Adiabatic Approximation in Hydrogen Atom

In summary, we are considering the case where Planck's constant is not constant but is a slowly varying function of time, and we are looking at the Hydrogen atom in this case. The condition for the adiabatic approximation to be valid is when the time scale of variation of the Hamiltonian ##T## is much larger than the typical energy level difference ##\Delta E##. We can use the adiabatic approximation to find the lowest energy term ##E_0(t)## and the corresponding electron wave function ##\psi_0(\textbf{r},t)##. To solve such problems, we can use the usual equations for hydrogen and substitute ##\hbar(t)## into them. The time
  • #1
CDL
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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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  • #2
CDL said:
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.
CDL said:
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.
 
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  • #3
mfb said:
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}##?

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

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

mfb said:
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?
 
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  • #4
CDL said:
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.
CDL said:
A typical energy would be -13.6eV, the ground state energy.
Good.
CDL said:
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.
CDL said:
n any case, by approximation using the taylor expansion of ##\hbar##
You don't need that.
CDL said:
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.
 
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  • #5
mfb said:
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}##

mfb said:
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,

mfb 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.
 
  • #6
CDL said:
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.
 
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  • #7
mfb said:
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.
 

Related to Adiabatic Approximation in Hydrogen Atom

1. What is the Adiabatic Approximation in Hydrogen Atom?

The Adiabatic Approximation in Hydrogen Atom is a theoretical model in which the electron and the nucleus of a hydrogen atom are treated as moving independently from each other. This approximation simplifies the mathematical calculations and allows for a better understanding of the energy levels and transitions in the atom.

2. How does the Adiabatic Approximation affect the energy levels in a hydrogen atom?

The Adiabatic Approximation assumes that the electron and the nucleus move slowly enough that they can be treated as stationary during calculations. This leads to a separation of the kinetic and potential energy terms, resulting in equally spaced energy levels in the atom.

3. Is the Adiabatic Approximation accurate for all energy levels in a hydrogen atom?

No, the Adiabatic Approximation is most accurate for low energy levels in the hydrogen atom. As the energy level increases, the approximation becomes less accurate due to the increasing speed of the electron and the nucleus.

4. How does the Adiabatic Approximation explain the spectral lines in hydrogen atom?

The Adiabatic Approximation predicts that when an electron transitions from one energy level to another, the energy difference between the levels causes a photon to be emitted or absorbed. This results in the observed spectral lines in the atom's emission or absorption spectrum.

5. Can the Adiabatic Approximation be applied to other atoms besides hydrogen?

Yes, the Adiabatic Approximation can be applied to other atoms with only slight modifications. However, it is most accurate for single-electron systems like hydrogen and becomes less accurate for multi-electron atoms due to the interactions between the electrons and the nucleus.

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