# Self adjoint operators in spherical polar coordinates

• JALAJ CHATURVEDI
In summary, the adjointness of an operator in spherical coordinates is dependent on the inner product used. For functions of ##\phi##, the inner product is defined as ##\langle \Phi_1 | \Phi_2\rangle=\int_0^{2\pi} d\phi \Phi_1^{*}\left(\phi\right)\Phi_2\left(\phi\right)##, and an operator is self-adjoint if it satisfies the condition ##\int_0^{2\pi} d\phi \Phi_1^{*}\left(\phi\right)\hat{O}\Phi_2\left(\phi\right)=\int_0^{2\pi} d
JALAJ CHATURVEDI
Hi, I have a general question. How do I show that an operator expressed in spherical coordinates is self adjoint ? e.g. suppose i have the operator i ∂/∂ϕ. If the operator was a function of x I know exactly what to do, just check
<ψ|Qψ>=<Qψ|ψ>
But what about dr, dphi and d theta

You need to know what your inner product is. The adjointness is defined relative to inner product, i.e. an operator may or may not be self-adjoint depending on which inner product you use.

Normally, for full 3d spherical polars ##\langle \psi_1 | \psi_2 \rangle = \int d^3 r \psi_2^{*} \left(\boldsymbol{r}\right)\psi_1 \left(\boldsymbol{r}\right)=\int_0^\infty r^2 dr \int_0^\pi \sin\theta d\theta \int_0^{2\pi} \psi_2^{*} \left(r,\,\theta,\,\phi\right)\psi_1 \left(r,\,\theta,\,\phi\right)##

It therefore makes sense to define the inner product for functions of ##\phi## as ##\langle \Phi_1 | \Phi_2\rangle=\int_0^{2\pi} d\phi \Phi_1^{*}\left(\phi\right)\Phi_2\left(\phi\right)##. Now an operator ##\hat{O}## on this Hilbert space (function of ##\phi## in the domain ##0\dots2\pi##) is self-adjoint if:

##\int_0^{2\pi} d\phi \Phi_1^{*}\left(\phi\right)\hat{O}\Phi_2\left(\phi\right)=\int_0^{2\pi} d\phi \Phi_2\left(\phi\right) \hat{O} \Phi_1^{*}\left(\phi\right)##

## 1. What are self adjoint operators in spherical polar coordinates?

Self adjoint operators in spherical polar coordinates are operators that satisfy certain mathematical properties, including being Hermitian (equal to their own complex conjugate) and having real eigenvalues. These operators are commonly used in quantum mechanics to describe the behavior of particles in a three-dimensional space.

## 2. How are self adjoint operators related to symmetry?

Self adjoint operators are closely related to symmetry because they represent physical quantities that remain unchanged under certain transformations, such as rotations or reflections. This property is essential in understanding the symmetries of physical systems and can provide valuable insights into their behavior.

## 3. What is the significance of self adjoint operators in quantum mechanics?

Self adjoint operators play a crucial role in quantum mechanics because they correspond to observable quantities, such as position, momentum, and energy. The eigenvalues and eigenvectors of these operators give us information about the possible outcomes of measurements on a quantum system, making them essential in predicting and understanding physical phenomena.

## 4. How are self adjoint operators represented mathematically?

In spherical polar coordinates, self adjoint operators can be represented as matrices that operate on a wave function. These matrices are typically diagonal, with the eigenvalues of the operator along the diagonal and the corresponding eigenvectors as the columns of the matrix.

## 5. Can any operator in spherical polar coordinates be self adjoint?

No, not all operators in spherical polar coordinates are self adjoint. Self adjoint operators must satisfy certain mathematical conditions, such as being Hermitian, and not all operators will meet these criteria. However, many physical quantities, such as angular momentum and the Hamiltonian, are represented by self adjoint operators in spherical polar coordinates.

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