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7.1 Dimensionless coordinates

The classical energy (Hamiltonian) is

E = $ {\frac{1}{2}}$m$\displaystyle \dot{x}^{2}$ + $ {\frac{1}{2}}$m$\displaystyle \omega^{2}_{}$x2 = $\displaystyle {\frac{1}{2m}}$p2 + $ {\frac{1}{2}}$m$\displaystyle \omega^{2}_{}$x2 (7.4)

The quantum Hamiltonian operator is thus

$\displaystyle \hat{H}$ = $\displaystyle {\frac{1}{2m}}$$\displaystyle \hat{p}^{2}_{}$ + $ {\frac{1}{2}}$m$\displaystyle \omega^{2}_{}$x2 = - $\displaystyle {\frac{\hbar^2}{2m}}$$\displaystyle {\frac{d^2}{d x^2}}$ + $ {\frac{1}{2}}$m$\displaystyle \omega^{2}_{}$x2. (7.5)

And we thus have to solve Schrödinger's equation

- $\displaystyle {\frac{\hbar^2}{2m}}$$\displaystyle {\frac{d^2}{d x^2}}$$\displaystyle \phi$(x) + $ {\frac{1}{2}}$m$\displaystyle \omega^{2}_{}$x2$\displaystyle \phi$(x) = E$\displaystyle \phi$(x). (7.6)

In order to treat this equation it is better to remove all the physical constants, and go over to dimensionless coordinates

y = $\displaystyle \sqrt{\frac{m\omega}{\hbar}}$x,        $\displaystyle \epsilon$ = $\displaystyle {\frac{E}{\hbar \omega}}$. (7.7)

Quiz What is the dimension of $ {\frac{m\omega}{\hbar}}$? and of $ \hbar$$ \omega$? (The dimension of $ \hbar$ is [$ \hbar$] = mass x length2/time.)

When we substitute these new variables into the Schrödinger equation we get, using

$\displaystyle {\frac{d}{dx}}$f (y) = $\displaystyle {\frac{dy}{dx}}$$\displaystyle {\frac{d}{dy}}$f (y) = $\displaystyle \sqrt{\frac{m\omega}{\hbar}}$$\displaystyle {\frac{d}{dy}}$f (y), (7.8)

that ( $ \phi$(x) = u(y))

- $\displaystyle {\frac{\hbar^2}{2m}}$$\displaystyle {\frac{m\omega}{\hbar}}$$\displaystyle {\frac{d^2}{dy^2}}$u(y) + $ {\frac{1}{2}}$m$\displaystyle \omega^{2}_{}$$\displaystyle {\frac{\hbar}{m\omega}}$y2u(y) = $\displaystyle \epsilon$$\displaystyle \hbar$$\displaystyle \omega$u(y). (7.9)

Cancelling the common factor $ \hbar$$ \omega$ we find

$\displaystyle {\frac{d^2}{dy^2}}$u(y) + (2$\displaystyle \epsilon$ - y2)u(y) = 0 (7.10)


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Next: 7.2 Behaviour for large Up: 7. The Harmonic oscillator Previous: 7. The Harmonic oscillator

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