Parity Of Spherical Harmonics p4366

Parity of Spherical Harmonics The eigenfunctions of the regular Hydrogen atom have party that goes as (−1)l . We want to show this. In spherical coordinates, the parity operator P changes r → r, θ → π − θ and φ → φ + π as we showed in a previous operator. The radial part of the wavefunction remained unchanged under party operation. Therefore, the parity of Φnlm will totally depend on the angular part, which is determined by the spherical harmonics Ylm (θ, φ). Consider for any fixed l. From the lecture, we know the parity of Ylm (θ, φ) remains unchanged by l−1 (θ, φ) are different. This operating L+ or L− on it. Also, we know that parity of Yll (θ, φ) and Yl−1 indicates that the parity of Ylm (θ, φ) is determined only by quantum number l. Let’s consider PYll (θ, φ), with Yll (θ, φ) = sinl θeilφ . PYll (θ, φ) = Yll (π − θ, φ + π) = sinl (π − θ)eil(φ+π) = sinl (θ)eilφ eilπ = eilπ Yll (θ, φ) = (eiπ )l Yll (θ, φ) = (−1)l Yll (θ, φ) Therefore we have proved the statement. Similarly, if we use the basis |nr , li for the 3-D SHO, the wavefunction could be written as l+ 1

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ψ(r, θ, φ) = rl e−αr Ln 2 (2αr2 )Ylm (θ, φ) The same argument holds here if you want to refer it back to the 2nd problem we covered in this week’s workshop. Parity of the wavefunction is (−1)l .

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