# How to Approximate Potential Energy for a Linear Harmonic Oscillator?

• neelakash
In summary, the linear harmonic oscillator approximation for the given potential energy function is obtained by finding the minimum at x = +-(a/b)^1/4 and expanding the function as a Taylor series around that point, retaining up to the x^2 term. The function is not defined at x = 0 and approaches infinity as x tends to zero.
neelakash

## Homework Statement

Find the linear harmonic oscillator approximation for potential energy function:

$$\ V=$$$$\frac{a}{x^2}+$$$$\ b$$$$\ x^2$$

## The Attempt at a Solution

The 2nd term will be present in the expression of V(approx).But what about the first term. Should we make it {1+(x-1)} and expand binomially?But that would involve two points of eqlbm---one is 0 and the other is 1...

Last edited:
You have to find the minimum first. If both a,b>0, then the minima occur at x= +-(a/b)^1/4. Take the +ve value, say. Expand the function as a Taylor series around that point and retain up to the x^2 term.

Note that the function is not defined at x=0, and approaches infinity as x tends to zero. Why were you thinking of x=0 as an equilibrium point?

Why were you thinking of x=0 as an equilibrium point?

Yes,I really made a mistake in undestanding the problem.Now, I can do it.Thank you very much.

## What is potential energy approximation?

Potential energy approximation is a method used in physics and engineering to estimate the potential energy of a system or object. It involves simplifying complex systems into simpler models, in order to make calculations and predictions easier.

## How is potential energy approximation used in science?

Potential energy approximation is used in various fields of science, such as thermodynamics, mechanics, and quantum mechanics. It is particularly useful in situations where exact calculations are not possible, such as in large scale systems or highly complex systems.

## What are the limitations of potential energy approximation?

One of the main limitations of potential energy approximation is that it can only provide an estimate of the potential energy of a system, rather than an exact value. It also assumes certain simplifications and idealizations, which may not always accurately reflect the real-world system.

## How does potential energy approximation differ from other methods of calculating potential energy?

Potential energy approximation differs from other methods, such as numerical integration or analytical solutions, in that it relies on simplifying assumptions and models rather than exact calculations. This makes it more suitable for certain types of systems and calculations.

## What are some examples of potential energy approximation in action?

Potential energy approximation is used in a variety of real-world applications, such as in the design of structures and machines, in modeling chemical reactions, and in understanding the behavior of complex systems like the Earth's atmosphere. It is also used in theoretical physics to study phenomena that cannot be observed directly.

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