Gauge-invariant momentum operators

The mechanical or kinematical momentum operator is therefore sometimes also called the gauge invariant momentum operator. 

The more general form of (5.10.2) is obtained on introducing field terms in the Hamiltonian via the gauge-invariant momentum operator , whose components (see Section 11.1) are... 

Finally we must verify that the energies and densities obtained from H are indeed gauge-invariant. To do this, we first show that the gauge-invariant momentum operator (taking first a one-electron system)... 

If the physical results are to remain unchanged under a unitary transformation, it is necessary to transform the operators as well as the state functions (eqns (8.28) and (8.29)). A gauge transformation has no effect on a coordinate operator but the momentum operator p is changed into p + [e/c) x- Thus, to maintain gauge invariance in properties determined by the momentum operator, p must be replaced by a new operator. In the presence of an electromagnetic field or just a magnetic field, the momentum operator is replaced by the expression... 

The form invariance of the Schrodinger equation will then lead to gauge invariant expectation values of the Hamiltonian. However, this will not be the case for an arbitrary operator. In particular, it turns out that expectation values of the canonical momentum operator, given in Eq. (2.45), are not gauge invariant, whereas expectation values of the mechanical or kinematical momentum operator, given in Eq. (2.97), are gauge invariant [see Exercise 2.15]... 

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