Expressing Quantum-Mechanical Operators in Second Quantization

EXPRESSING QUANTUM-MECHANICAL OPERATORS IN SECOND QUANTIZATION 

Having now seen how state vectors that are in one-to-one correspondence with /V-eIcctron Slater determinants can be represented in terms of Fermion creation and annihilation operators, it still remains for us to show how to express one- and two-electron operators in this language. The second-quantized version of any operator is obtained by simply demanding that the operator, when sandwiched between ket vectors of the form [ r vac , yield exactly the same result as arises in using the first quantized operator between corresponding Slater determinant wavefunctions. For an arbitrary one-electron operator, which in first-quantized language is 5] = i /( i) second quantized equivalent is 

Here represents the usual two-electron integral involving the 

When g l,2) = we often express these integrals in short-hand notation as r5 tM . It should be noted that the order of the creation and annihilation operators appearing in Eq. (1.16) must be as presented in order to guarantee that the proper sign will result when expectation and transition value matrix elements of such operators are formed. These spin orbitals are, in most practical applications, obtained as linear combinations of atomic orbital basis functions 

A few clarifying remarks are now in order. First, one should notice that the first-quantized forms of the above operators contain explicit reference to the number of electrons N, whereas the second-quantized operators do not. This means, for example, that the kinetic energy operator 

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B. EXPRESSING QUANTUM-MECHANICAL OPERATORS IN SECOND QUANTIZATION... 

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