# Physical interpretation of the constant coefficient appearing in solution to the Schrodinger equation

The product solution to the Schrodinger's equation is $$\Psi_{n} \left ( x,t \right )=\psi\left ( x \right )\phi\left ( t \right )$$

By superposition, the solution becomes

$$\Psi \left ( x,t \right )=\sum_{n=1}^{\infty}C_{n} \psi_{n} \left ( x \right )\phi_{n} \left ( t \right )$$

In the book "Introduction to Quantum Mechanics" by Griffiths, the author hand waved about the constant coefficient $C_n$ being the amount of $\psi_{n}$ that is contained within $\Psi$, but I'm not too sure if I am comfortable with this explanation, as it isn't very rigorous. Is there a better (and obviously more improved explanation) way to see what role that constant coefficient plays in the quantum system?

This is a much more general rule that works for any inner product space. When you describe a vector $v$ in terms of other "basis" vectors $v_n$, the sum can be written out as: $$v=\sum_n (v_n, v) v_n$$ Where the bracketed value is the inner product, or "how much" of $v_n$ is in $v$. This relationship holds for much more general objects than just vectors in $\mathbb {R}^n$.
In fact, if one views the set of continuous functions as a vector space, and one defines the inner product as: $$(g (x),f(x))=\int_0^\pi f (x)g(x)dx$$ And the basis vectors as $\sin (nx),\cos (nx)$, the relationship above becomes the definition of the Fourier series, and the coefficients the amount of a certain frequency within a function.
A similar idea is going on here withe the wave functions $\Psi_n$ replacing the sines and cosines.