# Video: Photon Wavelength Produced by a Transition between One-Dimensional Bound States of a Quantum Particle

An electron is confined to a box of width 0.250 nm. What is the wavelength of photons emitted when the electron transitions between the second excited state and the ground state?

03:40

### Video Transcript

An electron is confined to a box of width 0.250 nanometers. What is the wavelength of photons emitted when the electron transitions between the second excited state and the ground state?

Weโre told in this statement that the box has a width 0.250 nanometers which weโll call ๐ฟ. We want to solve for the wavelength of the photons emitted when the electron transitions from the second excited state to the ground state. Letโs start our solution by drawing a diagram. In this example, we have an infinitely deep box the electron is not able to escape out of. Weโre told the electron starts at the second excited state where ๐ equals three, and it then transitions down to the ground state where ๐ is one. In the process, the electron emits a photon with a wavelength ๐. Itโs that wavelength we want to solve for.

To do that, we can recall that for a particle in a box, like we have here, the energy of the particle at the ๐th energy level, ๐ธ sub ๐, equals ๐ squared times โ squared, where โ is Planckโs constant which weโll take to equal exactly 6.626 times 10 to the negative 34th joule seconds, all of which is divided by eight times the mass of the electron ๐ times the width of the box ๐ฟ squared.

In our case, we have an electron that starts out at ๐ equals three and then moves to ๐ equals one. By the particle in a box energy equation, that equals nine โ squared over eight ๐๐ฟ squared minus one โ squared over eight ๐๐ฟ squared, which simplifies to โ squared over ๐๐ฟ squared. Now thatโs the change in energy of the electron, but what about the photon thatโs emitted? We can remember that photon energy equals โ times the frequency ๐ or โ times ๐ over ๐. All the energy of the electronโs transition is delivered to the emitted photon. That means we can write: โ squared over ๐๐ฟ squared is equal to the energy of the emitted photon โ ๐ over ๐. We can cancel out a factor of Planckโs constant from each side and then rearrange this equation to solve for ๐. And we find itโs equal to ๐, the mass of the electron, times ๐, the speed of light, times the width of the box, ๐ฟ squared, divided by โ, Planckโs constant.

In this example, weโll treat the mass of the electron ๐ as exactly 9.1 times 10 to the negative 31st kilograms and the speed of light ๐ as exactly 3.00 times 10 to the eighth meters per second.

When we plug in for ๐, ๐, ๐ฟ, and โ being careful to write our value for ๐ฟ in units of meters, when we calculate ๐, we find that, to three significant figures, it is 25.8 nanometers. That is the wavelength of the emitted photon.