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A blue-green photon $(\lambda=486 \mathrm{nm})$ is absorbed by a free hydrogen atom, initially at rest. What is the recoil speed of the hydrogenatom after absorbing the photon?

0.816 $\mathrm{m} / \mathrm{s}$

Physics 103

Chapter 30

Quantum Physics

Rutgers, The State University of New Jersey

Simon Fraser University

University of Sheffield

McMaster University

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be equals m We divided by scare route off one minus we square divided by C square is the expression for the relativistic momentum off an object. Er NBC cretin uh, moss off object is M and speed off object is free. A c is the speed of light. Uh, not applying conservation of momentum. Uh, initial momentum equals final momentum. An initial momentum is ah, the some off momentum off hydrogen and a momentum off put on their four p. Initial equals momentum. Uh, five region hydrogen in issue. Bless momentum or foot on mo mentum or foot on initial. And this initial momentum is equal to final momentum. Ah, mo mentum Final Momenta move by Christian Ah, Final momentum of hydrogen Less final momentum or for dawn final Momenta Mall. What on here? Um, hydrogen atom is initially a crest. Came for, um, initial momentum off hydrogen atom. Ah, zero therefore Ah, here he is. Zero. You in issue be initial for hydrogen equals zero. And the final moment, um, off the photons is zero is It is embedded in the hydrogen atom. Therefore final momentum put on Ah, final momentum off foot on final equals zero and Therefore, we have, um mo mentum. Initial momentum off tone in issue momentum off. Put on equals. Um, final Momenta More fight region. Final momentum off. Ah, hydrogen, hydrogen and final momento. And further well went off. Uh, lesser eyes given in nanometers. Let's convert it into meters. So Lamda equals lambda babble and off the laser equals for And the six nanometers for 86 nanometers 4 86 nanometers equals four. It is six. Multiply by 10 to the power minus nine meters. And the expression for the Momenta move for Tony is given by, uh, being for tone. A momentum off for tone in issue is equal to etch divided by lambda and and here, uh, momentum off hydrogen. Um, minimal hydrogen final equals and not be. I am not re divided by, um, straight route off one minus. We squared, divided by C square Here. The wrist muscle hydrogen atom is I'm not and tow Calculate the record speed off the hydrogen atom. Let's create initial momentum off tone and final moment to move hydrogenated. Um, is we know that our initial momentum off put on initial momentum off for tone equals ah, final Momenta Mawf hydrogen, hydrogen, atom, hydrogen eight on Let's plug in the values Reno date Initial momentum overtone is h divided by lambda equals. Uh uh Then finally Momenta more hydrogenated recalls and not we divided by under route off one minus we squared, divided by C square. Now screening on both sides. Let's do squaring on both sides. We have HD mighty by lander hold scare equals I am not. We hold scare divided by, um one minus We squared divided by sees where and for dinner we have one minus we squared, divided by C square times each divided by lambda Whole sphere equals and not we hold square And finally we have relation for velocity we and re equals Sometimes it seems the speed of light in eighties blanks constant C times each divided by ah, square root off scared Rudolph I am not square and not square landed square Ah C square minus eight square No, let's plug in the values, uh, we equals be equals pre Multiply by 10 to the power eight into 6.63 multiplied by 10 to the power minus 34. Divided by divided buying. Uh, spare Rudolph scared Rudolph 1.6741 point 674 Multiply by 10 to the power minus 27. Hold skater into for 86 month supply by for 86 multiplied by 10 to the power minus mine all square into three. Multiply by 10 to the power eight old spare minus six point 63 Multiply by 10 to the power minus 34 whole square minus 34 whole square and therefore re equals zero point 816 meter per second. So the record speed of the hydrogen atom after it zapping the photon is 0.816 meters per second.

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