Wave Function for 1000 Particles in a Box

In summary, the equation for the number of particles in the right half of the well is (1), and the wave function for a particle in that half is given by (3).
  • #1
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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) [itex]\Psi(x,t) = \Sigma C_{n} \varphi_{n} (x) e^{-i n^{2}wt} [/itex] (1)

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

The Attempt at a Solution



i) [itex]\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} [/itex]

ii) I need to know if the formula (4) is right Thanks in advance.
 
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  • #2
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 ?​
 
  • #3
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?
 
  • #4
i) Yeah, well, the 1/9 is probably 1/10 . And the square of 1/9 is 1/81 (:smile: -- 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 ?
 
  • #5
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.
 
  • #6
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...
 
  • #7
1D infinitely deep well. I'll try to do some calculations on my own now.

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

Related to Wave Function for 1000 Particles in a Box

1. What is a "wave function" for a system of particles?

A wave function is a mathematical concept that describes the quantum state of a system of particles. It contains information about the position, momentum, and energy of each particle in the system.

2. What is a "box" in the context of a wave function for particles?

In this context, a "box" refers to a boundary or container that confines the particles within a certain region. This could be a physical box or an imaginary boundary used for mathematical calculations.

3. How does the number of particles affect the wave function in a box?

The wave function for a system of particles in a box is a complex mathematical function that becomes increasingly difficult to solve as the number of particles increases. This is because the interactions between particles become more complex and the number of possible states increases exponentially.

4. What is the significance of 1000 particles in a box?

Using 1000 particles in a box is a common example used in quantum mechanics to illustrate the behavior of a large number of particles in a confined space. It allows for the observation of emergent properties and phenomena, such as the relationship between particle density and energy.

5. Is the wave function for 1000 particles in a box a realistic model?

No, the wave function for 1000 particles in a box is a simplified model used for theoretical calculations and does not accurately represent a physical system. Real-world systems are much more complex and require more sophisticated models to accurately describe them.

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