Quantum Physics - I

Classical Mechanics Level 5

This problem is a little difficult version of the 1-D infinite potential problems.

$Ax(a-x) = \displaystyle\sum_{n=1}^∞ c_{n}\sqrt{\frac{2}{a}}sin(\frac{nπx}{a})$

where $f(x) = Ax(a-x) , 0 ≤ x ≤ a$ is the function of a particle in a 1-D infinite potential box such that -

$V= 0~~ for~~ 0 ≤x ≤ a$

$V = ∞ ~~for~~ -∞< x< 0~~ and~~ a < x< ∞$

Find $[1000c _{3}]$

where $[~~ ]$ represents the greatest integer function.

1 solution

Kevin Chiang
Jan 24, 2015

To find $c_n$ , we need to solve the integral

$c_n = \sqrt{\frac 2a} \int_0^a Ax(a - x) \sin (n\pi x) dx$

To do that we need to normalize the wave function.

$\int_0^a| f(x) |^2 dx = 1$

And we get $A = \sqrt{\frac{30}{a^5}}$

Plugging in that value of A and solving some integrals eventually leaves you with the equation:

$c_n = \frac{2\sqrt{15} (2 - 2cos(n\pi) - n\pi sin(n\pi))}{n^3\pi^3}$

which has a $sin(n\pi)$ and $cos(n\pi)$ term in it.

For positive integers values of n, this result reduces to $c_n = \frac{4\sqrt{15}}{n^3\pi^3} (1 - (-1)^n)$

For n = 3,

$c_3 = \frac{4\sqrt{15}}{3^3\pi^3} (1 - (-1)^3) = \frac{8\sqrt{15}}{27\pi^3} \approx 0.03701$

Floor of 1000 times that is $\boxed{37}$ .