What Are the Energy Values and Probabilities for Two Particles in a 1D Box?

[nemisis42]

Homework Statement

If there are 2 particles of mass m1 and m2 in a one dimensional box of width a, I'm trying to find 1)what values will be found if the energy is measured, and with what probability these values will take occur. and 2) what is the probability of finding particle 1 with mass m1 in the interval (0,a/2) (all of this is at time t=0) (the particles are not symmetric)The wave equation is:

Homework Equations

$$\Psi$$ (X1,X2,0)=(3$$\Phi$$₇(X1)*$$\Phi$$₆(X2)+8$$\Phi$$₃(X1)*$$\Phi$$₂(X2))/(sqrt(73))

The Attempt at a Solution

I ended up with energy E=(((n7)^2/(m1))+((n6)^2/(m2)))*$$\hbar$$^2*$$\pi$$^2/(2*a^2)+(((n3)^2/(m1))+((n2)^2/(m2)))*$$\hbar$$^2*$$\pi$$^2/(2*a^2))

With (9/73) chance for E_7,6 and (64/73) chance for E_3,2

Would anybody be able to tell me if what I have looks correct(and point me in the right direction if its not) and tell me where to start with the probability of finding particle 1 in the interval (0,a/2). I did change the values from the original equation. I'm really just interested in the principal behind this.

 

[vela]

I ended up with energy E=(((n7)^2/(m1))+((n6)^2/(m2)))*$$\hbar$$^2*$$\pi$$^2/(2*a^2)+(((n3)^2/(m1))+((n2)^2/(m2)))*$$\hbar$$^2*$$\pi$$^2/(2*a^2))

With (9/73) chance for E_7,6 and (64/73) chance for E_3,2

Your probabilities are correct, but what exactly is your expression above for E supposed to represent? Your E is a single number, but you have two probabilities. And what are n7, n6, n3, and n2?

Would anybody be able to tell me if what I have looks correct(and point me in the right direction if its not) and tell me where to start with the probability of finding particle 1 in the interval (0,a/2). I did change the values from the original equation. I'm really just interested in the principal behind this.

The probability of finding particle 1 in the interval a ≤ x₁ ≤ a+dx₁ and particle 2 in interval b ≤ x₂ ≤ b+dx₂ is given by

$$P(a\le x_1 \le a+dx_1 , b\le x_2 \le b+dx_2) = \Psi^*(a,b)\Psi(a,b)\,dx_1\,dx_2$$

You need to integrate this function over the appropriate ranges to find the total probability.

 

[nemisis42]

Your probabilities are correct, but what exactly is your expression above for E supposed to represent? Your E is a single number, but you have two probabilities. And what are n7, n6, n3, and n2?

The probability of finding particle 1 in the interval a ≤ x₁ ≤ a+dx₁ and particle 2 in interval b ≤ x₂ ≤ b+dx₂ is given by

$$P(a\le x_1 \le a+dx_1 , b\le x_2 \le b+dx_2) = \Psi^*(a,b)\Psi(a,b)\,dx_1\,dx_2$$

You need to integrate this function over the appropriate ranges to find the total probability.

Thanks alot. I assumed the probability for finding particle 1 was something like that, but for some reason whenever I came across that formula, it was written in weird notation. Sorry for the energy formula. I was just learning how to use the latex reference. Below is what I would have for E_7,6 with n₇=7 and so on. I was just curious if there was something else I could do to simplify it.


[tex]\frac{ n₇ ² } { m₁ }[/tex] + ( [tex]\frac{ n₆ ² }{ m₂ }[/tex])[tex]\frac{ \hbar ² \pi ² }{2 a²} [/tex]


I apologize again. This is the best I could do for the energy formula for the energy formula for the first one. the second part is hbar squared multiplied by Pi squared divided by 2 mulitplied by a squared, the first part is n subscript 7(which is 7) squared divided by mass 1 plus n subscript 6 divided by mass 2. Sorry for the confusion.

(n₇²/m₁+ n₆²/m₂)( $$\hbar$$ ²/(2a²)

 

[vela]

Subscripts would typically used to indicate which particle a variable describes, not the value of the variable. You'd have n₁=3 and n₂=2 or n₁=7 and n₂=6.

I'm still not sure why you're summing the energies for the different states though.

 

[paranormal]

I would say that your attempt at a solution appears to be on the right track. The wave equation you have provided seems to accurately describe the system of 2 particles in a 1-dimensional box. However, without knowing the specific values of n and \Phi for each particle, it is difficult to confirm the correctness of your calculations for the energy values and probabilities.

To determine the probability of finding particle 1 with mass m1 in the interval (0,a/2), you will need to use the wave function \Psi and integrate it over the given interval. This will give you the probability amplitude, which you can then square to get the actual probability.

It may also be helpful to visualize the system and the wave function to better understand the behavior and probabilities of the particles in the box. I would suggest consulting a textbook or seeking guidance from your professor or a TA for further clarification and assistance.