# Perturbed Potential Well

• ZeroScope
So, you'd want to add (pi/a)*x to (pi/a)*x - pi. This would give you a trough in the well. In summary, the eigenvalues of a particle in a well are not affected by the structure of the bottom of the well, even when it is deeper in the middle than at the sides. This can be shown by applying perturbation theory. To find the mathematical description of the potential at the bottom of the well, one can scale and shift a sine function to achieve the required shape. By shifting the function by pi, the trough of the sine wave can be aligned with the bottom of the well. The perturbed Hamiltonian can then be written down to further investigate this case.

#### ZeroScope

d) Show that the eigenvalues of a particle in a well aren’t affected by the structure of the bottom of the well by applying perturbation theory. As an example, investigate the case where the bottom of the well is deeper in the middle (x = a2) than at the sides (x = 0 and x= a). To do this:-
i. Find a mathematical description of the potential at the bottom of the well by scaling and shifting a
sine function to achieve the required shape. Illustrate this with a plot of your chosen function.
ii. Write down the perturbed Hamiltonian

With part (i) is this asking for a function that relates the sine wave to the potential. I know that the bottom of the well is in a shape such that it has the negative part of the sine curve as its base. I can't get an equation for the potential at the bottom of the well.

I have tried using sin((gamma + 1)*pi) where 0 < gamma < 1 but it doesn't seem to work through.

What should i do in order to find the function for the potential at the bottom of the well.

First you need to scale the sine function so that it's period is such that half a cycle fits just inside the well (corresponding to a "bowl-shaped" bottom surface of the well, with the bowl being a trough of the sine wave). If the argument is (pi/a)*x, then when x = a, we'll have the argument being pi (half a cycle, as desired).

Unfortunately, the half cycle currently in the well is a crest, not a trough (due to the nature of the sine function). So, you have to shift everything a half cycle to the left or right (by pi). Remember how you shift the function...by adding a constant to the phase.

## What is a perturbed potential well?

A perturbed potential well is a concept used in quantum mechanics to describe the behavior of a particle in a potential well that has been disturbed or altered in some way. This could be due to the introduction of a new force or potential, or a change in the shape of the well.

## How does a perturbed potential well affect the behavior of a particle?

The perturbed potential well can cause the particle to have different energy levels and wavefunctions compared to a regular potential well. This can result in changes in the probability of finding the particle in certain regions of space and the frequencies of its energy transitions.

## What is the significance of studying perturbed potential wells?

Studying perturbed potential wells allows us to gain a deeper understanding of the effects of external forces on the behavior of particles in quantum systems. This is important in fields such as condensed matter physics, where the behavior of particles in potential wells can be used to explain and predict the properties of materials.

## How can perturbed potential wells be experimentally observed?

There are various experimental techniques that can be used to observe perturbed potential wells, such as scanning tunneling microscopy and optical spectroscopy. These techniques allow us to directly measure the energy levels and wavefunctions of particles in a perturbed potential well.

## Are there any real-life applications of perturbed potential wells?

Yes, perturbed potential wells have many real-life applications, such as in the design of electronic devices and sensors. For example, the behavior of electrons in a perturbed potential well can be used to create transistors and other electronic components. They are also used in medical imaging techniques, such as magnetic resonance imaging (MRI).