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qft-lsn-04006
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I Lesson Overview
Objectives
To define normal product of operators and to give examples of computation of simple matrix elements.
Prerequisites
Quantization of Schrodinger Field; Definition of creation and annihilation operators;
Properties of $a_k, a^\dagger_k, N_k$
Multiparticle states
II Recall and Discuss
Expansion of Fields
In the interaction picture the total Hamiltonian is split into two parts \(H= H_0+ H{'}\). Let\(u_n\) denote the eigenfunctions of \(H_0\) with eigenvalues \(E_n\)
\begin{equation} H_0 u_n(\mathbf x)= E_n u_n(\mathbf x) \end{equation}
Taking the expansion of the field in terms of \(u_n(x)\) as \begin{equation} \psi(\mathbf x,t) = \sum_n a_n u_n(\mathbf x) e^{-iE_nt/\hbar}, \qquad \psi^\dagger(\mathbf x,t) = \sum_n a_n^\dagger u_n^*(\mathbf x) e^{iE_nt/\hbar}, \end{equation} We note that the operators \(a_n\) will be independent of time.
Multi particle states
The states corresponding to \(\nu_1,\nu_2,...\) particle in levels \(m_1, m_2,...\) are defined by \begin{equation} \ket{\nu_1, \nu_2, ...}= \prod_m\frac{(a_{k}^\dagger)^{\nu_k}}{\sqrt{\nu_k!}}\,\ket{0}. \end{equation}
Commutation Relations
The field operators obey equal time commutation relations. \begin{eqnarray} [\psi(\mathbf x,t),\psi^\dagger(\mathbf y,t)] = \delta(x-y) \end{eqnarray} The nonzero commutators of field with \(a_n, a_n^\dagger\) can be worked out \begin{eqnarray} [a_n, \psi^\dagger(\mathbf x,t)] = u_n^*(\mathbf x,t), \qquad [\psi(\mathbf x,t), a_n^\dagger] = u_n(\mathbf x,t) \end{eqnarray} where \(u_n(\mathbf x,t)= u_n(\mathbf x) e^{-iE_nt/\hbar}\).
III Main Topics
- \(\S1\) Normal Product and Matrix Elements
- \(\S2\) Examples-normal-product
IV EndNotes
The examples here are meant to illustrate the normal form and computation of matrix elements. Eventually. Wick's theorem and a few standard rules for handling expressions, like those in above examples, will enable us to write the answers directly.
Download: qft-lsn-04006.pdf (179.85 KB)
