Why can't we walk through a solid wall?

Try it and find out. Kidding, walking through a solid wall is impossible because the atoms are crammed too tightly to be separated by a human. If we were to want to walk through a wall, human atoms would have to pass through wall atoms.

If you're interested in more reasons why you can't walk through a wall, the Pauli Exclusion Principle will explain things pretty well. It basically states that two or more identical fermions can't occupy the same state within a quantum physics simultaneously. Fermions are considered protons, neutrons, and electrons; and no two electrons can have the same quantum number.

Theoretically, we could. It’s been done on an electron.

Quantum Mechanics is a helluva phenomenon. Scientists have seen an electron pass through an atom of a solid object. Quantum Mechanics isn’t governed by normal laws of physics and has its own laws to govern it. The process through which this was done is called quantum tunneling.

It is speculated that this occurs due to the wave nature of matter. But a normal human being passing through a wall is something straight out of a comic or a comic-book movie. It is perfectly possible through suppression of atomic barriers but the energy that would be required for an entire human is too much. Future humans or a ghost(reference intended) could surely do it.

Coming back to the present reality, Pauli Exclusion Principle will help you. The Pauli exclusion principle is the quantum mechanical principle which states that two or more identical fermions (particles with half-integer spin) cannot occupy the same quantum state within a quantum system simultaneously.(Source: Wikipedia).

That means if a human wants to walk through a wall, the atoms in human beings will have to pass the atoms of a wall. Now, it is arguable that the distance between interatomic particles is enough for another atom to pass through but it will have to violate the Pauli Exclusion Principle and Newtonian mechanics which is not possible.

Electrons have a particular spin that gives them a distinct energy field and this energy field will prevent other electrons from passing through the atom. It is a kind of repulsion that is not exactly explained uptil now. It cannot be classified as a force yet. Although if we can have a quantum field that can accommodate a human being then it should be possible to break those energy fields easily and help the human through.

To walk through a solid wall, we would have to be able to pass through another, different atom. To do this, the electrons of the first atom would have to exist in the same atomic space of the electrons of the second atom. It would not be possible. No two electrons in an atom can simultaneously be in the same state and configuration.

You can't have two electrons occupying the same space doing the same job. This is what makes an object solid and keeps them from passing through each other. This is the force between the formations, and this keeps atoms from touching other atoms.