An electron is bound in a square well of width 1.50 nm and depth U0 = 6E1-IDW. If the electron i

An electron is bound in a square well of width 1.50 nm and...

Key Concepts

Finite Potential Well

A finite potential well is a model in quantum mechanics where a particle such as an electron is confined to a region of space with a potential energy lower than that in the surrounding regions. Unlike an infinite well, the barriers here are finite, meaning that there is a non-zero probability for the particle to tunnel out if it has enough energy.

Quantized Energy Levels

In a confined system like a finite potential well, the allowed energy states are discrete rather than continuous. The electron’s energy is quantized, meaning it can only occupy specific energy levels determined by the dimensions and depth of the well.

Photon Absorption and Quantum Transitions

When an electron absorbs a photon, it gains energy equal to the photon’s energy, which can promote it from a lower to a higher energy state. In the context of a confined electron, the absorbed photon must supply at least the energy difference needed for the electron to overcome the well’s binding energy.

Ionization Energy

Ionization energy refers to the minimum energy required to liberate an electron from its bound state in a potential well. For the electron to escape from the well, the photon absorbed must provide at least enough energy to surpass the binding energy of the electron.

Planck’s Relation

Planck’s relation connects the energy of a photon to its wavelength through the equation E = hc/?, where h is Planck’s constant and c is the speed of light. This relation is crucial for determining the maximum wavelength of a photon that can provide the necessary energy for ionizing the electron from the well.

Transcript

00:01 Okay, so problem number 23 states that an electron is bound in a square well of 1 .50 nanometers and a depth of the potential u0 is equal to 6e1, the ground state of an electron, six times the ground state of an electron in an infinite one -dimensional square well.

00:22 If the electron is initially in the ground level and absorbs a photon, what maximum wavelength can the photon have and still liberate the electron from the well? okay, so let's start with what we know from the book.

00:39 So from figure 40, let me rewrite that, sorry, 40 .15b, we're given a square well with the potential u0 is equal to 6, e1, the one -dimensional well.

01:02 And in this finite potential well, the ground state electron has an energy of 0 .625 times the energy of a ground state electron in an infinite well.

01:24 So now we can say that the energy of a photon, i'll call it e gamma, is equal to hc over lambda.

01:33 So the question is really asking.

01:36 So the question wants to know what's the longest wavelength, but it can also basically, what's the lowest energy photon that can excite an electron from the ground state of this finite potential well outside of the potential well? so in order to do that, the photon e gamma must have energy equal to at least the potential of the well, the difference of the potential of the well and the ground state of the electron.

02:07 So this photon will, it needs to have at least this much energy to excite this electron out of the well.

02:15 So if you have a well here, and this is u0, and this is e1, in order to, you know, a photon comes in with h -bar h -mew, this energy, actually let me use.

02:36 The what i wrote here, e gamma, that e gamma must be equal to this, at least this energy difference.

02:46 So u0 minus e1 in order to excite an electron out.

02:51 So now if we plug that back in here, so let's solve for lambda because it wants to know what the longest wavelength is.

03:00 So we'll say lambda is equal to hc over e gamma, which is equal to h c over e gamma, which is equal to h c over u0 minus e1 where e1 is equal to pi squared h bar squared over 2m l or which is equal to h squared over 8 m l squared because h bar is equal to h over 2 pi.

03:35 Okay.

03:36 So now let's go ahead and plug that in here.

03:40 We can say, so we know e1 is equal to this, so that's equal to hc over six times, you know, yeah, that's fine...

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