A photon with wavelength 8.00 nm is absorbed when an electron in a three-dimensional cubical box

A photon with wavelength 8.00 nm is absorbed when an...

Key Concepts

Planck's Relation

Planck's relation, which states that the energy of a photon is given by E = hc/?, connects the wavelength of electromagnetic radiation to the energy exchanged during quantum transitions. This fundamental equation links the properties of light with electronic transitions in confined systems, allowing one to determine system parameters, such as the size of a potential well, from observed wavelengths.

Photon Absorption and Electronic Transitions

In quantum mechanics, an electron can transition between energy levels by absorbing or emitting a photon. The energy of the photon must match the difference between the quantized energy states, making this process a direct application of energy conservation at the quantum level. This principle underpins many phenomena in spectroscopy and helps explain transitions in confined systems.

Particle in a Box

This concept involves solving the Schrödinger equation for a particle confined in an infinitely deep potential well. The allowed wave functions must satisfy boundary conditions at the edges of the box, leading to discrete energy levels. The model is fundamental for understanding quantum confinement and how spatial restrictions lead to energy quantization.

Energy Quantization

Energy quantization in quantum systems arises because only specific wave function solutions that satisfy the boundary conditions are allowed, resulting in discrete energy levels. This concept explains why a particle in a confined region, such as a cubical box, can only have certain fixed energies determined by its quantum numbers and the dimensions of the confinement.

Transcript

00:02 Okay, well, this problem, we have a photon that is absorb when an electron makes a transition from the ground state for the second excited state.

00:13 So first of all, to make things clear, let's try to make a sketch in here.

00:18 This is the nucleus of the atom, for example.

00:24 This could be the first actually this could be the ground state and let's represent the second excited state as this second orbit here so we have an electron which is in the ground state and we know that when a photon hits the electron he makes a jump for the second excited state make sure that you do not miss the word second because the first excited state is one thing the second the second excited state is other thing and we must find the size of the box knowing only this and the wavelength of the photon okay so so basically we know that when we are in the ground state, when we are in the ground state, we know that the quantum numbers that describe the ground state could only be an x and y and nz equals 1.

01:55 But when we are in the second excited state, the quantum numbers should be.

02:05 Nx equals ny equals 2 and nz equals 1.

02:16 This is one possibility, of course.

02:19 We have the degeneracy.

02:21 But let's use this possibility, this configuration, because this is way better than the others.

02:31 So let's think about, we know how to describe, the configurations for the ground state and the second excited state.

02:44 How we're going to find the size of the box? well, first of all, let's remember how the energy for the electron is described in a state.

02:55 And we know that the energy, let's put this on blue.

03:01 So the energy of the electron is described by nx square plus ny square plus n z square, multiply by pi, a plank constant divided by 2m, l square, which is the size of the box.

03:27 So this is what we're looking for.

03:30 Okay, and how we go into solve this? well, we must remember that when the electron makes a transition, we have a change and energy.

03:44 So we have a delta e that we could calculate using this formula.

03:50 And we must also know that this change of energy is equal the energy of the photon.

03:57 So this change of energy must be equal the energy of the photon, which is described by the equation hc divided by the wavelength of the photon.

04:13 So this is basically what we should do.

04:17 We calculate the change of energy of the electron and equates with the energy of the photon.

04:27 So let's do this.

04:30 Looking for this equation here...

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