \rangle &= \frac12 \tag{2}\\
\langle i|u\rangle \langle u|i \rangle &= \frac12 \tag{3}\\
\langle i|d \rangle \langle d|i \rangle &= \frac12 \tag{4}\\
\end{align*}$$
where $u$ and $d$ are the up and down states
I am having a problem in understanding how the product of these "inner-products" equal $\frac12$
My attempt :
I know that $$\begin{align*}
\langle A|u\rangle \langle u|A\rangle &= P_u \\
\langle A|d \rangle \langle d|A \rangle &= P_d
\end{align*}$$
where $P_u$ and $P_d$ are the probabilities for measurements of up and down states and $|A\rangle$ is a generic state. So in the same way, $\langle o|u\rangle \langle u|o\rangle$ is the probability that the spin is up given that out is our generic state and here $|o\rangle$ plays the role of $|A\rangle$. The probability is $\frac12$ because we can choose between two states namely up and down and hence the probability is $\frac12$ and the same holds for the other $3$. Is my logic correct or is there something missing? Any help/hint is appreciated
Thanks in advance 🙂
Best Answer
$|A>$ is the state in which the spin was prepared. Susskind now does an experiment with "turning the apparatus". If you do this to measure $|u>$ (for example), then you get the probability $P_u$. If the prepared spin state $|A>$ is $|o>$, then you get your probability of 1/2. I think this is pretty much your understanding anyways, correct? In further chapters Susskind explains also what happens when the apparatus is turned around an arbitrary angle.