An electron that has an energy of approximately 6 eV moves between infinitely high walls 1.00 nm

step 1 We have a particle in a box with six electron volts. So you're going to start with the equation,

An electron that has an energy of approximately 6 eV moves...

An electron that has an energy of approximately 6 $\mathrm{eV}$ moves between infinitely high walls 1.00 $\mathrm{nm}$ apart. Find (a) the quantum number $n$ for the energy state the electron occupies and (b) the precise energy of the electron.

Key Concepts

Energy Quantization

Energy quantization refers to the concept that systems described by quantum mechanics can only occupy discrete energy levels rather than a continuous range. In the particle in a box scenario, the energy levels are quantified and depend on factors such as the size of the box and the mass of the particle, with each level corresponding to a particular quantum state.

Quantum Numbers

Quantum numbers are integers that arise from the solutions of the Schrödinger equation in quantum systems, and they serve to label and distinguish between the discrete energy states of particles. In the context of the infinite potential well, the principal quantum number determines the energy level, with higher quantum numbers corresponding to higher energy states.

Infinite Potential Well

The infinite potential well, often referred to as the particle in a box model, is a fundamental quantum mechanics framework where a particle is confined within an area bounded by impenetrable walls. The walls are considered to have infinitely high potential, which restricts the particle to a specific region in space and forces its energy to be quantized. This model is essential for understanding the principles of wave functions and boundary conditions in quantum systems.

Transcript

00:01 We have a particle in a box with six electron volts.

00:03 So you're going to start with the equation, the energy.

00:20 Now, one of the first things we're going to need to do is convert the energy in electron volts to joules because our other constants are in si units.

00:31 So the si unit for energy is joules, and so therefore we need the energy in that unit.

00:37 So six electron volts can be multiplied by 1 .6, 10 .6.

00:44 Times 10 to the minus 19, because you have that many joules per electron bolt.

00:51 That is the conversion factor.

00:54 The electron bolt unit cancels, and you end up with 9 .6 times 10 to the minus 19 joules.

01:05 And we now set that equal to this equation, knowing the other constants as well.

01:19 So for planck's constant, we have 6 .66 times 10 of the minus 34, joules seconds.

01:25 But i'm going to leave out the units, just to make this a little simpler.

01:29 We square plonks constant.

01:31 We're actually going to figure out what value of n this energy would correspond to, if roughly.

01:40 And down here we have eight.

01:42 We have the mass of an electron, so we place that here.

01:49 This is in kilograms.

01:51 And we know the length of this box that the particle is in.

01:55 It's a potential whale.

01:57 So we know that one nanometer is equal to 10 to the minus 9 meters.

02:05 So we can easily place that down here.

02:17 And this comes out, this basically will give us an equation for n squared.

02:23 And probably the easiest thing to do, we can calculate all this, or we can simply rearrange.

02:31 So we can multiply both sides by everything that is multiplying n squared.

02:39 Or rather the reverse of everything, the inverse of everything that is multiplying n squared...

Please give Ace some feedback