Two excited energy levels are separated by the very small energy difference . As atoms in these levels undergo quantum jumps to the ground state, the photons they emit have nearly identical wavelengths .
a. Show that the wavelengths differ by
b. In the Lyman series of hydrogen, what is the wavelength difference between photons emitted in the to transition and photons emitted in the to transition?
The change in wavelength is and the wavelength difference of two transitions is .
Use the relationship between the energy of the photon and wavelength.
The expression for the energy of the photon is,
Here, is the planck's constant and is the speed of light.
Differentiate the equation with respect to .
Rewrite the above equation for small change in energy corresponding to small change in wavelength. So,
Therefore, the changes in wavelength is .
Use the result of part (a) to find the difference of wavelength between two transitions.
The energy difference between the states and is,
The energy of emitting photon when transition between the states and is.
The wavelength of emitting photon when transition between the states and is.
Substitute for and for .
Here the wavelength of the emitted photon for transition is almost equal to the wavelength of the emitted photon for transition .
The difference between two wavelength is ,
Substitute for , for and for .
therefore, the wavelength difference of two transitions is 0.021 nm.
Suppose you put five electrons into a -wide one dimensional rigid box (i.e., an infinite potential well).
a. Use an energy-level diagram to show the electron configuration of the ground state.
b. What is the ground-state energy that is, the total energy of all five electrons in the ground-state configuration?
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