{"id":3968,"date":"2024-09-19T12:27:22","date_gmt":"2024-09-19T12:27:22","guid":{"rendered":"https:\/\/workhouse.sweetdishy.com\/?p=3968"},"modified":"2024-09-19T12:27:23","modified_gmt":"2024-09-19T12:27:23","slug":"an-experiment-to-illustrate-superpositions","status":"publish","type":"post","link":"https:\/\/workhouse.sweetdishy.com\/index.php\/2024\/09\/19\/an-experiment-to-illustrate-superpositions\/","title":{"rendered":"An Experiment to Illustrate Superpositions"},"content":{"rendered":"\n

An experiment such as that in Figure 2.6 represents a measurement out-<\/p>\n\n\n\n

come: the average value of the z-component of spin of the input beam is<\/p>\n\n\n\n

+1\/2, obtained by the weighted probabilities of spin 1\/2 at the up port and<\/p>\n\n\n\n

spin \u22121\/2 at the down port, which in quantum notation is:<\/p>\n\n\n\n

hS<\/p>\n\n\n\n

z<\/p>\n\n\n\n

i<\/p>\n\n\n\n

|ii<\/p>\n\n\n\n

=<\/p>\n\n\n\n

1<\/p>\n\n\n\n

2<\/p>\n\n\n\n

|h\u2191<\/p>\n\n\n\n

z<\/p>\n\n\n\n

|\u2191<\/p>\n\n\n\n

z<\/p>\n\n\n\n

i|<\/p>\n\n\n\n

2<\/p>\n\n\n\n

+ (\u2212<\/p>\n\n\n\n

1<\/p>\n\n\n\n

2<\/p>\n\n\n\n

)|h\u2193<\/p>\n\n\n\n

z<\/p>\n\n\n\n

|\u2191<\/p>\n\n\n\n

z<\/p>\n\n\n\n

i|<\/p>\n\n\n\n

2<\/p>\n\n\n\n

=<\/p>\n\n\n\n

1<\/p>\n\n\n\n

2<\/p>\n\n\n\n

.<\/p>\n\n\n\n

Similarly, the experiment in Figure 2.4 gives us the average value of the x-<\/p>\n\n\n\n

component of spin the input beam |\u2191<\/p>\n\n\n\n

z<\/p>\n\n\n\n

i, which is zero:<\/p>\n\n\n\n

hS<\/p>\n\n\n\n

z<\/p>\n\n\n\n

i<\/p>\n\n\n\n

|ii<\/p>\n\n\n\n

= +<\/p>\n\n\n\n

1<\/p>\n\n\n\n

2<\/p>\n\n\n\n

|h\u2191<\/p>\n\n\n\n

x<\/p>\n\n\n\n

|\u2191<\/p>\n\n\n\n

z<\/p>\n\n\n\n

i|<\/p>\n\n\n\n

2<\/p>\n\n\n\n

+ (\u2212<\/p>\n\n\n\n

1<\/p>\n\n\n\n

2<\/p>\n\n\n\n

)|h\u2193<\/p>\n\n\n\n

x<\/p>\n\n\n\n

|\u2191<\/p>\n\n\n\n

z<\/p>\n\n\n\n

i|<\/p>\n\n\n\n

2<\/p>\n\n\n\n

= 0.<\/p>\n\n\n\n

Consider a beam of spin-up electrons from the \ufb01lter SG<\/p>\n\n\n\n

z<\/p>\n\n\n\n

\u2191. We now set up<\/p>\n\n\n\n

a second Stern\u2013Gerlach machine, but rotated by an angle \u03b8 to z. Let us label<\/p>\n\n\n\n

this machine \u201cSG<\/p>\n\n\n\n

\u03b8<\/p>\n\n\n\n

\u201d. Into this machine we pass the beam of |\u2191<\/p>\n\n\n\n

z<\/p>\n\n\n\n

i electrons.<\/p>\n\n\n\n

What would be the measured output? The schematic setup is in Figure (2.7)<\/p>\n\n\n\n

and the outputs are |\u2191<\/p>\n\n\n\n

\u03b8<\/p>\n\n\n\n

i state at the up port and |\u2193<\/p>\n\n\n\n

\u03b8<\/p>\n\n\n\n

i state at the down port.<\/p>\n\n\n\n

We wish to predict the probabilities of each state.<\/p>\n\n\n\n

FIGURE 2.7: An experiment with the SG along an arbitrary direction \u03b8.<\/p>\n\n\n\n

Now the \u201cclassical\u201d projection cos \u03b8\u00d71\/2 of the spin of the incoming beam<\/p>\n\n\n\n

along the \u03b8 direction, gives us an average value for the measured spin, weighing<\/p>\n\n\n\n

in both the output ports. The intensity of the up beam gives an average<\/p>\n\n\n\n

spin of +1\/2 with probability P(\u2191<\/p>\n\n\n\n

\u03b8<\/p>\n\n\n\n

) = |h\u2191<\/p>\n\n\n\n

\u03b8<\/p>\n\n\n\n

|\u2191<\/p>\n\n\n\n

z<\/p>\n\n\n\n

i|<\/p>\n\n\n\n

2<\/p>\n\n\n\n

and \u22121\/2 with probability<\/p>\n\n\n\n

P(\u2193<\/p>\n\n\n\n

\u03b8<\/p>\n\n\n\n

) = |h\u2193<\/p>\n\n\n\n

\u03b8<\/p>\n\n\n\n

|\u2191<\/p>\n\n\n\n

z<\/p>\n\n\n\n

i|<\/p>\n\n\n\n

2<\/p>\n\n\n\n

. The total probability of this happening is cos \u03b8. So we have<\/p>\n\n\n\n

1<\/p>\n\n\n\n

2<\/p>\n\n\n\n

cos \u03b8 =<\/p>\n\n\n\n

1<\/p>\n\n\n\n

2<\/p>\n\n\n\n

P(\u2191<\/p>\n\n\n\n

\u03b8<\/p>\n\n\n\n

) \u2212<\/p>\n\n\n\n

1<\/p>\n\n\n\n

2<\/p>\n\n\n\n

P(\u2193<\/p>\n\n\n\n

\u03b8<\/p>\n\n\n\n

). (2.8)<\/p>\n\n\n\n

\"\"<\/p>\n\n\n\n

24 Introduction to Quantum Physics and Information Processing<\/p>\n\n\n\n

We also have an equation for conservation of probability:<\/p>\n\n\n\n

1 = P(\u2191<\/p>\n\n\n\n

\u03b8<\/p>\n\n\n\n

) + P(\u2193<\/p>\n\n\n\n

\u03b8<\/p>\n\n\n\n

). (2.9)<\/p>\n\n\n\n

From these two, we get the probabilities of the up and down spin states in the<\/p>\n\n\n\n

output:<\/p>\n\n\n\n

P(\u2191<\/p>\n\n\n\n

\u03b8<\/p>\n\n\n\n

) = cos<\/p>\n\n\n\n

2<\/p>\n\n\n\n

\u03b8<\/p>\n\n\n\n

2<\/p>\n\n\n\n

, P(\u2193<\/p>\n\n\n\n

\u03b8<\/p>\n\n\n\n

) = sin<\/p>\n\n\n\n

2<\/p>\n\n\n\n

\u03b8<\/p>\n\n\n\n

2<\/p>\n\n\n\n

. (2.10)<\/p>\n\n\n\n

Let\u2019s check this result by comparing with the special cases:<\/p>\n\n\n\n

\u2022 \u03b8 = 0: SG<\/p>\n\n\n\n

\u03b8<\/p>\n\n\n\n

= SG<\/p>\n\n\n\n

z<\/p>\n\n\n\n

,<\/p>\n\n\n\n

P(\u2191) = 1, P(\u2193) = 0.<\/p>\n\n\n\n

\u2022 \u03b8 = \u03c0: SG<\/p>\n\n\n\n

\u03b8<\/p>\n\n\n\n

= SG<\/p>\n\n\n\n

\u2212z<\/p>\n\n\n\n

or an SG<\/p>\n\n\n\n

z<\/p>\n\n\n\n

turned upside down,<\/p>\n\n\n\n

P(\u2191) = 0, P(\u2193) = 1.<\/p>\n\n\n\n

\u2022 \u03b8 = \u03c0\/2: SG<\/p>\n\n\n\n

x<\/p>\n\n\n\n

(or SG<\/p>\n\n\n\n

y<\/p>\n\n\n\n

)<\/p>\n\n\n\n

P(\u2191) =<\/p>\n\n\n\n

1<\/p>\n\n\n\n

2<\/p>\n\n\n\n

= P(\u2193)<\/p>\n\n\n\n

Mathematically we can write the state of the input electron as a superpo-<\/p>\n\n\n\n

sition of the output states of SG<\/p>\n\n\n\n

\u03b8<\/p>\n\n\n\n

:<\/p>\n\n\n\n

|\u2191<\/p>\n\n\n\n

z<\/p>\n\n\n\n

i = \u03b1|\u2191<\/p>\n\n\n\n

\u03b8<\/p>\n\n\n\n

i + \u03b2|\u2193<\/p>\n\n\n\n

\u03b8<\/p>\n\n\n\n

i,<\/p>\n\n\n\n

where<\/p>\n\n\n\n

|\u03b1|<\/p>\n\n\n\n

2<\/p>\n\n\n\n

= P(\u2191<\/p>\n\n\n\n

\u03b8<\/p>\n\n\n\n

) = cos<\/p>\n\n\n\n

2<\/p>\n\n\n\n

\u03b8<\/p>\n\n\n\n

2<\/p>\n\n\n\n

, |\u03b2|<\/p>\n\n\n\n

2<\/p>\n\n\n\n

= P(\u2193<\/p>\n\n\n\n

\u03b8<\/p>\n\n\n\n

) = sin<\/p>\n\n\n\n

2<\/p>\n\n\n\n

\u03b8<\/p>\n\n\n\n

2<\/p>\n\n\n\n

.<\/p>\n\n\n\n

The experiment only tells us the magnitudes of the complex amplitudes \u03b1 and<\/p>\n\n\n\n

\u03b2.<\/p>\n","protected":false},"excerpt":{"rendered":"

An experiment such as that in Figure 2.6 represents a measurement out- come: the average value of the z-component of spin of the input beam is +1\/2, obtained by the weighted probabilities of spin 1\/2 at the up port and spin \u22121\/2 at the down port, which in quantum notation is: hS z i |ii […]<\/p>\n","protected":false},"author":1,"featured_media":3940,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[488],"tags":[],"class_list":["post-3968","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-3-the-quantum-postulates"],"jetpack_featured_media_url":"https:\/\/workhouse.sweetdishy.com\/wp-content\/uploads\/2024\/09\/science-1.png","_links":{"self":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/3968","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/comments?post=3968"}],"version-history":[{"count":1,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/3968\/revisions"}],"predecessor-version":[{"id":3969,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/3968\/revisions\/3969"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/media\/3940"}],"wp:attachment":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/media?parent=3968"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/categories?post=3968"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/tags?post=3968"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}