- #1
raintrek
- 75
- 0
I'm trying to bridge the gap between several expressions describing the insertion of a constant perturbation:
[tex]a_{f}(t) = \frac{1}{i\hbar} V_{fi} \int^{t}_{0} e^{i(E_{f}-E_{i})t'/\hbar}dt' = \frac{1}{i\hbar}V_{fi}\frac{e^{i(E_{f}-E_{i})t/\hbar} - 1}{i(E_{f}-E_{i})/\hbar}[/tex]
so:
[tex]|a_{f}(t)|^{2} = \frac{1}{\hbar^{2}}|V_{fi}|^{2} 2 \frac{1 - cos(E_{f} - E_{i})t/\hbar}{(E_{f} - E_{i})^{2}/\hbar^{2}}[/tex]
^ I'm not sure how they arrive at that step... I assume it's something to do with the Euler equation, but I can't see how if it is where the sin terms disappear to...
Furthermore, they state that:
[tex]P_{if} = \frac{d}{dt}|a_{f}(t)|^{2} = \frac{2\pi}{\hbar}|V_{fi}|^{2}\frac{sin(E_{f} - E_{i})t/\hbar}{\pi(E_{f} - E_{i})} \stackrel{}{\rightarrow}\frac{2\pi}{\hbar}|V_{fi}|^2\delta(E_{f} - E_{i})[/tex] for large t
Which baffles me further - factors of [tex]\pi[/tex] have krept in somehow!
Would be more than thankful is someone would be able to explain these steps! Thanks in advance
[tex]a_{f}(t) = \frac{1}{i\hbar} V_{fi} \int^{t}_{0} e^{i(E_{f}-E_{i})t'/\hbar}dt' = \frac{1}{i\hbar}V_{fi}\frac{e^{i(E_{f}-E_{i})t/\hbar} - 1}{i(E_{f}-E_{i})/\hbar}[/tex]
so:
[tex]|a_{f}(t)|^{2} = \frac{1}{\hbar^{2}}|V_{fi}|^{2} 2 \frac{1 - cos(E_{f} - E_{i})t/\hbar}{(E_{f} - E_{i})^{2}/\hbar^{2}}[/tex]
^ I'm not sure how they arrive at that step... I assume it's something to do with the Euler equation, but I can't see how if it is where the sin terms disappear to...
Furthermore, they state that:
[tex]P_{if} = \frac{d}{dt}|a_{f}(t)|^{2} = \frac{2\pi}{\hbar}|V_{fi}|^{2}\frac{sin(E_{f} - E_{i})t/\hbar}{\pi(E_{f} - E_{i})} \stackrel{}{\rightarrow}\frac{2\pi}{\hbar}|V_{fi}|^2\delta(E_{f} - E_{i})[/tex] for large t
Which baffles me further - factors of [tex]\pi[/tex] have krept in somehow!
Would be more than thankful is someone would be able to explain these steps! Thanks in advance