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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

State whether each of the following statements is true or false a) In a one-dimensional quantum harmonic oscillator, the energy levels are evenly spaced. b) In an infinite one-dimensional potential well, the energy levels are evenly spaced. c) The minimum total energy possible for a classical harmonic oscillator is zero. d) The correspondence principle states that because the minimum possible total energy for the classical simple harmonic oscillator is zero, the expected value for the fundamental state \((n=0)\) of the one-dimensional quantum harmonic oscillator should also be zero. e) The \(n=0\) state of the one-dimensional quantum harmonic oscillator is the state with the minimum possible uncertainty \(\Delta x \Delta p\)

Determine the two lowest energies of a wave function of an electron in a box of width \(2.0 \cdot 10^{-9} \mathrm{~m} .\)

Think about what happens to infinite square well wave functions as the quantum number \(n\) approaches infinity. Does the probability distribution in that limit obey the correspondence principle? Explain.

An electron is confined between \(x=0\) and \(x=L\). The wave function of the electron is \(\psi(x)=A \sin (2 \pi x / L)\). The wave function is zero for the regions \(x<0\) and \(x>L\) a) Determine the normalization constant \(A\). b) What is the probability of finding the electron in the region \(0 \leq x \leq L / 3 ?\)

The probability of finding an electron in a hydrogen atom is directly proportional to a) its energy. b) its momentum. c) its wave function. d) the square of its wave function. e) the product of the position coordinate and the square of the wave function. f) none of the above.
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