Is the radial part of the Laplacian in spherical coordinates Hermitian?

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In summary, the Hermicity of a radial equation refers to a mathematical property that describes the symmetry of a function. It is important in physics because it is related to the conservation of physical quantities and is a fundamental property in quantum mechanics. Although a non-Hermitian radial equation can still have physical significance, Hermicity can be tested by checking if the equation satisfies the Hermitian symmetry condition.
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geoduck
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The radial part of the Laplacian in spherical coordinates goes as:

[tex]\frac{d^2}{dr^2}\psi+\frac{2}{r}\frac{d}{dr}\psi [/tex]

How can this be Hermitian? The first term can be Hermitian, but the second term, with its 2/r factor, seems like it's not Hermitian?
 
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2 Ψ = d2/dr2 Ψ + 2/r dΨ/dr = (1/r2) d/dr (r2 dΨ/dr)

The inner product is ∫Φ*Ψ r2 dr dΩ = 4π ∫Φ*Ψ r2 dr

Integrate twice by parts:
4π ∫Φ*∇2Ψ r2 dr = - 4π ∫ (dΦ*/dr) (r2 dΨ/dr) dr = 4π ∫ d/dr (r2 dΦ*/dr) Ψ dr = 4π ∫ (1/r2) d/dr (r2 dΦ*/dr) Ψ r2 dr
= 4π ∫∇2Φ* Ψ r2 dr
 

1. What is the Hermicity of a radial equation?

The Hermicity of a radial equation refers to a mathematical property that describes the symmetry of a function. In the context of a radial equation, Hermicity means that the equation is invariant under certain transformations, such as rotations or reflections.

2. Why is Hermicity important in physics?

In physics, Hermicity is important because it is closely related to the conservation of physical quantities, such as energy and momentum. If a physical system has a Hermitian operator, then the corresponding physical quantity will be conserved over time.

3. How is Hermicity related to quantum mechanics?

In quantum mechanics, Hermicity is a fundamental property of operators that represent physical observables. In order for a physical quantity to have a well-defined value in quantum mechanics, the corresponding operator must be Hermitian.

4. Can a non-Hermitian radial equation still have physical significance?

Yes, a non-Hermitian radial equation can still have physical significance. While Hermiticity is a desirable property for operators in quantum mechanics, there are cases where non-Hermitian operators can be used to model physical systems and still yield accurate predictions.

5. How is Hermicity tested in a radial equation?

Hermicity can be tested in a radial equation by checking if the equation satisfies the Hermitian symmetry condition, which involves taking the complex conjugate of the equation and comparing it to the original equation. If the two equations are equal, then the operator is Hermitian.

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