If the n = 3 to n = 4 transition for a certain particle-in-a-box system occurs at 4.00
x 10^13 s
-1
, find the frequency of the n =6 to n =9 transition in this system.
To find the frequency of the n = 6 to n = 9 transition, we can use the formula:
ν = (En - Em) / h
where ν is the frequency, En and Em are the energy levels of the respective transitions, and h is Planck's constant.
In the particle-in-a-box system, the energy levels are given by:
En = (n^2 * h^2) / (8mL^2)
where n is the quantum number, h is Planck's constant, m is the mass of the particle, and L is the length of the box.
Given that the transition from n = 3 to n = 4 occurs at 4.00 x 10^13 s^−1, we can find the energy difference (∆E) between these two levels:
∆E = E4 - E3 = (4^2 * h^2) / (8mL^2) - (3^2 * h^2) / (8mL^2)
Now, we can calculate the frequency of the n = 6 to n = 9 transition using the same formula:
ν = (En - Em) / h = (∆E) / h = [(6^2 * h^2) / (8mL^2) - (9^2 * h^2) / (8mL^2)] / h
Simplifying the expression, we get:
ν = [(6^2 - 9^2) * h^2] / (8mL^2 * h)
Since the mass (m) and the length of the box (L) are not provided, we cannot directly calculate the frequency of the n = 6 to n = 9 transition. Further information about the system is needed to find the specific values of mass and length in order to compute the frequency.