A particle is described by the wave function ψ ( x ) = { c e x / L x ≤ 0 mm c e − x / L x ≥ 0 mm \psi (x)=\begin{cases} ce^{x/L} & x\leq 0\text{ mm} \\ ce^{-x/L} & x\geq 0\text{ mm} \end{cases} where L = 2.0 mm. Sketch graphs of both the wave function and the probability density as functions of x.

Hello everyone. Let's go through this practice problem. The wave function of the hydrogen ground state is given by SI of R equals one divided by the square root of pi A knot cube all multiplied by E raised to the power of negative R divided by a knot where si of R represents the wave function. A knot is the bore radius and R is the distance from the proton plot. The wave function and the probability density as functions of R divided by a knot. OK. So this problem is pretty simple as long as you have a graphing calculator. So for the first plot we're plotting si of R and so a Y axis is one divided by the square root of pi A knot cubed. And our horizontal axis is R divided by a knot. So we're writing this plot as a function of R divided by R knot by a knot. Which means that if we look at the, at the SI equation, we were given, all we have to do is put this into a calculator. But sh R negative R divided by a knot as the, as the variable as the X as the independent variable of the equation that we're varying with the graphing calculator. And in doing that, what we find is that there is a peak on the Y axis at 1.0. But from there, the function goes down exponentially approaching the X axis. So at a point of one, we get pretty low and two, we get a little lower and then so on and so forth until it starts approaching the horizontal axis. So that's it for that plot. But the problem also asks us to find a plot of the probability density which is basically just sigh of R squared. So all you got to do for this one is take the function that you just put into your graphing calculator and then put in the squared version. But what that's going to accomplish, so we can see pi a knot cub. Now there's no square out here, we can see what that does is all it's gonna do since we've got decimals on the Y axis is it's going to look basically the same except it's going to travel downwards and much more steeply. So the plot travels downwards much more quickly and approaches and asymptotically approaches the horizontal axis much more quickly, just like that. So if the numbers are all the same, if we have the axes labeled the exact same way, this is what it'll look like. And really that is it for this problem, I hope this video helped you out and please check out some of 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

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35. Special Relativity

Inertial Reference Frames

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Problem 38a

Textbook Question

A particle is described by the wave function $ψ (x) = {\begin{matrix} c e^{x / L} & x \leq 0 mm \\ c e^{- x / L} & x \geq 0 mm \end{matrix}$ where L = 2.0 mm. Sketch graphs of both the wave function and the probability density as functions of x.

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Understand the given wave function: The wave function ψ(x) is piecewise-defined, with an exponential form that depends on whether x is less than or greater than 0. The constant c is a normalization constant, and L is given as 2.0 mm. The wave function is ψ(x) = c * e^(x/L) for x ≤ 0 and ψ(x) = c * e^(-x/L) for x ≥ 0.

Normalize the wave function: To ensure the total probability is 1, integrate the square of the wave function over all space and solve for c. The normalization condition is ∫|ψ(x)|² dx = 1, which becomes ∫[c² * e^(2x/L)] dx from -∞ to 0 + ∫[c² * e^(-2x/L)] dx from 0 to ∞ = 1.

Calculate the probability density: The probability density is given by |ψ(x)|². For x ≤ 0, |ψ(x)|² = c² * e^(2x/L), and for x ≥ 0, |ψ(x)|² = c² * e^(-2x/L). These expressions describe how the probability of finding the particle varies with position x.

Sketch the wave function ψ(x): Plot ψ(x) as a function of x. For x ≤ 0, the wave function increases exponentially as x approaches 0, and for x ≥ 0, it decreases exponentially as x moves away from 0. The graph should be continuous at x = 0.

Sketch the probability density |ψ(x)|²: Plot |ψ(x)|² as a function of x. For x ≤ 0, the probability density decreases exponentially as x becomes more negative, and for x ≥ 0, it decreases exponentially as x becomes more positive. The graph is symmetric about x = 0 and peaks at x = 0.

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Key Concepts

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

Wave Function

The wave function, denoted as ψ(x), is a fundamental concept in quantum mechanics that describes the quantum state of a particle. It contains all the information about the particle's position and momentum. The square of the wave function's absolute value, |ψ(x)|², gives the probability density of finding the particle at a specific position x.

Probability Density

Probability density is derived from the wave function and represents the likelihood of locating a particle within a given region of space. For a one-dimensional wave function, the probability density is calculated as |ψ(x)|². This concept is crucial for interpreting quantum mechanics, as it allows us to predict where a particle is likely to be found upon measurement.

Exponential Decay

Exponential decay is a mathematical function that describes how a quantity decreases over time or space at a rate proportional to its current value. In the context of the given wave function, the terms eˣ/ᴸ and e−ˣ/ᴸ illustrate how the wave function behaves differently in the regions x ≤ 0 mm and x ≥ 0 mm, leading to distinct probability densities that reflect the particle's confinement and behavior.

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A particle is described by the wave function $ψ (x) = {\begin{matrix} c e^{x / L} & x \leq 0 mm \\ c e^{- x / L} & x \geq 0 mm \end{matrix}$ mm where L = 2.0 mm. Determine the normalization constant c.

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A particle is described by the wave function ψ(x)={cex/Lx≤0 mmce−x/Lx≥0 mm\psi (x)=\begin{cases} ce^{x/L} & x\leq 0\text{ mm} \\ ce^{-x/L} & x\geq 0\text{ mm} \end{cases} where L = 2.0 mm. Sketch graphs of both the wave function and the probability density as functions of x.

Key Concepts

Wave Function

Probability Density

Exponential Decay

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A particle is described by the wave function $ψ (x) = {\begin{matrix} c e^{x / L} & x \leq 0 mm \\ c e^{- x / L} & x \geq 0 mm \end{matrix}$ where L = 2.0 mm. Sketch graphs of both the wave function and the probability density as functions of x.