Action of L+ and L- | Santa Fe College - Edubirdie

University:
Santa Fe College
Course:
PHY 2048 | General Physics 1 with Calculus
Academic year:

2024
Views:

87

Pages:

6
Author:

Jabin F.

Action of L+ and L- [L+, Lz] = ħL+ Lz|im> = mħ|im> [L-, Lz+]|im> = (ħL+ + ħLz)|im> L_z L_+ |l,m> = ħL_+ |l,m> + L_+(mħ)|l,m> (275) L_z L_+ |l,m> = (m+1)ħ L_+ |l,m> => L_z L_- |l,m> = (-ħL_- + L_- L_z) |l,m> = -ħL_- |l,m> + L_- L_z |l,m> L_z L_- |l,m> = (m-1)ħ L_- |l,m> L_+ |l,m> & L_- |l,m> are eigen states of L_z with eigen value. (m+1)ħ and (m-1)ħ respectively which are corresponding to (m+1)th state & (m-1)th state. L_+ |l,m> = ( ) |l,m+1> L_- |l,m> = ( ) |l,m-1> L_+ -> Raising operator. L_- -> lowering operator. for multiplicative factor -> L_+|l,m> = C_+|l,m+1> (i) =

= |C_+|^2 = |C_+|^2 δ_l,l δ_m+1,m+1 (λ - m^2 ħ^2 - m ħ^2) δ_l,l δ_m,m = |C_+|^2 δ_l,l δ_m+1,m+1 |C_+|^2 = λ - m^2 ħ^2 - m ħ^2 (ii) for a given l, m = -l to +l with diff of Unity = (2l+1) no. of values. m_max = +l, m_min = -l L_+|l,+l> = 0 L_-|l,-l> = 0 L_-L_+|l,+l> = 0 L_+L_-|l,-l> = 0 (L^2 - L_z^2 + ħL_z)|l,+l> = 0 (L^2 - L_z^2 - ħL_z)|l,-l> = 0 λ - l^2 ħ^2 - l ħ^2)|l,+l> = 0 (λ - l^2 ħ^2 + l ħ^2)|l,-l> = 0 l = l(l+1)ħ^2 l = l(l+1)ħ^2 put in (ii) |C_+|^2 = (l(l+1) - m(m+1))ħ^2 = (l^2 + l - m^2 - m)ħ^2 C_+ = √(l(l+1) - m(m+1))ħ = √((l-m)(l+m+1))ħ L_z |l,m> = mħ|l,m> L^2 |l,m> = l(l+1)ħ^2 |l,m> from (i) L_+ |l,m> = √((l-m)(l+m+1))ħ |l,m+1> 13 JULY 2020 => for L_- L_- |l,m> = C_- |l,m-1> = = |C_-|^2 = (l(l+1)-m^2+ħ^2+mħ)|l,m> = |C|^2 |C|^2 = (l(l+1)-m(m-1))ħ^2 |C|^2 = (l+m)(l-m+1)ħ^2 C = √((l+m)(l-m+1))ħ L_+|l,m> = √((l+m)(l-m+1))ħ|l,m+1> L_-|l,m> = √((l+m)(l-m+1))ħ|l,m-1> for l=1 |1m> -> |1,+1>, |1,0>, |1,-1> are Simultaneous eigenkets of L^2 & Lz => L^2|1,m> = l(l+1)ħ^2|1,m> L^2|1,+1> = 2ħ^2|1,+1> L^2|1,0> = 2ħ^2|1,0> L^2|1,-1> = 2ħ^2|1,-1> (2ħ^2 is 3-fold degenerate.) => Lz|1m> = mħ|1m> mħ = +ħ, 0, -ħ Non-degenerate.

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