Animated Solution for Physics - Dual Nature of Matter and Radiation: Comprehension Passage

Imagine a tiny particle, perhaps an electron, trapped inside a one-dimensional box of length $a$. The walls of this box are perfectly rigid, meaning the particle can never escape. In the quantum realm, this confinement forces the particle's wave function to be exactly zero at the boundaries ($x=0$ and $x=a$).

Because of these strict boundary conditions, the particle cannot just have any random wavelength. It must form standing waves, much like a guitar string plucked at both ends. For a standing wave to fit perfectly inside the box, the total length $a$ must be an integer multiple of half-wavelengths: where $n=1,2,3,\dots$ is the quantum number representing the number of loops. Rearranging this, we find the allowed wavelengths:

Now, let's connect this geometric picture to the particle's energy. According to the de-Broglie relation, the momentum $p$ of the particle is inversely proportional to its wavelength:

The kinetic energy $E$ of a non-relativistic particle is given by $E=\frac{p^2}{2m}$. Substituting our de-Broglie momentum into this energy equation yields:

Now, we bring in our standing wave condition $\lambda =\frac{2a}{n}$. Substituting this into the energy expression gives us the master equation for the energy levels of a particle in a 1D box:

Notice the beautiful relationship here! The allowed energy $E$ is inversely proportional to the square of the box length $a$. This perfectly answers our first question.

Let's put this formula to the test by calculating the ground state energy. The ground state corresponds to the lowest possible energy, which occurs when $n=1$.

We are given the mass $m=1.0\times 10^{-30}$ kg and the box length $a=6.6\times 10^{-9}$ m. Let's carefully substitute these values, along with Planck's constant $h=6.6\times 10^{-34}$ Js:

Notice how the problem setter has been kind to us! The $6.6^2$ terms in the numerator and denominator will elegantly cancel out:

To convert this energy from Joules to electron-volts (eV), we divide by the elementary charge $e=1.6\times 10^{-19}$ C: This gives us the answer to the second question.

Finally, let's investigate how the speed of the particle depends on its quantum state. We already established that the momentum is quantized:

Since momentum is simply mass times velocity ($p=mv$), we can write:

Because $h$, $m$, and $a$ are all constants for a given system, we can clearly see that the velocity $v$ is directly proportional to the quantum number $n$. As the particle jumps to higher energy states, it moves proportionally faster. This elegantly resolves our final question!