Reducing angular Schrodinger equation to eigenvalue problem

In summary, the angular part of the Schrodinger equation for a positron in the field of an electric dipole moment is reduced to a matrix eigenvalue problem by considering an ansatz and using relevant equations. The final equation is almost in the required form, except for an extra term that can be moved to the LHS in a diagonal form.
  • #1
perishingtardi
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1

Homework Statement


The angular part of the Schrodinger equation for a positron in the field of an electric dipole moment [itex]{\bf d}=d{\bf \hat{k}}[/itex] is, in spherical polar coordinates [itex](r,\vartheta,\varphi)[/itex],
[tex]\frac{1}{\sin\vartheta}\frac{\partial}{\partial\vartheta} \left( \sin\vartheta\frac{\partial Y}{\partial\vartheta} \right) + \frac{1}{\sin^2 \vartheta}\frac{\partial^2 Y}{\partial\varphi^2} - 2dY\cos\vartheta + \lambda Y=0.[/tex]
By considering the ansatz [tex]Y=Y_m (\vartheta,\varphi) = \sum_{l'=|m|}^{\infty} C_{l'} Y_{l'm}(\vartheta,\varphi),[/tex]
where [itex]Y_{l'm}[/itex] are spherical harmonics, show that the problem of finding [itex]\lambda[/itex] reduces to a matrix eigenvalue problem of the form [tex]\sum_{l'=|m|}^{\infty} A_{ll'}C_{l'} = \lambda C_{l}.[/tex]

Homework Equations


[tex]\frac{1}{\sin\vartheta}\frac{\partial}{\partial\vartheta} \left( \sin\vartheta\frac{\partial Y_{l'm}}{\partial\vartheta} \right) + \frac{1}{\sin^2 \vartheta}\frac{\partial^2 Y_{l'm}}{\partial\varphi^2} = -l'(l'+1)Y_{l'm}[/tex]
[tex]\int_0^{2\pi}\int_0^\pi Y_{lm}^* Y_{l'm} \sin\vartheta\,d\vartheta\,d\varphi = \delta_{ll'}[/tex]
[tex]Y_{lm}^* = (-1)^m Y_{l,-m}[/tex]
[tex]\int_0^{2\pi}\int_0^\pi Y_{l_1 m_1}Y_{l_2 m_2}Y_{l_3 m_3} \sin\vartheta \, d\vartheta \, d\varphi = \sqrt{\frac{(2l_1 +1)(2l_2 + 1)(2l_3 + 1)}{4\pi}}\begin{pmatrix}l_1&l_2&l_3\\ 0&0&0\end{pmatrix}\begin{pmatrix}l_1&l_2&l_3\\ m_1&m_2&m_3\end{pmatrix} [/tex] where the arrays in parentheses are Wigner 3jm symbols.
[tex]\cos\vartheta = \sqrt{\frac{4\pi}{3}}Y_{10}[/tex]

The Attempt at a Solution



By inserting the ansatz into the angular Schrodinger equation and using the first of the 'relevant equations' I got
[tex]\sum_{l'} C_{l'} \{ -l'(l'+1)Y_{l'm} - 2d\cos\vartheta\,Y_{l'm} + \lambda Y_{l'm} \}=0.[/tex]
Then by multiplying through by [itex]Y_{lm}^*[/itex] and integrating over the unit sphere and using the other equations given, I got (after some manipulation)
[tex]2d(-1)^m \sum_{l'=|m|}^{\infty} \sqrt{(2l+1)(2l'+1)} \begin{pmatrix}1&l&l'\\0&0&0\end{pmatrix}\begin{pmatrix}1&l&l'\\0&-m&m\end{pmatrix}C_{l'} = [\lambda - l(l+1)]C_l.[/tex]
This is almost in the form required, I think, except that there is an extra term of [itex]-l(l+1)[/itex] on the RHS, which I don't know how to get rid of. Any help would be very much appreciated.
 
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  • #2
perishingtardi said:
This is almost in the form required, I think, except that there is an extra term of [itex]-l(l+1)[/itex] on the RHS, which I don't know how to get rid of. Any help would be very much appreciated.
Just move it to the LHS in the form of a diagonal term (you can use ##\delta_{l,l'}## if you want to put it in the sum).
 
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1. What is the Schrodinger equation?

The Schrodinger equation is a fundamental equation in quantum mechanics that describes the behavior of a quantum system over time. It is used to calculate the wave function of a system, which represents the probability of finding a particle at a certain position.

2. What is an angular Schrodinger equation?

An angular Schrodinger equation is a version of the Schrodinger equation that is specifically used for systems with a spherical symmetry, such as atoms or molecules. It includes an additional term, the angular momentum operator, which accounts for the rotational motion of the system.

3. How is the angular Schrodinger equation reduced to an eigenvalue problem?

To reduce the angular Schrodinger equation to an eigenvalue problem, the equation is rearranged and solved for the eigenvalues and eigenfunctions of the angular momentum operator. This allows for the separation of variables, where the radial and angular components of the wave function can be solved separately.

4. What is an eigenvalue and why is it important in this context?

An eigenvalue is a number that represents the possible values of a physical quantity, such as energy or momentum, in a quantum system. In the context of the angular Schrodinger equation, the eigenvalues correspond to the allowed values of the angular momentum of the system. Solving for the eigenvalues allows for the determination of the energy levels and properties of the system.

5. How is the eigenvalue problem solved for the angular Schrodinger equation?

The eigenvalue problem for the angular Schrodinger equation can be solved using various mathematical techniques, such as separation of variables, substitution, or matrix diagonalization. The solution yields the eigenvalues and eigenfunctions of the angular momentum operator, which can then be used to solve for the wave function and properties of the system.

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