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.
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Transcribed Image Text:18.2.7
With n(x)=x²/2
an(x) =
Hn (x)
(2"n! ¹/2)1/2
x - ip
√2
1
verify that
==
+ √2/2 ( x + 4 )
dx
Vn(x) =n ¹/²yn-1(x),
ip
a* n(x) = = x + ₂ p = √/12 (x - d) Vn(x) = (n+1) ¹/2 √n+1(x).
√√2
dx
Note. The usual quantum mechanical operator approach establishes these raising and
lowering properties before the form of (x) is known.
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