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LSMOG
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What are the key things to look when solving Schrodinger equation for the particular system like Hydrogen atom
All I know so far is to solve partial differential equation, and to solve Schrodinger equation for the particle in a box situation, now for the atom is a different story.BvU said:Textbooks !
Corny, I know. Could you be more specific ? Where are you in the curriculum, what brings you to this question and what kind of answer do you expect ?
HAYAO said:What you should keep in mind when solving such system, is:
1) Use spherical coordinate for 3-dimensional case since it makes solving Schrodinger Equation much much easier. In an atom, we have electron orbitals that expand in a spherical way with the nucleus at the origin. Using Cartesian coordinate is extremely difficult and impractical.
You are absolutely right. I left that out because I wasn't thinking about harmonic oscillators. Thanks.stevendaryl said:In general, what you're saying is good advice: If there is spherical symmetry, then you should use spherical coordinates. I found out the hard way that there are a few exceptions to the general principle, though. The one that comes to mind is the 3-D harmonic oscillator: [itex]H = \frac{-\hbar^2}{2m} \nabla^2 + \frac{K}{2} r^2[/itex]. You can solve it using spherical coordinates, but it's actually easier to use cartesian coordinates, and assume that the wave function has the form: [itex]X(x) Y(y) Z(z)[/itex], then [itex]X, Y, Z[/itex] each satisfy the equation for a one-dimensional harmonic oscillator.
LSMOG said:All I know so far is to solve partial differential equation, and to solve Schrodinger equation for the particle in a box situation, now for the atom is a different story.
Schrodinger's equation is a mathematical equation that describes how the quantum state of a physical system changes with time. It is a fundamental equation in quantum mechanics and is used to calculate the behavior of particles at the atomic and subatomic level.
Solving Schrodinger's equation allows us to predict the behavior of particles at the quantum level, which is crucial for understanding and developing new technologies in fields such as electronics, chemistry, and materials science.
Schrodinger's equation is solved using mathematical techniques such as separation of variables, perturbation theory, and numerical methods. The exact method used depends on the specific system being studied and the level of accuracy required.
Schrodinger's equation has many applications in physics, chemistry, and engineering. For example, it is used to study the behavior of electrons in atoms, the structure of molecules, and the properties of materials. It is also used in developing quantum computers and understanding the behavior of complex systems.
While Schrodinger's equation is a powerful tool, it does have limitations. It cannot fully describe systems with more than a few particles, and it does not take into account the effects of relativity. Additionally, it only provides probabilistic predictions and cannot determine the exact location or behavior of particles.