Consider an electron confined to a diameter of 0.1 nm about the size of a hydrogen atom. If the

Consider an electron confined to a diameter of 0.1 nm about...

Key Concepts

Heisenberg Uncertainty Principle

The Heisenberg Uncertainty Principle is a cornerstone of quantum mechanics that establishes fundamental limits on the precision with which pairs of physical properties, such as position and momentum, can be simultaneously known. It implies that confining a particle to a very small space increases the uncertainty in its momentum and, therefore, its velocity, which is crucial for understanding quantum behaviors in confined systems.

Quantum Confinement

Quantum confinement refers to the restriction of particles, like electrons, within a very small region of space, which alters their energy levels compared to those in bulk materials. This concept is essential in quantum mechanics for explaining phenomena in systems such as atoms, quantum dots, or nanostructures where spatial constraints lead to increased uncertainties in momentum and energy.

Non-Relativistic Kinetic Energy

In quantum mechanics, when particles move with speeds that are much less than the speed of light, their kinetic energy can be accurately described using the classical expression, KE = ½mv². This non-relativistic approximation is valid under conditions where relativistic effects are negligible, enabling simpler calculations and analysis of the particle's energy.

Energy Unit Conversion

Converting energy between different units, such as Joules to electron volts, is an important procedure in physics to express results in a form that is most appropriate or commonly used in the field. This ensures that the calculated kinetic energies, derived from fundamental equations, can be compared with experimental data or related theoretical values in a clear and standardized manner.

Transcript

00:01 All right, so we have an electron here, and we have it confined into a diameter of 0 .1 nanometers.

00:11 And we need to figure out what the velocity is and what the classical newtonian energy of it would be without doing relativistic corrections.

00:22 So first thing we want to do is find v.

00:25 Well, we know v is part of momentum, and it says the uncertainty in v is going to be similar to the, actual v.

00:36 So if we know our uncertainty, we'll have a pretty good idea of the b.

00:40 So let's start there.

00:44 So we know that p is equal to mass times velocity, but we are looking for the uncertainty in them because we have a better equation for that.

00:57 So the uncertainty in the momentum is equal to the mass times the uncertainty in the velocity so now we just need to get that velocity.

01:20 So if we plug it into the one equation that we have for uncertainty, the delta x, delta p of x, that's greater than h.

01:37 So plugging this into here, we will get delta x times the mass times delta p is greater than planks constant.

01:59 I'm sorry, not p, that should be v.

02:05 We're putting in for the velocity.

02:10 And then we can find that uncertainty in the velocity by just rearranging a little bit.

02:17 So the uncertainty in the velocity is going to be equal to planck's constant divided by mass times uncertainty in position.

02:31 So now if we take uncertainty in position to be half of our diameter, so one radii, so we're going to say delta x is equal to 0 .5 nanometers, which is equal to 5 times 10 to the negative 11 meters.

02:57 So that's the version we will use...

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