A particle is described by the wave function

Question

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. Calculate the probability of finding the particle within 1.0 mm of the origin.

Accepted Answer

Hi everyone. Let's take a look at this practice problem dealing with quantum mechanics. This problem says consider a particle confined along the Y direction and characterized by the following wave function Phi of Y is equal to SI N multiplied by the square root of the quantity of one minus Y squared divided by four millimeters squared. If the absolute value of Y is less than or equal to two millimeters and P si of Y is equal to zero. If the absolute value of Y is greater than or equal to two millimeters, we need to determine the probability of locating the particle within a distance of 0.5 millimeter meters from Y equals zero millimeters. And we're given four possible choices as our answers. Choice A is 37%. Choice B is 52%. Choice C is 59% and choice D is 72%. Now, we need to find the probability of a particle within this distance. So we need to recall a formula for finding the probability of a particle. And that probability which will label as P is gonna be equal to the integral. And here the integral limits are going to be the uh region in which we need to find the probability. So in this case, I have the limits of integration going from negative 0.5 millimeters, 20.5 millimeters that's within a distance of 0.5 millimeters from the origin. And then inside the integral will have the modulus of size squared multiplied by dy. So the modules as I squared here is our probability density. And so when we integrate that over a region that gives us the probability of finding the particle in that region, so we have an equation for si and since it's a real function, I don't need to worry about taking the complex conjugate. So I can go ahead and plug that um function into my equation. So I have P is equal to the integral going from negative 0.5 millimeters to 0.5 millimeters of sin not squared, multiplying the quantity of one minus Y squared divided by four millimeters squared dy. So I noticed that this is an even function inside the integral here. So I can multiply the integral by two and change the limits of integration to go from zero to 0.5 millimeters. We'll have P is equal to two. And here I'm gonna go ahead and pull out the constant which is the sin out squared. And now I'm going to have that multiply in the interval going from zero to 0.5 millimeters of the quantity of one binus Y squared divided by four millimeters squared dy. And here I can go ahead and do the integration. So I'll have P is equal to two multiplied by sin not squared. And this is multiplying the quantity of when I integrate one, I get Y it'll have minus. And when I integrate Y squared, I get Y cubed divided by three. So I'll have three multiplied by four in the denominator. So it'll give me 12 millimeters squared in the denominator. This is going to be evaluated at the limits of zero and 0.5 millimeters. Now, at this point, I noticed that I do not know what the CNO value is. So that's gonna be my normalization constant. And so I need to do a different calculation to find that. So recall our normalization condition and that is the integral going from minus infinity to infinity of the modules of size squared dy is going to be equal to one. And what this is basically saying is that somewhere in space, the particle has to be located. And so if I integrate over all space, the probability of finding the particle is one. So I can go ahead and plug in uh my side value just like I did before. But here my limits of integration are gonna change. So I'm gonna be going from negative two millimeters to two millimeters. That's because we're told that size is equal to zero outside of the two millimeter range radius around the origin. Then inside the integral, I'll have the C knot squared multiplying the quantity of one minus Y squared divided by four millimeters squared that is multiplied by Dy and that is equal to one. We're in the same trick as we did before. We're going to change the limits of integration to go from 0 to 2 millimeters. So I'll have two and I'll pull out the constant, the sin knot squared. This is gonna be multiplied by the integral going from 0 to 2 millimeters of the quantity of one minus Y squared divided by four millimeters squared multiplied by Dy, that's gonna be equal to one. And we'll go ahead and do the integration and we'll get the same result as before. So we'll have two dino squared multiplying the quantity of Y minus Y cubed divided by 12 millimeters squared. This is evaluated at zero and two millimeters and this is gonna be equal to one. So here I can go ahead and plug in my limits of integration. I'll have two is thy not squared multiplying the quantity of two millimeters minus two millimeters cubed divided by 12 millimeters squared. And this is has has to equal to one. So here at this point I can solve for Psy not. So I have, why not? It's going to be equal to. So just divide it over by all the constants and um then take the square root, I'll have the square root of one divided by two, multiplying the quantity of two millimeters minus the quantity of two millimeters in quantity cubed divided by 12 millimeters squared. And this gives me a value for sin not of 0.612. And that has units of millimeters to the negative one half do not have my cy kot value. I can plug it back in to my probability equation and go ahead and plug in the limits of integration there. So I'll have my probability P, it's gonna be equal to two. And here I'll plug in for cy knot multiplying the quantity of 0.612 millimeters to the negative one half. That quantity is squared. This gets multiplied by the quantity of 0.5 millimeters minus the quantity of 0.5 millimeters in quantity CED divided by the 12 millimeters squared. And so I can plug those values into my calculator and get that the probability P is going to be equal to 0.37 just kept two significant figures. But if I look at my answer choices, they need to be in percentages. So I need to multiply this by 100%. And when I do so I get P is equal to 37%. And if I compare that to my answer choices, this corresponds to answer choice. A. So just to recap what we did, we um use our definition for the probability of a particle within that region to try to calculate the probability of the particle within the given region from the problem. But in order to do so, we needed to find the normalization constant or our wave function. And so we just use our normalization condition that says that the particle, the probability of finding the particle in anywhere has to be equal to one. So I hope that this has been useful and I'll see you in the next video.

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. Calculate the probability of finding the particle within 1.0 mm of the origin.

Key Concepts

Wave Function

Probability Density

Normalization of the Wave Function

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