We define a potential well having energies V as a function...

We define a potential well having energies $V$ as a function of position $x$, as follows: $V=\infty$ for $x=-0.5 \mathrm{~nm}$ to $0 ; V=0 \mathrm{eV}$ for $x=0$ to $5 \mathrm{~nm} ; V=10 \mathrm{eV}$ for $x=5$ to $6 \mathrm{~nm}$, and $\mathrm{V}=0$ for $x>6 \mathrm{~nm}$ and $x<-0.5 \mathrm{~nm}$. We put an electron with energy $7 \mathrm{eV}$ in the region $x$, between 0 and $5 \mathrm{~nm}$. What is the probability of finding the electron at $x<0 \mathrm{~nm} ?$ Is the probability of finding the electron at $x>6 \mathrm{~nm}$ zero or nonzero? What is this probability for $x>6 \mathrm{~nm}$ if the electron was described by classical mechanics and not quantum mechanics?

We define a potential well having energies $V$ as a function of position $x$, as follows:
$V=\infty$ for $x=-0.5 \mathrm{~nm}$ to $0 ; V=0 \mathrm{eV}$ for $x=0$ to $5 \mathrm{~nm} ; V=10 \mathrm{eV}$
for $x=5$ to $6 \mathrm{~nm}$, and $\mathrm{V}=0$ for $x>6 \mathrm{~nm}$ and $x<-0.5 \mathrm{~nm}$. We put an electron with energy $7 \mathrm{eV}$ in the region $x$, between 0 and $5 \mathrm{~nm}$. What is the probability of finding the electron at $x<0 \mathrm{~nm} ?$ Is the probability of finding the electron at $x>6 \mathrm{~nm}$ zero or nonzero? What is this probability for $x>6 \mathrm{~nm}$ if the electron was described by classical mechanics and not quantum mechanics?

Key Concepts

Quantum Tunneling

Quantum tunneling is the phenomenon where a particle has a finite probability of being found in a region that it classically should not access due to an energy barrier higher than its kinetic energy. This arises from the wave-like nature of matter described by the quantum mechanical wavefunction, which can extend into regions considered forbidden by classical physics.

Potential Well

A potential well is a region in space where the potential energy of a particle is lower than in surrounding regions. This concept is used to model scenarios where particles are confined and helps in analyzing their energy states and dynamics, including the possibility of tunneling through adjacent barriers.

Wavefunction and Probability Density

In quantum mechanics, the state of a particle is described by its wavefunction, and the square of its magnitude gives the probability density of finding the particle at a particular position. This principle leads to nonzero probabilities in classically forbidden regions, while an infinite potential barrier forces the wavefunction to be zero.

Classical versus Quantum Mechanics

In classical mechanics, a particle is strictly restricted to regions where its energy exceeds the potential energy, meaning it cannot penetrate barriers higher than its kinetic energy. Conversely, quantum mechanics allows for a nonzero probability of finding a particle in such regions, highlighting the fundamental differences between classical predictions and quantum phenomena.

Transcript

00:01 So we have over here an electron trapped a quantum dot which acts like an infinite square well.

00:13 And the infinite square well starts on the left at x equals to 0, and its length is about 1 nanometers.

00:27 What we want to find is for two different states, right, and it goes to 1 state and then it goes to 2.

00:34 What is the probability of finding the electron between x1 and x2 where x1 is at 0 .15 nanometers and x2 is at 0 .35 so regardless of the state it's what we have to do is to just take the probability density function which is just side square integrated cross dx from 0 .15 nanometers to 0 .35.

01:14 So what are the wave functions for the infinite square well? well in general, it's a square root of 2 over l times sine x over l.

01:32 So this is a general, sorry 2 pi.

01:36 And this not 2 pi sorry this should be n pi it's for general n wave function so the wave function pattern that we want to sketch out for p1 this would be sorry for si1 this would be when our function it has n equals to 1.

02:35 So throughout this entire length it will only have one half of the wavelength.

02:49 On the other hand for the second with function side 2 to be right n equals to 2 pi x then you have a full wavelength throughout the entire box.

03:09 Alright and so on the other hand for the probability density functions.

03:24 Okay at sight 2 square and sigh 1 square.

03:42 So for sight 2 square, when we absolute square it, we bring all the positive values up to the positive side.

03:53 So we'll get two picks and the middle one must reach 0, the 0 point, because basically it touches 0 at this point, right? so definitely 0 square is to 0 on the other end for side 1 because it's already all positive we are just basically increasing the magnitude right so that's all it's a probability density function for side 1 now we can find what is the actual probability for the n equals to 1 state the actual probability of finding the electron right we do the so remember that l is 1 nanometers, the length, and n equals to 1.

05:34 Now using the equation 41 .6, which is written in the question itself, it will help us with the integral.

05:48 Looking at sine square, sine square, a x, bx, right, where a, in this case it's just pi over 1 ,000, nanometer...

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