In an interference experiment with electrons, you find the...

In an interference experiment with electrons, you find the most intense fringe is at $x=7.0 \mathrm{cm} .$ There are slightly weaker fringes at $x=6.0$ and $8.0 \mathrm{cm},$ still weaker fringes at $x=4.0$ and $10.0 \mathrm{cm},$ and two very weak fringes at $x=1.0$ and $13.0 \mathrm{cm} .$ No electrons are detected at $x<0 \mathrm{cm}$ or $x>14 \mathrm{cm}$ a. Sketch a graph of $|\psi(x)|^{2}$ for these electrons. b. Sketch a possible graph of $\psi(x)$ c. Are there other possible graphs for $\psi(x) ?$ If so, draw one.

In an interference experiment with electrons, you find the most intense fringe is at $x=7.0 \mathrm{cm} .$ There are slightly weaker fringes at $x=6.0$ and $8.0 \mathrm{cm},$ still weaker fringes at $x=4.0$ and $10.0 \mathrm{cm},$ and two very weak fringes at $x=1.0$ and $13.0 \mathrm{cm} .$ No electrons are detected at $x<0 \mathrm{cm}$ or $x>14 \mathrm{cm}$
a. Sketch a graph of $|\psi(x)|^{2}$ for these electrons.
b. Sketch a possible graph of $\psi(x)$
c. Are there other possible graphs for $\psi(x) ?$ If so, draw one.

Key Concepts

Wavefunction

The wavefunction is a complex mathematical function that fully describes the quantum state of a particle. Its magnitude contains information about the probability of finding the particle in a particular location, while its phase carries information about the oscillatory and interference properties of the state. The fact that the wavefunction can be complex means that different wavefunctions can yield the same probability density when you take the modulus squared.

Probability Density

Probability density is obtained by taking the absolute square of the wavefunction, |?(x)|², which represents the likelihood of observing a particle at position x. In quantum experiments, the interference pattern observed on a detector is a direct manifestation of this probability density, as regions of constructive interference correspond to higher probability densities and vice versa.

Quantum Interference

Quantum interference arises when the wavefunctions corresponding to different paths or states overlap and superpose. This leads to regions of both constructive and destructive interference in the probability density, hence producing an interference pattern. Understanding these interference effects is crucial for explaining the distribution of detection events in experiments with electrons or other quantum particles.

Phase Ambiguity and Superposition

The phase of the wavefunction, which is not directly observable, plays a key role in determining the interference pattern when multiple components of the wavefunction superpose. Different choices of phase can result in different forms for ?(x) even if they yield the same probability density |?(x)|². This inherent freedom means that there are multiple valid wavefunction representations for a given probability density, as long as they differ only by a global or position-dependent phase that does not alter |?(x)|².

Transcript

00:01 So the first thing they tell us is that the most intense fringe or the peak of the way function is at 7 centimeters from here.

00:12 And then we have slightly weaker fringes at 6 and 8.

00:26 And then still weaker fringes at 4 and 10.

00:30 So 4 and 10.

00:36 And then the last ones are very weak.

00:40 So let's put them like this at 1.

00:43 And 13.

00:51 Now, the key here is to assume that there should be, or to realize that there should be some point for which the wave function, and therefore the square of the absolute value of the wave function is zero between each of the peaks.

01:08 So the shape could be something like here goes to zero, goes up, goes to zero, goes up, goes to zero, up, goes to zero.

01:23 And the size could be something like, here goes to zero.

01:24 And the side.

01:24 Same on the other side.

01:37 Okay, this was a.

01:39 Now they want us to draw a possible shape for the side itself, the way function.

01:48 What you need to know or to take into account is the position of the zeros of this first graph because they need to be the same.

02:05 So there's one here, one here, here, here, here, and here.

02:12 And here...

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