# Particle in a rectangular box.

• cgw
In summary, the question is to find the 6 lowest energy states of a particle of mass m in a box with edge lengths of L_{1}=L, L_{2}=2L, L_{3}=2L. The answer gives E_{0}=\frac{\pi^2\hbar^2}{8mL^2}, but the expected answer is E_{0}=\frac{3\pi^2\hbar^2}{4mL^2}. The difference is likely due to a mistake in the calculation or a different interpretation of the problem.
cgw
I am missing something.

The question is to find the 6 lowest energy states of a particle of mass m in a box with edge lengths of $$L_{1}=L$$, $$L_{2}=2L$$, $$L_{3}=2L$$.
The answer gives $$E_{0}=\frac{\pi^2\hbar^2}{8mL^2}$$.

I would have said $$E_{0}=\frac{3\pi^2\hbar^2}{4mL^2}$$ .

What am I missing?
(the answer given for the actual question is 6, 9, 9, 12, 14, 14)

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Finally figured out the latex.

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Well done with the latex. Why would you say what you say ?

I don't know what you mean by "6, 9, 9, 12, 14, 14".

The question asks for the energy of the six lowest states $$\frac{E}{E_{0}}$$. The textbook answer gives $$E_{0}=\frac{\pi^2\hbar^2}{8mL^2}$$

The way I see it is:

E = E_0 at n1=1, n2=1, n3=1

$$E_{0}=\frac{\pi^2\hbar^2}{2m}\left{\left(\frac{n_{1}}{L}\right)^2+\left(\frac{n_{2}}{2L}\right)^2+\left(\frac{n_{3}}{2L}\right)^2\right}$$

which should be $$E_{0}=\frac{3\pi^2\hbar^2}{4mL^2}$$

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## 1. What is a "Particle in a rectangular box?"

A particle in a rectangular box is a theoretical model used in quantum mechanics to study the behavior of a particle confined within a finite region of space, typically a rectangular box with impenetrable walls.

## 2. What is the significance of studying a particle in a rectangular box?

Studying a particle in a rectangular box allows scientists to understand the behavior of quantum particles, which have unique properties that differ from classical particles. It also helps in developing new technologies, such as quantum computing.

## 3. What is the Schrödinger equation and how is it used in the particle in a rectangular box model?

The Schrödinger equation is a fundamental equation in quantum mechanics that describes the behavior of a particle in terms of its wave function. In the particle in a rectangular box model, the Schrödinger equation is used to determine the allowed energy levels and corresponding wave functions of the particle within the box.

## 4. How are the energy levels of a particle in a rectangular box determined?

The energy levels of a particle in a rectangular box are determined by solving the Schrödinger equation for the specific boundary conditions of the box. The allowed energy levels are discrete and depend on the size and shape of the box.

## 5. Can a particle in a rectangular box exist in multiple energy states at the same time?

Yes, according to quantum mechanics, a particle in a rectangular box can exist in multiple energy states simultaneously. This is known as superposition and is a fundamental principle of quantum mechanics.

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