What is Parity of a Wave function? (Odd / Even parity examples)
New Section
This section discusses the concept of space reflection or parity operation and its application to wave functions in a quantum system.
Space Reflection and Parity Operation
- When looking at a mirror, our reflection appears inverted from left to right.
- In mathematics, this is known as space reflection or parity operation.
- The parity operation involves reflecting the coordinate axis with respect to the origin.
- The Schrödinger equation describes the behavior of particles in a quantum system.
- Under parity operation, the Schrödinger equation is modified by substituting X with -X.
Symmetry of Potentials under Parity Operation
- Most potentials encountered in physics share symmetry with respect to parity operation.
- These potentials are symmetric when reflected along the coordinate axis at the origin.
- If a potential is symmetric under parity operation, it remains unchanged after reflection.
- The solutions of the Schrödinger equation for such potentials have two distinct forms:
- Even-parity wave functions (ψ(-X) = ψ(X))
- Odd-parity wave functions (ψ(-X) = -ψ(X))
Probability Distribution and Wave Function Solutions
- For potentials symmetric under parity operation, probability distributions remain invariant.
- The probability distribution for even-parity wave functions is equal to that of odd-parity wave functions squared.
- This implies that both types of wave function solutions have the same form for symmetric potentials.
Example: Square Well Potential
- The square well potential is an example of a potential symmetric under parity operation.
- It consists of both even-parity and odd-parity wave function solutions.
- The ground state wave function of the infinite square well potential exhibits both types of solutions.
New Section
This section continues discussing the nature of wave function solutions in the context of symmetric potentials.
Continued Discussion on Wave Function Solutions
- The wave function solutions for symmetric potentials can be observed at positive and negative values of X.
- The even-parity wave function solution remains unchanged when reflected along the origin.
- The odd-parity wave function solution changes sign when reflected along the origin.
Importance of Symmetric Potentials
- Potentials that are symmetric under parity operation allow for both even-parity and odd-parity wave functions.
- Understanding the nature of these solutions is crucial in quantum mechanics.
- The concept of parity operation helps explain certain properties and behaviors in quantum systems.
New Section
This section discusses the symmetry properties of wave functions in an infinite square well potential and other systems.
Wave Function Symmetry
- The wave function solution for an infinite square well potential is anti-symmetric with respect to reflection across the origin.
- The ground state wave function has even parity, while the first excited state has odd parity.
- The wave functions for higher excited states alternate between even and odd parity.
- This alternating pattern of even and odd parity wave functions is a general property of potentials symmetric with respect to reflection along the origin.
Other Systems
- Similar symmetry patterns can be observed in other systems such as the harmonic oscillator and hydrogen atom potential.
- In nuclear physics, the shell model structure assumes a symmetric potential, leading to alternative even and odd parity wave functions for nucleons.
Parity Transformation
- Most physically observable quantities are invariant under parity transformation due to the invariance of probability distributions.
- Different kinds of interactions between particles, such as nuclear forces or electromagnetic forces, do not distinguish between left-handed and right-handed systems.
Exceptions
- Weak interaction violates parity conservation in certain situations, such as beta decay processes where there is a preference for specific directions of electron emission that are not conserved under parity operation.