FIGURE EX39.14 is a graph of |ψ(x)| 2 for an electron. What is the value of a?

Hello, let's go through this practice problem. The probability density of finding a particle at position X is shown below, determine the value of Phi not. Then we're given this graph and we have four multiple choice options. Option, a one half inverse millimeters, option B one third, inverse millimeters, C one quarter, inverse millimeters and D 1/6 inverse millimeters. Now, luckily for us, this problem is quite a bit simpler than it looks, we're talking about probability density, which usually means that we'll have to do some kind of integration over the area under the under the curve because that's going to be equal to one, the total probability of wherever it's going to be found. But in this case, we have a very simple square shaped graph. So really the only thing we need to consider is the fact that the normal, the normalization condition I mentioned it represents the area under the curve. So the area under the curve which has a height of C and a width of from negative three to positive three. So a width of six that is equal to one, the area under the curve C times six is equal to one. So all we got to do is solve for C by dividing both sides of the equation by six. And that is 1/6. So 1/6 is the answer to this problem which you'll notice corresponds to option D. So option D is the correct answer to this problem. And that is really it for the problem. I hope this video helped you out and please consider checking out our other tutoring videos which will give you more experience with these types of problems. That's all for now. And I hope you all have a lovely day. Bye bye.

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35. Special Relativity

Inertial Reference Frames

Physics

35. Special Relativity

Inertial Reference Frames

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Problem 14a

Textbook Question

FIGURE EX39.14 is a graph of |ψ(x)|² for an electron. What is the value of a?

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Understand the problem: The graph of |ψ(x)|² represents the probability density function of the electron's position. The value of 'a' is likely a parameter related to the spatial extent or characteristic length scale of the wavefunction. This could be inferred from the graph's features, such as its width or periodicity.

Analyze the graph: Examine the graph of |ψ(x)|² to identify key features such as the distance between peaks (if periodic), the width of the central peak, or any other characteristic length scale. These features are often related to 'a' in quantum mechanics problems.

Relate the graph to the wavefunction: Recall that |ψ(x)|² is the square of the wavefunction ψ(x). If the wavefunction is given by a specific form (e.g., a Gaussian or sinusoidal function), 'a' might represent a parameter such as the standard deviation of a Gaussian or the wavelength of a sinusoidal function.

Use the mathematical relationship: If the wavefunction is known or can be inferred, write down its mathematical form. For example, if ψ(x) = A * exp(-x² / (2a²)), then 'a' is the parameter that determines the width of the Gaussian. Alternatively, if ψ(x) = A * sin(πx / a), then 'a' is related to the wavelength or periodicity.

Determine 'a' from the graph: Measure the relevant feature from the graph (e.g., the full width at half maximum for a Gaussian or the distance between peaks for a sinusoidal function). Use this measurement in the mathematical relationship to solve for 'a'.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Wave Function (ψ)

The wave function, denoted as ψ, is a fundamental concept in quantum mechanics that describes the quantum state of a particle. It contains all the information about the system and is a complex-valued function of position and time. The square of the absolute value of the wave function, |ψ(x)|^2, represents the probability density of finding the particle at a given position x.

Probability Density

Probability density is a measure used in quantum mechanics to determine the likelihood of finding a particle in a specific region of space. It is derived from the wave function, where |ψ(x)|^2 gives the probability per unit length for a one-dimensional system. Understanding probability density is crucial for interpreting the results of quantum experiments and predicting particle behavior.

Normalization Condition

The normalization condition is a requirement in quantum mechanics that ensures the total probability of finding a particle in all space equals one. Mathematically, this is expressed as the integral of |ψ(x)|^2 over all space being equal to one. This condition is essential for the physical interpretation of the wave function and ensures that the probabilities derived from it are meaningful.

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FIGURE EX39.14 is a graph of |ψ(x)|2 for an electron. What is the value of a?

Key Concepts

Wave Function (ψ)

Probability Density

Normalization Condition

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FIGURE EX39.14 is a graph of |ψ(x)|² for an electron. What is the value of a?