Question

The differential equation
−
d
2
dx2
+ x
2
!
z = λz (2)
arises when treating the quantum mechanics of simple harmonic motion.
a. Show that making the substitution z = e
−x
2
/2
y transforms this equation into Hermite’s differential
equation
d
2
y
dx2
− 2x
dy
dx
+ (λ − 1)y = 0.
b. Show that if λ = 2n + 1 where n is a nonnegative integer, (2) has a solution of the form z =
e
−x
2
/2Pn(x), where Pn(x) is a polynomial.

Answer 1

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