One dimensional problems. More particles.

[LagrangeEuler]

If I have one dimensional problem with many particles that are all in same $|\psi\rangle$ state is it equal to one dimensional problem of one particle in state $|\psi\rangle$.
If I have for example 50 particles in some state $\psi(x)$ in infinite potential well and that state is symmetric around $\frac{a}{2}$ such that $\int^{\frac{a}{2}}_0|\psi(x)|^2dx=\frac{1}{2}$. Is in that case true statement that $25$ particles are in the region $0<x<\frac{a}{2}$ and 25 are in the region $\frac{a}{2}<x<a$?

 

[Jilang]

I would say not. The average may be 25 on each side though.

 

[Bill_K]

If I have one dimensional problem with many particles that are all in same $|\psi\rangle$ state is it equal to one dimensional problem of one particle in state $|\psi\rangle$.
If I have for example 50 particles in some state $\psi(x)$ in infinite potential well and that state is symmetric around $\frac{a}{2}$ such that $\int^{\frac{a}{2}}_0|\psi(x)|^2dx=\frac{1}{2}$. Is in that case true statement that $25$ particles are in the region $0<x<\frac{a}{2}$ and 25 are in the region $\frac{a}{2}<x<a$?

Think about the classical case. 50 particles, each independently with probability 1/2 to be on the left, probability 1/2 to be on the right. How many do you expect to find on each side? Hint: can you say "binomial distribution"?

The quantum case is no different.

 

[LagrangeEuler]

My question is if I have coin, probability to get head is $\frac{1}{2}$. If I throw the coin 100 times I could get for example 60 times head. Then in this case is it possible that $60$ particles be in the region $0<x<\frac{a}{2}$ and 40 in the region $\frac{a}{2}<x<a$?

 

[HomogenousCow]

You need to count all the permutations which are the same, hence use the binomial distribution.

 

[micromass]

My question is if I have coin, probability to get head is $\frac{1}{2}$. If I throw the coin 100 times I could get for example 60 times head. Then in this case is it possible that $60$ particles be in the region $0<x<\frac{a}{2}$ and 40 in the region $\frac{a}{2}<x<a$?

Yes, this is certainly possible.