The wavefunction of a quantum mechanical system can be evaluated in any gauge and can be expressed in terms of a complete set of eigenstates of a physical observable or of a mathematical operator. If the physical observable is expressed in the same gauge as the Hamiltonian from which the wavefunction is obtained, the expansion coefficients are gauge invariant and have definite experimental implications. This physical observable need not be the energy. If the basis set of vectors are eigenstates of a mathematical operator, the expansion coefficients are gauge dependent. This mathematical operator can be translated to a physical observable once the gauge under which the wavefunction is calculated is specified. The expansion coefficients can then be interpreted as probability amplitudes.