0
$\begingroup$

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?

Improve this question
$\endgroup$

1 Answer 1

Reset to default
1
$\begingroup$

You can't just exchange the particles in the first or the second term individually; you need to exchange them in both terms together, and see what sign the entire state acquires.

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.

Improve this answer
$\endgroup$
3
  • 1
    $\begingroup$ 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. $\endgroup$
    – BLAZE
    Commented Jun 6, 2017 at 5:09
  • 1
    $\begingroup$ @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). $\endgroup$
    – tparker
    Commented Jun 6, 2017 at 5:26
  • 1
    $\begingroup$ 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. $\endgroup$
    – BLAZE
    Commented Jun 6, 2017 at 5:34

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.