Two identical particles with spin 1/2

[the_doors]

hello guys , in this problem from zettili quantum mechanics that i attach , i think something is wrong , first the problem said two particles with spin 1/2 but didn't mention that the system is in singlet state or triplet state , so if the system be in triplet state then our spatial wave function must be untisymmetric .


what do you think guys ?

Best regards

 

[Simon Bridge]

first the problem said two particles with spin 1/2 but didn't mention that the system is in singlet state or triplet state

... does not say they are in any state. It just says two identical spin 1/2 particles in a box then it asks you to work some stuff out. It only asks about the ground state.

So which is the ground state?

ref.
http://galileo.phys.virginia.edu/classes/752.mf1i.spring03/IdenticalParticlesRevisited.htm

 

[the_doors]

two particles can be on first energy level with up and down spin direction so the space wave function is symmetric and spin wave function is in singlet state . what is wrong with this ?

 

[Simon Bridge]

I think I see the problem ... you mean: why is the singlet state not the ground state?

It's tricky to explore in various course notes: i.e.
http://physicspages.com/2013/03/08/infinite-square-well-2-particle-systems/
... where they say the ground state is n1=1, n2=2

vs this:
http://www.st-andrews.ac.uk/physics/quvis/embed_item_3.php?anim_id=48&file_sys=index_phys
GS has n1=n2=1 but spins are opposite.

A lecture that kinda covers both views is:
http://physics.uwyo.edu/~yurid/QM/Lecture%2017.pdf
... without considering spins, the |1,1> combined state does not exist - so the ground state is a triplet state.

Singlet state description is covered later.
It would be nice if the author made a definitive statement about the resulting ground state.

See also in these forums - pretty much the same question:
https://www.physicsforums.com/showthread.php?t=393603

I'm wondering if there is an unspoken assumption in the context of the problem.

One possibility is that some sources consider "noninteracting" to mean the fermions cannot see each other's spin - so the spin component of the wavefunction has no effect. In order for indistinguishable non-interacting spinless fermions to follow fermi-dirac statistics, the space wavefunction must be antisymmetric. It's when they start glibly referring to "electrons" that bothers me - atomic subshells clearly have 2 electrons each.

I don't see anything wrong with the GS being the singlet, off the top of my head.

 

[Simon Bridge]

After further checking - still don't see anything wrong with it.
A careful reading of the chapter that the exercise belongs to may let you know why the author has not considered the singlet state. Maybe the book has simply neglected the effect of the spinors to begin with - that's a common way to write these things these days - which would make the answer simply "wrong", and the author wants to get you used to the simpler form of the math before adding complications. Something like that.

May be worth taking up with the lecturer?