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## Homework Statement

Assume that Planck's constant is not actually constant, but is a slowly varying function of time, $$\hbar \rightarrow \hbar (t)$$ with $$\hbar (t) = \hbar_0 e^{- \lambda t}$$ Where ##\hbar_0## is the value of ##\hbar## at ##t = 0##. Consider the Hydrogen atom in this case.

a) Present a condition on ##\lambda## which guarrantees the applicability of the adiabatic approximation to the Hydrogen atom.

b) Using the adiabatic approximation find the lowest energy term ##E_{0} (t)##.

c) Find the electron wave function ##\psi_0 (\textbf{r},t)## for this term. [/B]

## Homework Equations

We are looking to solve the equation ##H(t) \psi_n (t) = E_n (t) \psi_n (t)##

The Hamiltonian for the Hydrogen atom is given by ## H = - \frac{\hbar^2}{2m} \nabla^2 - \frac{q^2}{4 \pi \epsilon_0 r}##

The condition for the adiabatic approximation to be valid is ##T >> \frac{\hbar}{\Delta E}## where ##T## is the time scale of variation of ##H## and ##\Delta E## is a typical energy level difference. (How does one define ##T## and ##\Delta E## in a quantum system?)

If a particle starts out in eigenstate of the initial Hamiltonian ##\psi_n##, then the wave function of the particle is given by $$\Psi (t) = e^{i \theta_n (t)} e^{i \gamma_n (t)} \psi_n (t)$$ where

[/B]

## \begin{align*} \theta_n &= \frac{1}{\hbar} \int_{0}^{t} E_n (t') dt' \\ \gamma_n &= i \int_{0}^{t} \langle \psi_n (t') | \frac{\partial}{\partial t} \psi_n (t') \rangle dt ' \end{align*} ##

## The Attempt at a Solution

a) How does one define ##T## and ##\Delta E## in a quantum system?

b) For the time-varying ##\hbar## situation, I used the equation above and substituted ##\hbar (t) = \hbar_0 e^{- \lambda t}##, yielding $$H = -\frac{\hbar_0 ^2 e^{-2\lambda t}}{2m} \nabla^2 - \frac{q^2}{4 \pi \epsilon_0 r}$$ at ##t = 0##, the Hamiltonian is that of the usual case, with ##\hbar = \hbar_0##.

I am not sure how to proceed. Am I meant to solve for the eigenfunctions ##\psi_n## of the time-dependent Hamiltonian? Is it possible to obtain an exact solution?

c) Once I have obtained eigenfunctions for the Hamiltonian, I can then substitute into the equation for ##\gamma_n## and substitute my answer for b) into the equation for ##\theta_n##. Then, substitute these into the equation for ##\Psi_n##.

More General Question:

What is the strategy for solving adiabatic approximation questions such as these, where some parameter is varying slowly? Is there a general strategy for finding conditions in which the adiabatic approximation is valid?

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