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To desceloa (ha 54 Marure 08 rzaled, de Broglie proposed Ihxt h Epnoion"Iv Awnew{aanf J the cquuton h dUroi utrEyncron" I?...

Question

To desceloa (ha 54 Marure 08 rzaled, de Broglie proposed Ihxt h Epnoion"Iv Awnew{aanf J the cquuton h dUroi utrEyncron" I?

To desceloa (ha 54 Marure 08 rzaled, de Broglie proposed Ihxt h Epnoion"Iv Awnew {aanf J the cquuton h dUroi utr Eyncron" I?



Answers

Which is the de-Broglie equation: (a) $h=p \lambda$ (b) $h=p \lambda^{-1}$ (c) $h=\lambda p^{-1}$ (d) $h=p+\lambda$

In this exercise we have a massive particle that has a kinetic energy ke in a mess in and our goes to find What is the wavelength associated with this particle? A za function off k and M So, first of all, I'm gonna start with the Deborra Lee formula to remember that P is equal to age divided by London H aspirants Constant Lunda is the wave going? I also noticed that for a massive particle, the momentum p can be written as in times V and the kinetic energy for massive particle iss m times B squared divided by two I'm right. That's right V as a function of the mo mentum and the Mets was that v convert nous limited p divided by the mess It's OK is equal to m divided by two times V square. Now these p divided by in So you get that kinetic energy is equal to be swear to read about to win. So basically, if you blood the there really formally into the area that we just found. So instead of pee wee right h Kovar Linda so have ish. Over on the square divided by two m, we get that K is equal to age Swear divided by two n. I love this weird which is looking at Kennedy, the function London's and our goal is to find Linda. So basically, I'm gonna invert this equation. So I have a judge Square is a Maryland, a square. It's equal to May Square to motivate to win. Okay, I just multiply both sides by lime the square and divide both sides by K So this means that Lunda is the square root off Be right inside. So this is aged divided by the square root of two m Kay, This is the answer to question eight in question be we have an electric that has an energy off 800 and actor in votes. Remember that the mass of the electric this is gonna be useful for us is 9.1 time sent to the minus 31 kilograms. And I have to find the wavelength from the This is equal to H. As we found in question A ah which is 6.63 Times Sentinel minus 34. Use second, invited by the square root of two times the mass of the last one. That's 9.1 time stand the minus 31 kg times the kinetic energy, which is 800 I go to have 800 electric goats, and I have to convert it to shoes since the That's because it is Jules, so we have to multiply it by 1.6 times. Center them on as 19 jewels. Electron volts so under is equal to 4.34 times 10 to the minus 11 m. This is the wavelength, often election that has a kinetic energy of 800 election woods.

Okay, In this problem, we have a non relativistic particle with mass. We are asked to find the wavelength of this particle in terms of its mass and its kinetic energy. And once we have this expression were asked to find specifically the wavelength of an electron with eight hundred TV of kinetic energy equations will use is the definition of a diverse glee wave way in terms of Asian momentum. Thanks. Constant age. We also need the simple definition for kinetic energy and momentum. All right, let's jump into it. So starting off with our definition of the Debra Glee Wavelength H overpay, we're going to replace the momentum with something involving Mm kay. To do that, we will look at our definitions for Kay and P. It will combine them to get K equals p squared over to him. We can do this by replacing V by solving for V in one of these cases, the velocity and then combining it with the other equation. Being up with this new expression for kinetic energy in terms of momentum and in one more step, we can then express P momentum in terms of connect energy and mass. He is equal to two AM Kay, you can take this momentum. Plug it into our definition for the boat. But broccoli wavelength. We get a chew over route to M. K. So that's our expression involving M and gay. Now we can get a specific actual result for an electron with eight hundred TV of energy. You take place constant to be six point six two six. Heave a negative thirty for Jules seconds. Divided by a big square root two times the electron mass of night, which is nine point one times ten to the negative, thirty one kilograms and eight hundred electron volts. Then we need to convert our electron volts into Jules using one point six. Tend the negative nineteen Jules per evey. And once we do this calculation, we end up with a value of four point three four times ten to the negative. Eleven meters are very small and that's it. We have the wavelength for an electron. We have an expression relating wavelength to mass in connect energy. Thank you

Here we have to find very violent and momentum of Electrons accelerated through a potential difference of 56 words. So when I look, loan is accelerated through a potential difference. Delta V the kinetic energy will be equal to in time still to be. Technology is half mv square equal to E times delta B delta V. The potential difference mass of electron is mhm speed is unknown. He is 1.6 and two 10 to the Power -19 column. And Delta, he is 56 world. Yeah. So this gives V speed equal to we use the calculator and we get speed equal to thank you. Four point 42 10 to the power six m/s. So momentum P is equal to M P Mm is Mas 9.11. Yeah. Yeah. Mhm. Mhm. So we get momentum equal to 4.04 Into 10 to the Power -24. Could you meter per second? This is a required momentum now prevalent is given by lambda equal to h upon B. It You plank's constant. Mhm. Yeah. Mhm, mm hmm. We use a calculator. Yeah. And we get to have a length equal to one 64 into turn to the power minus 10 meter. Mhm. In dynamic A. It will be 1.64 and 2 10 to the power minus 10 upon 10 to the power -9 nanometer, Which is simply 0.16 four nanometer. So we have found the prevalent in metal and nanometer. Both this completes the solution.

In this exercise, we have to find what is the liberally wavelength of electrons with different speeds. And we have to use the relativistic formula for the moment, Um, in order to find the Waveland. So first of all, I'm gonna just right here. Cem, uh, constants that are gonna be important for us. So first, Planck's constant H that 61 63 times 10 the minus 30 floor jewels second and the mass of the electron and is 9.1 times 10 to the minus 31 kg in question. A. We have to find the way bling for an election with a speed of 0.48 times the speed of light. Now, remember that the right to visit momentum is equal to for a massive article, of course, is equal to the mass times the speed divided by the square root of one. Mine is the Speed square, divided by C squared. And also, according to the Debra Lee former, the momentum of a particle can written as age divided by its wavelength. Lunda. So in our case, and just gonna relate the two equations here. So I'm gonna write age over the lumber for P and this is equal to envy. Divided by the square root of one minus V square overseas square. And this implies that if we multiply both sides by h I'm sorry if a from multiple it multiply both sides by ah Lunda and then divide both sides by envy. Divided by the square root of one minus b squared overseas square we get that London is equal to age times the square root of one miners v squared over C squared, divided by MZ Okay, and this is the form of that we're gonna use notice. You find the wavelength in both questions A and B So for for V equals 0.48 times the speed of light we have that loved a is equal to age that 61 63 times 10 to the minus 34 usual second times one minus. We swear that 0.48 See square, the site of a C square. Actually, the 40 0.48 truth always square. So the C's cancel out You have one minus. This guy would have one of mine is rippling for you squared, divided by the mass That's 9.1 times 10. It's the minus 31 kg times e that 0.48 times the speed of light which is three times 10 to the eighth meters per second. So Lunda is equal to 4.4 times 10 to the minus 12 m. This year is the answer to question A and question me. We have the same thing for but now, for the speed of 0.96 times is beautiful again, I'm gonna go back to the equation that I highlighted. And Boo Solana is equal to H, which is six weight 63 times 10 to the minus 34. Your second times the square root of one minus 0.96 square, divided by knife one times 10 to the minus 31 kg. Time. 0.96 times this bit of life just three times into the eight meters per second. So London is equal to 7.1 times two the minus 13 m which concludes or exercise


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