In Sakurai's book chapter 2, he has discussed the diffence between canonical momentum and kenitic momentum under gauge transformatiom. The former should be gauge-dependent, and the latter one would be gauge invariance. What confuses me is that when he try to construct operator that relates the transformed state $|\tilde{\alpha}\rangle$ to original state $|\alpha\rangle$, he requires two things, that the epectation of position and kenitic momentum should be preserved, that is, $$\langle\alpha |x|\alpha\rangle = \langle\tilde{\alpha}|x|\tilde{\alpha}\rangle\tag{2.365a}$$ and $$\langle\alpha |\left(\mathbf{p}-\frac{e\mathbf{A}}{c}\right)|\alpha\rangle = \langle\tilde{\alpha}|\left(\mathbf{p}-\frac{e\tilde{\mathbf{A}}}{c}\right)|\tilde{\alpha}\rangle\tag{2.365b}$$ Under gauge transformation $$\mathbf{A} \rightarrow \tilde{A} = \mathbf{A}+\nabla\Lambda$$ In the R.H.S of (2.365b), I don't know why the canonical momentum isn't $\tilde{\mathbf{p}}$ but still $\mathbf{p}$, that is, why here the canonical momentum seems to be same?
My thought, but I don't know if it is correct. It seems that we can also denote the canonical momentum as $\tilde{p}$, and then compute $\langle\tilde{\alpha}|\tilde{\mathbf{p}}|\tilde{\alpha}\rangle = \langle\alpha|\mathbf{p}|\alpha\rangle+\frac{e\nabla\Lambda}{c}$