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$\newcommand{\ket}[1]{| #1\rangle}$This question basically has two very related parts. This came up in the context of trying to verify something my professor said a while ago: that if the wave functions for two identical particles are well separated (i.e. if they are very peaked, and the peaks are macroscopically far apart), then you can model them as distinguishable particles. He reasoned that when we swap the wave functions, the term we get from swapping is very small and can be ignored. So, taking the norm of the symmetrized wave function reduces to taking the norm of the non-trivial amplitude in the symmetrization. This will be the amplitude you would get by modeling the particles as distinguishable.

Yet, when I do this procedure, the normalization from the symmetrization procedure screws me up. I am unsure where I am going wrong.

Suppose I have two bosons. I know that one is in the state $\ket{\psi_1}$ and one is in the state $\ket{\psi_2}$.

The symmetrized state then is

$$ \frac{1}{\sqrt{2}} \left(\ket{\psi_1 \psi_2} + \ket{\psi_2\psi_1} \right)$$

Suppose the wave functions for $\ket{\psi_1}$ and $\ket{\psi_2}$ are peaked at separated place, or have non-overlapping support.

Then, if I look at $x_1,x_2$, with $x_1 \in \text{supp}({\psi_1 (\cdot)})$ and $x_2 \in \text{supp}({\psi_2 (\cdot)})$, then won't the probability that I observe a particle near $x_1$ and another near $x_2$ be

$$ \frac{1}{2} |\psi_1(x_1)|^2 |\psi_2(x_2)|^2$$

which is half of what it would be for distinguishable particles? This result feels wrong. I thought maybe something was wrong with the normalization, but it doesn't seem to be that there is.

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  • $\begingroup$ Finallly, a physics question in the HNQ list which can't be answered with a school textbook or a Wikipedia article. +1. $\endgroup$ – Dmitry Grigoryev Jun 30 '17 at 12:58
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People often say "quantum indistinguishable particles that are very far-separated behave like distinguishable particles," but this is a bit misleading. It would be more accurate to say "quantum indistinguishable particles that are very far-separated behave like classical indistinguishable particles." The difference is subtle but important.

The problem is in your sentence "if I look at $x_1$, $x_2$, with $x_1 \in \text{supp}(\psi_1(\cdot))$ and $x_2 \in \text{supp}(\psi_2(\cdot))$ ...". There's no possible experiment that says "I've detected particle $x_1$ at location $x$" - only experiments that say "I've detected a particle at location $x$." So the probability amplitude that you measure a particle at point $x$ and another particle at point $y$ is

$$\langle x, y | \frac{1}{\sqrt{2}}(|\psi_1 \psi_2 \rangle + | \psi_2 \psi_1 \rangle) = \frac{1}{\sqrt{2}}(\psi_1(x) \psi_2(y) + \psi_2(x) \psi_1(y))$$

and the actual probability is the norm-squared

$$P(x, y) = \frac{1}{2} \left( |\psi_1(x)|^2 |\psi_2(y)|^2 + |\psi_2(x)|^2 |\psi_1(y)|^2 \right),$$

where the cross-terms are all zero because the states $|\psi_1\rangle$ and $|\psi_2\rangle$ are orthogonal. This is indeed the result for classical indistinguishable particles, and $\int P(x, y)\ dx\, dy = 1$ as it should (unlike your expression).

The only things that differ from the generic quantum case are (a) you can just use the normalization factor of $1/\sqrt{2}$ and (b) when you norm-square the amplitude out to an actual probability density, you can ignore the cross-terms.

If you want to work classically right from the beginning, then you don't symmetrize the ket at all, but work with the ket $|\psi_1 \psi_2 \rangle$ instead. Then your normalization issue doesn't come up.

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  • $\begingroup$ This does not explain the question. It works around the problem by redefining what $P(x,y)$ means. In order to obtain a probability (ignoring the fact it's really a density) of finding one particle at $x_1$ and another at $x_2$ one'd have to manually sum $P(x_1,x_2) + P(x_2,x_1)$ (the two terms being equal). What event does then $P(x,y)$ alone describe as probability? $\endgroup$ – The Vee Jun 30 '17 at 9:28
  • $\begingroup$ (a) This was not meant to address your answer, sorry, I just did not want to say "density" like three times so I said it once with a disclaimer. (b) Of course I don't understand quantum physics. (c) It's not and that is exactly the heart of my comment. It's one half of the probability in question because in order to obtain the probability you'd have to sum $P(x,y)$ with $P(y,x)$. $\endgroup$ – The Vee Jul 1 '17 at 7:37
  • $\begingroup$ (The latter being not a matter of quantum mechanics. In a minimal example, let two indistinguishable particles have two possible states, $a$ and $b$, each. Then all we can know is for example "they are in different states". If we require that $\sum_{x,y} P(x,y) = 1$, this means for $P$ that $P(a,b) = \frac12$, $P(b,a) = \frac12$ and $P(a,a) = P(b,b) = 0$. Yet the probability of finding one particle in $a$ and one in $b$ is one. That is not $P(x,y)$ for any $x$, $y$.) $\endgroup$ – The Vee Jul 1 '17 at 8:15
  • $\begingroup$ But they are one half of that. Otherwise your integral would need to avoid double summing by imposing a bound $x ≤ y$ or something. By the way, that $|a\rangle\otimes|b\rangle \not\in \mathcal{H}$ is basically what my own answer says, and the problem vanishes if that is resolved first. $\endgroup$ – The Vee Jul 1 '17 at 8:38
  • $\begingroup$ Oh, and "one in..." meant occupation numbers, just like in the OP. I don't consider them distinct. $\endgroup$ – The Vee Jul 1 '17 at 8:56
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Instead of writing:

$$ \frac{1}{\sqrt{2}} \left(\ket{\psi_1 \psi_2} + \ket{\psi_2\psi_1} \right)$$

It will be more instructive to see this as

$$ \Psi(x_1, x_2) =\frac{1}{\sqrt{2}} (\psi_1(x_1) \psi_2(x_2) + \psi_1(x_2)\psi_2(x_1)) $$

Thus when you find $|\Psi|^2$, you get three (technically 4) terms:

$$ \frac{1}{{2}} (|\psi_1(x_1) \psi_2(x_2)|^2 + (\text{negligable interaction terms}) + |\psi_1(x_2) \psi_2(x_1)|^2)$$

But for indistinguishable particles, $ \psi_1(x_j) = \psi_1(x_i)$. Although the interaction terms vanish at large distance, this fact holds true regardless of distance between $x_i$ and $x_j$!. Exploiting that fact, and ignoring the interaction terms, we get:

$$ \frac{1}{2}(2 \cdot |\psi_1(x_1) \psi_2(x_2)|^2)$$

And so the result ends up being the same as distinguishable particles, but it just takes a different route to get there.

Edit: Remember that it is $\Psi$ and NOT $\psi$ that must be symmetric/antisymmetric for bosons/fermions respectively under the exchange of particles $x_1$ and $x_2$.

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  • $\begingroup$ $\psi_1(x_j)$ can't be trivially equal to $\psi_1(x_i)$ in the OP's notation. Only one of the two points lies in the support of that function. $\endgroup$ – The Vee Jun 30 '17 at 9:14
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    $\begingroup$ @TheVee dumb comment, I knew what he meant but what he wrote and so did you. $\endgroup$ – Señor O Jun 30 '17 at 17:30
  • $\begingroup$ Are you serious? This is a blatant discrepancy. It's not about what who meant. For a function $\psi_1$ with support centered about 1 and $\psi_2$ centered about 2 and points $x_1 = 1$ and $x_2 = 2$ how could you ever claim $\psi_1(x_1) = \psi_1(x_2)$? $\endgroup$ – The Vee Jul 1 '17 at 7:40
  • $\begingroup$ @TheVee uh..... it's referring to the particle index and not arbitrary x values. $\endgroup$ – Señor O Jul 1 '17 at 17:04
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You should use the bosonic symmetrization consistently (with the measured state as well) and say that the probability of detecting one particle in $|\phi_1\rangle$ and one in $|\phi_2\rangle$, where $\langle\phi_1|\phi_2\rangle = 0$, is the magnitude squared of the scalar product between $$|\Psi\rangle = \frac1{\sqrt2} (|\psi_1\rangle|\psi_2\rangle + |\psi_2\rangle|\psi_1\rangle)$$ and $$|\Phi\rangle = \frac1{\sqrt2} (|\phi_1\rangle|\phi_2\rangle + |\phi_2\rangle|\phi_1\rangle).$$ That is, after expansion, the mag-square of $$\langle\Phi|\Psi\rangle = \langle\phi_1|\psi_1\rangle\langle\phi_2|\psi_2\rangle + \langle\phi_2|\psi_1\rangle\langle\phi_1|\psi_2\rangle$$ rather than of $$(\langle\phi_1|\langle\phi_2|)\,|\Psi\rangle = \frac1{\sqrt2}\left(\langle\phi_1|\psi_1\rangle\langle\phi_2|\psi_2\rangle + \langle\phi_2|\psi_1\rangle\langle\phi_1|\psi_2\rangle\right)$$ that you used.

If you bend this formula for use with spatially separated generalized position eigenstates you get the correct result.

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