18.2.7With n(x)=x²/2an(x) =Hn (x)(2"n! ¹/2)1/2x - ip√21verify that==+ √2/2 ( x + 4 )dxVn(x) =n ¹/²yn-1(x),ipa* n(x) = = x + ₂ p = √/12 (x - d) Vn(x) = (n+1) ¹/2 √n+1(x).√√2dxNote. The usual quantum mechanical operator approach establishes these raising andlowering properties before the form of (x) is known.

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With n(x) = e-x²/2 an(x) = Hn (x) (2"n! 1/2)1/2 verify that x-ip √√2 x + ip √√2 = = √/ 2 (x + 2x) Vn(x) = n²/² #₁-1(x), d = √/ 2 ( x − ²)√₁(x) = (n + 1) ¹/² #n+1(x). dx an(x)= Note. The usual quantum mechanical operator approach establishes these raising and lowering properties before the form of (x) is known.

With n(x) = e-x²/2 an(x) = Hn (x) (2"n! 1/2)1/2 verify that x-ip √√2 x + ip √√2 = = √/ 2 (x + 2x) Vn(x) = n²/² #₁-1(x), d = √/ 2 ( x − ²)√₁(x) = (n + 1) ¹/² #n+1(x). dx an(x)= Note. The usual quantum mechanical operator approach establishes these raising and lowering properties before the form of (x) is known.