An electron is confined to a box the size of a small atom, 0.10 nm across. (a) What's the uncert

An electron is confined to a box the size of a small atom,...

Key Concepts

Heisenberg Uncertainty Principle

The Heisenberg Uncertainty Principle states that it is fundamentally impossible to simultaneously know the exact position and momentum of a particle. This principle implies a reciprocal uncertainty between the position and momentum: the more precisely one is known, the less precisely the other can be determined. In quantum mechanics, this principle underpins the concept that confining a particle to a very small region (in positional space) results in a significant uncertainty in its momentum.

Particle in a Box Model

The particle in a box model is an idealized quantum mechanical system where a particle is confined within an infinitely deep potential well with rigid boundaries. Although it is a simplified model, it illustrates key concepts such as quantization of energy levels and the effect of spatial confinement on the particle’s momentum and energy. This model is especially useful in understanding the behavior of electrons in confined spaces, such as in atoms or quantum dots.

de Broglie Wavelength

The de Broglie wavelength concept describes the wave-like behavior of particles, linking a particle's momentum with its associated wavelength by the relation ? = h/p, where h is Planck's constant and p is the momentum. This relation is central to quantum mechanics and explains phenomena where particles exhibit both particle and wave characteristics, allowing the calculation of wavelengths corresponding to given momentum or energy values.

Kinetic Energy in Quantum Systems

In quantum mechanics, the kinetic energy of a particle is often derived from its momentum using the classical relation E = p^2/(2m), where m is the particle’s mass. When a particle is confined to a small region, as in the particle in a box model, the high uncertainty in momentum leads to significant kinetic energy. This relationship is critical when analyzing the behavior of electrons under confinement and comparing their energies to those of photons through their respective energy-wavelength relationships.

Transcript

00:01 We will be considering an electron that has been confined to a box that is 0 .1 nanometers across.

00:12 So the first thing i would like to do is to convert the nanometers into meters, and we can do that by multiplying with the prefix nano, which is equal to 10 to the negative 9.

00:26 And what we're left with is 0 .10 into 10 to the negative 9 meters, which is the same as 1 .0 into 10 to the negative 8 and scientific notation.

00:38 I just prefer this notation.

00:43 So it's confined to a box that has a side length, 1 into 10 to the negative 8 meters.

00:56 That means the electron can be anywhere inside this box, but it can't be outside the box.

01:03 So with know its location to within 1 into 10 to the negative 8 meters, that means the uncertainty delta x is equal to plus minus 1 into 10 to the negative rate, because we're certain that it's in the box, but it can be anywhere within the box.

01:31 Now to find the uncertainty in momentum, we need to consider the uncertainty principle, which states that delta x, delta b, always square than equal to the plank constant, the reduced plank constant divided by 2.

01:47 So in this way we can calculate the maximum uncertainty in the momentum as being the maximum uncertainty in momentum will be h -bar divided by 2 times delta x.

02:10 So we can substitute in the value for h -bar which is 1 .05 into 10 to the negative 34 joules seconds and delta x we just calculated above is 1 into 10 to the negative 8 meters and we can just simplify this expression using a calculator this just turns out to be 5 .25 into 10 to the negative 27 newton seconds the uncertainty in momentum so that's the answer to our first part.

03:01 For the second part, you're given that the uncertainty in momentum is about equal to the momentum.

03:12 So our momentum will be 5 .25 into 10 to the negative 27 newton seconds going forward with the calculations.

03:24 And we need to find the electrons kinetic energy and its de broil -wave length.

03:32 So the energy is given by the formula b squared divided by 2m if we're considering non -relativistic kinetic energy, but that's not that good of an idea at these speeds.

03:51 So we will instead take e as being equal to e -square equals m squared, c to the power 4 plus b squared times c squared.

04:06 This is the true relativistic momentum formula, and we'll be using that.

04:09 So the mass of an electron is 9 .11 into 10 to the negative 31 kilograms.

04:17 The speed of light is 3 into 10 to the 8 meters per second.

04:23 Thus the momentum we calculated earlier is 5 .25 into 10 to the negative 27.

04:36 And the speed of light again is 3 into 10 to the 8 meters per second.

04:41 And we can go ahead and just calculate the expression, which will give us east.

04:45 Squared, sorry i just missed square over here.

04:52 I'm going to go and correct that.

04:54 And using a calculator, this expression evaluates to 4 .73 into 10 to the negative 10...

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