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edited Apr 25 "12 at 15:07
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asked Apr 25 "12 in ~ 14:43
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1) Quantum mechanically waves are probability waves, i.e. the probability of recognize a particle has actually a functional dependence ~ above sines and cosines. It has nothing to do with amplitude as power or momentum or whatever, the crescents and also troughs room increased and also decreased probabilities of being found when an observation is made.
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2)Particles have actually spins. Particles that have integer rotate are referred to as bosons and can accounting the same room at the exact same time definition the probability of detect one in an (x,y,z) coordinate increases the more of them over there are. Bosons can occupy the very same quantum state in general. Particles with half integer spin are fermions and follow the fermi-dirac statistics , and also thus can not occupy the same space; i.e the probability of detect one in one (x,y,z) clues will always be the probability because that finding one particle; just one deserve to occupy a quantum state in ~ a time, in general.
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answered Apr 25 "12 at 15:26
anna vanna v
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$\begingroup$ Thanks. In an answer to your 1st point- what about electrons? they aren't particles-they're literally waves in the quantum field. Thus why can't they exist in the same ar at the same time? $\endgroup$
Apr 25 "12 at 16:15
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Particles are not waves, or at least not in the sense of water tide or electromagnetic waves. It"s true that particles are described by a wavefunction, yet in general the wavefunction is a complex function of countless variables. The an easy solutions to Schrodinger"s equation the you learn in elementary school QM classes tend to be waves, however these space special cases and not typical of the genuine world.
If you take two particles, e.g. Two electrons, then you can"t simply take separate wavefunctions because that each bit and add them, together you would v e.g. Water waves. This is due to the fact that the electron interact and also this interaction introduces a new term come the potential power term in the Schrodinger equation. This method you have actually to define the 2 particle system by a brand-new wavefunction the is not just the sum of the two original wavefunctions.
There is one elementary proof that 2 electrons can"t occupy precisely the very same quantum state. If you fix the Dirac equation (it has to be the Dirac equation due to the fact that the Schrodinger equation doesn"t explain spin) you uncover the result wavefunction is antisymmetric v respect come exchange of the 2 electrons i.e.
$$\Psi(e_1, e_2) = - \Psi(e_2, e_1)$$
where the readjust $(e_1, e_2)$ to $(e_2, e_1)$ is supposed to suggest we"ve swapped the two electrons. Currently suppose the two electrons space in identical states, that way there is no difference in between $e_1$ and $e_2$ i.e.
$$\Psi(e_1, e_2) = \Psi(e_2, e_1)$$
Combining these equations us get:
$$\Psi(e_1, e_2) = - \Psi(e_1, e_2)$$
and the only means this deserve to be correct is if $\Psi$ is zero as soon as $e_1$ and $e_2$ room the same, i.e. The probability the the two electrons deserve to be in similar states is zero.
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As Anna mentions, not all particles room fermions. Particles through zero rotate are described by a different equation, the Klein Gordon equation, and they have the right to occupy similar states. In reality this is the beginning of Bose Einstein condensation.