A beam of 40-eV electrons traveling in the +x- direction...

A beam of $40-\mathrm{eV}$ electrons traveling in the $+x-$ direction passes through a slit that is parallel to the $y$ -axis and 5.0$\mu \mathrm{m}$ wide. The diffraction pattern is recorded on a screen 2.5 $\mathrm{m}$ from the slit. (a) What is the de Broglie wavelength of the electrons? (b) How much time does it take the electrons to travel from the slit to the screen? (c) Use the width of the central diffraction pattern to calculate the uncertainty in the $y$ -component of momentum of an electron just after it has passed through the slit. (d) Use the result of part (c) and the Heisenberg uncertainty principle (Eq. 39.29 for $y$ ) to estimate the minimum uncertainty in the $y$ -coordinate of an electron just after it has passed through the slit. Compare your result to the width of the slit.

A beam of $40-\mathrm{eV}$ electrons traveling in the $+x-$ direction passes through a slit that is parallel to the $y$ -axis and 5.0$\mu \mathrm{m}$ wide. The diffraction pattern is recorded on a screen 2.5 $\mathrm{m}$ from the slit. (a) What is the de Broglie wavelength of the electrons? (b) How much time does it take the electrons to travel from the slit to the screen? (c) Use the width of the central diffraction pattern to calculate the uncertainty in the $y$ -component of momentum of an electron just after it has passed through the slit. (d) Use the result of part (c) and the Heisenberg uncertainty principle
(Eq. 39.29 for $y$ ) to estimate the minimum uncertainty in the $y$ -coordinate of an electron just after it has passed through the slit. Compare your result to the width of the slit.

Key Concepts

de Broglie Wavelength

This concept connects the wave-like properties of particles to their momentum by the relation ? = h/p, where h is Planck’s constant and p is the momentum. It is a foundational idea in quantum mechanics that illustrates how matter can exhibit interference and diffraction similar to classical waves.

Electron Diffraction

Electron diffraction is the phenomenon where electrons, when forced through a narrow opening like a slit, interfere with themselves to produce a diffraction pattern on a distant screen. This provides direct evidence of the wave-particle duality of matter and is used to study both microscopic structure and fundamental quantum behavior.

Time-of-Flight Calculation

In the context of quantum experiments, time-of-flight refers to the time it takes a particle to travel a known distance. It is determined using basic kinematics and helps in linking the particle’s kinetic energy with its observed spatial behavior on a detection screen.

Heisenberg Uncertainty Principle

The Heisenberg uncertainty principle states that certain pairs of physical properties, such as position and momentum, cannot be simultaneously measured to arbitrary precision. When an electron passes through a narrow slit, its position is better defined, which, by this principle, leads to an increased uncertainty in its momentum, particularly in the direction perpendicular to the slit.

Transcript

0:00 Okay.

00:02 And this problem.

00:02 We have an electron beam with electron energies of forty evey that passes through a single split with the width of five times in the night of six meters.

00:12 So my commuters and it travels beyond the slit two point five meters onto a screen increase.

00:18 It creates an interference pattern.

00:22 And because these electrons have wavelike properties with a too broadly a wavelength, they were going to make this interference pattern in the same way that light would so our first act find what this wavelength is for these electrons.

00:36 Where then asked, what's the time it takes for the electrons to travel to the screen? and we're finally asked for the uncertainty in the uae directed momentum of these electrons.

00:47 So start off we'll need to know the definition whether burghley wavelengths basic definitions for energy and momentum.

00:53 The single slit interference equation for a note for the first minimum.

01:01 Uh, the and constant is one that's so actuacion precipitously include that for our case and is going to you one.

01:10 And of course, we need heisenberg's uncertainty principle to relate position, a uncertainty in position and uncertainty, momentum to the point constant for constance.

01:20 We use the electron mass and reduce place constant.

01:24 Let's get started.

01:26 So it's all right.

01:27 Let's get this.

01:27 Ah, wavelength of a elektron.

01:33 So it's h over the momentum.

01:36 We can express a momentum.

01:39 Let's go off to the side.

01:40 So if momentum is envy and okay, the connect energy is one f n b squared.

01:51 We can combine these two expressions to say that kay is equal to oh, momentum squared over two am so we know the connect energy, these electrons.

02:07 It's for forty m e v.

02:11 Thus we can throw this in for momentum eliminated from the equation and express the wavelength in terms of energy.

02:19 Only ana being route two times m.

02:25 I'm scared and playing in the plant...

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