# How to use the exchange operator to determine the symmetry of a wavefunction?

I have just started learning about the exchange operator to determine whether wavefunctions are symmetric or antisymmetric and I have an example as follows:

What is the symmetry of the state $$\big|\downarrow\uparrow\,\rangle-\big|\uparrow\downarrow\,\rangle\tag{1}$$ with respect to exchange of the 2 particles?

If I swap the spins (particles) of the first component of the wavefunction I find that $$\big|\uparrow\downarrow\,\rangle-\big|\uparrow\downarrow\,\rangle=0$$

If I swap the spins (particles) of the second component of the wavefunction I find that $$\big|\downarrow\uparrow\,\rangle-\big|\downarrow\uparrow\,\rangle=0$$

For this example; the condition to be a symmetric wavefunction is: $$\big|\downarrow\uparrow\,\rangle=\big|\uparrow\downarrow\,\rangle$$ or $$\big|\uparrow\downarrow\,\rangle=\big|\downarrow\uparrow\,\rangle$$

For this example; the condition to be a anti-symmetric wavefunction is: $$\big|\uparrow\downarrow\,\rangle=-\,\big|\downarrow\uparrow\,\rangle$$ or $$\big|\downarrow\uparrow\,\rangle=-\,\big|\uparrow\downarrow\,\rangle$$

So given this information how do I tell if the state $(1)$ is symmetric or anti-symmtric?

Also, your equations under the break are wrong. If $\hat{P}$ is the particle exchange operator, the condition for a state $| \psi \rangle$ to be symmetric or antisymmetric is $\hat{P} | \psi \rangle = | \psi \rangle$ or $\hat{P} | \psi \rangle = -| \psi \rangle$, respectively. The state $| \uparrow \downarrow \rangle$ is neither equal to $| \downarrow \uparrow \rangle$ nor to $- | \downarrow \uparrow \rangle$, so that state is neither symmetric nor antisymmetric under particle exchange.
• Thank you for your answer, if I "exchange them in both terms together" I get $\big|\downarrow\uparrow\,\rangle-\big|\uparrow\downarrow\,\rangle\to \big|\uparrow\downarrow\,\rangle-\big|\downarrow\uparrow\,\rangle$ but where do I go from here. Clearly, I'm missing the point so would you please elaborate in your answer to explain what you mean by 'not exchanging the particles individually'? Also could you show me how to arrive at the correct answer [$(1)$ is antisymmetric]. I just need an model calculation using $\hat P$ to understand the logic. Many thanks. Commented Jun 6, 2017 at 5:09
• @BLAZE If $| \psi \rangle$ is symmetric under particle exchange, then $\hat{P} | \psi \rangle = + | \psi \rangle$. If it's antisymmetric under particle exchange, then $\hat{P} | \psi \rangle = - | \psi \rangle$. If it's neither, then $\hat{P} | \psi \rangle$ is not proportional to $| \psi \rangle$ at all. So first calculate $\hat{P} | \psi \rangle$ (which you have done correctly), then decide whether this new state is equal to $+1$ or $-1$ times the original state (or neither). Commented Jun 6, 2017 at 5:26
• Thank you, I now understand. In this case $\hat{P} | \psi \rangle= - (\big|\downarrow\uparrow\,\rangle-\big|\uparrow\downarrow\,\rangle)$. I see now why you gave hints instead of just computing the answer. That was very helpful, thanks. Commented Jun 6, 2017 at 5:34