11.3: Solutions independent of angular variables
- Page ID
- 14755
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Initially we shall just restrict ourselves to those cases where the wave function is independent of \(\theta\) and \(\varphi\), i.e.,
\[\phi(r, \theta, \varphi)=R(r) .\]
In that case the Schrödinger equation becomes (why?)
\[-\frac{\hbar^2}{2 m} \frac{1}{r^2} \frac{\partial}{\partial r}\left(r^2 \frac{\partial}{\partial r} R(r)\right)+V(r) R(r)=E R(r)\]
One often simplifies life even further by substituting \(u(r) / r=R(r)\), and multiplying the equation by \(r\) at the same time,
\[-\frac{\hbar^2}{2 m} \frac{\partial^2}{\partial r^2} u(r)+V(r) u(r)=E u(r)\]
Of course we shall need to normalise solutions of this type. Even though the solution are independent of \(\theta\) and \(\varphi\), we shall have to integrate over these variables. Here a geometric picture comes in handy. For each value of \(r\), the allowed values of \(x\) range over the surface of a sphere of radius \(r\). The area of such a sphere is \(4 \pi r^2\). Thus the integration over \(r, \theta, \varphi\) can be reduced to
\[\int_{\text {all space }} f(r) d x d y d z=\int_0^{\infty} f(r) 4 \pi r^2 d r .\]
Especially, the normalisation condition translates to
\[\int_0^{\infty}|R(r)|^2 4 \pi r^2 d r=\int_0^{\infty}|u(r)|^2 4 \pi d r=1\]