A system consists of two indistinguishable spin-1 fermions, both confined

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Homework Statement

A system consists of two indistinguishable spin-1 fermions, both confined inside the same box of length L centred on the origin. The particles do not interact with each other.

a) What is the total energy of the ground state of this system? (Use the standard formulas for the energy eigenvalues of a single particle in a box.)
b) What is the spin angular momentum of the system in the ground state?
c) How many distince and mutually orthogonal space-spin energy eigenstates of the whole system are there in which there is one particle in the single-particle ground state and one particle in the single-particle first exctied state? Choosing these eigenstates so that they are also eigenvectors of the square and the z-component of total spin angular momentum, write explicit formulas for all these space-spin eigenstates. (Use alpha, beta notation for spin-up and spin-down states.)
d) With no interaction between the particles, the energy eigenstates described in part c) all have exactly the same energy (they are degenerate). We now introduce a repulsive interaction between the two particles. Describe the qualitative effect that this interaction has on the relative energies of the singlet and triplet eigenstates.

The Attempt at a Solution

a)E(subscriipt n)=(h^2 n^2)/(8mL^2)

there are 2 spin 1 fermions, n=1, so the total energy is=(h^2)/(3mL^2)

b)spin angular momentum=(1/2)+(1/2)=1
=(1/2)+(-1/2)=0 is this also possible?
=(-1/2)+(-1/2)=-1 is this also possible?

I haven't tried c and d yet. But I just wanted to check that I got questions a and b right because I am worried that I have misunderstood que a and b. Thank you if you reply.

 

[vela]

Homework Statement

A system consists of two indistinguishable spin-1 fermions, both confined inside the same box of length L centred on the origin. The particles do not interact with each other.

a) What is the total energy of the ground state of this system? (Use the standard formulas for the energy eigenvalues of a single particle in a box.)
b) What is the spin angular momentum of the system in the ground state?
c) How many distince and mutually orthogonal space-spin energy eigenstates of the whole system are there in which there is one particle in the single-particle ground state and one particle in the single-particle first exctied state? Choosing these eigenstates so that they are also eigenvectors of the square and the z-component of total spin angular momentum, write explicit formulas for all these space-spin eigenstates. (Use alpha, beta notation for spin-up and spin-down states.)
d) With no interaction between the particles, the energy eigenstates described in part c) all have exactly the same energy (they are degenerate). We now introduce a repulsive interaction between the two particles. Describe the qualitative effect that this interaction has on the relative energies of the singlet and triplet eigenstates.

The Attempt at a Solution

a)E(subscriipt n)=(h^2 n^2)/(8mL^2)

there are 2 spin 1 fermions, n=1, so the total energy is=(h^2)/(3mL^2)

This isn't correct. How did you get a 3 in the denominator?

b)spin angular momentum=(1/2)+(1/2)=1
=(1/2)+(-1/2)=0 is this also possible?
=(-1/2)+(-1/2)=-1 is this also possible?

You have spin-1 particles, not spin-1/2 particles. Also, it's not exactly clear what you're calculating here. The angular momentum of a system is described by two quantum numbers, j and m_j. You need to specify the possible values of both. Look up addition of angular momenta in your textbook if you're not sure how to do that.

I haven't tried c and d yet. But I just wanted to check that I got questions a and b right because I am worried that I have misunderstood que a and b. Thank you if you reply.

 

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I meant to type '4' in the denominator. It was a typo. Sorry.

 

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actually that was a typo on the homewok. fermiions are spin 1/2. I have now looked at que c nd d and I have no idea how to do them. Please help. I have tried, really, but the reason that I can't even produce a line or two of the attempt at the solution for c and d is that I am seriously stuck and have no idea what to do.

 

[vela]

Learn about the addition of angular momentum to get (b) done correctly. Your textbook covers this, and it probably goes over the case of two spin-1/2 particles.

In particular, you should know what the four eigenstates of S² are, where S=S₁+S₂ and whether each state is symmetric or antisymmetric. From this, you can deduce which ones are allowed in the ground state.

 

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my notes say:

two electrons have spin eigenstates:

|0, 0>=(1/sqrt2) (|alpha 1>|beta2> -|beta1>|alph2>)
|1,0>=(1/sqrt 2)(|alpha1>|beta2>+|beta1>|alpha2>)
|1,-1>=|beta1>|beta2>
|1, +1>=|alpha1>|alpha>

Why would a state being symmetric or antisymmetric matter with respect to whether it is allowed in the ground state or not?

What is a 'space-spin' eigenstate? I've heard of 'spin eigenstates', but I have not seen the terminology 'space-spin' eigenstate used anywhere.

 

[vela]

You have two fermions, so they obey the Pauli exclusion principle.

The state of a particle consists of a spatial part and a spin part. For example, you might say a particle is in the n=1 state, which describes its spatial state, with its spin pointing up. A space-spin eigenstate would be one where both the spatial and spin parts are eigenstates of the Hamiltonian.

In this problem, the Hamiltonian depends only on the spatial part, so all spin states are eigenstates of the Hamiltonian. If the Hamiltonian contained terms that depended on the spin of the particle, then you'd find distinct spin eigenstates.