# Delta potential in classical mechanics

• hilbert2
In summary, in quantum mechanics there are some systems where the potential energy is represented by a Dirac delta function of position. This can also be approximated by a Gaussian with nonzero width, and in classical mechanics, this would result in a particle either having enough kinetic energy to get over the barrier, reaching a point of unstable equilibrium, or bouncing back. However, when both A and k tend towards infinity, only the option of bouncing back would apply, as classical particles cannot cross barriers higher than their kinetic energy.
hilbert2
Science Advisor
Gold Member
In quantum mechanics, there exist some systems where the potential energy of some particle is a Dirac delta function of position: ##V(x) = A\delta (x-x_0 )##, where ##A## is a constant with proper dimensions.

Is there any classical mechanics application of this? It would seem that if I approximate the delta with a Gaussian of nonzero width

##V(x) = Ae^{-k(x-x_0 )^2}##,

then a particle coming from the left with velocity ##v## could either

1. Have enough kinetic energy to get over the barrier and continue to right with same velocity ##v##

2. Have exactly the right amount of kinetic energy to get on the top of the barrier and stay there in unstable equilibrium

3. Have less kinetic energy than needed to get over the barrier and bounce back, returning to the left direction with velocity ##-v##.

Here I'm assuming that ##A>0##. Is there any reason why this wouldn't also hold when ##A\rightarrow\infty## and ##k\rightarrow\infty## ?

Last edited:
In the latter situation only 3. would apply and hold. No tunneling in classical mechanics.

hilbert2
Ok, now I got it - the classical particle can't cross a barrier higher than its kinetic energy, no matter how narrow the barrier is.

## 1. What is a delta potential in classical mechanics?

A delta potential is a type of potential energy function used in classical mechanics to describe the interaction between particles. It is also known as a Dirac delta function and is represented by the symbol δ(x).

## 2. How is a delta potential different from other types of potentials?

A delta potential is unique in that it is a point-like potential, meaning it only affects particles at a specific point in space. This is in contrast to other potentials, such as the harmonic potential, which have a finite range of influence.

## 3. What are the applications of a delta potential in classical mechanics?

A delta potential is commonly used in quantum mechanics to model interactions between particles, such as in the study of bound states and scattering processes. It is also used in classical mechanics to describe point-like interactions, such as the Coulomb potential between two charged particles.

## 4. How is a delta potential mathematically defined?

A delta potential is mathematically defined as a function that is zero everywhere except at a single point, where it is infinite. It is represented by the Dirac delta function, which has the properties of being infinite at x=0 and zero everywhere else, and has an area under the curve of 1.

## 5. What are the limitations of using a delta potential in classical mechanics?

One limitation of using a delta potential is that it is a simplified model and may not accurately represent the true interactions between particles. Additionally, it can lead to mathematical difficulties, such as non-integrable infinities, which may require regularization techniques to overcome.

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