Fock Space and Slater Determinants

Constructing States from the Vacuum

The foundation of Fock space is the vacuum state, denoted $|0⟩$. This is the state containing no particles, and it is defined by the property that it is annihilated by any annihilation operator:

From this empty state, we can build the entire Fock space. A state containing a single fermion in the single-particle state ${\nu }_1$ is created by simply acting on the vacuum with the corresponding creation operator:

To create a two-particle state with fermions in states ${\nu }_1$ and ${\nu }_2$, we apply two creation operators:

Here, the operator algebra from our last lesson immediately reveals its power. What if we had created the particles in the opposite order? Using the anticommutation relation $\{c_{{\nu }_1}^{†},c_{{\nu }_2}^{†}\}=0$, we find:

The state vector acquires a minus sign, precisely capturing the antisymmetry required for exchanging two identical fermions. Furthermore, if we try to create two particles in the same state, say ${\nu }_1$, the anticommutation relation implies $(c_{{\nu }_1}^{†}{)}^2=0$, so $c_{{\nu }_1}^{†}{c}_{{\nu }_1}^{†}|0⟩=0$. The formalism automatically enforces the Pauli exclusion principle.

Generalizing this, a normalized N-particle state corresponding to the occupation of N distinct single-particle states $\{{\nu }_1,{\nu }_2,\dots ,{\nu }_N\}$ is constructed as:

It is crucial to establish a fixed, canonical ordering for the single-particle basis states (e.g., by increasing energy). The sign of the N-particle state vector then depends on the order in which the creation operators are written. This convention must be maintained consistently throughout any calculation.

The following video segment summarizes this construction process.

The Connection to Slater Determinants

You'll recall from your quantum mechanics courses that in the first quantization formalism, the state of N identical fermions is represented by a Slater determinant. This is an explicitly antisymmetrized wavefunction constructed from the single-particle wavefunctions ${\varphi }_{\nu }(x)$.

The following reading from a set of lecture notes by Henrik Johannesson provides a concise review of the Slater determinant before explicitly linking it to the second-quantized state construction we've just discussed.

The central message is that the state $|{\Psi }_N⟩=c_{{\nu }_N}^{†}\cdots c_{{\nu }_1}^{†}|0⟩$ is not just an analogy; it is the state represented by the Slater determinant. The operator algebra of second quantization provides an elegant and compact way to handle the combinatorics of signs and permutations that are explicit in the determinant form.

This next video segment makes this connection even more explicit.

Orthonormality of Fock States

A crucial property of the N-particle states we construct is that they form an orthonormal basis. States built from different sets of single-particle orbitals are orthogonal. A state built from a specific set of N creation operators is normalized to one.

Let's prove the normalization for a two-particle state $|\Psi ⟩=c_2^{†}{c}_1^{†}|0⟩$, assuming the vacuum is normalized, $⟨0|0⟩=1$. The norm-squared is $⟨\Psi |\Psi ⟩$.

Now we use the CAR, $\{c_{\alpha },c_{\beta }^{†}\}={\delta }_{\alpha \beta }$, to move the annihilation operators to the right until they act on the vacuum.

1. Move $c_2$ past $c_2^{†}$: $c_2{c}_2^{†}=-c_2^{†}{c}_2+\{c_2,c_2^{†}\}=-c_2^{†}{c}_2+1$. $$⟨\Psi |\Psi ⟩=⟨0|c_1(-c_2^{†}{c}_2+1)c_1^{†}|0⟩=⟨0|c_1{c}_1^{†}|0⟩-⟨0|c_1{c}_2^{†}{c}_2{c}_1^{†}|0⟩$$
2. The second term is zero because we can move $c_2$ to the far right (it anticommutes with $c_1$ and $c_1^{†}$), where it will give $c_2|0⟩=0$.
3. We are left with $⟨0|c_1{c}_1^{†}|0⟩$. We use the CAR again: $c_1{c}_1^{†}=-c_1^{†}{c}_1+1$. $$⟨\Psi |\Psi ⟩=⟨0|(-c_1^{†}{c}_1+1)|0⟩=⟨0|1|0⟩-⟨0|c_1^{†}{c}_1|0⟩$$
4. The second term is again zero because $c_1|0⟩=0$. This leaves us with: $$⟨\Psi |\Psi ⟩=⟨0|0⟩=1$$

This calculation demonstrates that the state is properly normalized. The procedure can be generalized by induction to show that any state $c_{{\nu }_N}^{†}\cdots c_{{\nu }_1}^{†}|0⟩$ with distinct ${\nu }_i$ is normalized, and that states built from different sets of orbitals are orthogonal.

A Quick Contrast with Bosons

To highlight what makes fermions special, it is useful to contrast them with bosons. Bosonic creation ($b^{†}$) and annihilation ($b$) operators obey commutation relations:

The fact that creation operators for different states commute ($b_{\alpha }^{†}{b}_{\beta }^{†}=b_{\beta }^{†}{b}_{\alpha }^{†}$) means the resulting many-boson state is symmetric under particle exchange. Furthermore, since $(b_{\alpha }^{†}{)}^2\ne 0$, multiple bosons can occupy the same single-particle state.

Conclusion

In this lesson, we have solidified our understanding of the Fock space representation. We have established a clear and concrete procedure for constructing many-fermion states.

Here are the key takeaways:

• An N-fermion basis state is created by applying a string of N distinct creation operators to the vacuum state: $|{\Psi }_N⟩=c_{{\nu }_N}^{†}\cdots c_{{\nu }_1}^{†}|0⟩$.
• This construction is the direct equivalent of the Slater determinant in first quantization. The antisymmetry and the Pauli exclusion principle are automatically enforced by the canonical anticommutation relations of the operators.
• The resulting basis of Fock states is orthonormal, a property which also follows directly from the operator algebra.

We now know how to represent many-body states. The next logical step is to represent the physical observables and interactions that act on these states. In the upcoming lesson, we will learn how to express single-particle operators, such as kinetic energy, in the language of second quantization. This will bring us one step closer to writing down our first many-body Hamiltonian.