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The Pauli exclusion Principle states that two Fermions of the same type cannot exist in the same state at the same time. This means that two electrons cannot both exist in the same spin state and be in the same location at the same time. This is also the reason that solids tend to be unable to pass through each other.

I was wondering if there were two pieces of frozen hydrogen in which all the electrons of one piece were spin up and all the electrons of the other were spin down would these two pieces of frozen hydrogen be able to pass through each other considering that all the electrons in one would have an opposite spin from all the electrons from the other.

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    $\begingroup$ Frozen hydrogen means by definition dihydrogen, i.e. $\mathrm{H_2}$. In that molecule the two electrons are in a $\sigma$ molecular orbital, with one spin 'up', one spin 'down' to respect the Pauli Exclusion Principle. $\endgroup$ – Gert Nov 7 '15 at 17:58
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The Pauli exclusion Principle states that two Fermions of the same type cannot exist in the same state at the same time. This means that two electrons cannot both exist in the same spin state and be in the same location at the same time. This is also the reason that solids tend to be unable to pass through each other.

This is a misleading statement. Two electrons cannot occupy the same state implies a quantum mechanical wavefunction defining the state. The QM wavefunction has quantum numbers that separate the solutions. The Pauli exclusion does not allow two electrons to fill the same energy level : the levels have to differ at least by one quantum number. Thus chemistry is born, otherwise all the electrons would be at the lowest ground state of the nucleus and no chemistry as we know it would develop.

As DilithiumMatrix states in the answer the primary reason atoms do not pass through each other is the repulsion of the external electron levels. Even if the Pauli exclusion were not there, the electron clouds would repel each other so again no free pass.

I was wondering if there were two pieces of frozen hydrogen in which all the electrons of one piece were spin up and all the electrons of the other were spin down would these two pieces of frozen hydrogen be able to pass through each other considering that all the electrons in one would have an opposite spin from all the electrons from the other.

Even if this could be done, the repulsion of the electric charges would still be at work.

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Your understanding of the Pauli Exclusion principle is good. There are two main issues with what you have described.

1) 'freezing' is a macroscopic phenomenon describing the structure of many atoms/molecules which have formed a solid (lattice) structure. It isn't meaningful to think of a single 'frozen' atom. Similarly, particles will always have some residual thermal motion because you can never actually reach absolute zero. Another way of thinking about this is the uncertainty principle --- which, in this context, means that a hydrogen atom (and its electron(s)) can never be perfectly fixed in space with zero velocity.

2) The primary reason why 'solid' materials don't pass through each-other is due to electromagnetic interactions - not the exclusion principle. When you bring two atoms/molecules/solids together, their respective electrons will repel each other. If you had two hydrogen cations (i.e. each just a proton), then they would similarly repel each other.

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The Pauli exclusion Principle states that two Fermions of the same type cannot exist in the same state at the same time.

And place. It's to do with atomic orbitals. However you sometimes hear people saying it applies to fermions throughout the universe. That's a popscience myth. See this question

This means that two electrons cannot both exist in the same spin state and be in the same location at the same time.

Good stuff. I think it helps to think of spinors, and remember that two ocean waves can ride over one another, but two whirlpools cannot overlap.

This is also the reason that solids tend to be unable to pass through each other.

I'd say it's part of the reason. Co-rotating vortices attract, counter-rotating vortices repel. And electrons repel one another.

I was wondering if there were two pieces of frozen hydrogen in which all the electrons of one piece were spin up and all the electrons of the other were spin down would these two pieces of frozen hydrogen be able to pass through each other considering that all the electrons in one would have an opposite spin from all the electrons from the other.

They wouldn't. You can see a depiction of spin up and spin down on Rod Nave's hyperphysics website:

enter image description here

Imagine you had two hydrogen atoms next to each, one spin up, one spin down. Their electrons will still repel one another. But see Gert's comment. They can also make a molecule.

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