Adiabatic Approximation in Hydrogen Atom

[CDL]

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?

 

[mfb]

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

What is the timescale where $\hbar$ varies? What are typical energies and energy differences in a hydrogen atom? A factor 2 does not matter here, the order of magnitude is sufficient.

Am I meant to solve for the eigenfunctions $\psi_n$ of the time-dependent Hamiltonian? Is it possible to obtain an exact solution?

Here you can use the slow change. In particular, you can neglect it to first order and find the energy at a specific t assuming a constant Planck constant.

 

[CDL]

What is the timescale where $\hbar$ varies?

I am not sure how to proceed with this. How does one figure out such a time scale? Does it have to do with the derivative $\frac{d}{d t} \hbar(t) = -\lambda \hbar_0 e^{- \lambda t}$?

What are typical energies and energy differences in a hydrogen atom?

A typical energy would be -13.6eV, the ground state energy.

Here you can use the slow change. In particular, you can neglect it to first order and find the energy at a specific t assuming a constant Planck constant.

Why can we neglect to first order? In any case, by approximation using the taylor expansion of $\hbar$, I get $\hbar(t) = \hbar_0 (1- \lambda t)$. But where do I go from here?

I managed to get an answer to the problem by using all of the usual equations for hydrogen, and then replacing all $\hbar$'s with $\hbar(t)$. Why is / isn't this a valid thing to do?

 

[mfb]

I am not sure how to proceed with this. How does one figure out such a time scale?

What is the time for $\hbar$ to change by 10%? Or to be reduced to a factor 1/2? Or 1/e? Or 1/10? It doesn't matter which of these numbers you pick because they all just differ in a prefactor that is not too far away from 1.

A typical energy would be -13.6eV, the ground state energy.

Good.

Why can we neglect to first order?

You are interested in the limit of slow changes. You cannot ignore that the value is different at different times, but you can ignore the fact that it is still changing at a given point in time.

n any case, by approximation using the taylor expansion of $\hbar$

You don't need that.

I managed to get an answer to the problem by using all of the usual equations for hydrogen, and then replacing all $\hbar$'s with $\hbar$(t).

Good! That is the energy level as function of time with the approximation you were asked to do.

 

[CDL]

What is the time for $\hbar$ to change by 10%? Or to be reduced to a factor 1/2? Or 1/e? Or 1/10? It doesn't matter which of these numbers you pick because they all just differ in a prefactor that is not too far away from 1.

I used the fact that $\hbar$ will be $\frac{1}{e}$ of the initial value $\hbar_0$ at $t = \frac{1}{\lambda}$, and that $\frac{\hbar_0}{13.6\text{eV}} \approx 10^{-17}$ Hence, we can write $\frac{1}{\lambda} << 10^{-17} \Leftrightarrow \lambda << 10^{17}$

Good! That is the energy level as function of time with the approximation you were asked to do.

Does this arise from the fact that we can ignore that $\hbar$ is changing at some time $t$, so we can substitute $\hbar(t)$ into all of the 'static' equations for hydrogen? Hence, as you said,

You cannot ignore that the value is different at different times, but you can ignore the fact that it is still changing at a given point in time.

 

[mfb]

I used the fact that $\hbar$ will be $\frac{1}{e}$ of the initial value $\hbar_0$ at $t = \frac{1}{\lambda}$, and that $\frac{\hbar_0}{13.6\text{eV}} \approx 10^{-17}$ Hence, we can write $\frac{1}{\lambda} << 10^{-17} \Leftrightarrow \lambda << 10^{17}$

That has missing units.

Does this arise from the fact that we can ignore that $\hbar$ is changing at some time $t$, so we can substitute $\hbar(t)$ into all of the 'static' equations for hydrogen? Hence, as you said,

Yes.

 

[CDL]

That has missing units.

Sorry! $\frac{1}{\lambda} >>10^{-17} s$ and so $\lambda << 10^{17} Hz$. I made a mistake with the direction of the first inequality sign in my previous post. Thank you very much for your help.