The identity principle

The previous seven chapters dealt almost exclusively with one particle: an electron in a potential well, in a hydrogen atom, hopping between levels under a perturbation. On the rare occasions two particles appeared (the entangled states of section 3.10, for instance), we took it for granted that we could call them “particle 1” and “particle 2”.

This chapter has to face an awkward fact: when the two particles are of the same kind, the labels “1” and “2” simply will not stick. And this is not a technical difficulty — it is a matter of principle. Its consequences are astonishingly large: without it there would be no chemistry, no solids, no floor that fails to collapse under your feet.

A thought experiment: labelling billiard balls

Start in the classical world. On the table sit two perfectly identical white billiard balls, factory specifications exactly the same. Are they “identical”?

In one sense, yes. But you can still tell them apart, and the method is simple: keep watching. At $t=0$ name the one on the left A and the one on the right B, then track their trajectories the whole time. However they collide and swap places, at every instant you can say “the one on the left right now is A” (or B). Classical particles have continuous, definite trajectories, and a trajectory is a built-in label. Even without a pen, the label is always there.

Now replace them with two electrons. Trouble arrives:

1. Electrons have no trajectories. As chapter 2 explained, an electron is described by a wavefunction, and its position takes a value only when measured. Between two measurements, “which path did it take?” has no answer (think of the double slit).
2. Wave packets overlap. When two electrons come close, their wavefunctions merge into one blur in space. When the next measurement finds an electron somewhere, it is impossible in principle to answer “which of the two is this?” — not because the apparatus is too crude, but because the question has no counterpart in the theory at all.
3. Electrons carry no distinguishing “serial number”. Every electron has exactly the same mass, charge, and spin; experiments have checked the uniformity of the electron mass to better than one part in $10^{8}$. No electron is even slightly “older” than any other.

This gives us the definition of identical particles: particles whose intrinsic properties (mass, charge, spin, and so on) are all exactly equal, so that no physical procedure whatsoever can tell them apart. Any two electrons in the universe are identical; an electron and a proton are not (their masses differ by a factor of 1836); even a proton and a neutron are not (their charges differ).

Where the existing formalism breaks

By the rules of chapter 3, the state of a two-particle system is a wavefunction $\psi(\vec{r}_1,\vec{r}_2)$ — ignoring spin for now, with $\vec{r}_1$ the coordinate of “particle 1” and $\vec{r}_2$ that of “particle 2”.

But we have just argued that for identical particles the labels “particle 1” and “particle 2” carry no physical meaning — they are mere bookkeeping symbols we use when writing formulas. So we must demand: no observable prediction may depend on how the labels are assigned.

The probability density is observable, so at the very least

$$|\psi(\vec{r}_1,\vec{r}_2)|^2=|\psi(\vec{r}_2,\vec{r}_1)|^2\tag{8.1.1}$$

The probability of “finding one particle at $a$ and another at $b$” must not change just because you swapped the labels around. Note that a generic function of two variables can easily violate this — for example $\psi\propto\ee^{-r_1}\,\ee^{-2r_2}$ (one particle in an inner shell, one in an outer) plainly does not satisfy it. Identity is a genuine restriction on the state space, not empty talk.

The exchange operator

Promote “swapping labels” to an operator. Define the exchange operator $\hat{P}_{12}$:

$$\hat{P}_{12}\,\psi(\vec{r}_1,\vec{r}_2)=\psi(\vec{r}_2,\vec{r}_1)\tag{8.1.2}$$

The eigenvalues of the exchange operator can only be ±1basic~5 minexpand

Step 1: exchanging twice is the same as doing nothing.

$$\hat{P}_{12}^2\,\psi(\vec{r}_1,\vec{r}_2)=\hat{P}_{12}\,\psi(\vec{r}_2,\vec{r}_1)=\psi(\vec{r}_1,\vec{r}_2)\tag{8.1.3}$$

This holds for arbitrary $\psi$, so $\hat{P}_{12}^2=\hat{1}$.

Step 2: find the eigenvalues.

Suppose $\hat{P}_{12}\psi=\lambda\psi$. Apply $\hat{P}_{12}$ once more to both sides:

$$\hat{P}_{12}^2\psi=\lambda\hat{P}_{12}\psi=\lambda^2\psi\tag{8.1.4}$$

But the left-hand side is just $\psi$, so $\lambda^2=1$, that is

$$\lambda=+1\quad\text{or}\quad\lambda=-1\tag{8.1.5}$$

Step 3: the two families of eigenstates.

• $\lambda=+1$: $\psi(\vec{r}_2,\vec{r}_1)=+\psi(\vec{r}_1,\vec{r}_2)$, called symmetric;
• $\lambda=-1$: $\psi(\vec{r}_2,\vec{r}_1)=-\psi(\vec{r}_1,\vec{r}_2)$, called antisymmetric.

Both satisfy $|\psi(\vec{r}_1,\vec{r}_2)|^2=|\psi(\vec{r}_2,\vec{r}_1)|^2$ — the modulus squared swallows the sign.

Step 4: this property is never lost in time.

The Hamiltonian of identical particles is necessarily symmetric in the two coordinates (equal masses, and a potential energy unchanged when the coordinates are swapped), for example $\hat{H}=\frac{\hat p_1^2}{2m}+\frac{\hat p_2^2}{2m}+V(\vec{r}_1)+V(\vec{r}_2)+U(|\vec{r}_1-\vec{r}_2|)$. For such an $\hat{H}$, exchanging then evolving gives the same result as evolving then exchanging:

$$[\hat{P}_{12},\hat{H}]=0\tag{8.1.6}$$

By the result of section 3.6, $\hat{P}_{12}$ is a conserved quantity: a state that starts symmetric stays symmetric forever, and one that starts antisymmetric stays antisymmetric. The symmetry is a particle’s “identity for life”.

Why it must be one or the other

All we have proved so far is: if a state is an eigenstate of $\hat{P}_{12}$, its eigenvalue is ±1. But a general state can be a superposition of the two, say $\psi=c_S\psi_S+c_A\psi_A$, which under exchange becomes $c_S\psi_S-c_A\psi_A$ — a different state. The modulus squared of such a superposition is not invariant under exchange (the cross term flips sign), so it makes label-dependent predictions and must be ruled out.

Quantum mechanics therefore adds one more postulate (sometimes called the symmetrization postulate):

The physical states of a system of identical particles must be eigenstates of the exchange operator: either fully symmetric under every pairwise exchange, or fully antisymmetric.

This cannot be derived from the earlier postulates. It has the same standing as the five postulates of section 3.7: an independent assumption, tested by experiment again and again.

Identity changes the counting: a concrete example

The most direct observable consequence of identity is that the number of states changes. Take 3 single-particle levels $\varepsilon_1,\varepsilon_2,\varepsilon_3$ and put 2 particles in them:

Particle type	Allowed states	Count
Distinguishable	$\phi_i(1)\phi_j(2)$, any $i,j$	$3\times3=9$
Identical, symmetric	symmetric combinations of $\phi_i\phi_j$ ($i\le j$, including $i=j$)	$6$
Identical, antisymmetric	$\psi_-$, requiring $i\ne j$	$3$

The three countings correspond to three statistical distributions: Maxwell–Boltzmann (classical), Bose–Einstein, and Fermi–Dirac. Look at the antisymmetric row: when $i=j$, $\psi_-\equiv0$ — the state with two particles in the same level has simply vanished. That is the embryo of the Pauli exclusion principle, two sections from now. Notice also that for symmetric particles the fraction of “crowded into the same level” states ($3/6$) is higher than in the distinguishable case ($3/9$): symmetric particles have a statistical tendency to bunch, and both lasers and Bose–Einstein condensation run on it.

∎Key formulas

• Exchange operator

  $$\hat{P}_{12}\,\psi(\vec{r}_1,\vec{r}_2)=\psi(\vec{r}_2,\vec{r}_1),\qquad \hat{P}_{12}^2=\hat{1}$$

  Its eigenvalues can only be ±1

• Symmetrization postulate

  $$\hat{P}_{12}\ket{\psi}=\pm\ket{\psi}$$

  Physical states must pick one; the sign is fixed by the particle species

• Conservation

  $$[\hat{P}_{12},\hat{H}]=0$$

  The symmetry is an identity for life; evolution never changes it

• Basic two-particle combinations

  $$\psi_\pm=\frac{1}{\sqrt2}\bigl[\phi_a(\vec{r}_1)\phi_b(\vec{r}_2)\pm\phi_b(\vec{r}_1)\phi_a(\vec{r}_2)\bigr]$$

  The antisymmetric combination vanishes identically when a = b

?Self-check4 questions

1. 1.

   Why can classical particles be "labelled" while identical quantum particles cannot?

   • Classical particles have continuous, definite trajectories that serve as labels; once quantum wave packets overlap, "which one is this" corresponds to no observable in the theory
   • Classical particles are big, so it is easy to mark them
   • Quantum particles are too small for instruments to resolve
   • Quantum particles are constantly being created and annihilated

   Submit
2. 2.

   Which statements about the exchange operator P̂₁₂ are correct? (Select all that apply.)

   Select all that apply

   • P̂₁₂² = 1, so its eigenvalues can only be +1 or −1
   • The Hamiltonian of identical particles satisfies [P̂₁₂, Ĥ] = 0, so the symmetry is conserved
   • Every two-particle state is automatically an eigenstate of P̂₁₂
   • Invariance of |ψ|² under exchange is guaranteed once ψ is an eigenstate of P̂₁₂

   Submit
3. 3.

   Place 2 identical particles in 4 single-particle levels. If the wavefunction must be antisymmetric (and only this degree of freedom counts), how many distinct allowed states are there?

   个0% relative tolerance

   Submit
4. 4.

   An electron on Earth and an electron on the Moon are also identical. Why can a lab calculation on Earth ignore the Moon electron?

   • Their wavefunctions do not overlap, so the exchange term contributes exactly zero to every observable
   • The distance is so large that the Coulomb force is negligible
   • The Moon electron is in a mixed state
   • Identity holds only for electrons within the same atom

   Submit

What comes next

We have proved that physical states must be “symmetric or antisymmetric — pick one”, but not answered: who picks which?

The answer is startlingly tidy: the particle’s spin decides, with no exceptions — half-integer spin is always antisymmetric, integer spin always symmetric. The next section introduces the names of these two great families, fermions and bosons, and the theorem that welds spin to statistics.