# Particle in a box, what's the most likely states for particle to jump to?(by calculation)

I'm doing homework that use "The absorption spectrum of this molecule displays a maximum around" some length, and to estimate the length of box with 1d box. So I assume that it was asking "Using the energy difference between n=1 to n=2 of a 1d particle in a box to calculate the length of the box."

However, I did have questions that:

1. Why the particle in a box at ground states mostly likely jump to the "next excited states"?(In this case, $n=2$) what's the math behind it? I tried to calculate it by using the wave function $\psi_n=\sqrt{\frac{2}{a}}sin(\frac{n\pi}{L}x)$ but I couldn't get any clues.

2. (A further question but not necessary) I also thought of thing I learned that the change of excited states must follow some rules like: A rise of energy states could only goes to value in range of values and was not arbitrary, where the lower seemed to be able to just go straight back to ground states. Could you explain it a bit by theory and best accompany with math?

• Hints : 1. Jumping to higher states require energy. Calculate the states for minimum energy difference. 2. Jumping to lower states releases energy. Calculate the maximum energy difference. – Yuzuriha Inori Mar 11 '18 at 4:27
• your situation of calculating the probability of transition has been perhaps discussed in ..see ><web.stanford.edu/~ajlucas/143aSection3SOLN.pdf> – drvrm Mar 11 '18 at 10:24