[LSN/QFT-04003] The Physical Interpretation of States in Number Representation

The canonical quantization of Schr\"{o}dinger field is completed by postulating the equal time commutator relation (ETCR) \begin{equation}\label{EQ01} \big[\psi(x,t), \pi(y,t)\big]= i\hbar \delta(x-y) \end{equation} is the first and last step towards quantization.

 The creation operators \(a^\dagger_\mathbf k\) and annihilation operators \(a_\mathbf k\) and number operators obey the commutation relations \begin{eqnarray} [a_\mathbf k, a^\dagger_{\mathbf k{'}}]&=&\delta_{\mathbf k,\mathbf k{'}}; \\{} [a_\mathbf k,a_{\mathbf k{'}}]=0 &\quad& [a_\mathbf k^\dagger,a_{\mathbf k{'}} ^\dagger ]=0; \end{eqnarray}

1. The operators \(N_\mathbf k=a^\dagger_\mathbf k a_\mathbf k\) form a commuting set of hermitian operators and obey commutation relations. \[[a_\mathbf k^\dagger, N_\mathbf k] =1, \qquad [a_\mathbf k, N_\mathbf k]=-1 \] all other commutators, \(N_\mathbf p\) with \(a_\mathbf q\) or \(a^\dagger_\mathbf q\) are zero for \(\mathbf p\ne \mathbf q\).

 We had defined number operators \(N_k\) which are hermitian and commute
pairwise. Their eigenvalues can be worked out. In fact their algebra is the
same as one encounters in the treatment of the harmonic oscillator. The
eigenvalues are all non-negative integers.  

The fact that \(N_k, k=1,2,3..\) commute pairwise, their eigenvectors form a complete orthonormal set. This means that the basis vectors are specified by a sequence of corresponding eigenvalues \(\nu_1,\nu_2,\ldots\). The details of interpretation of states \(\ket{\nu_1,\nu_2, \ldots, \nu_k,\ldots}\) depends on the choice of the orthonormal set \(\{u_n(x)|n=1,2,\ldots\}\)