Question About Change Of Variables

Introduction

In physics, the change of variables is a common technique used to simplify complex problems by transforming the variables in a differential equation. This technique is often used in conjunction with symmetry principles, such as time reversal symmetry, to solve problems in quantum mechanics and other areas of physics. However, the validity of this technique has been questioned by some, who argue that it is not mathematically rigorous. In this article, we will explore the mathematical validity of the change of variables technique and its applications in physics.

Mathematical Background

To understand the change of variables technique, we need to start with some basic mathematical concepts. A differential equation is a mathematical equation that involves an unknown function and its derivatives. The change of variables technique involves substituting a new variable into the differential equation, which can simplify the equation and make it easier to solve.

One common example of the change of variables technique is the substitution of $x$ with $-x$. This substitution is often used to simplify problems involving symmetry principles, such as time reversal symmetry. However, as we will see, this substitution is not always mathematically valid.

The Problem with Substitution

The problem with substitution is that it can change the domain of the differential equation. In other words, the substitution can introduce new solutions that were not present in the original equation. This can lead to incorrect results and a loss of physical meaning.

To see why this is the case, let's consider a simple example. Suppose we have a differential equation of the form:

$$\frac{d^2y}{dx^2} + \lambda y = 0$$

where $\lambda$ is a constant. This equation has a solution of the form:

$$y(x) = A \cos(\sqrt{\lambda}x) + B \sin(\sqrt{\lambda}x)$$

where $A$ and $B$ are constants.

Now, suppose we substitute $x$ with $-x$. This gives us a new differential equation of the form:

$$\frac{d^2y}{d(-x)^2} + \lambda y = 0$$

which is equivalent to:

$$\frac{d^2y}{dx^2} - \lambda y = 0$$

This new equation has a solution of the form:

$$y(x) = C \cosh(\sqrt{\lambda}x) + D \sinh(\sqrt{\lambda}x)$$

where $C$ and $D$ are constants.

As we can see, the substitution of $x$ with $-x$ has introduced a new solution that was not present in the original equation. This is because the substitution has changed the domain of the differential equation, introducing new solutions that were not present in the original equation.

Mathematical Rigor

So, is the change of variables technique mathematically valid? The answer is no. While the technique can be useful in simplifying complex problems, it is not mathematically rigorous. The substitution of variables can change the domain of the differential equation, introducing new solutions that were not present in the original equation.

However, there are some cases where the change of variables technique is mathematically valid. For example, if the differential equation is invariant under the of variables, then the substitution is mathematically valid. This is because the substitution does not change the domain of the differential equation, and the new solutions are equivalent to the original solutions.

Applications in Physics

Despite the lack of mathematical rigor, the change of variables technique is widely used in physics. This is because the technique can be useful in simplifying complex problems and making them easier to solve.

One common application of the change of variables technique is in the study of symmetry principles, such as time reversal symmetry. By substituting $x$ with $-x$, physicists can simplify complex problems and make them easier to solve.

Another application of the change of variables technique is in the study of quantum mechanics. By substituting $x$ with $-x$, physicists can simplify complex problems and make them easier to solve.

Conclusion

In conclusion, the change of variables technique is not mathematically valid in all cases. While the technique can be useful in simplifying complex problems, it is not mathematically rigorous. The substitution of variables can change the domain of the differential equation, introducing new solutions that were not present in the original equation.

However, there are some cases where the change of variables technique is mathematically valid. For example, if the differential equation is invariant under the substitution of variables, then the substitution is mathematically valid.

Despite the lack of mathematical rigor, the change of variables technique is widely used in physics. This is because the technique can be useful in simplifying complex problems and making them easier to solve.

References

• [1] Mathematical Methods in the Physical Sciences by Mary L. Boas
• [2] The Feynman Lectures on Physics by Richard P. Feynman
• [3] Quantum Mechanics by Lev Landau and Evgeny Lifshitz

Further Reading

• Symmetry Principles in Physics by Gerald E. Brown
• Mathematical Methods for Physicists by George B. Arfken
• Quantum Mechanics and Path Integrals by Richard P. Feynman
  Q&A: Change of Variables in Physics =====================================

Q: What is the change of variables technique in physics?

A: The change of variables technique is a mathematical method used to simplify complex problems in physics by transforming the variables in a differential equation. This technique is often used in conjunction with symmetry principles, such as time reversal symmetry, to solve problems in quantum mechanics and other areas of physics.

Q: Is the change of variables technique mathematically valid?

A: No, the change of variables technique is not mathematically valid in all cases. While the technique can be useful in simplifying complex problems, it is not mathematically rigorous. The substitution of variables can change the domain of the differential equation, introducing new solutions that were not present in the original equation.

Q: When is the change of variables technique mathematically valid?

A: The change of variables technique is mathematically valid when the differential equation is invariant under the substitution of variables. This means that the new solutions obtained by substituting variables are equivalent to the original solutions.

Q: What are some common applications of the change of variables technique in physics?

A: The change of variables technique is widely used in physics to simplify complex problems and make them easier to solve. Some common applications include:

• Studying symmetry principles, such as time reversal symmetry
• Solving problems in quantum mechanics
• Simplifying complex differential equations

Q: Can the change of variables technique be used to introduce new solutions to a differential equation?

A: Yes, the change of variables technique can introduce new solutions to a differential equation. This is because the substitution of variables can change the domain of the differential equation, introducing new solutions that were not present in the original equation.

Q: How can I determine if the change of variables technique is mathematically valid for a given differential equation?

A: To determine if the change of variables technique is mathematically valid for a given differential equation, you need to check if the differential equation is invariant under the substitution of variables. This can be done by substituting the new variables into the differential equation and checking if the new solutions are equivalent to the original solutions.

Q: What are some common mistakes to avoid when using the change of variables technique?

A: Some common mistakes to avoid when using the change of variables technique include:

• Failing to check if the differential equation is invariant under the substitution of variables
• Introducing new solutions that were not present in the original equation
• Not properly transforming the boundary conditions

Q: Can the change of variables technique be used to solve problems in other areas of physics, such as classical mechanics?

A: Yes, the change of variables technique can be used to solve problems in other areas of physics, such as classical mechanics. However, the technique is most commonly used in quantum mechanics and other areas of physics where symmetry principles are important.

Q: Are there any alternative methods to the change of variables technique for simplifying complex differential equations?

A: Yes, there alternative methods to the change of variables technique for simplifying complex differential equations. Some common alternative methods include:

• Using symmetry principles to simplify the differential equation
• Using numerical methods to solve the differential equation
• Using approximation methods to simplify the differential equation

Q: Can the change of variables technique be used to introduce new physical phenomena or effects?

A: Yes, the change of variables technique can be used to introduce new physical phenomena or effects. This is because the substitution of variables can change the domain of the differential equation, introducing new solutions that were not present in the original equation.

Q: How can I learn more about the change of variables technique and its applications in physics?

A: To learn more about the change of variables technique and its applications in physics, you can:

• Read textbooks on mathematical methods in physics
• Take courses on quantum mechanics and other areas of physics
• Read research papers on the change of variables technique and its applications in physics

References

• [1] Mathematical Methods in the Physical Sciences by Mary L. Boas
• [2] The Feynman Lectures on Physics by Richard P. Feynman
• [3] Quantum Mechanics by Lev Landau and Evgeny Lifshitz

Further Reading

• Symmetry Principles in Physics by Gerald E. Brown
• Mathematical Methods for Physicists by George B. Arfken
• Quantum Mechanics and Path Integrals by Richard P. Feynman