Question

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

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

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).

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.

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.

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

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.

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.

Key concepts

Photon Emission

Photon emission is a fascinating process in which an electron transitions between different energy levels, emitting a photon as a result. This phenomenon is governed by quantum mechanics. When an electron drops from a higher energy state to a lower one, the energy difference between these states is released as a photon. This emitted photon's energy is directly related to its wavelength and frequency.
In quantum systems like atoms, electrons occupy discrete energy levels. During a quantum jump, an electron leaps from one energy level to another. This leap is known as a transition. Each transition corresponds to emission or absorption of a photon with a specific wavelength.

• The larger the energy gap, the higher the energy of the emitted photon.
• The energy of a photon can be calculated using Planck’s equation, which describes this relationship between energy and wavelength.

Understanding photon emission helps explain many natural and technological phenomena, from the glow of neon lights to the colors of stars.

Harmonic Potential Well

Imagine a tiny pit where a particle, like an electron, can bounce back and forth. This idea visualizes a harmonic potential well, a concept used to describe how particles behave in quantum mechanics. A harmonic potential well is characterized by a force that pulls the particle toward a central position. The mathematical model resembles a spring that causes oscillations. The term 'harmonic' refers to the potential energy function, which resembles simple harmonic motion.
In simple terms, electrons in atoms are often modeled as particles in a harmonic potential. In quantum terms, these electrons can only occupy certain areas of the potential well corresponding to permitted energy levels.

• These energy levels are quantized, meaning they have specific, discrete values.
• The concept helps in understanding electron behavior in molecules and solids.

Quantum jumps within a harmonic potential well depict transitions between these quantized energy levels, accompanied by the emission or absorption of photons.

Energy Levels

In quantum mechanics, energy levels refer to the specific, quantized states that an electron can occupy in an atom or molecule. Unlike classical systems where energy can take any value, quantum systems consist of discrete levels. Electrons orbiting an atom reside in these fixed energy levels. Moving to a lower energy level, they emit energy in the form of photons. Each quantum jump between levels corresponds to a unique energy difference.

• The magnitude of these differences determines the properties of the emitted photon.
• Energy levels determine atomic behaviors and chemical properties, influencing how atoms bond and emit light.

Understanding energy levels is crucial for interpreting phenomena like laser operation, spectroscopy, and atomic bonding.

Planck's Equation

Planck's equation is the link between energy and frequency of radiation. Formulated by Max Planck, it explains how energy is quantized and relates directly to the theory of quantum jumps. The equation is given by:\[ E = h \cdot u \]Where:

• \( E \) is the energy of the photon.
• \( h \) is Planck's constant (6.63 \times 10^{-34} \text{ J s}).
• \( u \) is the frequency of the radiation.

Planck's equation also appears in a form involving the wavelength:\[ E = \frac{h \cdot c}{\lambda} \]Where:

• \( c \) is the speed of light (3 \times 10^8 \text{ m/s}).
• \( \lambda \) is the wavelength of the photon.

This equation underpins many applications of quantum mechanics by connecting the abstract concept of energy levels to measurable physical quantities like wavelength and frequency. It provides a framework for understanding phenomena like emission spectra and the photoelectric effect.