Discussion
The normalized wave function of a particle can be written as
\[ \begin{aligned} \psi(x) = N\sum_{n=0}^{\infty}&\left(\frac{1}{\sqrt{7}}\right)^{n} \\ &\phi_{n}(x) \end{aligned} \]
where \(\phi_{n}(x)\) are the normalized energy eigenfunctions of a given Hamiltonian. The value of N is :
Answer: A
The normalisation constant is \(N = \sqrt{6/7}\).
The eigenfunctions are orthonormal, so the norm of \(\psi\) is the sum of the squared moduli of its coefficients:
\[ \begin{aligned} 1 &= N^{2}\sum_{n=0}^{\infty}\left(\frac{1}{7}\right)^{n} \\ &= N^{2}\cdot\frac{1}{1 - 1/7} = N^{2}\cdot\frac{7}{6} \end{aligned} \]
so \(N^{2} = 6/7\) and \(N = \sqrt{6/7}\).
The coefficients \((1/\sqrt{7})^{n}\) must be squared before summing, giving a geometric series of ratio 1/7. Forgetting to square them, or stopping after the first term (which gives \(N = 1\)), leads to the wrong values; \(\sqrt{1/7}\), \(\sqrt{3/7}\) and \(\sqrt{13/7}\) do not make the sum equal to 1.
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