# Simplification in Schrodinger derivation

• TheFerruccio
In summary, the conversation discusses the derivation of the Schrodinger equation, specifically how the relativistic total energy is approximated to arrive at the equation. The use of the binomial approximation is introduced and can be explored further through plots. It is noted that the concept of binomial approximation is not well-known, even among those who have tutored math for years.
TheFerruccio
This is not a homework question. This is not for a course. However, I got a warning for posting such questions elsewhere, so, I suppose I must post them here.

## Homework Statement

The following is an excerpt of the derivation of the Schrodinger equation. After deriving the Klein-Gordon equation, the relativistic total energy is approximated to arrive at the Schrodinger equation.

## Homework Equations

$E = mc^2\sqrt{1+\frac{p^2}{m^2c^2}}$
$\approx mc^2\left(1+\frac{1}{2}\frac{p^2}{m^2c^2}\right)$

## The Attempt at a Solution

Well, frankly, I do not see how they went from the first step to the second step. Where did the 1/2 come from? How does the removal of the square root effectively approximate this? I am not seeing it. Was a conjugate used and multiplied somehow?

It may also help you to play around with some of these plots to get a feel for how well the approximations work. For example, letting ##x = p/(mc)##, take a look at a comparison of the two curves, plotted with wolframalpha:

http://www.wolframalpha.com/input/?i=Plot[{Sqrt[1+x^2],1+x^2/2},{x,0,1}]

Play around with some other examples of the binomial approximation as well.

Thank you! This is perfect. I do not know how I managed to get this far without having heard of the binomial approximation, and having tutored math for years. I guess I am one of the lucky ten thousand.

I can provide some explanation for the simplification in the derivation of the Schrodinger equation.

The first step in the derivation involves using the relativistic energy equation, which includes the square root term. This term accounts for the relativistic effects of momentum on the total energy of a particle. However, in the context of quantum mechanics, we are dealing with particles of very small masses and velocities, so the relativistic effects are negligible. Therefore, we can approximate the relativistic energy equation by dropping the higher order terms, which is where the 1/2 comes from in the second step.

To better understand this, let's look at the Taylor series expansion of the square root term:

\sqrt{1+\frac{p^2}{m^2c^2}} \approx 1 + \frac{1}{2}\frac{p^2}{m^2c^2} - \frac{1}{8}\frac{p^4}{m^4c^4} + ...

As you can see, the first term is just 1, and the second term is the one we are interested in, which is where the 1/2 comes from. The higher order terms are much smaller and can be neglected. So, by dropping these higher order terms, we effectively approximate the relativistic energy equation to the non-relativistic form, which is the second step in the derivation.

I hope this helps in understanding the simplification in the derivation of the Schrodinger equation.

## What is the Schrodinger equation?

The Schrodinger equation is a fundamental equation in quantum mechanics that describes the evolution of a quantum system over time. It is named after Austrian physicist Erwin Schrodinger and is used to calculate the probability of finding a particle in a certain state at a given time.

## What is the process of simplification in the derivation of the Schrodinger equation?

The process of simplification in the derivation of the Schrodinger equation involves applying certain assumptions and approximations to the more complex equations of quantum mechanics. This includes assuming that the particle is non-relativistic, that it is in a stationary state, and that its energy is constant, among others.

## Why is simplification important in the derivation of the Schrodinger equation?

Simplification is important in the derivation of the Schrodinger equation because it allows us to solve complex quantum mechanical problems in a more manageable way. By simplifying the equations, we can focus on the most relevant aspects of the system and obtain more practical and useful solutions.

## What are some common simplifications made in the derivation of the Schrodinger equation?

Some common simplifications made in the derivation of the Schrodinger equation include ignoring relativistic effects, considering only one particle in the system, and assuming that the potential energy is constant. These simplifications allow us to solve the equation more easily and obtain more practical results.

## Are there any limitations to the use of simplified Schrodinger equations?

Yes, there are limitations to the use of simplified Schrodinger equations. As with any simplification, there is a trade-off between accuracy and practicality. While simplified equations may be easier to solve, they may not accurately represent all aspects of a quantum system. Therefore, it is important to carefully consider the limitations of the simplifications used in the derivation of the Schrodinger equation.

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