Examples of unitary transformations

The variation of the functional [lA, fl] for a stationary state can be equated to an infinitesimal times the flux in the current density of the generator through the surface of the atom (eqn (5.99)). The corresponding statement for the variation of the atomic Lagrangian integral, obtained by using eqnS (8.141) and (8.145), is 

The action of the generator of a unitary transformation, as represented by the Hermitian operator F, can be determined through the use of the operator U defined as 

The same expression applies if F is an infinitesimal Hermitian operator equal to eG, for, as e - 0, the series expansion for an exponential yields, to first-order in , 

As demonstrated in Chapter 6 for a stationary state, the atomic force law is obtained by setting the generator equal to the momentum conjugate to the coordinate r, the electronic coordinate that is integrated over the basin of the atom It follows immediately from eqn (8.35), which shows the effect of an infinitesimal unitary transformation on an operator, that the generator F = e-p induces an infinitesimal uniform translation of the electronic coordinate r by — e. Each component of the vector e is an arbitrary, infinitesimal real number, fixed for all r and, from eqn (8.35), 

It is easily shown that the same result is obtained for finite e through the use of the generator p in a unitary transformation. This is most easily done by parametrizing U as 

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The Hilbert space for one qubit has only two dimensions. Quantum information processing requires unitary transformations operating on states of one and two qubits, called logic gates. Some important examples of unitary transformations of one qubit are the Pauli ma-... 

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