Highly excited states for hydrogen like atoms (also called Rydberg states) with nuclear charge Ze are…

Highly excited states for hydrogen like atoms (also called Rydberg states) with nuclear charge Ze are defined by their principal quantum number n, where n >> 1. Which of the following statement(s) is (are) true?
  1. Relative change in the radii of two consecutive orbitals does not depend on Z
  2. Relative change in the radii of two consecutive oribitals varies as 1/n
  3. Relative change in the energy of two consecutive orbitals varies as 1/ n3
  4. Relative change in the angular momenta of two consecutive orbitals varies as 1/n

Solution

As radius r ∝n2z0
⇒  ∆rr=n+1z2-nz2nz2=2n+1n2≈2n∝1n
As energy E ∝z2n2
⇒  ∆EE=z2n2-z2n+12z2n+12=n+12-n2n2.n+12. n+12
⇒  ∆EE=2n+1n2≃2nn2∝1n
As angular momentum L=nh2π
⇒  ∆LL=n+1h2π-nh2πnh2π=1n ∝1n .

Asked in: JEE Advanced 2016 (Paper 1)

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