- #1
Salmone
- 101
- 13
If I have two identical particles of ##1/2## spin, for Pauli's exclusion principle all physical states must be antysimmetrical under the exchange of the two particles, so ##\hat{\Pi}|\alpha\rangle=-|\alpha\rangle##. Now, let's say for example this state ##\alpha## is an Hamiltonian eigenfunction and we have a superposition of different eigenstates, what about the antysimmetric condition? I thought this applies: since the exchange operator ##\Pi## is linear and all the states must be antysimmetric, even a states which is a superposition of eigenstates need to be antysimmetric so it must be a superposition of antysimmetric eigenstates. Example: ##|\psi\rangle=\frac{1}{\sqrt{2}}|\alpha_1\rangle+\frac{1}{\sqrt{2}}|\alpha_2\rangle## I apply the exchange operator: ##\Pi|\psi\rangle=\Pi(\frac{1}{\sqrt{2}}|\alpha_1\rangle+\frac{1}{\sqrt{2}}|\alpha_2\rangle)=\Pi\frac{1}{\sqrt{2}}|\alpha_1\rangle+\Pi\frac{1}{\sqrt{2}}|\alpha_2\rangle=-\frac{1}{\sqrt{2}}|\alpha_1\rangle-\frac{1}{\sqrt{2}}|\alpha_2\rangle=-(\frac{1}{\sqrt{2}}|\alpha_1\rangle+\frac{1}{\sqrt{2}}|\alpha_2\rangle)=-|\psi\rangle## and so it is antysimmetric. Is this right?