Analytic Solutions to Schroedinger Equation

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In summary, the conversation discusses the impossibility of solving certain differential equations, specifically the radial Schrodinger equation for potential energy functions other than the ones mentioned. There is a question about whether this impossibility has been proven or if there is still the possibility of finding solutions using custom-made functions or orthogonal polynomials.
  • #1
YAHA
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Everyone knows that the above equation cannot be solved for anything, with the exception of some simple cases(infinite well, harmonic oscillator, hydrogenic atoms, etc).

Now, my question is whether we know that it is impossible to solve such differential equation for other potential energy functions or is it simply that we haven't discovered a way of solving it? In other words, is there some proof of the impossiblity of finding solutions for certain functions?
 
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  • #2
Soluble in terms of what? For the radial Schrodinger equation, for example, for any V(r), one can always define a new custom-made special function and/or a set of orthogonal polynomials in terms of which the solution can be exactly represented.
 

1. What is the Schrödinger equation?

The Schrödinger equation is a fundamental equation in quantum mechanics that describes how the wave function of a physical system evolves over time. It is named after Austrian physicist Erwin Schrödinger and is a central tool for understanding the behavior of particles at the atomic and subatomic level.

2. What are analytic solutions to the Schrödinger equation?

Analytic solutions to the Schrödinger equation refer to exact mathematical expressions that describe the behavior of a quantum system. These solutions can be used to calculate the probabilities of different outcomes and make predictions about the behavior of the system.

3. Why are analytic solutions important in quantum mechanics?

Analytic solutions are important in quantum mechanics because they provide a precise and rigorous description of how a quantum system will evolve over time. They allow scientists to make accurate predictions and test the validity of the theory.

4. How are analytic solutions to the Schrödinger equation obtained?

Analytic solutions to the Schrödinger equation are obtained through mathematical techniques such as separation of variables, perturbation theory, and the use of special functions. These methods allow for the calculation of the wave function, which describes the probability distribution of a particle in a given system.

5. Are there limitations to using analytic solutions in quantum mechanics?

Yes, there are limitations to using analytic solutions in quantum mechanics. These solutions are only applicable to simple systems and cannot accurately predict the behavior of highly complex systems. Additionally, some systems may not have analytic solutions and require numerical methods for analysis.

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