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d² Let  = Consider the orthonormal basis: dx² | 1) = $1(x) = √√sin (x) and 2 [2) = ₂(x) = sin (2x). (a) Find Â1) and Â12). The operator A can be expressed in a matrix form as follows:

d² Let  = Consider the orthonormal basis: dx² | 1) = $1(x) = √√sin (x) and 2 [2) = ₂(x) = sin (2x). (a) Find Â1) and Â12). The operator A can be expressed in a matrix form as follows:

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d² Let  = Consider the orthonormal basis: dx² |1) = ₁(x) = [2) = ₂(x) = √sin(x) L 2π sin and (a) Find Â1) and Â12). The operator  can be expressed in a matrix form as follows:  = a₁1)(1| + a₁21)(2 + a21 2)(1| + a2212X<21. (b) Use part (a) to compute: amn= (m|Â\n); m, n = 1,2.
Transcribed Image Text:d² Let  = Consider the orthonormal basis: dx² |1) = ₁(x) = [2) = ₂(x) = √sin(x) L 2π sin and (a) Find Â1) and Â12). The operator  can be expressed in a matrix form as follows:  = a₁1)(1| + a₁21)(2 + a21 2)(1| + a2212X<21. (b) Use part (a) to compute: amn= (m|Â\n); m, n = 1,2.
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Part a)

Given that |1> = ϕ1(x)  = 2LsinπxL.................(1)|2> = ϕ2(x)  = 2Lsin2πxL.................(2)A^ = d2dx2A^|1> = d2dx22LsinπxLA^|1> = 2LsinπxLLπ2 = Lπ2 |1>A^|2> = d2dx22Lsin2πxLA^|2> = 2LsinπxLL2π2 = L2π2 |2>

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