Ground State Wave Function for Identical Spin 1/2 Particles in a Potential Well

In summary, the problem involves finding the wave function of the ground energy state for a system of two non-interacting, identical spin 1/2 particles in a potential well. The particles must have an antisymmetric wave function due to being fermions, and the wave function will be a combination of a spatial and spin component. The spin state can be either symmetric or antisymmetric, with the symmetric state corresponding to the particles occupying a state in the spin triplet and the antisymmetric state corresponding to the singlet. To determine which component of the wave function must be antisymmetric, we must consider which state has the lowest energy. Since we want the ground state, the single-particle spatial wave functions must be the same, indicating that the spatial
  • #1
lion8172
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Homework Statement


Find the wave function of the ground energy state for a system of two non-interacting, identical spin 1/2 particles in a potential well extending from x=0 to x=a. Don't forget to consider spin.

Homework Equations


The Attempt at a Solution



Since the particles are fermions, they must occupy a state that is antisymmetric with respect to particle exchange. Their wave function will be of the form \psi (r) \chi(s), where \chi is a spinor. Either the spatial part or the spin part of the wave function must be antisymmetric (and the other part must be symmetric). If the spin state is symmetric, then the particles will occupy a state in the spin triplet. Otherwise, they will occupy the singlet. At this point I am confused, however. How do we know which component of the wave function must be anti-symmetric (spatial or spin)? Moreover, if the particles are in the spin triplet, how do we know which of the three possible spin states corresponds to the lowest energy?
 
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  • #2
Since we want the ground state, we want to put each particle into the single-particle state of lowest energy. So their single-particle spatial wave functions are the same. What does that tell you about spatial symmetry?
 

What are identical particles?

Identical particles are particles that cannot be distinguished from one another based on their physical properties such as mass, charge, and spin. This means that they have the same quantum state and cannot be individually identified.

Why do we study identical particles?

We study identical particles because they play a crucial role in understanding many physical phenomena, such as the behavior of atoms, molecules, and subatomic particles. They also have applications in fields such as quantum computing and cryptography.

What is the Pauli exclusion principle?

The Pauli exclusion principle states that no two identical particles can occupy the same quantum state simultaneously. This means that two identical particles cannot occupy the same position and have the same momentum, spin, and energy at the same time.

What is the difference between bosons and fermions?

Bosons and fermions are two types of identical particles. Bosons have integer spin and follow Bose-Einstein statistics, meaning that they can occupy the same quantum state simultaneously. Fermions have half-integer spin and follow Fermi-Dirac statistics, meaning that they cannot occupy the same quantum state simultaneously due to the Pauli exclusion principle.

How do identical particles behave in different systems?

The behavior of identical particles depends on the system they are in. In a classical system, identical particles can be distinguished and behave like distinguishable particles. In a quantum system, identical particles are indistinguishable and follow the rules of quantum mechanics, such as the superposition principle and the Pauli exclusion principle.

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