Imagine a system of SternGerlach apparatuses that allow posi

Imagine a system of Stern-Gerlach apparatuses that allow positive spins through, and block negative spins. The first apparatus filters in the X direction, the second silters in the Y direction, and the third filters in the X direction again. To be clear, particles with a positive x spin are allowed through the first and third apparatuses, while particles with a positive y spin are allowed through the second.

Let\'s say we are sending some spin-half particles through these apparatuses. What is the likelihood of any given particle being allowed through all three apparatuses and why?

Solution

spin-half particle will have spin +1/2 or -1/2 with equal probability

when a particle passes through the first filter (X+) it has spin +1/2 in x-direction

the probability it passes throught he first filter =0.5

This is equal to a particle having spin 1/2 along +45 deg xy and -45deg x-y

The probability it passes through the second filter aligned alongy-axis is 0.5

This particle after passing through the y-fiter will have a spin of +1/2 in y-direction

When it is passed through the third filter x+ again as above the input particle is equal to have a spin of +1/2 and -1/2 with an axis that makes 45 deg with x and y axis. Hen cthe probabality it passes through the third filter is again 0.5

The probabality that the particle passes through all the three filters is

= 0.5*0.5*0.5  = 0.125