2 Rules of the game

As said, the math­e­mat­ical tools of quantum mechanics are prob­a­bility algo­rithms. They serve a single pur­pose, which is to assign prob­a­bil­i­ties to the pos­sible out­comes of mea­sure­ments, and they are gov­erned by two over­ar­ching rules.

Sup­pose that you want to cal­cu­late the prob­a­bility of a par­tic­ular out­come of a mea­sure­ment M2 (per­formed at the time t2), given the out­come of a mea­sure­ment M1 (per­formed at an ear­lier time t1). Here is what you have to do:

Choose any sequence of mea­sure­ments that may be made in the meantime.

Refer to a pos­sible sequence of out­comes as an alter­na­tive.

Assign to each alter­na­tive a com­plex number and refer to it as its ampli­tude.

Apply either of these rules:

  • Rule A: If the inter­me­diate mea­sure­ments are made (or if it is pos­sible to find out what their out­comes would have been if they had been made), first square the mag­ni­tudes of the ampli­tudes of the alter­na­tives and then add the results.
  • Rule B: If the inter­me­diate mea­sure­ments are not made (and if it is impos­sible to find out what their out­comes would have been if they had been made), first add the ampli­tudes of the alter­na­tives and then square the mag­ni­tude of the result.

To get the idea, let’s apply these rules to some exper­i­mental situations.

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