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2 (18pts). An electron in a hydrogen atom can be considered as if it is confined in a region of 0.1 nm (size of the atom):(a.6pts) Use the uncertainty principle (wi...

Question

2 (18pts). An electron in a hydrogen atom can be considered as if it is confined in a region of 0.1 nm (size of the atom):(a.6pts) Use the uncertainty principle (with h ) to show that it cannot be exactly at rest and has minimum non-zero kinetic energy. Compare this energy with the ground state energy of the hydrogen atom: (b.6pts) Calculate the uncertainty in the orbital speed of the electron(c.6pts) Calculate the orbital speed V, of the electron in the ground state using the quantization of an

2 (18pts). An electron in a hydrogen atom can be considered as if it is confined in a region of 0.1 nm (size of the atom): (a.6pts) Use the uncertainty principle (with h ) to show that it cannot be exactly at rest and has minimum non-zero kinetic energy. Compare this energy with the ground state energy of the hydrogen atom: (b.6pts) Calculate the uncertainty in the orbital speed of the electron (c.6pts) Calculate the orbital speed V, of the electron in the ground state using the quantization of angular momentum. Compare this value with the result of part (b) by calculating



Answers

The energy of an electron mass $m$ charge $e$ circling a proton at radius $r$ is $$ E=\frac{p^{2}}{2 m}-\frac{e^{2}}{4 \pi \varepsilon_{0} r} $$ where $p$ is its momentum. Use Heisenberg's Uncertainty Principle in the form $\Delta p \Delta r \approx \hbar$ to show that the minimum energy $\left(\mathrm{H}_{2}\right.$ atom ground state) is $$ E_{0}=\frac{-m e^{4}}{8 \varepsilon_{0}^{2} h^{2}} $$ at a Bohr radius $$ r=\frac{\varepsilon_{0} h^{2}}{\pi m e^{2}} $$

Considering a hydrogen atom in the ground state. We want to find the orbital speed. So the radius that I'm gonna be using our is bores radius its 5.29 times 10 to the minus 11 meters. So due to the orbit of the electron around the adamant for assuming boards model than the centripetal force is going to be equal to the electrostatic force that's trying to pull the Adam and the nucleus of the atom in the electron together versus this in triple Force was just keeping it apart. If the electron isn't collapsing in on the Adam, these two must be equal. So what we're gonna do is we're gonna set F sub sea, which is the centripetal force equal to FCB the electrostatic force. So, doing this, this gives us 1/4 Pi Absalon, not well supplied by the charge of the electron e squared over r squared is equal to MV Squared, where V is the speed that we want to find. And M is the mass of the electron. Since that's what's going around the Adam divided by R so we can cancel out one of the ours on the left side of the square goes away with this are in the denominator on the right side, they're rearranging this equation to solve. For the we find that V is equal to the square root. Since its squared, we have to square root e squared, divided by four pi Absolute not times the mass of the electron which were using kilograms here. So this is 9.11 times 10 to the minus 31 kilograms multiplied by are the board Brady's playing these values in. We find that the speed is equal to 2.19 times 10 to the six meters per second making box that it is our solution for a part. B wants us to find the kinetic energy of this electron. So kinetic energy which we're gonna call K e is of course, equal to 1/2 MV squared, where m is the mass of the electron once again and kilograms and the speed is the speed that we found in part a so plugging these values. And we find that the kinetic energy is equal to 2.18 times 10 to the minus 18 jewels. And since the energy of Adams is normally written an electron volts. We can convert this from jewels to electron volts, and this gives us 13.6 electron volts, which just happens to be the ground state of the hydrogen atoms. That's a nice little check that we did this correctly so we can box both of those and as the solution for Part B and then lastly, Parsi says, calculate the potential energy Well, I'm going to do the electrostatic potential energy to calculate the potential energy, and I find that using this equation this is equal to Negative K, where K is just the electrostatic constant 1/4 pi absolute not Times e squared over our again. He's the charge of the electron in our is the Boers. Radius K is equal to 8.99 times 10 of the nine Newton meters per Coolum playing these values, and we find that the potential energy is equal to negative 27.2 electron volts making boxes and is the solution for Parsi

We have to consider an electron in the ground state of the hydrogen orbit and it is separated from the proton by a distance small eight equals to 0.5299 m. Okay, so for the part A we have to calculate the speed of the electron in this orbit. So since there is only electric force acting so we can write that electric force, it will be equals two. This fc. So electric forces K E squared divided by a square and it will be equals two M electron mass of electron multiplied by the square. There will be a. So from here we get V equals two under root of K. E squared divided by M. E. Minor player by so substituting values so we get B equals two. And the root of Okay. Which is 19 to 10 to the power nine but cleared by charge on proton or electron. 1.602 into 10. To the power minus 19. Holy square They were by masson electron 9.11 into 10. To the power minus 31 kg and A. Which is 0.5 to nine. Molecular weight 10 to the power minus night. So from here, after solving we get speed. V equals two 2.1 nine 2.19 into 10. To the power six m per second. Okay, so this becomes the answer for the part eight. Okay, now moving to the part B in which we have to calculate the effective escape speed for the electron so we can write that if elected escapes the orbit, it needs in afghan ethnic energy to counter its potential energy so we can write kato will be equals to you. So this will be equals 21 by two massive electronic Larbi escapee Spirit square. This become K E squared whereby a. So from here we get escape speed. We equals two K E square divided by mass of electron multiplier by distance. A. So this is equal to understood to manipulate by visa spill, which is obtained in the part. So escapist till we comes out to be under tumor player by we which is 2.19 to 10 to the power six m per second. So from here we get escapist pill E equals to 3.9 multiplied by 10 to the power six m per second. So this becomes the answer for the part. Okay now moving to the party, moving to the part C. In which we have to calculate the energy of the left own having this speed. Okay, so the additional energy delta. It will be equal to change in kinetic energy. The phase equals two K final minus K. Initial and final is equal to the energy of the escape. So one by two. M. E. Square. We'ii square minus one by two M. E. Multiplied by the square. Okay, so since we have we equals two. Since we have V equals two under two times a week. So we get we ve square equals to two. We square. So substituting this value here. So we will get delta equals 21 by two M. E. This can be taken out so we eat, this will be equal to the square minus we square. So we get one by two. M. A medical herb. We square. So now substituting values. So we get delta equals 21 by two. Molecular by uh multiplied by M. E. Which is 9.11 into 10. To the power minus 31 kg. And we which is equal to 2.19 into 10. To the power six m per second. Holy Square. So from here Delta, he comes out to be 13.6 electron world. Okay, so this will be the answer for this question. Okay. This value will come out into jewels. So by dividing 1 16 to 10 to the power minus 19, we will get answer into the electoral vote. Okay?

For this problem. On the topic of the shooting the equation We have an electron in an infinite square. Well with l equal to 10 to the -12 m. And it is moving at relativistic speed. And so we want to use the uncertainty principle to verify that the speed is firstly relativistic. Then did I have an expression for the allowed energy levels for the electron And then compute the one and then finally find the fraction by which the one differs from the non relativistic value for everyone. So we know that the uncertainty delta P bye dot X is approximately H bar which means that um dot tv by data X is approximately H bar and delta V is approximately H bar divided by M does A. Which is 1.055 Times 10 to the -34 jewel seconds Over 9.11 Times 10 to the minus 31 kg time. Start A which is 10 to the minus 12 m. And so we get out to be to be 1.6 times 10 to the power eight meters per second, Which is 0.39 c. Which is a speed close to or comparable to the speed of light. And hence this is a domestic now for part B. The weight of the world Allison into the number of half wave lands. So al is equal to and lambda over to. And the broccoli celebration gives our equal to NH over two p But he is not given by the squares of two M. E. But by the relativistic equation P is equal to e squared minus M C squid all squared all to the power half divided by C. And then if we substitute we get L two B N H C divided by to into e squared minus and C squared all squared all to the power half. And so from here we get that e squared minus EMC squared all squared is equal to NHC divided by to l squid. And so we get the allowed energies Eaton to be N edge see over to L all squared plus and see squid. Yeah, yeah, squid all to the power 1/2. And so we have the allowed energies for part C. We can now calculate he won. Anyone is getting anybody to what in the equation above. Hc over for L squared all squared plus MC 2aredared all to the power half. And so we're putting in our values, we get this to be 1240 electron volt nanometers squared divided by for times 10 to the minus three nanometers Squared plus zero 511 Times 10 to the power six electron walls squared all to the power half. And so calculating we get E one to be 8.03 Times 10 to the power of five electron volts. Now for party We want to find the non relativistic E one and this the one is eight squared over heat AM L squared which is h see all squared divided by eight into M C squid times al squid. Again, putting in values this is 1240 electron volts electra involved nanometers squared, Divided by eight into 0 511 Times 10 to the power six electron volts times 10 to the minus three nanometers. All of this squid. And so calculating, we get the Hundreds artistic value to be 3.76 Times 10 to the power five electron volts. Mhm.

Already. So first in our equation as equals H C or H grade. Whoever ate pie serious is minus. There were a pie and r squared and e a going to and square. So we want all of this. Do you become minus concert? So there are a creation is then e equals minus constant times one over and agree. And we actually knew all these like, this is Plank's constant. This is a price. Great, Aaron. And, you know, I am of you in a row just looking up your tax book. So these are our constant. So we go ahead and plug all these in. Then our new constant is 1.16 hires into the my stray sex drawers. So now we want to know what? Yes, where any close to you. So it's gonna be negative or employment. 16 Well, actually bounced it, okay. And because so the negative ends of actually canceling out see, you don't end up using it. So, uh, this ends up being even one is We're gonna call it 1.16 tires into the minus 26 times wind over two squared and said, our is 7.25 tires tend to the last 28 rules and then every day her and equals three. Then we get one point you ate. Time is 10 to the minus 27 rules. It's now another change in energy, which is gonna be you, too, minus E one. So that's this number minus this number. And so that ends up being 5.5 tires, 10 to the minus 28 jewels. It's not for want wavelength. We know labeling is HC over the changing heat. And so now we know everything, So you literally just plug it in a just plank's constant. She is to be the light in a vacuum and that you just from So you multiply all that out, you end up getting away links of 361.3 meters. So this is your


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