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Show that for the hydrogenic orbitals, Ĥynem (r, 0,0) = (-1) Unem (r,0,0).

Show that for the hydrogenic orbitals, Ĥynem (r, 0,0) = (-1) Unem (r,0,0).

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I'm trying to answer this question but I'm stuck. also I'm not sure if it makes sense do:

Plm(-cos theta) = (-1)^(l+m)Plm(cos theta)

specifically the ^(l+m) 

thanks!

Show that for the hydrogenic orbitals, Înem (r, 0,0) = (-1) Vnem (r, 0, 0). That is, nem is an eigenstate of the parity operator, with eigenvalue (-1). Note: This result actually applies to the stationary states of any central potential V (r) = V (r). For a central potential, the eigenstates may be written in the separable form Rnt (r) Y (0, 0) where only the radial function Rwhich plays no role in determining the parity of the state-depends on the specific functional form of V (r).
Transcribed Image Text:Show that for the hydrogenic orbitals, Înem (r, 0,0) = (-1) Vnem (r, 0, 0). That is, nem is an eigenstate of the parity operator, with eigenvalue (-1). Note: This result actually applies to the stationary states of any central potential V (r) = V (r). For a central potential, the eigenstates may be written in the separable form Rnt (r) Y (0, 0) where only the radial function Rwhich plays no role in determining the parity of the state-depends on the specific functional form of V (r).
CYING PARITY OPERATOR π Rue (r) 4) 25). m ÎY" (0,0) = π (1)" 슈 +Siksne Lose - DOESN'T DEPEND ON PARITY STAT (2l+1) (l-m)! 411 Pm (coso) e imp (lim)! P tim AP (cose) = PM (- cose ) = (-1)^² PM (cose) Pr Î CUZ DEPENDS Temp Ĉ img im T e EACH e l+m π Y₁" (0₁9) = (-1) (-₁) ²+ (-1) m ON IF ANG mem EVEN OR OPD 1/2 2+1) (l-m)! pm (case) Что (l+m)! ●
Transcribed Image Text:CYING PARITY OPERATOR π Rue (r) 4) 25). m ÎY" (0,0) = π (1)" 슈 +Siksne Lose - DOESN'T DEPEND ON PARITY STAT (2l+1) (l-m)! 411 Pm (coso) e imp (lim)! P tim AP (cose) = PM (- cose ) = (-1)^² PM (cose) Pr Î CUZ DEPENDS Temp Ĉ img im T e EACH e l+m π Y₁" (0₁9) = (-1) (-₁) ²+ (-1) m ON IF ANG mem EVEN OR OPD 1/2 2+1) (l-m)! pm (case) Что (l+m)! ●
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