27.1 Canonical anticommutation relations and the fermionic oscillator

27.1 Canonical anticommutation relations and the fermionic oscillator

Concept links · terms present in this machine draft; source roles are unverified: vector space · eigenvalue · irreducible representation · Pauli matrices

Recall that the Hamiltonian for the quantum harmonic oscillator system in $d$ degrees of freedom (setting $\hslash =m=\omega =1)$ is

$$H=\sum _{j=1}^d\frac{1}{2}(Q_j^2+P_j^2)$$

and that it can be diagonalized by introducing number operators $N_j=a_j^{†}{a}_j$ defined in terms of operators

$$a_j=\frac{1}{\sqrt{2}}(Q_j+i{P}_j),a_j^{†}=\frac{1}{\sqrt{2}}(Q_j-i{P}_j)$$

that satisfy the so-called canonical commutation relations (CCR)

$$[a_j,a_k^{†}]={\delta }_{jk}1,[a_j,a_k]=[a_j^{†},a_k^{†}]=0$$

The simple change in the harmonic oscillator problem that takes one from bosons to fermions is the replacement of the bosonic annihilation and creation operators (which we’ll now denote $a_B$ and $a_B{}^{†})$ by fermionic annihilation and creation operators called $a_F$ and $a_F{}^{†}$ , and replacement of the commutator

$$[A,B]\equiv AB-BA$$

of operators by the anticommutator

$$[A,B{]}_{+}\equiv AB+BA$$

The commutation relations are now (for $d=1$ , a single degree of freedom)

$$[a_F,a_F^{†}{]}_{+}=1,[a_F,a_F{]}_{+}=0,[a_F^{†},a_F^{†}{]}_{+}=0$$

with the last two relations implying that $a_F^2=0$ and $(a_F^{†}{)}^2=0$

The fermionic number operator

$$N_F=a_F^{†}{a}_F$$

now satisfies

$$N_F^2=a_F^{†}{a}_F{a}_F^{†}{a}_F=a_F^{†}(1-a_F^{†}{a}_F)a_F=N_F-a_F^{†2}{a}_F^2=N_F$$

(using the fact that $a_F^2={a_F^{†}}^2=0)$ . So one has

$$N_F^2-N_F=N_F(N_F-1)=0$$

which implies that the eigenvalues of $N_F$ are just 0 and 1. We’ll denote eigenvectors with such eigenvalues by |0⟩ and |1⟩. The simplest representation of the operators $a_F$ and $a_F^{†}$ on a complex vector space ${\mathcal{H}}_F$ will be on ${\mathbf{C}}^2$ , and choosing the basis

$$|0⟩=\left(\genfrac{}{}{0ex}{}{0}{1}\right),|1⟩=\left(\genfrac{}{}{0ex}{}{1}{0}\right)$$

the operators are represented as

$$a_F=\left(\begin{array}{cc}0& 0\\ 1& 0\end{array}\right),a_F^{†}=\left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right),N_F=\left(\begin{array}{cc}1& 0\\ 0& 0\end{array}\right)$$

Since

$$H=\frac{1}{2}(a_F^{†}{a}_F+a_F{a}_F^{†})$$

is just $\frac{1}{2}$ the identity operator, to get a non-trivial quantum system, instead we make a sign change and set

$$H=\frac{1}{2}(a_F^{†}{a}_F-a_F{a}_F^{†})=N_F-\frac{1}{2}1=\left(\begin{array}{cc}\frac{1}{2}& 0\\ 0& -\frac{1}{2}\end{array}\right)$$

The energies of the energy eigenstates |0⟩ and |1⟩ will then be $\pm \frac{1}{2}$ since

$$H|0⟩=-\frac{1}{2}|0⟩,H|1⟩=\frac{1}{2}|1⟩$$

Note that the quantum system we have constructed here is nothing but our old friend the two-state system of chapter 3. Taking complex linear combinations of the operators

$$a_F,a_F^{†},N_F,1$$

we $\mathrm{get}$ all linear transformations of ${\mathcal{H}}_F={\mathbf{C}}^2$ (so this is an irreducible representation of the algebra of these operators). The relation to the Pauli matrices is

$$a_F^{†}=\frac{1}{2}({\sigma }_1+i{\sigma }_2),a_F=\frac{1}{2}({\sigma }_1-i{\sigma }_2),H=\frac{1}{2}{\sigma }_3$$

Chapter contents · Next section: 27.2 Multiple degrees of freedom · 中文版本

来源与版本

正文：英文 · OCR 机器稿 · 待校对

核对状态：OCR 机器稿 · 待校对

原书 PDF · 印刷页 305、306、307、308、309、310

来源版本：2025-10-20

来源 PDF SHA-256：5a1941b2443b54d5db3d055f1e5ba390429b7a728475258017aaac87ee85a837

OCR 来源 SHA-256：c5f60a28e528298c225022ad97788868b73f69d85fe61d2f4d5175af6c239f1f

OCR 产物 SHA-256：c5f60a28e528298c225022ad97788868b73f69d85fe61d2f4d5175af6c239f1f