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I am trying to understand the Berry phase through the evolution of a system that evolves adiabatically.

Schrodinger's equation is: \begin{equation} H(\lambda)|n(\lambda)\rangle=E_n|n(\lambda)\rangle \tag{1} \end{equation}

where $n$ labels the eigenstates. If $\lambda$ doesn't change with time, the wave function is: \begin{equation} \psi(t)=e^{-iE_nt/\hbar}|n\rangle \tag{2} \end{equation}

If $\lambda$ is slowly changing in time and if we approximated it as constant in each interval $\Delta t$, the phase evolution would be: \begin{equation} \prod e^{-iE_n\Delta t/\hbar}=e^{-i\sum E_n(t)\Delta t/\hbar} \tag{3} \end{equation}

In the continuum limit the sum turns into an integral, and we expect the phase evolution to be of the form $|\psi(t)\rangle=e^{-i\gamma(t)}|n(t)\rangle$ with \begin{equation} \gamma(t)=\frac{1}{\hbar}\int_{0}^{t} E_n(t')dt' \tag{4} \end{equation}

Questions:

  • How did they get equation number (2)?
  • Where did the exponential come from?
  • Why is the phase in equation (4) called a dynamic phase?
  • What physical meaning does $\lambda$ have?
  • What does $\lambda$ symbolize?

References: David Vanderbit - "Berry Phases in Electronic Structure Theory - Electric Polarization, Orbital Magnetization and Topological Insulators" (2018, Cambridge University Press)

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    $\begingroup$ There's 5 questions: The first 2 are about Eq. 2, the 3rd is about Eq. 4, and the last 2 are about Eq. 1. This could have been 3 questions: (1) What is the lambda in Eq. 1? (2) How do I derive Eq. (2)? (3) Why is gamma in Eq. 4 called a "dynamic phase" ? I think this time there's no point in separating the question into 3, but in general I think this is something we should be mindful of. When there's several questions in one question, it can make things messy in terms of answering and commenting. $\endgroup$ – Nike Dattani Jul 27 at 3:09
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The exponential comes from solving a linear differential equation:

\begin{align} \frac{\textrm{d}|\psi(t)\rangle}{\textrm{d}t} &= -\frac{\textrm{i}}{\hbar}H|\psi(t)\rangle\tag{1}\\ |\psi(t)\rangle &=e^{-\frac{\rm{i}}{\hbar}Ht}|\psi(t=0)\rangle\tag{2}\label{eq:matrixDynamics}. \end{align}

Now if you diagonalize $H$ then instead of $H$ you have a matrix with $n$ diagonal entries: $E_n$. The matrix exponential of a diagonal matrix is just the matrix of scalar exponentials of the diagonals, so $e^{-\frac{\rm{i}}{\hbar}Ht}$ becomes the diagonal matrix with $n$ diagonal entries of $e^{-\frac{\rm{i}}{\hbar}E_nt}$.

Now we can write the matrix equation of Eq. \eqref{eq:matrixDynamics} as $n$ scalar equations which are exactly like your Eq. 2, except, are you possibly missing a subscript $n$ for your $\psi(t)$? If not, maybe the textbook has a typo (most textbooks have several of them). Unless $|n\rangle$ is the initial wavefunction at $t=0$, in which case I guess the equation is fine the way it is.

I have answered your questions:

  • How did they get equation number (2)?
  • Where did the exponential come from?

But your next question seems to be about a different equation:

  • Why is the phase in equation (4) called a dynamic phase?

I would guess it's called dynamic phase because it's a phase (exponent of a complex number) and it's dynamic (changing with respect to time). Jun has offered another possible explanation in his answer: they couldn't just call it the "phase" because they want to distinguish it from the Berry phase which is a geometric phase, so they called it something else, and since it's changing with time, I think it makes sense that they called it the "dynamic" phase.

As for your last two questions:

  • What physical meaning does λ have?
  • What does λ symbolize?

It's just a parameter of the Hamiltonian. Remember that in your other question we established that $\lambda$ can be $R$ which is a nuclear coordinate, but ProfM used the wave-vector $\textbf{k}$ instead, because he wanted to speak in the context of a Brillouin zone.

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Here are some basic comments:

  • I don't think you should call Eq(1) as the Schrödinger equation. It's just defining the energy eigen states/values, or in other words, establishing the relation of those with the Hamiltonian.
  • The Schrödinger equation is $i\hbar\frac{d}{d t} |n(\lambda)\rangle = \hat{H}(\lambda)|n(\lambda)\rangle$, telling you how the quantum state evolves in time.
  • So you can write down the solution in the form of Eq(2), since it's just a first-order differential equation. This is standard, and Nike has answered nicely in great detail.

It's tricky to answer your second question "why is the dynamical phase factor called so?" because after all, it's just a matter of history. But my understanding is that because the phase in Eq(4) shows up as a natural consequence of time evolution (i.e. dynamics). There's also the Berry phase (geometric phase) that comes from the geometry of the parameter space, so people had to name the phase of Eq(4) to distinguish it with the Berry/geometric phase, and well, I guess there simply was no better wording people came up with.

$\lambda$ can basically be any parameter of the Hamiltonian. The strength of the interaction, strength of a magnetic field, the strength of some potential energy... etc. I think the easiest way to imagine things is to just think of it as the size of the box you're containing the particle or whatever. Having a box around your particle is basically the same as having a huge potential energy that surrounds your particle so the particle doesn't go outside. Then, changing the size of the box can be expressed in terms of changing the parameters of your potential energy function. Does $\lambda$ seem more intuitive now? Of course, this "box" is just one example, and $\lambda$ can be virtually any parameter in the Hamiltonian as long as you have the technology to control/change it in some way.

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