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7.2 Behaviour for large | y|

Before solving the equation we are going to see how the solutions behave at large | y| (and also large | x|, since these variable are proportional!). For | y| very large, whatever the value of $ \epsilon$, $ \epsilon$ $ \ll$ y2, and thus we have to solve

$\displaystyle {\frac{d^2u }{dy^2}}$ = y2u(y). (7.11)

This has two type of solutions, one proportional to ey2/2 and one to e-y2/2. We reject the first one as being not normalisable.

Question: Check that these are the solutions. Why doesn't it matter that they don't exactly solve the equations?

Substitute u(y) = H(y)e-y2/2. We find

$\displaystyle {\frac{d^{2}u}{dy^{2}}}$ = $\displaystyle \left(\vphantom{H''(y)-2yH'(y)+y^{2} H(y)}\right.$H''(y) - 2yH'(y) + y2H(y)$\displaystyle \left.\vphantom{H''(y)-2yH'(y)+y^{2} H(y)}\right)$e-y2/2 (7.12)

so we can obtain a differential equation for H(y) in the form

H''(y) - 2yH'(y) + (2$\displaystyle \epsilon$ - 1)H(y) = 0. (7.13)

This equation will be solved by a substitution and infinite series (Taylor series!), and showing that it will have to terminates somewhere, i.e., H(y) is a polynomial!


next up [*]
Next: 7.3 Taylor series solution Up: 7. The Harmonic oscillator Previous: 7.1 Dimensionless coordinates

© 1998 Niels Walet, UMIST
Email Niels.Walet@umist.ac.uk