Two-particle QM Infinite Well

In summary, the expectation value for the spatial displacement between two non-interacting particles in the infinite square well can be calculated using the formula <(x1-x2)^2> = <x1^2> + <x2^2> - 2<x1><x2>, but for identical particles, an additional term must be added to account for the indistinguishability of the particles.
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Homework Statement



Find the expectation value: <(x1-x2)^2> for two non-interacting particles in the infinite square well. If one is in state [tex]\psi_1( n \neq l) [/tex] and the other in state [tex]\psi_n[/tex] find the expectation value for a) distinguishable particles, b) bosons, c) fermions

Homework Equations



[tex]\psi_n = \sqrt{\frac{2}{a}}sin(\frac{n \pi x}{a})[/tex]
[tex]\psi_1 = \sqrt{\frac{2}{a}}sin(\frac{ \pi x}{a})[/tex]

The Attempt at a Solution



verification for a):

[tex]<{\Delta x}^2> = <x^2>_1 + <x^2>_n - 2<x>_1<x>_n[/tex]

after integration I get:

[tex]<{\Delta x}^2> = 4a(\frac{1}{3} + \frac{1+\delta_{n,m=1}}{2\pi^2} - 4n\delta_{n,m=1})[/tex]

My Problem

While I don't know whether the above is correct, assuming it is, when I do this for identical particles, I get the extra term (in addition to the above):

[tex]\mp 2|<x>_1,n|^2 = \int \frac{4}{a^2} sin(\frac{\pi x}{a})sin(\frac{n \pi x}{a})dx[/tex]

which (using http://integrals.wolfram.com/index.jsp") comes to:
http://integrals.wolfram.com/Integrator/MSP?MSPStoreID=MSPStore188022599_0&MSPStoreType=image/gif [Broken]

evaluating this integral, the sins go to zero, but the cosines leave a 1 and a couple +/- 1's

Unfortunately, in the denominator, one of the terms has a (n-1) term. Does this mean that n cannot equal 1 even for Bosons? That is to say that it's meaningless to measure the spatial displacement between particles in the same state?
 
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  • #2


Hi, thank you for your interesting forum post. The expectation value for the spatial displacement between two non-interacting particles in the infinite square well can be calculated using the formula you provided, <(x1-x2)^2> = <x1^2> + <x2^2> - 2<x1><x2>. However, the calculation for identical particles requires a slight modification. For distinguishable particles, the probability of finding one particle at position x and the other at position x' is given by |psi1(x)|^2|psi2(x')|^2. However, for identical particles, we must take into account the fact that the particles are indistinguishable, so the probability is given by |psi1(x)|^2|psi2(x')|^2 + |psi1(x')|^2|psi2(x)|^2. This leads to an additional term in the expectation value calculation, which is the term you mentioned. This term does not necessarily mean that n cannot equal 1 for bosons, but it does indicate that there is a difference in the expectation value for bosons compared to distinguishable particles. In fact, for bosons, this term will be positive, while for fermions it will be negative. I hope this helps clarify the calculation for identical particles.
 

1. What is the "Two-particle QM Infinite Well" system?

The "Two-particle QM Infinite Well" system is a theoretical model in quantum mechanics that describes the behavior of two particles confined within an infinitely deep potential well. This model is often used to study the interactions between two particles in a confined space.

2. How is the potential well defined in the "Two-particle QM Infinite Well" system?

In this system, the potential well is defined by two infinite potential barriers, creating a box-like structure. The particles are confined within this box and cannot escape due to the infinitely high potential barriers.

3. What are the allowed energy states in this system?

The allowed energy states in the "Two-particle QM Infinite Well" system are determined by the Schrödinger equation, and are discrete and quantized. The energy levels are labeled by quantum numbers and can be calculated using the particle's mass, the width of the well, and the potential energy of the barriers.

4. How do the particles behave in this system?

The particles in this system behave as both particles and waves, and their behavior is described by wave functions. These wave functions determine the probability of finding the particles at a certain location within the potential well. The particles can also interact with each other through their wave functions, leading to interesting phenomena such as quantum entanglement.

5. What are some real-world applications of the "Two-particle QM Infinite Well" system?

This system has many applications in fields such as quantum computing, quantum cryptography, and quantum information theory. It is also used in studying the behavior of electrons in atoms and molecules, as well as in understanding the properties of superconductors and superfluids.

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