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Chapter 41: Find three electrons (page 1208)

Three electrons are in a one-dimensional rigid box (i.e., an infinite potential well) of length $0.50 n m$ . Two are in the $n = 1$ state and one is in the $n = 6$ state. The selection rule for the rigid box allows only those transitions for which $鈭� n$ is odd.
a. Draw an energy-level diagram. On it, show the filled levels and show all transitions that could emit a photon.
b. What are all the possible wavelengths that could be emitted by this system?

Short Answer

Expert verified

For one dimensional rigid box the energy levels are as follows.

$E_{n} = n^{2} \frac{h^{2}}{8 m l^{2}}$

Length of the box is,

role="math" localid="1648742877385" $= 1.51 n^{2} e V$

Step by step solution

01

For one dimensional rigid box the energy levels are as follows.

$E_{n} = n^{2} \frac{h^{2}}{8 m l^{2}} L e n g t h o f t h e b o x i s, L = 0.5 n m = (0.5 n m) (10^{- 9} \frac{m}{n m}) = 0.5 x 10^{- 9} m S u b s t i t u t i n g 6.63 x 10^{- 34} j . s f o r h 9.1 x 10^{- 24} k g f o r m . 0.5 x 10^{- 9} m f o r L E_{n} = n^{2} (\begin{matrix} {(6.63 x 10^{- 34} j s)}^{2} \\ (8) (9.1 x 10^{- 31} k g) {(0.5 x 10^{- 9} m)}^{2} \end{matrix}) = (2.415 x 10^{- 19} j) n^{2} = \frac{(2.415 x 10^{- 19} j) n^{2}}{(1.6 x 10^{- 19} \frac{J}{e V})} = 1.51 n^{2} e V$

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In fluorescence microscopy, an important tool in biology, a laser beam is absorbed by target molecules in a sample. These molecules are then imaged by a microscope as they emit longer-wavelength photons in quantum jumps back to lower energy levels, a process known as fluorescence. A variation on this technique is two-photon excitation. If two photons are absorbed simultaneously, their energies add. Consequently, a molecule that is normally excited by a photon of energy $E_{p h o t o n}$ can be excited by the simultaneous absorption of two photons having half as much energy. For this process to be useful, the sample must be irradiated at the very high intensity of at least $10^{32} \frac{p h o t o n s}{m^{2} s}$ . This is achieved by concentrating the laser power into very short pulses ( $100 f s$ pulse length) and then focusing the laser beam to a small spot. The laser is fired at the rate of $10^{8}$ pulses each second. Suppose a biologist wants to use two-photon excitation to excite a molecule that in normal fluorescence microscopy would be excited by a laser with a wavelength of $420 n m$ . If she focuses the laser beam to a $2.0 - m m$ -diameter spot, what minimum energy must each pulse have?

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