2 particle exchange operator P

In summary, to derive two functions that are eigenfunctions of the Hamiltonian for two identical and indistinguishable particles, as well as the 2-particle exchange operator, you need to solve the time-independent Schrodinger equation for the Hamiltonian H(1,2). From there, depending on whether the particles are bosons or fermions, the wave function must be symmetric or anti-symmetric under the exchange operator. Without knowing the expression for the potential, it is difficult to determine the complete wave function.
  • #1
jl29488
10
0
Trying to derive two functions which are eigenfunctions of the hamiltonian of 2 identical and indistinguishable particles and also eigenfunctions of the 2-particle exchange operator P.

Need some help with my workings I think.Have particle '1' and particle '2' in a hamiltonian given as:

[tex] \hat{H} = \frac{p_1^2}{2m} + \frac{p_2^2}{2m} +V(x_1,x_2) [/tex]

And for particles 1,2 to be indistinguisable, H must be symmetric w.r.t particle interaction, so

[tex]\hat{H}(1,2)=\hat{H}(2,1)[/tex]

So the time independent S.E can be:

[tex]\hat{H}(1,2)\psi(1,2)=E\psi(1,2) \\

and\\\hat{H}(2,1)\psi(2,1)=E\psi(2,1)[/tex]

[itex]\psi(2,1)[/itex] is also an eigenfunction of H(1,2) belonging to the same eigenvalue E.

And for a linear hermitian exchange operator P:

[tex]\hat{P}f(1,2)=f(2,1)[/tex]

Therefore, can write:

[tex]\hat{P}[\hat{H}(1,2)\psi(1,2)]=\hat{H}(2,1)\psi(2,1)=\hat{H}(1,2)\psi(2,1) = \hat{H}(1,2)\psi(1,2)[/tex]

Not sure where to go from here...??
 
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  • #2
You can't really derive the complete wave function if you don't know explicetely the expression for the potential. However, on general grounds, suppose that you are able to solve the S.E. for the Hamiltonian H(1,2) and hence find the normalized wave function, ##\psi(1,2)##. Now you have two cases: (a) your particles are identical bosons and (b) they're identical fermions. In the first case the wave function must always be symmetric under the action of the exchange operator P. Now if ##\psi(1,2)=\psi(2,1)## you're lucky and that's automatically satisfied, otherwise you will have to symmetrize your wave function as ##\psi_{symm}=(\psi(1,2)+\psi(2,1))/\sqrt{2}##.
For fermions, instead, the complete wave function must be anti-symmetric under P, so either ##\psi(1,2)=-\psi(2,1)## or you have to define ##\psi_{anti-symm}=(\psi(1,2)-\psi(2,1))/\sqrt{2}##.

I believe that if you don't know the form of the potential you can't really go much further.
 

Related to 2 particle exchange operator P

1. What is the purpose of a 2 particle exchange operator P?

The 2 particle exchange operator P is used in quantum mechanics to describe the exchange of two identical particles. This operator helps us understand the behavior of identical particles and their interactions.

2. How is the 2 particle exchange operator P represented mathematically?

In quantum mechanics, the 2 particle exchange operator P is represented by a permutation matrix. This matrix rearranges the wave function of the two particles to account for the exchange of their positions.

3. What is the significance of the 2 particle exchange operator P in quantum mechanics?

The 2 particle exchange operator P plays a crucial role in understanding the symmetry properties of quantum systems. It helps us distinguish between bosons and fermions, and explains the Pauli exclusion principle for fermions.

4. Can the 2 particle exchange operator P be applied to any two particles?

No, the 2 particle exchange operator P is only applicable to two identical particles. This is because the exchange of two distinct particles does not result in any change in the overall system.

5. How does the 2 particle exchange operator P affect the overall wave function of a system?

The 2 particle exchange operator P changes the sign of the wave function when two fermions are exchanged, while it leaves the wave function unchanged for two bosons. This has important implications for the behavior and properties of quantum systems.

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