Fermions must be described by antisymmetric and bosons by symmetric

[Mr confusion]

friends, we know that fermions must be described by antisymmetric and bosons by symmetric wavefunctions. but i was wondering why a particle of certain class behaves like that for ever? ie. say, an electron will never behave like a boson ??

my book says that there is a spin statistics theorem that ensures that this be so. ie. spin determines the statistics. but thinking this again, when i consider one dimensional problems, no spin can arise, yet i have never heard of a boson electron in one dimension??

i just hope i am not appearing mad here!

 

[ansgar]

I think the opposite is true, you should tell us why the electron should change to become boson in minkowski-space.

Heard of supersymmetry? If supersymmetry existed as an exact symmetry then we could have such interaction terms in the lagrangian changing the spins of the particles from bosons to fermions and vice versa.

 

[Mr confusion]

no friend, i do not know what supersymmetry is. my course has just started. but i am not telling that an electron acts as boson. i am just saying why it doesn't do so in one dimension problems where the concept of spin does not arise, -thus there is no spin statistics theorem playing in our minds...

 

[meopemuk]

when i consider one dimensional problems, no spin can arise, yet i have never heard of a boson electron in one dimension??

We live in 3D space, where particles do have spin. There are no true 1-dimensional systems in the world. All 1D models are just approximations of 3D objects (e.g., nanowires).

Eugene.

 

[peteratcam]

no friend, i do not know what supersymmetry is. my course has just started. but i am not telling that an electron acts as boson. i am just saying why it doesn't do so in one dimension problems where the concept of spin does not arise, -thus there is no spin statistics theorem playing in our minds...

Remember that spin space is different from real space. Spin is an extra degree of freedom for quantum particles - even if the particle moves in 1 spatial dimension, it can still have a spin degree of freedom. The directions in spin space only couple to real space with relativistic effects or when considering magnetic fields.

 

[meopemuk]

Spin is an extra degree of freedom for quantum particles - even if the particle moves in 1 spatial dimension, it can still have a spin degree of freedom.

The existence of spin is closely related to the 3-dimensional character of space, in particular to the presence of the rotational group of symmetry. This has been shown by Wigner in his works on unitary representations of the Poincare group. In 1 spatial dimension there can be no rotations, so particles cannot have spin.

Eugene.

 

[Mr confusion]

thank you , friends. i will remember what you said.

but i am still finding it hard to understand how a particle can have an extra spin degree of freedom in one dimension. am i missing something fundamental?

 

[Mr confusion]

moepemuk, although i did not fully understand those things , but can you tell the same thing in a easier way?
i understand that we live in 3D , but still when i sit in classroom and solve a problem where an electron is coming from left to a barrier, it is still called an 'electron', ie . a fermion.

complicacy- the same problem if says 2 electrons in a box ,say, then what state vector will i use? obviously the antisymmetric one. But i cannot detergent why? i mean, what is now ensuring the electrons are described by antisymmetric function since there is no spin now? considerating an infinite well in 1dimension.

 

[Mr confusion]

is my question understandable, friends? please tell me, i will place it more clearly then. actually my stock of english words is not much. sorry friends.

 

[meopemuk]

moepemuk, although i did not fully understand those things , but can you tell the same thing in a easier way?
i understand that we live in 3D , but still when i sit in classroom and solve a problem where an electron is coming from left to a barrier, it is still called an 'electron', ie . a fermion.

1-dimensional barriers are not real. There are no true 1-dimensional systems in nature. These textbook problems are grossly simplified. Their purpose is to teach you some fragments of the formalism of quantum mechanics, not to model realistic physical systems. So, there is definitely a contradiction when one speaks about fermions/bosons in 1 dimension.

Eugene.

 

[Mr confusion]

many thanks, moepemuk. and the complicacy? ok. its also1D. THANKS!

 

[peteratcam]

There is no contradiction in talking about 1-d fermions.
A sufficiently strong potential can confine an electron so that it only has one translational degree of freedom available. Nevertheless it will still have the spin degree of freedom.

Spin-statistics may be a theorem of QFT, but in non-relativistic QM it is just taken as a fact. There is no contradiction at all in assuming spin-statistics and then doing 1-D QM.

Mr confusion: electrons *are* fermions which means they must have an antisymmetric wavefunction. This is true, even if you confine them to 1-dimension.

 

[Mr confusion]

peteratcram thank you.
well, ok. but can you please confirm me one thing??
in my chemistry course, i learned that spin does not arise due to a spinning electron. it is something inherent.
now, when i think this again, i know electron is a particle. and like all particles, can it not spin in reality? i mean , like a real spinning ball? if it does, then why is the angular momentum resulting from that spin NOT added to the inherent spin of fermions?

 

[meopemuk]

in my chemistry course, i learned that spin does not arise due to a spinning electron. it is something inherent.
now, when i think this again, i know electron is a particle. and like all particles, can it not spin in reality? i mean , like a real spinning ball? if it does, then why is the angular momentum resulting from that spin NOT added to the inherent spin of fermions?

I don't think that textbooks explain these things correctly. Electron's spin has exactly the same physical origin as spinning of macroscopic particles. The only difference is that due to quantum effects, projection of electron's spinning momentum on any axis can be only $$\hbar/2$$ or $$-\hbar/2$$. By applying a force to the electron one cannot "stop" its spinning motion, one can only change the probabilities of finding the spin "up" or "down".

Eugene.

 

[SpectraCat]

Electron's spin has exactly the same physical origin as spinning of macroscopic particles.

Really? I thought that the spin of the electron was a phenomenological model introduced by Pauli to explain experimental observation of electronic wavefunctions of atoms. I also thought that Dirac then showed how it emerged naturally as a consequence of making the Shcrodinger equation consistent with relativity.

That seems like a different physical origin than macroscopic angular momentum, which can be completely described in 3-space, without the need for two-valued solutions (which is why Pauli introduced spin).

 

[ansgar]

Really? I thought that the spin of the electron was a phenomenological model introduced by Pauli to explain experimental observation of electronic wavefunctions of atoms. I also thought that Dirac then showed how it emerged naturally as a consequence of making the Shcrodinger equation consistent with relativity.

That seems like a different physical origin than macroscopic angular momentum, which can be completely described in 3-space, without the need for two-valued solutions (which is why Pauli introduced spin).

yes but Lorentz group includes rotations so there is a truth with modification :)

 

[meopemuk]

Really? I thought that the spin of the electron was a phenomenological model introduced by Pauli to explain experimental observation of electronic wavefunctions of atoms. I also thought that Dirac then showed how it emerged naturally as a consequence of making the Shcrodinger equation consistent with relativity.

Pauli and Dirac "derivations" of spin were just heuristic guesses. The true justification for the spin degrees of freedom in quantum relativistic particles comes from Wigner's work

E. P. Wigner, "On unitary representations of the inhomogeneous Lorentz group", Ann. Math.,40 (1939), 149.

This work makes no assumptions except postulates of quantum mechanics and the Poincare symmetry group.

Eugene.

 

[Mr confusion]

moepemuk, can you please tell me what are the prerequisites to understand wigner's work?? i want to study and find this myself.

 

[meopemuk]

moepemuk, can you please tell me what are the prerequisites to understand wigner's work?? i want to study and find this myself.

You need to learn quantum mechanics and the theory of group representations. The only group that is important is the Poincare group (or the Galilei group in the non-relativistic case). The best modern presentation is in first few chapters of S. Weinberg "The quantum theory of fields" vol. 1. For a beginner this could be a difficult read. So, you may find it easier to start with L. E. Ballentine "Quantum mechanics. A modern development".

Eugene.