the exponential form $$\exp{\left(\alpha a^{\dag} - \alpha^{*} a \right)}$$ which is unitary. This expression can then be written as $$e^{-\alpha\alpha^*/2}
\left[ \exp{ \left(\alpha a^{\dag}\right) }  \right ]
\left[\exp{\left(\alpha^* a \right)} \right],$$ according to the Baker-Campbell-Hausdorff (BCH) relation [@miller72; @hall03]. If this is applied to the ground state, the last bracket can be dropped, and the result is $$e^{-\alpha\alpha^*/2} \exp{\left[\alpha a^{\dag} \right]} ,$$ which is the unitary operator with the normalization constant $$e^{-\alpha\alpha^*/2}.$$

Likewise, we can conclude that the series of Eq.(\[seri12\]) is different from that of Eq.(\[seri00\]) due to the difference between the unitary operator of Eq.(\[exp01\]) and the non-unitary operator of Eq.(\[exp05\]). It may be possible to derive the normalization factor using the BCH formula, but it seems to be intractable at this time. The best way to resolve this problem is to present the exact calculation of the unitary operator leading to the normalized series of Eq.(\[gau00\]). We shall return to this problem in Sec. \[excited\], where squeezed excited states are studied.

Sheared Gaussian Function
-------------------------

In addition, there is a transformation called “shear,” where only one of the two coordinates is translated as shown in Fig. \[ctran11\]. This transformation takes the form $$\label{shr01}
\pmatrix{x' \cr y'} = 