36.1 Multi-particle quantum systems as quanta of a harmonic oscillator

36.1 Multi-particle quantum systems as quanta of a harmonic oscillator

It turns out that quantum systems of identical particles are best understood by thinking of such particles as quanta of a harmonic oscillator system. We will begin with the bosonic case, then later consider the fermionic case, which uses the fermionic oscillator system.

36.1.1 Bosons and the quantum harmonic oscillator

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A fundamental postulate of quantum mechanics (see chapter 9) is that given a space of states ${\mathcal{H}}_1$ describing a bosonic single particle, a collection of N identical such particles has state space

$$S^N({\mathcal{H}}_1)=(\underset{N-times}{\underbrace{{\mathcal{H}}_1\otimes \cdots \otimes {\mathcal{H}}_1}}{)}^S$$

where the superscript $S$ means we take elements of the tensor product invariant under the action of the group $S_N$ by permutation of the N factors. To describe states that include superpositions of an arbitrary number of identical particles, one should take as state space the sum of these

$$S^{\ast }({\mathcal{H}}_1)=\mathbf{C}\oplus {\mathcal{H}}_1\oplus ({\mathcal{H}}_1\otimes {\mathcal{H}}_1{)}^S\oplus \dots \text{\hspace{1em}(36.1)}$$

This same symmetric part of a tensor product occurs in the Bargmann-Fock construction of the quantum state space for a phase space $M={\mathbf{R}}^{2\bar{d}}$ , where the Fock space ${\mathcal{F}}_d$ can be described in three diferent but isomorphic ways. Note that we generally won’t take care to distinguish here between ${\bar{\mathcal{F}}}_d^{fin}$ (superpositions of states with finite number of quanta) and its completion ${\mathcal{F}}_d$ (which includes states with an infinite number of quanta). The three diferent descriptions of ${\mathcal{F}}_d$ are:

${\mathcal{F}}_d$ has an orthonormal basis

$$|n_1,n_2,\dots ,n_d⟩$$

labeled by the eigenvalues $n_j$ of the number operators $N_j$ for $j=1,\cdots ,d$ Here $n_j\in \{0,1,2,\cdot \cdot \cdot \}$ and

$$n=\sum _{j=1}^d{n}_j$$

is finite. This is called the “occupation number” basis of ${\mathcal{F}}_d$ . In this basis, the annihilation and creation operators are

$$\begin{array}{rl}{a}_j|n_1,n_2,\dots ,n_d⟩& =\sqrt{n_j}|n_1,n_2,\dots ,n_j-1,\dots ,n_d⟩\\ a_j^{†}|n_1,n_2,\dots ,n_d⟩& =\sqrt{n_j+1}|n_1,n_2,\dots ,n_j+1,\dots ,n_d⟩\end{array}$$

${\mathcal{F}}_d$ is the space of polynomials $\mathbf{C}[z_1,z_2,\cdots ,z_d]$ in $d$ complex variables $z_j.$ , with inner product the $d$ dimensional version of equation 22.4 and orthonormal basis elements corresponding to the occupation number basis the monomials 1

$$\frac{1}{\sqrt{n_1!\cdots n_d!}}{z}_1^{n_1}{z}_2^{n_2}\dots z_d^{n_d}\text{\hspace{1em}(36.2)}$$

Here the annihilation and creation operators are

$$a_j=\frac{\partial }{\partial z_j},a_j^{†}=z_j$$

${\mathcal{F}}_d$ is the algebra

$$S^{\ast }({\mathcal{M}}_J^{+})=\mathbf{C}\oplus {\mathcal{M}}_J^{+}\oplus ({\mathcal{M}}_J^{+}\otimes {\mathcal{M}}_J^{+}{)}^S\oplus \dots \text{\hspace{1em}(36.3)}$$

with product the symmetrized tensor product given by equation 9.3. Here ${\mathcal{M}}_J^{+}$ is the space of complex linear functions on $M$ that are eigenvectors with eigenvalue +$i$ for the complex structure $J.$ . Using the isomorphism between monomials and symmetric tensor products (given on monomials in one variable by equation 9.4), the monomials 36.2 provide an orthonormal basis of this space. Expressions for the annihilation and creation operators acting on $S^{\ast }({\mathcal{M}}_J^{+})$ can be found using this isomorphism, using their action on monomials as derivative and multiplication operators.

For each of these descriptions of ${\mathcal{F}}_d.$ , the choice of orthonormal basis elements given above provides an inner product, with the annihilation and creation operators $a_j,a_j^{†}$ each other’s adjoints, satisfying the canonical commutation relations

$$[a_j,a_k^{†}]={\delta }_{jk}$$

We will describe the basis state $|n_1,n_2,\cdots ,n_d⟩$ as one containing $n_1$ quanta of type 1, $n_2$ quanta of type 2, etc., and a total number of quanta n.

Comparing 36.1 and 36.3, we see that these are the same state spaces if ${\mathcal{H}}_1={\mathcal{M}}_J^{+}$ This construction of multi-particle states by taking as dual classical phase space $M$ a space of solutions to a wave equation, then quantizing by the Bargmann-Fock method, with ${\mathcal{H}}_1={\mathcal{M}}_J^{+}$ the quantum state space for a single particle, is sometimes known as “second quantization”. Choosing a (complex) basis of ${\mathcal{H}}_1$ , for each basis element one gets an independent quantum harmonic oscillator, with corresponding occupation number the number of “quanta” labeled by that basis element. This formalism automatically implies indistinguishability of quanta and symmetry under interchange of quanta since only the numbers of quanta appear in the description of the state. The separate symmetry postulate needed in the conventional quantum mechanical description of multiple identical particles by tensor products is no longer needed.

In chapter 43 we’ll see that in the case of relativistic scalar quantum field theory the dual phase space $M$ will be the space of real solutions of an equation called the Klein-Gordon equation. The J needed for Bargmann-Fock quantization will be determined by the decomposition into positive and negative energy solutions, and ${\mathcal{H}}_1={\mathcal{M}}_J^{+}$ will be a space of states describing a single relativistic particle.

In this chapter, we’ll consider a non-relativistic theory, with wave equation the Schr¨odinger equation. Here the dual phase space $M$ of complex solutions will already be a complex vector space, and we can use the version of Bargmann-Fock quantization described in section 26.4, so ${\mathcal{H}}_1=\mathcal{M}$ . The “second quantization” terminology is appropriate, since we take as (dual) classical phase space a quantum state space, and quantize that.

36.1.2 Fermions and the fermionic oscillator

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For the case of fermionic particles, if ${\mathcal{H}}_1$ is the state space for a single particle, an arbitrary number of particles will be described by the state space

$${\Lambda }^{\ast }({\mathcal{H}}_1)=\mathbf{C}\oplus {\mathcal{H}}_1\oplus ({\mathcal{H}}_1\otimes {\mathcal{H}}_1{)}^A\oplus \dots$$

where (unlike the bosonic case) this is a finite sum if ${\mathcal{H}}_1$ is finite dimensional. One can proceed as in the bosonic case, using instead of ${\mathcal{F}}_d$ the fermionic oscillator state space ${\mathcal{H}}_F={\mathcal{F}}_d^{+}$ . This again has three isomorphic descriptions:

${\mathcal{F}}_d^{+}$ has an orthonormal basis

$$|n_1,n_2,\dots ,n_d⟩$$

labeled by the eigenvalues of the number operators $N_j$ for $j=1,\cdots ,d$ where $n_j\in \{0,1\}$ . This is called the “occupation number” basis of ${\mathcal{F}}_d^{+}$

${\mathcal{F}}_d^{+}$ is the Grassmann algebra $\mathbf{C}[{\theta }_1,{\theta }_2,\cdots ,{\theta }_d]$ (see section 30.1) of polynomials in $d$ anticommuting complex variables, with orthonormal basis elements corresponding to the occupation number basis the monomials

$${\theta }_1^{n_1}{\theta }_2^{n_2}\dots {\theta }_d^{n_d}\text{\hspace{1em}(36.4)}$$

${\mathcal{F}}_d^{+}$ is the algebra of antisymmetric multilinear forms

$${\Lambda }^{\ast }({\mathcal{V}}_J^{+})=\mathbf{C}\oplus {\mathcal{V}}_J^{+}\oplus ({\mathcal{V}}_J^{+}\otimes {\mathcal{V}}_J^{+}{)}^A\oplus \dots$$

discussed in section 9.6, with product the wedge-product (see equation $9.6)$ . Here ${\mathcal{V}}_J^{+}$ is the space of complex linear functions on a vector space $V={\mathbf{R}}^{2d}$ (the pseudo-classical phase space), eigenvectors with eigenvalue +$i$ for the complex structure J.

For each of these descriptions of ${\mathcal{F}}_d^{+}$ we have basis elements we can take to be orthonormal, providing an inner product on ${\mathcal{F}}_d^{+}$ . We also have a set of d annihilation and creation operators $a_{Fj},a_{F_j^{†}}$ that are each other’s adjoints, and satisfy the canonical anticommutation relations

$$[a_{Fj},a_{Fk}^{†}{]}_{+}={\delta }_{jk}$$

We will describe the basis state $|n_1,n_2,\cdots ,n_d⟩$ as one containing $n_1$ quanta of type 1, $n_2$ quanta of type 2, etc., and a total number of quanta

$$n=\sum _{j=1}^d{n}_j$$

Analogously to the bosonic case, a multi-particle fermionic theory can be constructed using ${\mathcal{F}}_d^{+}$ , by taking ${\mathcal{V}}_J^{+}={\mathcal{H}}_1$ This is a fermionic version of second quantization, with the multi-particle state space given by quantization of a pseudo-classical dual phase space $V$ of solutions to some wave equation. The formalism automatically implies the Pauli principle (no more than one quantum per state) as well as the antisymmetry property for states of multiple fermionic quanta that is a separate postulate in our earlier description of multiple particle states as tensor products.

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