Changes

In quantum mechanics, we can no longer know what the outcome of an experiment will be. It only makes sense to ask what the ''probability'' of a particular outcome is. These probabilities are absolute squares of certain complex numbers called ''amplitudes'' associated to each possible outcome.

In quantum mechanics, we can no longer know what the outcome of an experiment will be. It only makes sense to ask what the ''probability'' of a particular outcome is. These probabilities are absolute squares of certain complex numbers called ''amplitudes'' associated to each possible outcome.

Now the fundamental difference between quantum behavior and classical behavior is the following: Suppose that a particular outcome, A, of an experiment can happen in two ways, with amplitudes <math>z</math> and <math>w</math> (so that the corresponding probabilities are <math>||z||^2</math> and <math>||w||^2</math>. In classical physics, we would predict that the probability that A happens is just the sum of the probabilities of the two ways that A can happen--that is: <math>||z||^2+||w||^2</math>. Strangely, nature, for reasons unknown, does not appear to work this way. Instead, to get the probability for the outcome A, we have to add the amplitudes first, and only then square it. That is, the true probability for A is given by <math>||z+w||^2</math>. This means that the two amplitudes associated to the two possible ways A can happen can constructively or destructively interfere with each other. This gives rise the wave-like behavior of particles observed in situations such as the double-slit experiment.

Now the fundamental difference between quantum behavior and classical behavior is illustrated by the following example: Suppose that a particular outcome, A, of an experiment can happen in two ways, with amplitudes <math>z</math> and <math>w</math> (so that the corresponding probabilities are <math>||z||^2</math> and <math>||w||^2</math>. In classical physics, we would predict that the probability that A happens is just the sum of the probabilities of the two ways that A can happen--that is: <math>||z||^2+||w||^2</math>. Strangely, nature, for reasons unknown, does not appear to work this way. Instead, to get the probability for the outcome A, we have to add the amplitudes first, and only then square it. That is, the true probability for A is given by <math>||z+w||^2</math>. This means that the two amplitudes associated to the two possible ways A can happen can constructively or destructively interfere with each other. This gives rise the wave-like behavior of particles observed in situations such as the double-slit experiment.

 

==Mathematics==

==Mathematics==

The mathematics of Quantum mechanics can be formulated in a number of ways: the "matrix mechanics" of Werner Heisenberg, the "path integrals" of [[Richard Feynman]], or the "wave mechanics" of Erwin Schrodinger. Wave mechanics is the most common formulation. It uses the language of infinite dimensional [[Hilbert Space]]s; observables such as position and momentum are [[operator]]s on such Hilbert Spaces.

The mathematics of Quantum mechanics can be formulated in a number of ways: the "matrix mechanics" of Werner Heisenberg, the "path integrals" of [[Richard Feynman]], or the "wave mechanics" of Erwin Schrodinger. Wave mechanics is the most common formulation. It uses the language of infinite dimensional [[Hilbert Space]]s; observables such as position and momentum are [[operator]]s on such Hilbert Spaces.