1. (a) Ladder operators are defined as L LatiLy. Suppose A, m) is the eigenfunction of L₂ and L2 with eigenvalue mħ and

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1 A Ladder Operators Are Defined As L Latily Suppose A M Is The Eigenfunction Of L And L2 With Eigenvalue Mh And 1 (93.06 KiB) Viewed 49 times

1. (a) Ladder operators are defined as L LatiLy. Suppose A, m) is the eigenfunction of L₂ and L2 with eigenvalue mħ and Aħ² respectively. Hence prove the following = i. L+ (L+) increases (decreases) the m value by one unit. ii. L² = L+L‡ + L² FħL₂ iii. Now choosing highest weight state show λ = 1(1+1), where is the maximum value of m. iv. is either integer or half-integer. (b) Wave function of the electron in a hydrogen atom is given by 1 1 $(F) = √5 $200(7) + √²3 $21_1(7) – 6 $100(7), √6 where Onlm (r) are the eigenstates of the Hamiltonian in the standard notation. Hence calculate the expectation value of the energy in this state. Energy of the n-th eigenstate is given by En = -(13.6/n²) eV (4+4+3+3)+6=20