L24.1 Symmetrizer and antisymmetrizer for N particles

TL;DR

The professor explains the concept of symmetric and anti-symmetric states in quantum mechanics and introduces operators that project states onto these subspaces.

Transcript

PROFESSOR: We spoke last time of the existence of symmetric states. And for that we were referring to states psi S that belonged to the M particle Hilbert space. And V is the vector space that applies to the states of the M particles. And we constructed some states. So within the construction, we postulated, there will be some states that [AUDIO OU... Read More

Key Insights

• 👾 Symmetric and anti-symmetric states form subspaces of the vector space in quantum mechanics.
• 🤘 Symmetric states remain unchanged under any permutation, while anti-symmetric states change sign.
• 🇦🇬 The symmetrizer and anti-symmetrizer operators project states onto the symmetric and anti-symmetric subspaces.
• 👂 Hermitian conjugation maps a list of permutations to the same list, scrambled in a different order.

Questions & Answers

Q: What are symmetric and anti-symmetric states in quantum mechanics?

Symmetric states are those that remain unchanged under any permutation of particles, while anti-symmetric states change sign based on the parity of the permutation.

Q: How are symmetric and anti-symmetric states related to the concept of transpositions?

Symmetric states can be built with any number of transpositions, while anti-symmetric states require an odd number of transpositions.

Q: Are all permutations unitary operators?

Yes, all permutations are unitary operators, and transpositions, which are a type of permutation, are also Hermitian operators.

Q: What is the relationship between the number of even and odd permutations in a group?

The number of even permutations in any permutation group of n objects is equal to the number of odd permutations.

Summary & Key Takeaways

• The professor discusses the existence of symmetric and anti-symmetric states in quantum mechanics and how they relate to permutations of particle states.

• Symmetric states are invariant under any permutation, while anti-symmetric states change sign based on the parity of the permutation.

• The professor introduces the symmetrizer and anti-symmetrizer operators, which are defined as sums of permutations, and explains their role in projecting states onto the symmetric and anti-symmetric subspaces.