# Wavefunctions for Indistinguishable and Distinguishable particles -

Wavefunctions for Indistinguishable and Distinguishable particles - URGENT

## Homework Statement

A one-dimensional potential well has a set of single-particle energy eigenstates Un(x) with energies En=E_o n^2 where n=1,2,3... Two particles are placed in the well with three possible sets of properties.
a)2 distinguishable spin 0 particles
b)2 identical spin 0 particles
c)2 identical spin 1/2 particles

Write down the spatial part of the two-particle wave functions at t=0 for the two lowest energy states of the two-particle system, and hence give the degeneracies of these energy states and explain how these two-particle wavefunctions depend on time

## The Attempt at a Solution

I am getting incredibly confused by two-particle wavefunctions and between the spatial and spin states....

a) If there are 2 indenticle spin 0 particles: The wavefunction must be symmetric then would the wavefunction just be
(|0>|1> + |1>|0>)/sqrt(2)

b) 2 identicle spin 1/2 particles: The wavefunction must be antisymmetric due to Pauli Exclusion Principle so
(|0>|1> - |1> |0>)/sqrt(2)

c) 2 indistinguishable particles (boson or fermion) would it just be a wavefunction with 6 dimensions
ie phi(r1,r2)

How would you then write the spin part of the wavefunction...and how do they depend on time....

Realise this might be all wrong but I have an exam coming up and would REALLY appreciate someone clarifying this!

Spin is one of those concepts that seems really hard to understand. The simple answer is that essentially all that is required is that opposite spins are orthogonal to each other in a particular basis.

So if you have spin up and spin down you might represent them as matrices like:

Spin up:$$\left[ \begin{array}{cc}1 \\ 0 \end{array} \right]$$

Spin down:$$\left[ \begin{array}{cc}0 \\ 1 \end{array} \right]$$

Does that mean that my wavefunctions look correct?

How would they differ for the two lowest energy states?