Consider a wave function given by ψ ( x ) = A s i n k x ψ(x) = A sinkx , where k = 2 π / λ k = 2π/λ and A A is a real constant. (a) For what values of x x is there the highest probability of finding the particle described by this wave function? Explain. (b) For which values of x x is the probability zero? Explain.

Hey, everyone. So this problem is dealing with quantum mechanics. Let's see what it's asking us. We have a wave function described by the equation si of Y is equal to B multiplied by the sine of two ky. Part one says using K is equal to two pi over lambda. And given that B is a positive real number, determine the first three nonne values least of Y that have the highest probability of finding the particle that obeys the wave function. And we're asked to justify a reasoning. And then part two, we are asked to determine the first three nonne least values of Y that have zero probability. And again, asked to justify our reasoning. Our multiple choice answers are given below. So what we are working with here is a probability function. So we have our wave function here si of Y is equal to bs of two ky. So we can recall that our probability density function or row is equal to the absolute value of SI squared. And so when we square this equation, we get B squared multiplied by sine squared of two KY. And it tells us that our wave number K is equal to two pi over lambda. So for part one, we are looking for our highest probability and given our sine functions and our trigonometry rules, we know that the s of a given angle theta is the maximum value of sign is one. And that happens when theta is equal to as the max. And that happens when beta is equal to and pie divided by two when N is a positive odd integer. And so for the first, for this first part, we can equate our angle. So two ky two that and pie divided by two. And we'll plug in this two pie over lambda for K and multiplied by Y is equal to N pi divided by two. So the pies here cancel solving for Y we get and lambda divided by eight. So it asks us for the first three. And so the first three NS that are positive and odd integers R one N or N equals one. So that would just be lambda over eight, N equals three. So three lambda divided by eight and then lambda or sorry, N equals five. So five lambda divided by eight. And so these, this is our answer for the first part of the problem. And again, our justification is because this two Ky is equal to is high. The our probability is highest when two Ky is equal to N pi divided by two. And so looking at our multiple choice answers, the only one that is correct is for part one is answer choice B, but we're going to solve through part two as well to make sure that we are on the right track. So for part two, very similar um thought process, but we want the probability to be the lowest or the probability of zero. And so when the sign of theta is equal to zero, that is when data is equal to and pi where N is any integer including zero. And so again, we will set our two ky, which we rewrite as two multiplied by two pi multiplied by Y all divided by lambda to this N pi and the pies cancel solving for why we get Y is equal to and lambda divided by four. And so any integer including zero looking for our first three. So we'll have and equal to zero so that Y will equal zero when N equals one, Y will equal lambda divided by four. And when N equals two, Y will equal lambda divided by two. And so that is the answer to the second part of our question again, with the justification that it, that the equation goes to zero when theta is equal to N pi N multiplied by pi and that two ky is equal to N multiplied by pi. And so we can see then that yes, that is correct. B is the correct answer for this problem. That's all we have for this one. We'll see you in the next video.

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Problem 5

Textbook Question

Consider a wave function given by $ψ (x) = A s i n k x$ , where $k = 2 π / λ$ and $A$ is a real constant.
(a) For what values of $x$ is there the highest probability of finding the particle described by this wave function? Explain.
(b) For which values of $x$ is the probability zero? Explain.

Verified step by step guidance

Step 1: Understand the wave function ψ(x) = A sin(kx). The probability density of finding the particle is proportional to the square of the wave function, |ψ(x)|². Since ψ(x) involves a sine function, its square will depend on the amplitude of the sine wave.

Step 2: Recall the properties of the sine function. The sine function, sin(kx), oscillates between -1 and 1. The square of the sine function, sin²(kx), will oscillate between 0 and 1. The highest probability corresponds to the maximum value of sin²(kx), which is 1.

Step 3: To find the values of x where the probability is highest, identify where sin²(kx) = 1. This occurs when sin(kx) = ±1. For sin(kx) = ±1, kx must equal (2n+1)π/2, where n is an integer. Substitute k = 2π/λ to express x in terms of λ: x = (2n+1)λ/4.

Step 4: To find the values of x where the probability is zero, identify where sin²(kx) = 0. This occurs when sin(kx) = 0. For sin(kx) = 0, kx must equal nπ, where n is an integer. Substitute k = 2π/λ to express x in terms of λ: x = nλ/2.

Step 5: Summarize the results. The highest probability of finding the particle occurs at x = (2n+1)λ/4, and the probability is zero at x = nλ/2, where n is an integer. These positions correspond to the maxima and nodes of the sine wave, respectively.

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Key Concepts

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

Wave Function

A wave function, denoted as ψ(x), describes the quantum state of a particle in quantum mechanics. It contains all the information about the system and is used to calculate probabilities of finding a particle in a particular state. The square of the absolute value of the wave function, |ψ(x)|², gives the probability density of finding the particle at position x.

Probability Density

Probability density is a measure that describes the likelihood of finding a particle in a specific location in space. For a wave function ψ(x) = A sin(kx), the probability density is given by |ψ(x)|² = A² sin²(kx). The peaks of this function indicate where the particle is most likely to be found, while the zeros indicate locations where the probability of finding the particle is zero.

Nodes and Antinodes

In the context of wave functions, nodes are points where the wave function equals zero, resulting in zero probability of finding the particle. Antinodes, on the other hand, are points where the wave function reaches its maximum value, indicating the highest probability of finding the particle. For the function ψ(x) = A sin(kx), nodes occur at integer multiples of λ/2, while antinodes occur at odd multiples of λ/4.

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Consider a wave function given by ψ(x)=Asinkxψ(x) = A sinkx, where k=2π/λ k = 2π/λ and AA is a real constant.(a) For what values of xx is there the highest probability of finding the particle described by this wave function? Explain.(b) For which values of xx is the probability zero? Explain.

Key Concepts

Wave Function

Probability Density

Nodes and Antinodes

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Consider a wave function given by $ψ (x) = A s i n k x$ , where $k = 2 π / λ$ and $A$ is a real constant.
(a) For what values of $x$ is there the highest probability of finding the particle described by this wave function? Explain.
(b) For which values of $x$ is the probability zero? Explain.