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Problem 53 Hard Difficulty

Use Bohr’s model of the hydrogen atom to show that when the atom makes a transition from the state n to the state $n-1,$ the frequency of the emitted light is given by
$$f=\frac{2 \pi^{2} m k_{e}^{2} e^{4}}{h^{3}}\left[\frac{2 n-1}{(n-1)^{2} n^{2}}\right]$$

Answer

$f=\left(\frac{2 \pi^{2} m_{c} k^{2} e^{4}}{h^{2}}\right)\left(\frac{2 n-1}{(n-1)^{2} n^{2}}\right)$

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Top Physics 103 Educators
Christina K.

Rutgers, The State University of New Jersey

LB
Liev B.

Numerade Educator

Farnaz M.

Other Schools

Meghan M.

McMaster University

Video Transcript

In this exercise, we have to find a formula for the frequency off a photon that's emitted due to the transition from the end level of the hydrogen atom to the n minus one level of the hydrogen atom. So what we need to remember is that the energy off the emitted Fulton from conservation of energy is equal to the energy of Ian's level, minus the energy of the an minus one level. And also we have of the energy of the level of the hydrogen atom from Morse Theory is equal to minus K square, each of the fourth M divided by four. It's hard to h r squared in square. So from here we can get that the energy of the emitted photon is minus case where into the fourth, I am divided by two h bar, square and square plus K squared into the fourth. I am divided by two h bar square, endless in in minus one, actually, and a minus one square. Okay, so this is equal to I'm that affect your old the Constance. So this is the cotton dis constant times one over and minus one. Squared, minus one over and squared Okay, so this could be rewritten as this Constance here. Times and square minus and minus one squared, divided by an minus one squared times in square. So we have the constant times and squared minus and squared. So this the first term cancels out because, uh, let me write it explicitly. Here we have and square in the inside the square brackets we have and squared minus and squared. Plus two in minus one. It's on the first and second cancel out. So you have to and minus one divided by and minus one squared and square. So this here is the energy, other fortune. And remember that the energy of a Fulton can be written us plus constant f I'm sorry. Blanks constant h times f So have a h f is equal to, uh I'm gonna write H bar as H bar as age over two pi. So you get four. Hi square K squared into the fourth AM divided by to age square. Okay, Just substituted age bar for by age over two pi and then the same thing. Two and minus one or an minus one squared and square. Okay. So f is equal to four Pi Square case where eat the fourth. I am actually the four here cancels with the two. So I have to find squared case square into the fourth M divided by age cubed, uh, times tu minus. Want two in minus one, Divided by and swear and minus one square. Okay, this is the equation that we wanted to find.

Universidade de Sao Paulo
Top Physics 103 Educators
Christina K.

Rutgers, The State University of New Jersey

LB
Liev B.

Numerade Educator

Farnaz M.

Other Schools

Meghan M.

McMaster University