If um and un are the wavefunctions corresponding to two...

If $u_{m}$ and $u_{n}$ are the wavefunctions corresponding to two energy states of a particle confined to a one-dimensional box with infinite sides, show that $$ \int_{-\infty}^{\infty} u_{n} u_{m} d x=0 \quad \text { if } n \neq m $$ This is an example of 'orthogonality' which will be discussed in chapter $4 .$

If $u_{m}$ and $u_{n}$ are the wavefunctions corresponding to two energy states of a particle confined to a one-dimensional box with infinite sides, show that
$$
\int_{-\infty}^{\infty} u_{n} u_{m} d x=0 \quad \text { if } n \neq m
$$
This is an example of 'orthogonality' which will be discussed in chapter $4 .$

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Key Concepts

Orthogonality

Orthogonality is the concept where two functions have an inner product equal to zero, meaning that they are independent and do not overlap in the context of the function space. In quantum mechanics, this property ensures that different energy states do not interfere with each other, which is essential for correctly describing the probability distributions associated with each state.

Inner Product in Hilbert Space

The inner product is a mathematical operation that quantifies the overlap between two functions in a Hilbert space. It is used in quantum mechanics to determine the normalization and orthogonality of wavefunctions, ensuring that the states form a valid set of basis functions for describing physical systems.

Eigenfunctions and Hermitian Operators

Eigenfunctions of a Hermitian operator, such as the Hamiltonian in quantum mechanics, corresponding to different eigenvalues, are orthogonal to each other. This property stems from the mathematical structure of Hermitian operators, which guarantees that the eigenfunctions form an orthogonal basis, simplifying the analysis and interpretation of quantum states.

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