Question

Question 1: Completeness of Eigenfunctions (3.1) Use the orthonormality condition of eigenfunction wave functions to prove eqn 3.24. \[ \sum_{n}\left|c_{n}\right|^{2}=1 . \]

Answer 1

To prove equation 3.24, which states that the sum of the squared absolute values of the coefficients of eigenfunctions is equal to 1, we can use the orthonormality condition of eigenfunction wave functions.

In quantum mechanics, eigenfunctions play a crucial role in describing physical systems. Equation 3.24 states that the sum of the squared absolute values of the coefficients of eigenfunctions is equal to 1: ∑ₙ |cₙ|²=1.

To prove this equation, we can utilize the orthonormality condition of eigenfunction wave functions. Eigenfunctions corresponding to different eigenvalues are orthogonal to each other. This implies that their inner product is zero:
`\int\limits\psi_(n)^(*) \psi_(m) dx=0` for n≠m.

We can express a general wave function as a linear combination of eigenfunctions:
`\psi(x)=\Sigma_(n) c_(n) \psi_(n) (x)`.

Using the orthonormality condition, we can take the inner product of the wave function with itself: