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Problem 30 Medium Difficulty

Using the concept of standing waves, de Broglie was able to derive Bohr’s stationary orbit postulate. He assumed a confined electron could exist only in states where its de Broglie waves form standing wave patterns, as in Figure 28.6. Consider a particle confined in a box of length $L$ to be equivalent to a string of length $L$ and fixed at both ends. Apply de Broglie's
concept to show that (a) the linear momentum of this particle is quantized with $p=m v=n h / 2 L$ and $(\mathrm{b})$ the allowed states correspond to particle energies of $E_{n}=n^{2} E_{0},$ where $E_{0}=h^{2} /\left(8 m L^{2}\right)$.

Answer

a) $\frac{n h}{2 L}$
b) $E_{n}=\frac{n^{2} h^{2}}{4 L^{2}(2 m)}$

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Top Physics 103 Educators
Zachary M.

Hope College

Aspen F.

University of Sheffield

Jared E.

University of Winnipeg

Meghan M.

McMaster University

Video Transcript

in this exercise, we have to assume that the electric is behaves like a standing wave inside a one dimensional box of length L and in question A. We want to show that the momentum p of the election can be redness and h divided by two l. So the first thing I need to remember here is that for a standing wave l the length of the box is equal to end times the wavelength of the wave divided by two If you don't remember where the come where that comes from I just noticed that four and that the first standing wave possible for n equals one is this one that I'm drawing here. OK, notice that the Here we have l that's the size of the box. But l is equal to 1/2 the wavelengths. Okay, because the distance between two nodes is 1/2 the wavelength. Okay, so in this case, l equals Linda over to for for n equals two. The second possible sending wave is this one When they dried Let it a little better. Is this one when l this distance here is able to London Okay, so notice that our equation here. The equation for l that I just wrote is a function of N and Linda holds. Okay, if you check for for larger values of N, you're going to see that Ah, this equation always holds. So let's start with this equation. And from here l equals and Linda over two. And we know that Lunda, according to the release formula, is equal to age over the momentum. P. So I just need to isolate WPI this world, and we get P equals sue and h over to l. Okay, that's what we wanted to show and we have. So let's go on to question be question, Do you have to show that the energy of the inthe level is given by and square age squared, divided by eight m squared L. And in order to do that, we need to remember that the energy of the kinetic energy of the electoral is P square divided by to win. So I'm gonna take the answer Depression eight, and swear it. So I have be square and P is an age over too. Well, then I squared times one over to em. So this is and square H squared, divided by eight l m Square. And that's exactly what we wanted to show

Universidade de Sao Paulo
Top Physics 103 Educators
Zachary M.

Hope College

Aspen F.

University of Sheffield

Jared E.

University of Winnipeg

Meghan M.

McMaster University