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Two objects with half spin would consist of the elementary particles (i.e. quarks, fermions etc.) which are waves. Therefore all objects consist of several waves. Waves can exist at the same place at the same time. Depending on whether the wave is at a trough or a crest and the amplitude, the waves either cancel each other out or amplify each other. Therefore why can't objects exist at the same place at the same time, and just cancel or amplify each other depending on whether they are at a crescent or a trough?

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2 Answers

Particles are not waves, or at least not in the sense of water waves or electromagnetic waves. It's true that particles are described by a wavefunction, but in general the wavefunction is a complicated function of many variables. The simple solutions to Schrodinger's equation that you learn in elementary QM classes tend to be waves, but these are special cases and not typical of the real world.

If you take two particles, e.g. two electrons, then you can't just take separate wavefunctions for each particle and add them, as you would with e.g. water waves. This is because the electrons interact and this interaction introduces a new term to the potential energy term in the Schrodinger equation. This means you have to describe the two particle system by a new wavefunction that is not simply the sum of the two original wavefunctions.

There is an elementary proof that two electrons can't occupy exactly the same quantum state. If you solve the Dirac equation (it has to be the Dirac equation because the Schrodinger equation doesn't describe spin) you find the resulting wavefunction is antisymmetric with respect to exchange of the two electrons i.e.

$$\Psi(e_1, e_2) = - \Psi(e_2, e_1)$$

where the change $(e_1, e_2)$ to $(e_2, e_1)$ is supposed to indicate we've swapped the two electrons. Now suppose the two electrons are in identical states, that means there is no difference between $e_1$ and $e_2$ i.e.

$$\Psi(e_1, e_2) = \Psi(e_2, e_1)$$

Combining these equations we get:

$$\Psi(e_1, e_2) = - \Psi(e_1, e_2)$$

and the only way this can be correct is if $\Psi$ is zero when $e_1$ and $e_2$ are the same, i.e. the probability that the two electrons can be in identical states is zero.

As Anna mentions, not all particles are fermions. Particles with zero spin are described by a different equation, the Klein Gordon equation, and they can occupy identical states. In fact this is the origin of Bose Einstein condensation.

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1) Quantum mechanical waves are probability waves, i.e. the probability of finding a particle has a functional dependence on sines and cosines. It has nothing to do with amplitude as energy or momentum or whatever, the crescents and troughs are increased and decreased probabilities of being found when an observation is made.

2)Particles have spins. Particles that have integer spin are called bosons and can occupy the same space at the same time meaning the probability of finding one in an (x,y,z) coordinate increases the more of them there are. Bosons can occupy the same quantum state in general. Particles with half integer spin are fermions and follow the fermi-dirac statistics , and thus cannot occupy the same space; i.e the probability of finding one in an (x,y,z) spot will always be the probability for finding one particle; only one can occupy a quantum state at a time, in general.

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Thanks. In response to your 1st point- what about electrons? They aren't particles-they're literally waves in the quantum field. Therefore why can't they exist in the same place at the same time? –  user8791 Apr 25 '12 at 16:15
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Why do you think electrons are not particles? Of course they are particles, fermions and obey fermi-dirac statistics. Again, those are probability waves, and and one cannot find a probability for two fermions to exist at the same time and place. –  anna v Apr 25 '12 at 16:26
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