Understanding the Antisymmetric Nature of Wavefunctions for Fermions

In summary, the conversation discusses the concept of wave functions for fermions and the importance of their antisymmetric nature. The explanation includes the Pauli exclusion principle and how it relates to the behavior of fermions in terms of occupying the same states. The conversation also suggests additional resources for further understanding.
  • #1
saray1360
57
1
Hi,

I have always read the texts in which they have mentioned that for the electrons which are fermions the wave function should be antisymmetric, but I have not yet found a good proof for that.

In some books they have mentioned the pauli's exclusion principle and some relations, but still the concept is not clear to me. I hope you would kindly help me understand this matter.

Regards,
 
Physics news on Phys.org
  • #2
It's quite simple. Assume you have a two-particle wave function [tex]\psi(r_1,r_2)[/tex], where rx are the coordinates, including spin for the respective particles. What happens, or rather, what can happen to that wave function if the particles are interchanged? I.e. how does [tex]\Psi(r_1,r_2)[/tex] relate to [tex]\Psi(r_2,r_1)[/tex]? Well, if the two particles are indisinguishable, the two wave functions should be physically equivalent; identical with respect to all measurable properties. Which means the only difference is at most a difference in the phase of the wavefunction. We can write:
[tex]\Psi(r_1,r_2)=x\Psi(r_2,r_1)[/tex]
where 'x' is a phase factor (e^i*w). Particle interchange is the equivalent of multiplying by 'x'.

If you change the particles back again, you multiply by x again. But obviously this is the same as our original wavefunction so:
[tex]\Psi(r_1,r_2)=x^2\Psi(r_1,r_2)[/tex] and [tex]x^2 = 1[/tex].

So x = ±1. If x = 1, the wave function is symmetric with respect to particle interchange ([tex]\psi(r_1,r_2)=\psi(r_2,r_1)[/tex]) then that's what's defined as a boson. Correspondingly if x = -1, the wave function is antisymmetric ([tex]\psi(r_1,r_2)=-\psi(r_2,r_1)[/tex]) and that's what's called a fermion.

Now, if you form a multi-particle wave function in terms of single-particle wave functions, then it's a simple product in the case of bosons:
[tex]\Psi(r_1,r_2) = \psi_1(r_1)\psi_2(r_2)[/tex]
The two particles have no problems being in the same state, since obviously if [tex]\psi_1=\psi_2[/tex] the condition [tex]\Psi(r_1,r_2)=\Psi(r_2,r_1)[/tex] is still obeyed.

But this simple product wavefunction does not work for fermions since it doesn't follow the anti-symmetry condition. The simplest two-particle wave function that does, in terms of single-particle wave functions is:
[tex]\Psi(r_1,r_2)=\frac{1}{\sqrt{2}}(\psi_1(r_1)\psi_2(r_2) - \psi_1(r_2)\psi_1(r_1))[/tex]
(where the 1/sqrt(2) is for normalization)
More generally, for any number of particles, this takes the form of a Slater determinant.

This obeys the anti-symmetry condition, since swapping r1 and r2 leads to a sign change:
[tex]\psi_1(r_1)\psi_2(r_2) - \psi_1(r_2)\psi_2(r_1) = -(\psi_1(r_2)\psi_2(r_1) - \psi_1(r_1)\psi_2(r_2))[/tex]

But can the particles be in the same state now? No. If [tex]\psi_1=\psi_2[/tex] then the two products are identical and the wave function becomes zero. This is the Pauli exclusion principle: Bosons can occupy the same states while having the same coordinates, Fermions can not.

(Now that I've spent my time writing an explanation that's already given in hundreds of textbooks, maybe it should be added to the PF "library"?)
 
  • Like
Likes shafque19 and hercules68
  • #3
you can add the post in PF library if you want :-)

Also, please refer to some reference books and/or free additional web-references :-)
 
  • #4
The sentences you have written about the pauli exclusion really made me feel and understand the antisymmetric nature of the wavefunctions for the fermions. Thanks so much for putting time on my answer.
 

1. What is an antisymmetric wavefunction?

An antisymmetric wavefunction is a mathematical description of a quantum system that obeys the Pauli exclusion principle. It is a function that describes the state of a system of two or more identical particles, where the wavefunction changes sign when the positions of any two particles are interchanged.

2. How is the antisymmetric wavefunction different from the symmetric wavefunction?

The symmetric wavefunction describes a system of particles that can have the same quantum state, while the antisymmetric wavefunction describes a system where particles cannot have the same quantum state. In other words, the symmetric wavefunction allows for identical particles to be in the same position, while the antisymmetric wavefunction does not.

3. What is the significance of the antisymmetric wavefunction?

The antisymmetric wavefunction is important in quantum mechanics because it helps to explain the behavior of identical particles, such as electrons. It also plays a crucial role in understanding the properties of atoms, molecules, and other complex systems.

4. How is the antisymmetric wavefunction used in quantum chemistry?

In quantum chemistry, the antisymmetric wavefunction is used to calculate the electronic structure and properties of molecules. By solving the Schrödinger equation with an antisymmetric wavefunction, chemists can predict the energies and spatial distributions of electrons in a molecule, which is essential for understanding its chemical and physical properties.

5. Can the antisymmetric wavefunction be experimentally observed?

No, the antisymmetric wavefunction cannot be directly observed. It is a mathematical construct that helps to describe the behavior of quantum systems. However, the predictions made using the antisymmetric wavefunction have been experimentally verified, providing strong evidence for its existence and accuracy.

Similar threads

  • Atomic and Condensed Matter
Replies
2
Views
1K
  • Quantum Physics
Replies
2
Views
702
  • Quantum Physics
Replies
11
Views
2K
Replies
17
Views
2K
  • Atomic and Condensed Matter
Replies
10
Views
4K
  • Atomic and Condensed Matter
Replies
5
Views
1K
Replies
15
Views
2K
  • Introductory Physics Homework Help
Replies
2
Views
1K
  • Atomic and Condensed Matter
Replies
4
Views
2K
  • Atomic and Condensed Matter
Replies
1
Views
2K
Back
Top