A hydrogen atom in a weak time-dependent perturbation

In summary, the conversation discusses a homework problem involving a hydrogen atom in its ground state being subjected to a weak time-dependent electric field. The perturbation potential and transition probabilities are calculated using lowest-order time-dependent perturbation theory. The conversation also includes a discussion on the selection rules for transitions from the ground state and how to calculate the probability of transition from the ground state to a specific state. The conversation also mentions using spherical harmonics and C-G coefficients to aid in the calculations.
  • #1
Urvabara
99
0

Homework Statement



A hydrogen atom, which is in its ground state 1s (i.e. [tex]\|1,0,0\rangle[/tex]), is put into a weak time-dependent external electric field, which points into the z direction:
[tex]\boldsymbol{E}(t,\boldsymbol{r}) = \frac{C\hat{\text{\boldsymbol{e}}}_{z}}{t^{2}+\tau^{2}}[/tex], where C and [tex]\tau > 0[/tex] are constants. This gives rise to a perturbation potential [tex]V(t) = C\frac{e\hat{z}}{t^{2}+\tau^{2}}[/tex], where e denotes the electron charge.

(a) Using lowest-order time-dependent perturbation theory, find the selection rules for transitions from the ground state, i.e. find out which final state values for the quantum numbers n, l and m are possible in transitions from the ground state.

(b) Calculate the probability of transition from the ground state 1s to to the state 2p during an infinitely long period of time, setting [tex]t_{0}\rightarrow -\infty[/tex] and [tex]t\rightarrow\infty[/tex].

Homework Equations


This is useful, isn't it?
[tex]P_{fi}(t,t_{0})\equiv |\langle\phi_{f}|\psi(t)\rangle|^{2}\approx \frac{1}{\hbar^{2}}\left|\int_{t_{0}}^{t}\text{d}t_{1}\langle\phi_{f}|V_{S}(t_{1})|\phi_{i}\rangle \text{e}^{\text{i}(E_{f}-E_{i})t_{1}/\hbar}\right|^{2}.[/tex]


The Attempt at a Solution



(b) I probably should use the equation given in the section "2. Homework Equations " and "sandwich" perturbation potential between the states [tex]\langle 2,1,m|[/tex] and [tex]|1,0,0\rangle[/tex], shouldn't I? What m-value should I use?

Any hints to (a)?
 
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  • #2
Urvabara said:

Homework Statement




(b) I probably should use the equation given in the section "2. Homework Equations " and "sandwich" perturbation potential between the states [tex]\langle 2,1,m|[/tex] and [tex]|1,0,0\rangle[/tex], shouldn't I? What m-value should I use?

Any hints to (a)?

what kind of spherical tensor is z? What is its rank?
Then you will get your selection rules, so you know what m values you should use for the matrix elements (i.e the set if final states that can be reached)

You might want to look in the chapter of tensor operators in your textbook.


Or you can write z as a spherical harmnonics and use the wave functions for the hydrogen-states (that is a radial part times a spherical harmonic) and use some well known spherical harmonics identities.

EDIT: You might also use parity relations for the Spherical harmonics, if you are not allowed to use collection of formulas for spherical harmonics.
 
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  • #3
Urvabara said:
Any hints to (a)?

How's it going?
 
  • #4
malawi_glenn said:
How's it going?

Thanks for your reply!

I think that [tex]z = r_{0} = r\cos (\theta)[/tex]. Right?

I also think that [tex]\langle 2,1,m|r_{0}|1,0,0\rangle = \sqrt{\frac{4\pi}{3}}\langle 2,1|r_{0}|1,0\rangle\langle 1,m|Y_{10}(\theta ,\phi)|0,0\rangle[/tex] Right?

The radial part is easy to calculate: [tex]\langle 2,1|r_{0}|1,0\rangle = \int_{0}^{\infty}r^{3}R^{*}_{21}(r)R_{10}(r)\,\text{d}r = ...[/tex] Right?

The angular part is: [tex]\langle 1,m|Y_{10}(\theta ,\phi)|0,0\rangle = \sqrt{\frac{3(2*0+1)}{4\pi (2*1+1)}}\langle 0,1;0,0|1,0\rangle\langle 0,1;0,0|1,m\rangle[/tex] Right?

The C-G coefficients [tex]\langle 0,1;0,0|1,m\rangle[/tex] vanish unless [tex]m=0+0=0[/tex] and [tex]0-1\leq 0 \leq 0+1[/tex] or [tex]\Delta m = m-0 = 1,0,-1[/tex] and [tex]\Delta l = 1-0=1[/tex] Hmm. Right?

PS. Sorry, because my reply is so late. I was somewhat busy.
 
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  • #5
Urvabara said:
Thanks for your reply!

I think that [tex]z = r_{0} = \cos (\theta)[/tex]. Right?

I also think that [tex]\langle 2,1,m|r_{0}|1,0,0\rangle = \sqrt{\frac{4\pi}{3}}\langle 2,1|r_{0}|1,0\rangle\langle 1,m|Y_{10}(\theta ,\phi)|0,0\rangle[/tex] Right?

The radial part is easy to calculate: [tex]\langle 2,1|r_{0}|1,0\rangle = \int_{0}^{\infty}r^{3}R^{*}_{21}(r)R_{10}(r)\,\text{d}r = ...[/tex] Right?

The angular part is: [tex]\langle 1,m|Y_{10}(\theta ,\phi)|0,0\rangle = \sqrt{\frac{3(2*0+1)}{4\pi (2*1+1)}}\langle 0,1;0,0|1,0\rangle\langle 0,1;0,0|1,m\rangle[/tex] Right?

The C-G coefficients [tex]\langle 0,1;0,0|1,m\rangle[/tex] vanish unless [tex]m=0+0=0[/tex] and [tex]0-1\leq 0 \leq 0+1[/tex] or [tex]\Delta m = m-0 = 1,0,-1[/tex] and [tex]\Delta l = 1-0=1[/tex] Hmm. Right?

PS. Sorry, because my reply is so late. I was somewhat busy.

You have some errors that you need to fix.

[tex] z = r\cdot \cos \theta [/tex]

and the "selection" rules from CG are:
[tex] m = m_1 + m_2 [/tex]
and
[tex] |j_1 - j_2| \leq j \leq j_1 + j_2 [/tex]

so z is essential [tex] r \cdot Y_{l=1}^{m=0} [/tex], which is a spherical tensor of rank (k) = 1, with q = 0
The so called m-selection rule is:
[tex] <\alpha , j_1,m_1 |T^{(k)}_{m}|\beta , j_2, m_2> = 0[/tex] ; unless m_1 = q + m_2

So now you know which final state your pertubation can take the initial state (the ground state in this case).

Then you just evaluate the integrals, you can look up explicit wave functions for the hydrogen atoms in your textbook and just do the math, straighforward but lenghty. Remember to use spherical coordinates when you integrate.
 
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1. What is a hydrogen atom in a weak time-dependent perturbation?

A hydrogen atom in a weak time-dependent perturbation refers to a hydrogen atom that is being influenced by a small external force that changes with time. This perturbation can arise from electromagnetic fields or other external influences, and it can cause the atom's energy levels to shift or its wave function to change over time.

2. How does a weak time-dependent perturbation affect a hydrogen atom?

A weak time-dependent perturbation can cause the energy levels of a hydrogen atom to shift, which can result in changes to the atom's spectral lines. It can also cause the atom's wave function to change, leading to changes in its physical properties and behavior.

3. What are some real-life applications of studying a hydrogen atom in a weak time-dependent perturbation?

Studying the effects of weak time-dependent perturbations on a hydrogen atom can help scientists understand and predict the behavior of atoms in various environments, such as in the presence of electromagnetic fields or under extreme conditions. This knowledge can be applied in fields such as quantum mechanics, atomic and molecular physics, and astrophysics.

4. What techniques are used to study a hydrogen atom in a weak time-dependent perturbation?

Some commonly used techniques for studying a hydrogen atom in a weak time-dependent perturbation include perturbation theory, time-dependent perturbation theory, and numerical simulations using computer programs. These techniques allow scientists to calculate the effects of the perturbation on the atom's energy levels and wave function, and to analyze its behavior over time.

5. What are the main challenges in studying a hydrogen atom in a weak time-dependent perturbation?

One of the main challenges in studying a hydrogen atom in a weak time-dependent perturbation is accurately modeling and calculating the effects of the perturbation on the atom's energy levels and wave function. This often requires complex mathematical equations and computer simulations. Another challenge is obtaining experimental data that can confirm the theoretical predictions and models.

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