Superposition and Pauli's principle

1. Jun 30, 2012

Mukilab

Does Pauli's exclusion principle apply to superposition or only collapsed states?

I.e. could a particle be 'superposed' - if that is the word in the same state as another electron in a localized region?

Also doesn't Pauli's exclusion principle create a form of measurement because particles have certain possible states eliminated as we observe other particles in that localized area.

2. Jun 30, 2012

tom.stoer

One should reformulate the Pauli exclusion principle in terms of creation and annihilation operators. Suppose we have some quantum number n=0,1,2,... and for each state we have either zero and one fermion; two fermions are forbidden by the Pauli principle.

So one allowed state is

$$|0,1,1,\ldots\rangle$$

where we have 0 fermions for n=0, 1 fermion for n=1 and 1 fermion for n=2.

The state is created from the vacuum state by using two creation operators, i.e.

$$|0,1,1,\ldots\rangle = a_1^\dagger\,a_2^\dagger|0\rangle$$

The Pauli principle is guarantueed by the fact that if we act with a second creation operator for n on a state where we already have one fermion for this specific n, the state is annihilated.

$$a_0^\dagger |0,1,1,\ldots\rangle = |1,1,1,\ldots\rangle$$

$$a_1^\dagger |0,1,1,\ldots\rangle = 0$$

$$a_2^\dagger |0,1,1,\ldots\rangle = 0$$

or in general for a '1' at position n

$$|\ldots,1,\ldots\rangle \to a_n^\dagger|\ldots,1,\ldots\rangle = 0$$

Now using these states with either 0 or 1 fermions for every n we can write down arbitrary superpositions.

If now use a superposition of states and act on it with a creation operator, this operator annihilates all states states with one fermion at n, and it creates a fermion in all states with zero fermions at n, i.e.

$$|0,0,\ldots\rangle + |0,1,\ldots\rangle + |1,1,\ldots\rangle \to a_0^\dagger(|0,0,\ldots\rangle + |0,1,\ldots\rangle + |1,1,\ldots\rangle) = |1,0,\ldots\rangle + |1,1,\ldots\rangle + 0$$

In this mathematical sense the Pauli principle applies to all states, not only to specific "one particle states" or "collspsed states".

Last edited: Jun 30, 2012
3. Jun 30, 2012

StevieTNZ

From what Tom has written, I gather that two fermions cannot occupy the same quantum number in each state part of the overall superposition state.

4. Jun 30, 2012

Mukilab

Thank you, as much as I appreciate the time Tom took to write that I didn't understand any of it apart from a few words.

I hope that some time in the future I can return to this thread and comprehend what he has written.

Thank you for this succinct summary.

5. Jun 30, 2012

tom.stoer

As you can see the Pauli principle is implemented in terms of states containing an arbitrary number of particles; the only restriction is that for a quantum number n there cannot be more than one particle with this quantum number. Here quantum number means a collection of all properties like momentum, spin, flavour, color, ...

As you can see the Pauli principle has nothing to do with measurements but with allowed states. It affects the preparation of states, not the measurement.

6. Jul 1, 2012

Mukilab

Thank you for this clarification.

I'm sorry but I still don't understand what you mean by the 'preparation of states', would you please be able to explain this in more layman's term? I understand that you are already stretching yourself by explaining it without mathematics, so I understand if it is not possible to fully explain this without maths, and if so, please just say.

Thank you again for your time.

7. Jul 1, 2012

tom.stoer

What I am saying is that the Pauli principle acts as a 'constraint on the Hilbert space'; it restricts the states that you can 1) prepare mathematically and 2) it restrictes the states that you can prepare in the lab.

1) mathematically you create all states by using creation operators (see above); once you use the same creation operators twice you get zero!
2) in the lab it restricts the experimental preparation of states; putting two electrons in the same state will never work, they will always sit in two different states, e.g. different energy or spin or something like that.

That's why I say that Pauli's principle has something to do with preparation of states, not with their measurement (the reason why you never observe an object moving faster than light has nothing to do with measurement; it is due to the fact that you can't prepare an object moving faster than light)

Last edited: Jul 1, 2012
8. Jul 1, 2012

Mukilab

Thank you, that clears everything up.