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The following 6 questions pertain to this figure and associated concepts. It shows a quantum particle in a box (the box is defined by the red boundary): This figure...

Question

The following 6 questions pertain to this figure and associated concepts. It shows a quantum particle in a box (the box is defined by the red boundary): This figure shows the situation at t = 0, when there is a probability of exactly 1.0 (100%) that the particle is in the left half of the boxThe box has potential energy "barrier" running vertically down its middle (represented by the white vertical line): Using Schrodinger s Equation the time-evolution of this situation can be calculat

The following 6 questions pertain to this figure and associated concepts. It shows a quantum particle in a box (the box is defined by the red boundary): This figure shows the situation at t = 0, when there is a probability of exactly 1.0 (100%) that the particle is in the left half of the box The box has potential energy "barrier" running vertically down its middle (represented by the white vertical line): Using Schrodinger s Equation the time-evolution of this situation can be calculated and examined.



Answers

A quantum particle of mass $m$ is placed in a one- dimensional box of length $L$ . What If? Assume the box is so small that the particle's motion is relativistic and $K=$ $p^{2} / 2 m$ is not valid. (a) Derive an expression for the kinetic energy levels of the particle. (b) Assume the particle is an electron in a box of length $L=1.00 \times 10^{-12} \mathrm{m} .$Find its lowest possible kinetic energy. By what percent is
the nonrelativistic equation in error? Suggestion: See Equation 39.23

We start with the relativistic equation for the total energy of a particle, and this is given by the square root of P. C. Squared, plus Tennessee squared squared where he is the momentum of the particle and M is the mass of the particle at rest when it's at rest. And we know that we can write the kinetic energy of the particle, the relativistic kinetic energy as the total energy minus the rest energy of the particle, which is the rest mass of the particle times the speed of light squared. So this basically is just the difference between the total energy and the rest energy. We get kinetic energy now proceeding. We can take the mo mentum of a particle in a box and substituted into the equation for the relativistic kinetic energy. And we get the following expression. So here we see that we substituted. So now we have a term inside the square root that depends on the quantum number as well as the dimension of the box. And it also has Planck's constant the speed of light. So the length of the box that we have here is one times 10 to the minus 12 meters. We're dealing with an electron. So we have a massive 9.11 times 10 to minus 31 kilograms. And so we'll go ahead and we will put in for the case of in equals. One will do a calculation for the what would be the relativistic kinetic energy. So we go ahead and we substitute in our numbers and we get kind of ah, somewhat lengthy calculation. Ah, just in terms of the number of Constance that we have. So that's for the all the quantities under the square root. Then we subtract the mass of the electron times, the square of the speed of light. So you put this in probably into a calculator, and the result you obtain is now the square root of 1.6601 times 10 to the minus 26 times 10 to the minus 26 minus 8.199 times 10 to the minus 11 or minus 14. And this So, of course, the square root the quantity under the square root is this first entire quantity under square root. In this 8.11 or 8.199 times in the minus 14 is this quantity. That's the mass of the electron Times Square, this beetle like Then you have Jane Finally, 4.68 times 10 to the minus 11 or rather, 14 jewels. So we get this for the relativistic kinetic energy. Now we look at the non relativistic energy that we would obtain, and we make a comparison. So we used the equation for the energy of a particle in a box for the case of in equal to one. And that's just eight squared equals eight or h squared over a m l squared and we substitute in our Constance and we do the calculation. We actually get a larger value for the energy in the non relativistic case. So we want to do a percent error calculation. Just make a comparison between the relativistic and non relativistic energies, and we see that the non relativistic energy is actually 28.6% higher than the relativistic, So the non relativistic is actually two large by 29%

The given problem. We have an electron. It's trapped in a one dimensional, infinite potential. The electron is present in the Crown Street. It sets and equal to want. Suppose the crown set image is not equal to zero. You should not right at the base, so that's wrong. So that's five taken DeSisto base, and this is the pumps and energy. And it was one. We need to find out the probability off finding the electron at certain locations. So let's take the 1st 1 gate between X equal to cereal and ex Equilar to 25. In this case, let's say this is cereal and this is hell. And then time Daniel your things. So you have excess. Our video with probably density is defined as Mark's Square off the way function anarchists for the electron in a one dimensional potential. Well, the wave function is Cuban, as who do over signed empire. Excellent. Since N is equal to one, I can simply say the big function is so you fix equal to two over a signed by Excellent l'd let's try to find out the property of the first case, So this is a net probability should be equal to be need to integrate this. So we integrate from Syria to point to fight off l was Cy Square, which is the square of the function that is nothing but signed square and I x over eight e x. We have to integrate because, as we see, the problem is dependent upon the variable X Because size dependent upon the variable X, the real function is dependent on very bullets. So in order to find the net probability between 0.25 we have to integrate. It's no. We can either even liberated easily or we can try to make a substitution. So let's make a substitution off end by X k. You just say that this is equal to buy in anarchist and is equal to one. So I like this is I eggs Or in this case, then the Y should be equal to this to a constant. So it will simply be by you held off the X so d X is equal to tell Fight Do you so in the girl now becomes too over. Dale, this d x p up. It's absurd for DX as a lullaby. So have L. A wanted by A by the great was CEO. Because if I substitute for X is equal to zero, why is also CEO? But that's absurd for X is equal to point to file, which is any over four. I can't Why as equal by or and Insane Daggett Same square. What delight Begin integrity this on, then the value that we can this point all nine It is with the values to find. Now, similarly, we can do the calculation for the next part which request from 0.75 So this is between on seven by and X equals. But before we start doing all the calculations, let us realize that open 752 x is equal to well, isn't this similar to x equal toe Fine to fire because X is equal to zero and exit this case is a lower four. So that means it's between this burnt and one over. For this is half. This is wonderful. So this region is a And if I look at B which is spawned seven file. This is three or four. So this is half. Then this should be three or four and this is three or four l and L. So these are entirely symmetrical. And because of a function is signed and signed a symmetry function that case the probability that you have between Sue and I know a lawful should be same as probably between three and lower for and l So he should directly died that this value is also equal to zero nine. Here we have argued solely on the basis of symmetry. For the last section, we have X as L o r four do x it just see work for help again, we do not need to make any calculations. This has been a wonderful and three years ago for, uh, it is a little four and three for all of this region. We already know that the electron will either be between zero. It has to be between zero and L. So the net probability of finding electron between zero and L A is equal to one. You will always find electron somewhere between zero and one. The probability of finding electron between zero and l by four a little forthis 109 and the probability of finding electron between see a lower floor and L is also 0.9 If I had the Stroop abilities, I can't 0.1 hit. And and I know that the overall probability of finding the lectern between zero and l should be one. So the probability of finding electron between L. A war four and three yellow for which is the remaining, should be equal to one minus that some of these two, as you can see from the Magnum so I can simply say that this probability should be equal to one. Linus elected separately 0.9 l linus buying CDO nine and partisan. That's equal to plane it who head. So again, we have made use of symmetry to prove this be made use of symmetry and broke this one be made use up the fact that the electron should be there somewhere between zero and also the net. Probably this one justice on the program. You're finding it in the remaining vision should be the subtraction off one, minus the other probabilities. I should say it is

Okay. Eso first of all, the probability on finding a quantum particle is described by Let's put in here the sigh, which is the way function is square. So side effects is the way function that describes the state of a particle. And the probability of finding this particle in the given position is given by Sigh Square. Okay, so what is wrong with this reasoning? It's This person is thinking that the behavior or quantum particles is like the behavior or classical mechanical particles. So he's thinking that the particle can cross the meat point. So I have here, This is the X. This is the site, the axe. Okay. And this is something like this. So the meat point here is zero. But the particle is not moving. It's not necessarily moving pro left to right. Okay, In quantum mechanics doesn't make sense talking about movement of a particle before moving this particle. So if the probability on finding this is the probability the probability of finding this particle is zero at the point is not coming that she can cross this section justice. This only means that we can never see the particle a disposition. But we only Reus will know what is the position of the particle when we measure. Therefore, we need experiment first to see this. So what is wrong? Where the following reason is thinking that quantum mechanical behavior is the same of the classical mechanical behavior. The critical is not moving from the left to site. Okay, Radical exists in a superposition off this way function. And we only know what is the position in the rial road. When we measure this particle, that's the answer. Thanks for watching.

UNESCO. Sure, we're looking every particle in a box, but, uh, a book say so. Small debts the emotion off the particle. It's no relativistic, and therefore we cannot use P square, But, um, that's our kinetic energy, but rather be after you see full relativistic former Davis using E sa p square scrap. Plus, I'm score. See full and subjecting this energy 50 rests. Energy do que fusty kinetic energy. Right? So in order to find he taught energy, however, we will need to find what is Pete. And we can derive p actually using T Deepu, please wait. Flick. Right. So PC goes toe ish number, But where do we get the number? So the Lunda comes from our boundary conditions, where the wave function has to be a standing wave, right? A full standing wave inside the boundaries off the box, right way. It starts and ends with zero. Right? So they can only be a fixed number off way flings within this, uh, this book, it can't be either 1/2 or full wavelength. Well, you can't be just half wavelength and can see this actually and into your multiple off halfway flings. Right? So you could be half can be fully frank. Can't be true with two heavy to so and so far. So this tells us that the full length off the box must be equals two It multiple off huff reflex looks to end over two time slander. Using this condition you can substitute are Lunda IHS to wear over. All right, so there is a condition for any part. Equitable's right. We call us with a small big. So using this, we can substitute back in into the expression for the total energy so they see tort energy and therefore, or kinetic energy expression Just be t total energy minus away The rest energy now to find the actual kinetic energy for some for an electron in a box with a certain length 10 actually one pickle meters. So tell ministry you wanna find what I see. Little wistful school cannot to energy. So the lowest possible would be when NBC close to one and it goes to one to be the lowest. So we had to substitute in any coastal one for this equation. Get close to so if there is a substitute for em, mess off the electron safety speed off light l The length of the boxes. Tent pole. Ministry off. Just ich Steve. Thanks, Constant. Right. So we could just put his FedEx in. This is all the various that you need. And we should be able to get to kinetic energy to be about four point 68 Stenholm minutes 14 with yours. No, I want to compare this with a non relativistic equation. And the non relativity equation will be the one day Steve arrived using piece go over to em, Right. The total energy being each square over it and it'll squid. This is from the typical particle in a box formula for the total energy, which is C care to energy. And, uh, if the calculate is putting in 6.626 and the l is also 10 apartments shelf, we will actually get around six points through to who said for minus 14 and comparatives to together. All right, if it takes explains your to this 4.68 actually see that this is about 28.6% larger, then the actual calculation using the relativistic equations. So it be overestimating the kinetic energy. If we didn't use the relativistic calculations


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