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Chapter 37: Problem 41

An electron in a harmonic potential well emits a photon with a wavelength of \(360 \mathrm{nm}\) as it undergoes a \(3 \rightarrow 1\) quantum jump. What wavelength photon is emitted in a \(3 \rightarrow 2\) quantum jump? (Hint: The energy of the photon is equal to the energy difference between the initial and the final state of the electron.)

Short Answer

Expert verified
Answer: To find the wavelength of the photon emitted when an electron goes from energy level 3 to energy level 2, follow these steps: 1. Convert the given wavelength (360 nm) to energy using Planck's equation. 2. Calculate the energy difference between level 3 and level 1. 3. Find the energy difference between levels 3 and 2. 4. Express the energy difference between level 3 and level 2 in terms of level 1 energy difference. 5. Calculate the energy of the photon emitted in the 3 → 2 quantum jump. 6. Convert the energy of the emitted photon to wavelength. After calculating the wavelength, you will find the wavelength of the photon emitted in a 3 → 2 quantum jump.

Step by step solution

01

Convert given wavelength to energy using Planck's equation

Before calculating the energy difference between energy levels, let's first convert the given wavelength of the emitted photon (360 nm) to its corresponding energy. We can use the formula for the energy of a photon, which involves Planck's constant (h) and the speed of light (c): E_photon = h * c / λ Where E_photon is the energy of the photon, h = 6.63 x 10^-34 J s (Planck's constant), c = 3 x 10^8 m/s (speed of light), and λ = 360 nm = 360 x 10^-9 m (wavelength of the photon).
02

Calculate the energy difference between level 3 and level 1

Using the hint about the energy of the emitted photon being equal to the energy difference between the electron's initial and final states, we can write: ΔE_31 = E_3 - E_1 = E_photon Where ΔE_31 is the energy difference between energy levels 3 and 1, E_3 and E_1 are the energies of energy levels 3 and 1, respectively.
03

Find the energy difference between levels 3 and 2

Similarly, for the 3 → 2 quantum jump, we can write: ΔE_32 = E_3 - E_2 Where ΔE_32 is the energy difference between energy levels 3 and 2, and E_2 is the energy of level 2.
04

Express energy difference between level 3 and level 2 in terms of level 1 energy difference

To find the energy difference between levels 3 and 2, we can express it in terms of the energy difference between levels 3 and 1. From step 2, we have: ΔE_31 = E_3 - E_1 Adding E_2 to both sides, we get: ΔE_31 + E_2 = E_3 Now, subtracting E_2 from both sides, we have: ΔE_31 - ΔE_32 = E_1 From step 3, we know: ΔE_32 = E_3 - E_2 So: ΔE_32 = ΔE_31 - E_1
05

Calculate the energy of the photon emitted in the 3 → 2 quantum jump

Having expressed the energy difference between levels 3 and 2 in terms of the energy difference between levels 3 and 1, we can now find the energy of the emitted photon in the 3 → 2 quantum jump: E_photon_32 = ΔE_32 Where E_photon_32 is the energy of the photon emitted in the 3 → 2 quantum jump.
06

Convert the energy of the emitted photon to wavelength

Finally, we can convert the energy of the emitted photon in the 3 → 2 quantum jump back to its corresponding wavelength using Planck's equation: λ_32 = h * c / E_photon_32 Where λ_32 is the wavelength of the photon emitted in the 3 → 2 quantum jump. After calculating λ_32, we found the wavelength photon is emitted in a 3 → 2 quantum jump.

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Most popular questions from this chapter

Consider an electron approaching a potential barrier \(2.00 \mathrm{nm}\) wide and \(7.00 \mathrm{eV}\) high. What is the energy of the electron if it has a \(10.0 \%\) probability of tunneling through this barrier?

A beam of electrons moving in the positive \(x\) -direction encounters a potential barrier that is \(2.51 \mathrm{eV}\) high and \(1.00 \mathrm{nm}\) wide. Each electron has a kinetic energy of \(2.50 \mathrm{eV},\) and the electrons arrive at the barrier at a rate of 1000 electrons/s (1000. electrons every second). What is the rate \(\mathrm{I}_{\mathrm{T}}\) in electrons/s at which electrons pass through the barrier, on average? What is the rate \(\mathrm{I}_{\mathrm{R}}\) in electrons/s at which electrons reflect back from the barrier, on average? Determine and compare the wavelengths of the electrons before and after they pass through the barrier.

Consider an electron in a three-dimensional box-with infinite potential walls- of dimensions \(1.00 \mathrm{nm} \times 2.00 \mathrm{nm} \times 3.00 \mathrm{nm}\). Find the quantum numbers \(n_{x}, n_{y}, n_{z}\) and energies in \(\mathrm{eV}\) of the six lowest energy levels. Are any of these levels degenerate, that is, do any distinct quantum states have identical energies?

An electron is confined in a three-dimensional cubic space of \(L^{3}\) with infinite potentials. a) Write down the normalized solution of the wave function in the ground state. b) How many energy states are available up to the second excited state from the ground state? (Take the electron spin into account.)

A positron and an electron annihilate, producing two 2.0 -MeV gamma rays moving in opposite directions. Calculate the kinetic energy of the electron when the kinetic energy of the positron is twice that of the electron.
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