Where Did I Go Wrong? Solving the 3D Hydrogen Atom

In summary, the 3D hydrogen atom is a theoretical model used to understand the behavior of a single hydrogen atom in three-dimensional space. Solving this model is important for gaining insight into the principles of quantum mechanics and atomic structure, and it is achieved through mathematical calculations and simulations using quantum mechanics principles. However, solving the 3D hydrogen atom poses challenges due to the complex mathematical equations involved and the 3D nature of the problem. Potential applications of solving this model include improving our understanding of atomic and molecular structures, developing new materials and technologies, and aiding in the design of more efficient energy sources.
  • #1
touqra
287
0
In solving the 3D hydrogen atom, we obtain a spherical harmonic, Y such that,
[tex] Y_{lm}(\theta,\phi) = \epsilon\sqrt{\frac{(2l+1)}{(4\pi)}}\sqrt{\frac{(l-|m|)!}{(l+|m|)!}}e^{im\phi}P^m_l(cos \theta)[/tex]
where [tex]\epsilon = (-1)^m [/tex] for [tex] m \geq 0 [/tex] and [tex] \epsilon = 1 [/tex] for [tex] m \leq 0 [/tex].
In quantum, m = -l, -l+1, ..., l-1, l.
But according to the formula above, when m = l, we should have zero and not a finite value, since [tex] l - |m| = 0 [/tex]. Which means the wavefunction should be zero when m = l.
Where did I go wrong?
 
Last edited:
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  • #2
The factorial of 0 is 1. :)
 

1. What is the 3D hydrogen atom?

The 3D hydrogen atom is a theoretical model that describes the behavior of a single hydrogen atom in three-dimensional space. It takes into account the electron's position in all three dimensions, rather than just the radial distance from the nucleus.

2. Why is it important to solve the 3D hydrogen atom?

Solving the 3D hydrogen atom is important because it can provide a deeper understanding of the fundamental principles of quantum mechanics and atomic structure. It can also help in the development of new technologies and materials.

3. How do scientists solve the 3D hydrogen atom?

Scientists solve the 3D hydrogen atom through mathematical calculations and simulations using quantum mechanics principles. This involves solving the Schrödinger equation, which describes the behavior of quantum particles such as electrons.

4. What are the challenges in solving the 3D hydrogen atom?

One of the main challenges in solving the 3D hydrogen atom is the complexity of the mathematical equations involved. The Schrödinger equation is a partial differential equation that requires advanced mathematical techniques to solve. Additionally, the 3D nature of the problem adds another layer of complexity.

5. What are the potential applications of solving the 3D hydrogen atom?

Solving the 3D hydrogen atom has many potential applications, such as improving our understanding of atomic and molecular structures, developing new materials and technologies, and aiding in the design of more efficient energy sources. It can also have implications in fields such as chemistry, biology, and materials science.

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