# Is Ballentine's description of the Berry phase (in his book _Quantum Mechanics_) flawed?

Ok, so I'm looking at Ballentine's Quantum Mechanics right now, 7th reprint (2010).

On page 363, he starts with 12.7 Adiabatic Approximation and quickly moves on to explain Berry's phase on page 365.

In equation $(12.90)$, he gives a formula for the time evolution of a certain, up to now seemingly "unimportant", phase, namely $$\tag{1} \dot{\gamma}_n(t)=\iota\langle n(R(t))|\dot{n}(R(t))\rangle,$$ where $|n(R(t))\rangle$ is the $n$-th Eigenstate of the time-dependent Hamiltonian $\hat{H}(R(t))$ for some curve $R(t)$ in the parameter space.

Next, he states that we may rewrite this equation as $$\tag{2} \dot{\gamma}_n(t)=\iota\langle n|\nabla_R\dot{n}\rangle \cdot \dot{R}(t).$$ Comparing this to Berry's original equation $(4)$, which is $$\tag{3} \dot{\gamma}_n(t)=\iota\langle n|\nabla_Rn\rangle \cdot \dot{R}(t),$$ you might already see where my problem arises: The dot over the $n$ or lack thereof. My intuition tells me that Berry is right and that it's just an error in Ballentine's book. Which would kind of make sense, since Berry's paper is peer-reviewed and I would interpret $\nabla_R{n}$ as $$|\dot{n}(R(t))\rangle=|\frac{\partial n}{\partial R^i}\frac{\partial R^i}{\partial t}\rangle=|\frac{\partial n}{\partial R^i}\rangle\dot{R^i}\equiv|\nabla_Rn\rangle\cdot\dot{R}.$$ But Ballentine's error is consistent: On the next page, we can see it 3 more times, and it is quite hard to believe that such an error occurs this impertinently.

Could you please tell me whose side the error is on and whether my interpretation of $\nabla$ is right?

• You are right, this is a typo. In my version (2010) there's no dot. – Mateus Sampaio Oct 20 '14 at 20:39
• How quaint. The version I'm looking at is 2010 as well. – Jo Be Oct 20 '14 at 20:43