An electron confined to a box absorbs a photon with wavelength λ. As a result, the electron make

An electron confined to a box absorbs a photon with...

An electron confined to a box absorbs a photon with wavelength $\lambda .$ As a result, the electron makes a transition from the $n=1$ state to the $n=3$ state. (a) Find the length of the box. (b) What is the wavelength $\lambda^{\prime}$ of the photon emitted when the electron makes a transition from the $n=$ 3 state to the $n=2$ state?

Key Concepts

Energy Transitions and System Dimensions

The differences between quantized energy levels can be used to deduce physical properties of the system, such as the dimensions of a confining potential. By equating the energy difference with the energy of the photon involved in a transition, one can derive relationships that connect the quantized nature of energy with measurable quantities like the size of the potential well.

Photon Absorption and Emission

When a particle makes a transition between quantized energy levels, it must absorb or emit a photon whose energy matches the difference between the initial and final states. This process is governed by energy conservation and is described by the relation E = hc/?, linking the energy change with the photon’s wavelength.

Quantized Energy Levels

In confined systems, energy is quantized, meaning that the particle can only occupy specific, discrete energy states. The energy levels typically depend on the square of the quantum number, and this discretization is essential for understanding how particles transition between states.

Particle in a Box

This model describes a particle confined in a region with infinitely high potential barriers, leading to discrete energy states. It serves as a basic example of quantum confinement where the allowed energies depend on the size of the box and the quantum number, illustrating fundamental quantum mechanical principles.

Transcript

00:01 So we start with the equation for the photon wavelength, as well as the equation for the energy levels of the electron.

00:12 We'll calculate the change in energy for the electron and then substitute it into the equation for the photon wavelength.

00:31 So first of all, this change in energy can be written as 3 squared times h squared over 8 ml squared, minus h squared over 8ml squared.

00:51 What i've done is i have a, i've placed a term here for the n equals three state, and i've subtracted a term for the n equals one state.

01:01 So then we can factor 3 squared minus 1 and place h squared over 8ml squared over on the right.

01:15 And this gives us 9 minus 1 times h squared divided by 8 ml squared.

01:27 Scrolling down so that we have some room, this can now be written as 8h squared over 8 ml squared.

01:38 And we see that these 8s will cancel each other, and we now have for the change in energy, h squared over ml squared.

01:46 So we'll now substitute this result into the wavelength equation.

01:53 So we have lambda equals hc over delta e, and this is now hc over h squared, ml squared, ml squared.

02:09 This fraction, h squared over ml squared in the denominator for the wavelength equation, this fraction will now flip since it's in the denominator, and we have hc times ml squared over h3.

02:25 And this will reduce as mcl squared over h.

02:34 We can now get an expression for the length l.

02:38 So what we're gonna do, i'm gonna write this lambda equation down here, and what we'll do is multiply by h and divide by mc on both sides.

02:55 And we see that on the right side, these will cancel.

03:00 And now we have an equation l squared equals h lambda over mc.

03:11 We can then take the square root, and this is the equation for l.

03:17 So the length is equal to the square root of h lambda divided by mc.

03:23 Now we want to consider a transition for the electron from n equal to three to the state n equal to two, and then we're going to end up comparing the wavelengths for the two cases.

03:40 So what we'll do is, we'll first of all get the energy, the change in energy for this transition.

03:45 We're gonna call it delta e prime to distinguish it from the other change in energy that we were looking at earlier.

03:52 So this will be, this change in energy will be three squared, eight squared over eight ml squared, minus two squared, eight squared, over eight ml squared...

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