Vita Mentis Challenge for Week 5:
Draw a Graph Highlighting the Difference Bell Found between 'Normal' Probabilities and those Provided by Quantum Mechanics
Get out a pencil with a good eraser, and a blank sheet of paper
Draw a vertically stretched dollar sign
Draw its mirror image
Erase the original dollar sign
Identify 3 points on your reversed, stretched dollar sign:
the 'top point' where the vertical slash meets the top of the reversed S
the 'middle point' where the vertical slash crossed the middle of the reversed S
the 'bottom point' where the vertical slash meets the bottom of the reversed S
Erase the portions of the slash which are above the top point and below the bottom point
Erase the tail end portions of the S which are to the left of the top point and to the right of the bottom point
Draw a circle around the remaining figure using the middle point as the center
Label the circle as if it is a clock, that is:
Label the point on the circle that is straight up above the 'top point' as 12
Label the point on the circle that is straight down below the 'bottom point' as 6
Label the mid point of the arc on the right hand side of the circle as 3
Label the point 1/3 of the way along the arc between 12 and 3 as 1
Label the point half way between 1 and 3 as 2
You can keep going, but that's all the points we will need, and we are going to end up erasing this 'clock' so...
Place your drawing on a flat surface (if it is not already on one)
Place something on your flat surface to point to where the 12 is, but keep it clear of your sheet of paper (we are going to rotate the paper in the next step, so we'll want to know in a bit where the 12 was (that is, where it 12 is right now--maybe a friend can just point to the spot).
Use your pencil to hold the 'middle point' in place, and rotate your page counterclockwise until the 1 is where the 12 used to be (30 degrees)
Draw a line down from the 1 straight through the 'middle point' and to the other side of the circle
Draw a line through the 'top point' that is parallel to the line you just drew through the 'middle point'.
Draw a line from the 'bottom point' so that it crosses, at right angles, the two lines that you have just drawn
Now we can erase the 'clock'
Label the 'bottom point' as 90°
Label the 'top point' as 1
Label the point where our third line crosses the second line as 45°
Label the point where our third line crosses the first line as 0
The third line should now be labled with three points, 0, 45° and 90°, and should be horizontal. Let's call this our x axis. It shows the angle of relative alignment of our detectors (polarizers in the CHSH experiment)
Our first line should be vertical and labled with two points, 0 and 1. Let's call this our y axis. This shows the range of probabilities from 0 to 1 (you could express these as percentages by multiplying by 100, so 1 is the same as 100%)
Draw a line parallel to the x axis that passes through the middle point and intersects with the y axis. Lable the point it hits on the y axis as 0.5 (50%).
Find the point halfway between 0 and 45° on the x axis, lable it as 22.5°.
Draw a line parallel to the y axis that passes through the 22.5° point and intersects with both the slash and the curve of the tilted, inverted, stretched, and partially erased dollar sign. The distance between those two points was one of the things that Bell was interested in.
Find the point halfway between 45° and 90° on the x axis, lable it as 67.5°.
Draw a line parallel to the y axis that passes through the 67.5° point and intersects with both the slash and the curve of the tilted, inverted, stretched, and partially erased dollar sign. The distance between those two points was another thing that Bell was interested in.
Notice that for the line extended from the 22.5° point first crosses the slash and then the curve of the dollar sign. This means that, for this angle of rotation between our detectors, the probability predicted by the slash (that is, regular probability theory) is less than that predicted by the curve (quantum mechanics)
Notice that for the line extended from the 67.5° point first crosses the curve and then the slash of the dollar sign. This means that, for this angle of rotation between our detectors, the probability predicted by the slash (that is, regular probability theory) is greater than that predicted by the curve (quantum mechanics)
What Bell and CHSH do is add when the values are above the middle point (.5 probability) and subtract when the values are below it, so quantum mechanics keeps 'winning' because it either adds more (when its above .5) or subtracts less (when it is below).
For the purposes of making the point clear, the diagram we have produced is not quite to scale. If you would like to see the real thing, use a graphing calculator (e.g., Desmos's) to map out these two equations:
y=(cos(θ))^{2} and
y=(-2x/π)+1. NOTE: If you use the Desmos graphing calculator, set the x axis to have radians as units by hiting the wrench icon and type 'pi' in the blank for "step" next to the range for the x axis.