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what exactly is meant by the separation of variables in the schrodinger equation? also what co-ordinate system would i use to solve for an electron in a hydrogen-like ion?

thanks

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In summary, separation of variables is a technique used to solve differential equations and for solving the Schrödinger equation for hydrogen-like ions, it is best to use a spherical coordinate system.

- #1

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what exactly is meant by the separation of variables in the schrodinger equation? also what co-ordinate system would i use to solve for an electron in a hydrogen-like ion?

thanks

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Welcome to Physics Forums, Sonko.Sonko said:

what exactly is meant by the separation of variables in the schrodinger equation? also what co-ordinate system would i use to solve for an electron in a hydrogen-like ion?

thanks

Separation of variable is a technique used to solve differential equations. In terms of partial differential equations one would assume that a function of two variables, say [itex]\Psi\left(x,t\right)[/itex] can be written as a product of two functions of single variables, say [itex]\psi\left(x\right)[/itex] and [itex]\phi\left(t\right)[/itex]. If this assumption is applicable, i.e. if the equation is separable, you can reduce the partial differential equation for two variables into two ordinary differential equations for one variable - one equation for each function and associated variable. Try search for "separation of variables" for more information.

In terms of solving the Schrödinger equation for hydrogen-like ions, you will have a spherically symmetric potential. Therefore, it will be best to solve Schrödinger's equation in spherical coordinates.

The Schrodinger equation is a fundamental equation in quantum mechanics that describes the behavior of a quantum system over time. It takes into account the wave-like nature of particles and allows for the prediction of their position and energy.

Seperation of variables is a mathematical technique used to solve the Schrodinger equation. It involves separating the equation into two parts: one that is dependent on time and one that is dependent on space. This allows for the solution to be expressed as a product of two functions.

Seperation of variables allows for the simplification of the Schrodinger equation, making it easier to solve and understand. It also allows for the separation of the time-dependent and time-independent parts of the equation, which is important in understanding the behavior of quantum systems.

The steps involved in seperation of variables in the Schrodinger equation are as follows: 1) Write the Schrodinger equation in its general form, 2) Separate the equation into two parts: time-dependent and time-independent, 3) Substitute the time-independent part with the product of two functions, 4) Solve each part of the equation separately, 5) Combine the solutions to get the final solution.

Seperation of variables in the Schrodinger equation is used in various fields such as quantum mechanics, physics, and chemistry. It is used to solve for the energy levels and wave functions of quantum systems, which are important in understanding the behavior of atoms, molecules, and other particles on a microscopic scale.

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