4. One to Hundred, Part I
Basic ProbabilityConsider the following state
$$ |\psi\rangle = \sum_{n=1}^{100} \sqrt{n} |n\rangle. $$
To normalize this wavefunction we can multiply it with a factor $1/\sqrt{N}$. Calculate $N$.
Solution
To normalize the state we can first calculate its inner product with itself
$$ \langle\psi | \psi \rangle = \sum_{n=1}^{100} n = \frac{100(101)}{2} = 5050. $$
Therefore, $N = 5050$ since
$$ \frac{1}{\sqrt{N}} = \frac{1}{\sqrt{\langle\psi | \psi \rangle}}. $$