QM Puzzle: Can One Particle Have Multiple States?

[aaronsky12]

I was taught that a particle is assigned to a unique quantum state. As a specific example, two bound electrons can't have the same quantum numbers in an atom. And likewise one and only bound electron is assigned to one quantum state in an atom. Yet, I am reading several solid state books and they are saying that an electron moving in a conductive material (crystal lattice) can be described by a localized wavepacket with a group velocity and central energy... That makes sense intuitively... but wavepackets are themeselves superpositions of sinusoidal traveling wave solutions to shrodinger's equation (each with a wavevector k)... This makes it sound like one particle is assigned several wavefunctions (each with their own quantum numbers)... How can this be true?

 

[bhobba]

Mate its the principle of superposition at work:
http://en.wikipedia.org/wiki/Quantum_superposition

Thanks
Bill

 

[andrien]

Have not you heard that an arbitrary wave can be expressed by superposition of many plane waves(provided complete sets) and also Schrödinger eqn is linear so it should hold.

 

[aaronsky12]

Thanks bhobba and andrien! OK, here is a related follow up question... First let me quote the wiki pages for the superposition principle and Pauli Exclusion principle state:

"[Quantum superposition] holds that a physical system—such as an electron—exists partly in all its particular, theoretically possible states (or, configuration of its properties) simultaneously; but, when measured, it gives a result corresponding to only one of the possible configurations (as described in interpretation of quantum mechanics)."

"The Pauli exclusion principle is the quantum mechanical principle that no two identical fermions (particles with half-integer spin) may occupy the same quantum state simultaneously."

My question is: Can two electrons "share" states and not violate the exclusion principle. It makes sense that one electron can exist in state A and another electron in B. But can one electron exist partially in state A and B, while the other also exists in partially state A and B at the same time?

 

[andrien]

Thanks bhobba and andrien! OK, here is a related follow up question... First let me quote the wiki pages for the superposition principle and Pauli Exclusion principle state:

"[Quantum superposition] holds that a physical system—such as an electron—exists partly in all its particular, theoretically possible states (or, configuration of its properties) simultaneously; but, when measured, it gives a result corresponding to only one of the possible configurations (as described in interpretation of quantum mechanics)."

"The Pauli exclusion principle is the quantum mechanical principle that no two identical fermions (particles with half-integer spin) may occupy the same quantum state simultaneously."

My question is: Can two electrons "share" states and not violate the exclusion principle. It makes sense that one electron can exist in state A and another electron in B. But can one electron exist partially in state A and B, while the other also exists in partially state A and B at the same time?

A state is the complete information for a particle.So how is it possible that an electron can be in two states simultaneously.

 

[Bill_K]

But can one electron exist partially in state A and B, while the other also exists in partially state A and B at the same time?

Yes. For example take a spin-1/2 system like an electron, and consider the eigenstates of S_z, call them |z,+> and |z,->, and compare these to the eigenstates of S_x, call them |x,+> and |x,->. The relationship is |z,+> = (1/√2)(|x,+> + |x,->) and |z,-> = (1/√2)(|x,+> - |x,->).

So if you have electron A in the state |z,+> and electron B in the state |z,->, you can say that they are both partially in states |x,+> and |x,->.

 

[Maui]

My question is: Can two electrons "share" states and not violate the exclusion principle.

Electrons are fermions and cannot occupy the same quantum state. This is taken care of by the PEP. Photons(bosons) on the other hand can, hence light is not solid.

 

[Bill_K]

Electrons are fermions and cannot occupy the same quantum state.

Quite right. Nonetheless, two fermions may occupy orthogonal linear combinations of the same states, as I have just described.

 

[mfb]

Quite right. Nonetheless, two fermions may occupy orthogonal linear combinations of the same states, as I have just described.

Even more: They have to occupy linear combinations. Electrons are indistinguishable - you cannot have "electron A in state a" and "electron B in state b", as "electron A" and "electron B" do not exist as different objects.

 

[andrien]

Yes. For example take a spin-1/2 system like an electron, and consider the eigenstates of S_z, call them |z,+> and |z,->, and compare these to the eigenstates of S_x, call them |x,+> and |x,->. The relationship is |z,+> = (1/√2)(|x,+> + |x,->) and |z,-> = (1/√2)(|x,+> - |x,->).

So if you have electron A in the state |z,+> and electron B in the state |z,->, you can say that they are both partially in states |x,+> and |x,->.

But is not the state |z,+> and |z,-> represents two different state.so will it be safe to say that electron existing in one state is definitely independent of the other,as is the case here.