Symmetric, antisymmetric and parity

In summary: Basically, it has to do with the wavefunction and the way that particles interact with each other. If particles are identical, then the wavefunction is the same for each particle and they will have the same separation. However, if particles are different, then the wavefunction will be different for each particle and they will be repelled from each other. This is called an exchange effect.
  • #1
Sacroiliac
13
1
Problem 5.5 In David Griffiths “Introduction to Quantum Mechanics” says:

Imagine two non interacting particles, each of mass m, in the infinite square well. If one is in the state psin and the other in state psim orthogonal to psin, calculate < (x1 - x2) 2 >, assuming that (a) they are distinguishable particles, (b) they are identical bosons, (c) they are identical fermions.

(a) a2 [1/6 – 1/(2*pi2)(1/n2 + 1/m2)

(b) The answer to (a) - (128*a2*m2n2) / pi4(m2 - n2) 4

But this last term is present only when m,n have opposite parity.

(c) The answer to (a) plus the term added in (b) with the same stipulation as in (b)

What does this mean? It seams to be saying that all three particles would have the same separation unless their states have opposite parity. Is this correct? Bosons and Fermions would have the same separation unless their states have odd parities? I never heard of this before, how does this work?
 
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  • #2
bosons, fermions, and distinguishons ?

Hi,

The way to solve the problem goes as follows:
if the particles are distinguishable, then the wave function is psi_n(x1) psi_m(x2). If they are fermions, then you have to use the antisymmetric form: 1/sqrt(2) (psi_n(x1) psi_m(x2) - psi_n(x2)psi_m(x1))

and if they are bosons, then you have to use the symmtric form:
1/sqrt(2) (psi_n(x1) psi_m(x2) + psi_n(x2)psi_m(x1))

This for the technical part. Now the interpretational part is more difficult, and the effects are called "exchange effects" But in general, fermions tend to "repulse" each other and bosons tend to "attract" each other. But not with a term in the hamiltonian, but purely through these exchange effects.

cheers,
Patrick.
 
  • #3
In case anyone's interested...

I finally found the answer to this question in Quantum Physics of Atoms molecules etc. by Eisberg and Resnick on page 315.

I'd try to explain it but I don't think I can pull it off.
 

1. What is symmetry?

Symmetry is a property in which an object or system is unchanged when it undergoes a transformation, such as a rotation, reflection, or translation.

2. What is the difference between symmetric and antisymmetric?

In symmetric systems, the property remains unchanged under any transformation, while in antisymmetric systems, the property changes sign (positive to negative or vice versa) under certain transformations.

3. What is parity?

Parity is a property that describes whether an object or system is unchanged under spatial inversion, where all coordinates are flipped to their opposite values.

4. How are symmetric and antisymmetric systems related to parity?

Symmetric systems have even parity, meaning they remain unchanged under spatial inversion, while antisymmetric systems have odd parity and change sign under spatial inversion.

5. What are some examples of symmetric, antisymmetric, and parity systems?

Examples of symmetric systems include a circle, a cube, or a perfect sphere. Examples of antisymmetric systems include a dipole magnet or a Möbius strip. Parity systems can be seen in the behavior of subatomic particles, where some are unchanged under spatial inversion (even parity) and others change sign (odd parity).

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