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9.3 Eigenfunctions of $ \hat{H}$ through ladder operations

If we start with the ground state we would expect that we can't go any lower,

$\displaystyle \hat{a}$u0(y) = 0. (9.17)

This can of course be checked explicitly,
$\displaystyle \hat{a}$e-y2/2 = $\displaystyle {\frac{1}{\sqrt 2}}$$\displaystyle \left(\vphantom{ \hat y +
\frac{d}{dy}}\right.$$\displaystyle \hat{y}$ + $\displaystyle {\frac{d}{dy}}$ $\displaystyle \left.\vphantom{ \hat y +
\frac{d}{dy}}\right)$e-y2/2  
  = $\displaystyle {\frac{1}{\sqrt 2}}$$\displaystyle \left(\vphantom{ y e^{-y^{2}/2}- y e^{-y^{2}/2} }\right.$ye-y2/2 - ye-y2/2$\displaystyle \left.\vphantom{ y e^{-y^{2}/2}- y e^{-y^{2}/2} }\right)$  
  = 0 (9.18)

Quiz Can you show that $ \epsilon_{0}^{}$ = 1/2 using the operators $ \hat{a}$?

Once we know that $ \epsilon_{0}^{}$ = 1/2, repeated application of Eq. (9.15) shows that $ \epsilon_{n}^{}$ = n + 1/2, which we know to be correct from our previous treatment.

Actually, once we know the ground state, we can now easily determine all the Hermite polynomials up to a normalisation constant:

u1(y) $\displaystyle \propto$ a$\scriptstyle \dagger$e-y2/2  
  = $\displaystyle {\frac{1}{\sqrt 2}}$$\displaystyle \left(\vphantom{ \hat y -
\frac{d}{dy}}\right.$$\displaystyle \hat{y}$ - $\displaystyle {\frac{d}{dy}}$ $\displaystyle \left.\vphantom{ \hat y -
\frac{d}{dy}}\right)$e-y2/2  
  = $\displaystyle {\frac{1}{\sqrt 2}}$$\displaystyle \left(\vphantom{ y e^{-y^{2}/2}+ y e^{-y^{2}/2} }\right.$ye-y2/2 + ye-y2/2$\displaystyle \left.\vphantom{ y e^{-y^{2}/2}+ y e^{-y^{2}/2} }\right)$  
  = $\displaystyle \sqrt{2}$ye-y2/2. (9.19)

Indeed H1(y) $ \propto$ y.

From math books we can learn that the standard definition of the Hermite polynomials corresponds to

Hn(y)e-y2/2 = ($\displaystyle \sqrt{2}$)n$\displaystyle \left(\vphantom{ \hat a^{\dagger}}\right.$$\displaystyle \hat{a}^{\dagger}_{}$ $\displaystyle \left.\vphantom{ \hat a^{\dagger}}\right)^{n}$e-y2/2. (9.20)

We thus learn H1(y) = 2y and H2(y) = (4y2 - 2).

Question: Prove this last relation.


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Next: 9.4 Normalisation and Hermitean Up: 9. Ladder operators Previous: 9.2 The operators and

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