Linearity and nonlinear theories. Schrödinger's equation

TL;DR

Linear theories are simpler than non-linear theories, and classical mechanics is non-linear, while quantum mechanics is linear.

Transcript

PROFESSOR: So here is of something funny. You might say, OK, what is simpler? A theory that is linear or a theory that is not linear? And the answer, of course, a linear theory is much simpler. General-- Maxwell's equations are linear. Einstein's theory of relativity is very nonlinear, very complicated. How about classical mechanics? Is classical m... Read More

Key Insights

• 👻 Linear theories, such as quantum mechanics, are simpler and allow for scaling, superpositions, and the discovery of new solutions.
• 🦾 Classical mechanics, despite being considered simpler, is actually non-linear and cannot solve the three-body problem.
• 🦾 Quantum mechanics, introduced by Schrodinger's equation, is linear and relies on complex numbers to represent probabilities.
• 💦 The wave function in quantum mechanics can depend on time and other variables, with its interpretation as probabilities coming later through Max Born's work.
• 👋 Complex numbers are essential in quantum mechanics to represent the complex-valued wave function and capture interference phenomena.
• ❓ Linearity in physics has significant implications for the simplicity and elegance of mathematical formulations.
• 🦾 The necessity of complex numbers in quantum mechanics reflects the inherent complexity and probabilistic nature of quantum phenomena.

Questions & Answers

Q: What is the difference between linear and non-linear theories in physics?

Linear theories in physics are simpler because they can be superposed and scaled, while non-linear theories, like Maxwell's equations and Einstein's theory of relativity, are more complicated and do not have these properties.

Q: Why is classical mechanics considered non-linear?

Classical mechanics is non-linear because Newton's equations cannot solve the three-body problem, where three bodies interact simultaneously. This inability to superpose solutions makes classical mechanics complicated.

Q: What is the equation that describes dynamics in quantum mechanics?

Schrodinger's equation, formulated in 1925, describes the dynamics of quantum systems. It is a linear equation, with the wave function as the dynamical variable, dependent on time and possibly other variables.

Q: Why are complex numbers necessary in quantum mechanics?

Complex numbers are necessary in quantum mechanics because the wave function, described by Schrodinger's equation, can be complex-valued. The complex nature of the wave function allows for the representation of probabilities and interference effects.

Summary & Key Takeaways

• Linear theories are simpler than non-linear theories, such as Maxwell's equations and Einstein's theory of relativity being non-linear.

• Classical mechanics is also non-linear, as Newton's equations cannot solve the three-body problem due to the inability to superpose solutions.

• Quantum mechanics, on the other hand, is linear and described by Schrodinger's equation, which allows for scaling, superpositions, and the discovery of new solutions.