First- and second-quantization operators compared

In Box 1.2, we summarize some of the characteristics of operators in the first and second quantizations. The dependence on the spin-oibital basis is different in the two representations. In first quantization, the Slater determinants depend on the spin-orbital basis whereas the operators are independent of the spin orbitals. In the second-quantization formalism, the ON vectors are basis vectors in a linear vector space and contain no reference to the spin-orbital basis. Instead, the reference to the spin-orbital basis is made in the operators. We also note that, whereas the first-quantization operators depend explicitly on the number of electrons, no such dependence is found in the second-quantization operators. 

The fact that the second-quantization operators are projections of the exact operators onto the spin-orbital basis means that a second-quantization operator times an ON vector is Just another 

The projected nature of the second-quantization operators has many ramifications. For exan le, relations that hold for exact operators such as the canonical commutation properties of the coordinate and momentum operators do not necessarily hold for projected operators. Similarly, the projected coordinate operator does not commute with the projected Coulomb repulsion operator. It should be emphasized, however, that these problems are not peculiar to second quantization but arise whenever a finite basis is employed. They also arise in first quantiztttion. but not until the matrix elements are evaluated. 

Second quantization treats operators and wave functions in a unified way - they are all expressed in terms of the elementary creation and annihilation operators. This property of the second-quantization formalism can, for example, be exploited to express modifications to the wave function as changes in the operators. To illustrate the unified description of states and operators afforded by second quantization, we note that any ON vector may be written compactly as a string of creation operators working on the vacuum state (1.2.4) 

Matrix elements may therefore be viewed as the vacuum expectation value of an operator 

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