Show that the total energy eigenfunctions ψ100(r) and ψ200(r) are orthogonal.

Name: Problem 26, Chapter 20: Physical Chemistry
Uploaded: 2021-01-12T06:12:48-08:00
Duration: 6 min 40 s
Channel: Rashmi Sinha
Description: Show that the total energy eigenfunctions ψ100(r) and ψ200(r) are orthogonal.

step 1 The total energy of again function, psi 100 and psi 200 are Si, psi 200 is equal to 1 upon root

Show that the total energy eigenfunctions ψ100(r) and...

Key Concepts

Hermitian Operators

In quantum mechanics, observables such as the Hamiltonian are represented by Hermitian operators. This property ensures that eigenfunctions corresponding to distinct eigenvalues are orthogonal under the appropriate inner product. This is a key result guaranteeing the mathematical consistency and physical interpretability of quantum states.

Orthogonality in Hilbert Space

Quantum states are defined within a Hilbert space equipped with an inner product. Orthogonality of wavefunctions implies that the inner product of two distinct eigenstates vanishes, reflecting their mutual independence. This principle is essential not only for simplifying calculations but also for the probabilistic interpretation of quantum mechanics.

Inner Product and Integration Measure

The inner product for wavefunctions is typically defined as an integral over all space, often taking the form ??* ? d?, where the integration measure depends on the coordinate system being used. In systems with spherical symmetry, the measure includes factors like r² sin(?), which are critical in verifying that different wavefunctions are orthogonal.

Separation of Variables and Radial Components

In problems with spherical symmetry, the Schrödinger equation is usually solved via separation of variables, leading to solutions that can be decomposed into radial and angular parts. The orthogonality of the total eigenfunctions thereby follows from the orthogonality of the individual radial and angular components, with the radial functions playing a crucial role in the full solution.

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