Why Can't We Solve the Schr?dinger Equation for Larger Systems?

One of the fundamental questions in quantum physics is why we cannot solve the Schr?dinger equation for systems larger than hydrogen. This article delves into the complexities behind this issue and explores whether it is due to a lack of computational power or a more fundamental limitation.

Introduction to the Challenges

The inability to solve the Schr?dinger equation for systems larger than hydrogen is primarily attributed to the nonlinearity and complexity of the equations involved. These equations become significantly harder to solve as the system size increases, necessitating the use of simplifying assumptions and numerical methods. (Keyword: Schr?dinger Equation)

Nonlinearity and Computational Complexity

Traditional solutions to the Schr?dinger equation rely heavily on symmetry arguments, which simplify the problem when substantial symmetries are present. For the hydrogen atom, rotational symmetry and a hidden symmetry reduce the number of degrees of freedom, allowing for an analytic solution using special functions. However, for more complex systems, these symmetries are insufficient to provide a complete solution. (Keyword: Symmetry Arguments)

For systems with many degrees of freedom, such as benzene, the problem becomes computationally intractable. The number of points required to numerically specify a solution grows exponentially with the number of degrees of freedom. Even with advanced computational techniques, the complexity of these problems can overwhelm current computing capabilities. (Keyword: Computational Complexity)

Many-Body Problem and Fundamental Limitations

The Schr?dinger equation is not inherently flawed; the challenge lies in the many-body problem. Solving the equations for systems with two or more interacting particles (like electrons) is extremely difficult. This is not a problem unique to the Schr?dinger equation but is also faced when computing the orbits of planets in the solar system. (Keyword: Many-Body Problem)

Although we do not have exact solutions for most many-body systems, we can approximate them using numerical methods and perturbation theory. These methods provide very good approximations, even if they are not exact. The finite computing power we possess allows us to make significant advancements in both quantum mechanics and celestial mechanics. (Keyword: Approximations)

Conclusion

The inability to solve the Schr?dinger equation for larger systems is a reflection of the limitations of many-body problems. While current computational power is a significant factor, the core issue stems from the inherent complexity of these systems. As computational technology continues to advance, we will likely see improvements in our ability to handle these problems, although exact solutions may remain out of reach for complex systems like benzene or multiple-electron atoms.