# Antisymmetry of final State Requiring L = 1

• Mr. Lapage
In summary, The example on wikipedia uses the fact that the deuteron has spin one and the pion has spin zero, along with the antisymmetry of the final state, to conclude that the two neutrons in the reaction d + pion- ---> n + n must have an orbital angular momentum of 1. This is because parity requires that the parity of the pion is equal to the parity of the neutrons, and this can only be achieved with an angular momentum of 1.
Mr. Lapage
I'm currently trying to solidify the notion of parity conservation in my head and saw this example on wikepdia, and am just wondering why in the reaction d + pion- ---> n + n (where d is a deuteron)

"Using the fact that the deuteron has spin one and the pion spin zero together with the antisymmetry of the final state they concluded that the two neutrons must have orbital angular momentum L=1"

Why does the antisymmetry of the final state require that the angular momentum = 1?

See the example on wikipedia here, http://en.wikipedia.org/wiki/Parity_(physics)#Parity_of_the_pion

parity = (-1)^L

we must have L = 1,3,5...

but only L = 1 is compatible with angular momentum conservation

I can explain that the principle of antisymmetry in quantum mechanics states that the wave function of a system must be antisymmetric under the exchange of identical particles. In the given reaction, the final state consists of two identical particles (neutrons) and the deuteron, which can be considered as a combination of a proton and a neutron.

Since the deuteron has a spin of 1 and the pion has a spin of 0, the total spin of the final state must be conserved, which means that the two neutrons must have a total spin of 1. This can only be achieved if the orbital angular momentum (L) of the two neutrons is also 1, as the total angular momentum (J) is given by the sum of spin and orbital angular momentum (J = L + S).

Moreover, the principle of antisymmetry requires that the wave function of the two neutrons must be antisymmetric, meaning that it must change sign under the exchange of the two particles. This can only be achieved if the orbital angular momentum is odd (L=1) for two identical particles.

Therefore, the antisymmetry of the final state, along with the conservation of spin, leads to the conclusion that the two neutrons must have an orbital angular momentum of 1. This is a fundamental principle of quantum mechanics and is essential for understanding the behavior of particles at the subatomic level.

## 1. What is meant by "Antisymmetry of final State Requiring L = 1"?

Antisymmetry of final State Requiring L = 1 refers to a quantum mechanical property of particles that dictates the allowed quantum states of a system. It states that the overall wave function of the system must be antisymmetric with respect to the exchange of identical particles when the total orbital angular momentum (L) is equal to 1.

## 2. Why is Antisymmetry of final State Requiring L = 1 important in physics?

This principle is important in physics because it explains the behavior of identical particles in a system and helps in determining the allowed quantum states of the system. It also has implications in determining the stability and behavior of atoms, molecules, and other quantum systems.

## 3. How does Antisymmetry of final State Requiring L = 1 relate to the Pauli exclusion principle?

The Pauli exclusion principle states that no two identical fermions can occupy the same quantum state simultaneously. Antisymmetry of final State Requiring L = 1 is a consequence of this principle, as it ensures that the overall wave function of the system is antisymmetric in order to satisfy the exclusion principle.

## 4. Are there any exceptions to the Antisymmetry of final State Requiring L = 1 principle?

There are some exceptions to this principle, such as particles with spin, where the overall wave function must be symmetric. This is known as the symmetry of the spin-statistics theorem. Additionally, in some cases, the Pauli exclusion principle may be violated, such as in the case of superconductivity.

## 5. How is Antisymmetry of final State Requiring L = 1 used in practical applications?

This principle is used in various fields of physics, such as quantum mechanics, nuclear physics, and solid-state physics, to understand and predict the behavior of particles and systems. It is also used in the development of technologies such as superconductors and quantum computers.

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