Identical Particle Exercises

These exercises practice the exchange-symmetry layer added after forming a tensor-product space. The labels $1,2,\ldots$ below are slot labels, not names of persistent particles. We use ordinary Bose or Fermi statistics and complete one-particle states, including spin when relevant. Bosonic states are symmetric under slot exchange; fermionic states are antisymmetric. Exchange symmetry is a constraint on the full state, not an additional interaction.

Required background. Symmetrization and symmetric or antisymmetric wavefunctions give the projection rules. Pauli exclusion, Slater determinants, and spin–spatial wavefunctions supply the later examples.

Exchange symmetry and normalization

1. Two orthogonal states

Let $\lvert a\rangle$ and $\lvert b\rangle$ be distinct orthonormal one-particle states. Construct the normalized symmetric and antisymmetric two-particle states, and verify their exchange parity.

Solution

The normalized symmetric state is

$$\lvert a,b\rangle_S = \frac{1}{\sqrt2} \bigl( \lvert a\rangle_1\lvert b\rangle_2 +\lvert b\rangle_1\lvert a\rangle_2 \bigr).$$

The normalized antisymmetric state is

$$\lvert a,b\rangle_A = \frac{1}{\sqrt2} \bigl( \lvert a\rangle_1\lvert b\rangle_2 -\lvert b\rangle_1\lvert a\rangle_2 \bigr).$$

The exchange operator $P_{12}$ swaps the two slots. Therefore

$$P_{12}\lvert a,b\rangle_S = \lvert a,b\rangle_S, \qquad P_{12}\lvert a,b\rangle_A = -\lvert a,b\rangle_A.$$

The first state is allowed for identical bosons; the second is allowed for identical fermions.

2. Repeated occupation

What happens if the two one-particle states in Exercise 1 are the same state $\lvert a\rangle$?

Solution

For bosons, two particles can occupy the same one-particle state. The normalized two-slot state is simply

$$\lvert a\rangle_1\lvert a\rangle_2.$$

Using the distinct-state formula with a plus sign would give $\sqrt2\,\lvert a\rangle_1\lvert a\rangle_2$, so it is not the correct normalized expression in the same-state limit.

For fermions, the antisymmetric projection vanishes:

$$\frac{1}{\sqrt2} \bigl( \lvert a\rangle_1\lvert a\rangle_2 -\lvert a\rangle_1\lvert a\rangle_2 \bigr) = 0.$$

There is no normalized state with two identical fermions in the same complete one-particle state.

3. Overlapping orbitals

Let $\lvert a\rangle$ and $\lvert b\rangle$ be normalized but not necessarily orthogonal, with

$$s=\langle a\vert b\rangle.$$

Find the normalization factors for

$$\lvert a\rangle_1\lvert b\rangle_2 \pm \lvert b\rangle_1\lvert a\rangle_2.$$

Solution

Let

$$\lvert\Phi_\pm\rangle = \lvert a\rangle_1\lvert b\rangle_2 \pm \lvert b\rangle_1\lvert a\rangle_2.$$

Then

$$\begin{aligned} \langle\Phi_\pm\vert\Phi_\pm\rangle &= 1+1 \pm \langle a\vert b\rangle\langle b\vert a\rangle \pm \langle b\vert a\rangle\langle a\vert b\rangle\\ &= 2(1\pm\lvert s\rvert^2). \end{aligned}$$

Therefore the normalized states are

$$\lvert\Phi_\pm\rangle_{\mathrm{norm}} = \frac{ \lvert a\rangle_1\lvert b\rangle_2 \pm \lvert b\rangle_1\lvert a\rangle_2 }{ \sqrt{2(1\pm\lvert s\rvert^2)} }.$$

Only $\lvert s\rvert^2$ appears, so changing the arbitrary phase of either orbital cannot change the normalization. If $\lvert s\rvert=1$, the antisymmetric vector is zero and cannot be normalized; the two orbitals describe the same ray.

4. Two electrons in one spatial orbital

Two electrons occupy the same spatial orbital $\phi(\mathbf x)$. Which spin state is allowed? Explain using exchange symmetry.

Solution

The spatial part is

$$\psi_{\mathrm{space}}(\mathbf x_1,\mathbf x_2) = \phi(\mathbf x_1)\phi(\mathbf x_2),$$

which is symmetric under exchange of the two electron slots. Electrons are fermions, so the total state must be antisymmetric. Therefore the spin part must be antisymmetric:

$$\chi_{\mathrm{singlet}} = \frac{1}{\sqrt2} \bigl( \lvert\uparrow\downarrow\rangle -\lvert\downarrow\uparrow\rangle \bigr).$$

The triplet spin states are symmetric and would make the total state symmetric, so they are not allowed for two electrons in the same spatial orbital.

5. Spatial and spin exchange parity

Suppose two electrons occupy distinct orthonormal spatial orbitals $a$ and $b$. Define

$$\psi_\pm(\mathbf x_1,\mathbf x_2) = \frac{1}{\sqrt2} \bigl[ a(\mathbf x_1)b(\mathbf x_2) \pm b(\mathbf x_1)a(\mathbf x_2) \bigr].$$

Which spin symmetry must accompany $\psi_+$ and which must accompany $\psi_-$?

Solution

The state $\psi_+$ is spatially symmetric. To make the total two-electron state antisymmetric, it must be multiplied by the antisymmetric spin singlet.

The state $\psi_-$ is spatially antisymmetric. It must be multiplied by a symmetric triplet spin state. Symbolically,

$$\begin{array}{ccl} \text{symmetric spatial} &\times& \text{antisymmetric spin},\\ \text{antisymmetric spatial} &\times& \text{symmetric spin}. \end{array}$$

The total spatial-spin product is antisymmetric in both cases. These are symmetry-adapted product states; a general two-electron state can be a superposition of singlet and triplet sectors and need not factor into one spatial state times one spin state.

6. A three-boson occupation state

Translate the bosonic occupation state $\lvert2_a,0_b,1_c\rangle_B$ into a normalized first-quantized slot wavefunction using orthonormal modes $a,b,c$.

Solution

The occupation state has two bosons in mode $a$ and one boson in mode $c$. The normalized symmetric slot state is

$$\lvert2_a,0_b,1_c\rangle_B = \frac{1}{\sqrt3} \bigl( \lvert a\rangle_1\lvert a\rangle_2\lvert c\rangle_3 +\lvert a\rangle_1\lvert c\rangle_2\lvert a\rangle_3 +\lvert c\rangle_1\lvert a\rangle_2\lvert a\rangle_3 \bigr).$$

There are three distinct slot arrangements, and they are orthonormal because $a$ and $c$ are orthonormal.

7. Fermionic mode order

Translate the fermionic occupation state $\lvert1_a,0_b,1_c\rangle_F$ into a normalized two-slot state, assuming the mode order is $a,b,c$.

Solution

With modes ordered as $a,b,c$, the state with modes $a$ and $c$ occupied corresponds to

$$\lvert1_a,0_b,1_c\rangle_F = \frac{1}{\sqrt2} \bigl( \lvert a\rangle_1\lvert c\rangle_2 -\lvert c\rangle_1\lvert a\rangle_2 \bigr).$$

The minus sign is the two-particle form of fermionic antisymmetry. In occupation notation the same state is recorded compactly as the bitstring with occupations $(1,0,1)$.

Continue with Fock Space Exercises to act on these occupation states with creation and annihilation operators.

References

• L. E. Ballentine, Quantum Mechanics: A Modern Development, 2nd ed., World Scientific, 2014.
• P. A. M. Dirac, The Principles of Quantum Mechanics, 4th ed., Oxford University Press, 1958.
• A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Systems, McGraw-Hill, 1971.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.