- #1
svletana
- 20
- 1
Hello! I have the following problem I'm trying to solve:
An Hydrogen atom in the state |100> is found between the plates of a capacitor, where the electric field (weak and uniform) is: E(t) = \epsilon e^{-\alpha t / \tau}.
Calculate the parameters of the potential (\epsilon, \alpha, \tau) so that for a time t \gg \tau the transition probability to any of the n=2 states is equal to 0.1.
The field is asumed to be in an arbitrary r = (x,y,z) direction, so that W = \epsilon e^{-\alpha t / \tau} r.
The formula for transition probability is (using atomic units):
P = \left| \int_0^{T \gg \tau} e^{i \omega t} <100 | r | 21m> \epsilon e^{-\alpha t / \tau}dt \right|^2
where \omega = \frac{E_{21m} - E_{100}}{\hbar} = \frac{-3}{4}.
For the <100|r|21m> elements we have the results, for each m:
<100|r|200> = 0
<100|r|210> = \frac{2^7 \sqrt{2} \hat{z}}{3^5}
<100|r|21\pm1> = \frac{2^7}{3^5}(\mp\hat{x} - i\hat{y})
I solved the integral for an arbitrary m and l=1, calling the result of the <100|r|21m>=\gamma. Since it does not depend on t we can take it out of the integral along with all the constants. The integral is then:
P = \gamma^2 \epsilon^2 \left| \int_0^{T \gg \tau} e^{-\left( \frac{3i}{4} + \frac{\alpha}{\tau}\right)t} dt \right|^2
For which I solved and took the limit T \rightarrow \infty and got the result:
\frac{\gamma^2 \epsilon^2}{\frac{9}{16} + \frac{\alpha^2}{\tau^2}}
Now, \gamma^2 = \frac{2^{15}}{3^{10}} for any value of m. And that result must be equal to 0.1 according to the guidelines.
I don't see how I could possibly calculate 3 parameters from this equation, what am I missing? Thanks in advance!
Homework Statement
An Hydrogen atom in the state |100> is found between the plates of a capacitor, where the electric field (weak and uniform) is: E(t) = \epsilon e^{-\alpha t / \tau}.
Calculate the parameters of the potential (\epsilon, \alpha, \tau) so that for a time t \gg \tau the transition probability to any of the n=2 states is equal to 0.1.
Homework Equations
The field is asumed to be in an arbitrary r = (x,y,z) direction, so that W = \epsilon e^{-\alpha t / \tau} r.
The formula for transition probability is (using atomic units):
P = \left| \int_0^{T \gg \tau} e^{i \omega t} <100 | r | 21m> \epsilon e^{-\alpha t / \tau}dt \right|^2
where \omega = \frac{E_{21m} - E_{100}}{\hbar} = \frac{-3}{4}.
For the <100|r|21m> elements we have the results, for each m:
<100|r|200> = 0
<100|r|210> = \frac{2^7 \sqrt{2} \hat{z}}{3^5}
<100|r|21\pm1> = \frac{2^7}{3^5}(\mp\hat{x} - i\hat{y})
The Attempt at a Solution
I solved the integral for an arbitrary m and l=1, calling the result of the <100|r|21m>=\gamma. Since it does not depend on t we can take it out of the integral along with all the constants. The integral is then:
P = \gamma^2 \epsilon^2 \left| \int_0^{T \gg \tau} e^{-\left( \frac{3i}{4} + \frac{\alpha}{\tau}\right)t} dt \right|^2
For which I solved and took the limit T \rightarrow \infty and got the result:
\frac{\gamma^2 \epsilon^2}{\frac{9}{16} + \frac{\alpha^2}{\tau^2}}
Now, \gamma^2 = \frac{2^{15}}{3^{10}} for any value of m. And that result must be equal to 0.1 according to the guidelines.
I don't see how I could possibly calculate 3 parameters from this equation, what am I missing? Thanks in advance!