2 particles in a 1-dimensional box

In summary: What is the point? Your energies are summed to find the total energy of the system. The point is to find the total energy of the system, not just the energies of the particles. In summary, if there are 2 particles of mass m1 and m2 in a one dimensional box of width a, the energy is found with 9 out of 73 chance and can be written as (n7)^2/(m1)+((n6)^2/(m2))*\hbar^2*\pi^2/(2*a^2).
  • #1
nemisis42
6
0

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



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




The Attempt at a Solution




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

With (9/73) chance for E7,6 and (64/73) chance for E3,2

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.
 
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  • #2
nemisis42 said:
I ended up with energy E=(((n7)^2/(m1))+((n6)^2/(m2)))*[tex]\hbar[/tex]^2*[tex]\pi[/tex]^2/(2*a^2)+(((n3)^2/(m1))+((n2)^2/(m2)))*[tex]\hbar[/tex]^2*[tex]\pi[/tex]^2/(2*a^2))

With (9/73) chance for E7,6 and (64/73) chance for E3,2
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

[tex]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[/tex]

You need to integrate this function over the appropriate ranges to find the total probability.
 
  • #3
vela said:
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

[tex]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[/tex]

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 E7,6 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)( [tex]\hbar[/tex] 2/(2a2)
 
Last edited:
  • #4
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.
 
  • #5


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.
 

1. What is a 1-dimensional box in the context of particle physics?

A 1-dimensional box refers to a theoretical model used in quantum mechanics to study the behavior of particles confined to a one-dimensional space. It can be thought of as a container with infinitely high potential walls on either side, creating a confined space for the particles to exist in.

2. What are the properties of a particle in a 1-dimensional box?

A particle in a 1-dimensional box has a discrete energy spectrum, meaning it can only have certain energy levels. The particle is also confined to a specific region within the box, and its position and momentum are described by wave functions.

3. How is the energy of a particle in a 1-dimensional box calculated?

The energy of a particle in a 1-dimensional box is calculated using the Schrödinger equation, which takes into account the particle's mass, the dimensions of the box, and the potential energy within the box. The resulting energy levels are quantized, meaning they can only take on specific values.

4. What is the significance of a particle in a 1-dimensional box in quantum mechanics?

The 1-dimensional box model is a simplified version of more complex quantum systems, but it is still useful in understanding fundamental principles of quantum mechanics, such as quantization of energy levels and the wave-like nature of particles. It is also a building block for more advanced models, such as the particle in a 3-dimensional box.

5. Can a particle in a 1-dimensional box exist in multiple energy levels simultaneously?

According to the principles of quantum mechanics, a particle can exist in multiple energy levels simultaneously, with each energy level having a certain probability of being observed. This is known as superposition, and it is a fundamental concept in understanding the behavior of particles in 1-dimensional boxes and other quantum systems.

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