# Quantam Mechanic - Particle in a rigid one-d box (PDF)

• MPKU
In summary, the problem involves finding the probability distribution for the second excited state (n=3) of a particle in a rigid box of length a. The probabilities of finding the particle in the intervals [0.50a, 0.51a] and [0.75a, 0.76a] are calculated using the wave-function squared and the distance between the two points. In the first interval, an approximation is made using sin^2(3∏/2) = 1, while in the second interval, a more general expression, sin^2(9∏/4), is used.
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## Homework Statement

Write down and sketch the probability distribution for the second excited state (n=3) of a particle in a rigid box of length a.

What are the probabilities of finding a particle in the intervals [0.50a, 0.51a] and [0.75a, 0.76a]?

## Homework Equations

|ψ(x)|^2 = (2/L)sin^2(n∏x/L)

## The Attempt at a Solution

I've found the solution online: http://www.uic.edu/classes/phys/phys244ma/p244hw7.html

However, I'm not understanding how they are getting the answers.

Multiply the wave-function squared times the distance between the two points, but I'm not sure why the sine squared term in the first part is one, while in the second part you are to use sin^2(9∏/4)?

Thank you.

In the first part, the author is making an approximation. In the range $x \in [ 0.5a,0.51a]$ :

$$\sin^2{\frac{3\pi}{a}x} \approx \sin^2{\frac{3\pi}{2}} = 1$$

In the second part, that approximation does not hold, so the author uses a more general expression.

## 1. What is a rigid one-d box in quantum mechanics?

A rigid one-d box, also known as an infinite potential well, is a theoretical construct used in quantum mechanics to model a particle confined to a one-dimensional space with infinitely high potential barriers at the boundaries. This means that the particle is unable to escape the box and its energy is quantized into discrete levels.

## 2. How does the particle's energy change in a rigid one-d box?

In a rigid one-d box, the particle's energy is quantized and can only take on certain discrete values. As the particle's position within the box changes, its energy levels also change. The energy increases as the particle moves towards the center of the box, and decreases as it moves towards the boundaries.

## 3. What is the Schrödinger equation for a particle in a rigid one-d box?

The Schrödinger equation for a particle in a rigid one-d box is a simplified version of the time-independent Schrödinger equation. It takes into account the infinite potential barriers at the boundaries of the box and is used to determine the allowed energy levels and corresponding wavefunctions for the particle.

## 4. How do the energy levels in a rigid one-d box compare to those in a particle in a box with finite potential barriers?

In a rigid one-d box, the energy levels are evenly spaced and the energy difference between each level is the same. In a particle in a box with finite potential barriers, the energy levels are not evenly spaced and the energy difference between each level decreases as the particle's position approaches the boundaries of the box.

## 5. What are some real-world applications of a particle in a rigid one-d box?

Rigid one-d boxes are mainly used as a theoretical model to study and understand quantum mechanical systems. However, they can also be used to model various physical systems, such as electrons in a crystal lattice or atoms in a molecule. They are also used in technologies such as transistors and lasers, which rely on the principles of quantum mechanics.

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