An electron first has an infinite wavelength and then after...

An electron first has an infinite wavelength and then after it travels through a potential difference has a de Broglie wave-length of $1.0 \times 10^{-10} \mathrm{m} .$ What is the potential difference that it traversed? Draw a picture of the situation and an energy bar chart, and explain why the electron's wavelength decreased.

Key Concepts

de Broglie Wavelength

This concept expresses the wave-particle duality of matter, stating that any particle with momentum exhibits a wavelength given by ? = h/p, where h is Planck’s constant. An electron at rest (with zero momentum) would have an infinite wavelength, and as it gains momentum through acceleration, its wavelength becomes finite and decreases.

Potential Difference and Energy Conversion

When an electron is accelerated by a potential difference, it loses electric potential energy which is converted into kinetic energy, quantified by the relation KE = eV, where e is the electron’s charge and V is the voltage difference. This process increases the electron’s momentum, which in turn affects its de Broglie wavelength.

Momentum and Its Relation to Wavelength

The momentum of a particle, which increases as the particle accelerates, is inversely related to its de Broglie wavelength. An increase in momentum due to the kinetic energy gained from a potential difference results in a decrease in the de Broglie wavelength, illustrating the interconnected relationship between energy, momentum, and wavelength.

Visualization with Energy Diagrams

Energy diagrams or bar charts are tools used to visually represent the changes in energy levels of a system. For an electron being accelerated by a potential difference, these diagrams help illustrate how the initial state of zero kinetic energy (infinite wavelength) transitions to a higher kinetic energy state (finite, decreased wavelength) after energy is imparted to the electron.

Transcript

00:01 For this question, we have an electron that has an infinite wavelength.

00:04 And after it travels through a potential difference, it has a wavelength lambda f of 1 times 10 of the minus 10.

00:09 And it says, what's the potential difference that's transverse? we're going to draw a picture.

00:12 So this picture shows the momentum initially is p0 because momentum is equal to, for double j is equal to planks constant divided by the wavelength.

00:22 If the wavelength is infinite, then the momentum is zero.

00:24 And it travels from a zero point potential to v1.

00:27 Okay.

00:29 And so because there's a potential difference between these two points, there's going to be an electric field flowing in the opposite direction, which will cause the electron to gain energy.

00:37 So this final momentum is lambda f, and so therefore, or piece of f, because it has a final wavelength of lambda f.

00:44 So to solve for v1 in this situation, we can use energy.

00:48 So assuming energy is conserved here, we have delta e from the electric potential equal to e times delta v.

00:58 Where delta v1 minus v0, but v0 is 0, so this is just e times v1.

01:04 We also have delta e equal to 1⁄2m times v final squared minus v initial squared.

01:18 You can write this in terms of momentum, since momentum is just mass times velocity.

01:23 This would be 1 over 2m because we're putting an m squared through the equation multiplied by p final squared...

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