Wave Function for 1000 Particles in a Box

[rmfw]

Homework Statement

At the instant t=0, there's a system with 1000 particles in a box of length a. It is known that 100 have energy 4E1 and 900 have energy 225E1, where E1 is the energy of the fundamental state.

i) Build a wave function that can represent the state of a particle

ii) How many particles are in the right half of the well? [a/2 , a]

Homework Equations

i) $\Psi(x,t) = \Sigma C_{n} \varphi_{n} (x) e^{-i n^{2}wt}$ (1)

$P(E_{n}) = |C_{n}|^{2}$ (2)ii) $P(t) = \int |\Psi|^{2}| dx$ (3)
$Number of particles = N_{total} P(t)$ (4)

The Attempt at a Solution

i) $\Psi (x,t) =\frac{1}{3} \sqrt{\frac{2}{a}} sin(\frac{4 \pi x}{a}) e^{i 16 wt} + \frac{3}{\sqrt{10}} \sqrt{\frac{2}{a}} sin(\frac{225 \pi x}{a}) e^{i 50625 wt}$

ii) I need to know if the formula (4) is right Thanks in advance.

 

[BvU]

i) What is the ${1\over 3}$ and the ${3\over \sqrt{10}}$ ?

ii) Is there a way there can be an asymmetry that causes the answer to differ from 500 ?

You know the $\phi_n$ are orthonormal, right ?​

 

[rmfw]

i) the squares of 1/9 and 9/10, which are the probabilities of a particle having energy 4E1 or 225E1.

ii) I just have to integrate (3) with the limits a/2 and a, no?

 

[BvU]

i) Yeah, well, the 1/9 is probably 1/10 . And the square of 1/9 is 1/81 ( -- it pays to be a nitpicker in physics...)

ii) Yes. But you already know that the cross term gives 0. And squaring gives you even functions, so a/2 -- a should be the same as 0 -- a/2 Or am I wrong ?

 

[rmfw]

i) I meant square root, sorry. I don't know where the hell I got that 1/9 for the probability from, it should be 1/10 yes.

ii) yes you are right , that means what? there are 250 particles on each half?

Thank you for the help so far.

 

[BvU]

I wouold guess that with a total of 1000 particles there are 500 in the right half. PS is it a 3D box or a 1 D infinitely deep well ?

Would be interesting to calculate the standard deviation in that 500 for an observation period of 1 femtosecond...

 

[rmfw]

1D infinitely deep well. I'll try to do some calculations on my own now.

Thank you, have a good night(or day) sir.