Question

(a) A green laser pointer emits light with a wavelength of 532 nm. What is the frequency of this light? (b) What is the energy of one of these photons? (c) The laser pointer emits light because electrons in the material are excited (by a battery) from their ground state to an upper excited state. When the electrons return to the ground state, they lose the excess energy in the form of 532 -nm photons. What is the energy gap between the ground state and excited state in the laser material?

Short answer

(a) The frequency of the light emitted by the green laser pointer is approximately \(5.64 × 10^{14}\) Hz. (b) The energy of one of these photons is approximately \(3.74 × 10^{-19}\) J. (c) The energy gap between the ground state and excited state in the laser material is approximately \(3.74 × 10^{-19}\) J.

Step-by-step solution

(a) Calculate the frequency of the light

To find the frequency of the light, we can use the speed of light equation: \(c = f × λ\). First, we need to convert the wavelength to meters: \(\lambda = 532 nm = 532 × 10^{-9} m\). Now, we can rearrange the equation and solve for frequency: \(f = \frac{c}{λ}\) The speed of light, \(c\), is approximately \(3 × 10^8 m/s\). \(f = \frac{3 × 10^8 m/s}{532 × 10^{-9} m} \approx 5.64 × 10^{14} Hz\) So, the frequency of the light is approximately \(5.64 × 10^{14}\) Hz.

(b) Calculate the energy of one photon

To find the energy of one photon, we can use the energy of a photon equation: \(E = h × f\), where Planck's constant, \(h\), is approximately \(6.63 × 10^{-34} Js\). We already found the frequency in the previous step, so we can now calculate the energy: \(E = (6.63 × 10^{-34} Js)(5.64 × 10^{14} Hz) \approx 3.74 × 10^{-19} J\) Therefore, the energy of one photon is approximately \(3.74 × 10^{-19}\) J.

(c) Determine the energy gap between the ground state and excited state

In this situation, the energy gap between the ground state and the excited state in the laser material is equal to the energy of the photon emitted. So, the energy gap is the same as the energy of the photon calculated in part (b): Energy gap = energy of one photon = \(3.74 × 10^{-19}\) J. Thus, the energy gap between the ground state and the excited state in the laser material is approximately \(3.74 × 10^{-19}\) J.

Key concepts

Frequency of Light

Understanding the frequency of light is crucial when studying the nature of photons. The frequency, denoted as \( f \), refers to the number of wave cycles that pass a given point per second. It is measured in hertz (Hz), which is equivalent to one cycle per second. Light, being an electromagnetic wave, has a wide range of frequencies, making up the electromagnetic spectrum. The frequency is inversely related to the wavelength (\( \lambda \)), which is the distance between successive peaks of the wave. Higher frequency means shorter wavelength, and vice versa.

When dealing with problems related to frequency, such as determining the frequency of a green laser pointer, the speed of light equation comes into play. With the speed of light being a constant (approximately \( 3 \times 10^8 \) m/s), knowing the wavelength allows one to calculate the frequency using the relationship \( f = \frac{c}{\lambda} \). In the case of our example, a green laser with a 532 nm wavelength translates to a frequency approximately \( 5.64 \times 10^{14} \) Hz, placing it in the visible spectrum.

Speed of Light Equation

The speed of light equation is a fundamental concept in physics. It dictates that the speed of light in a vacuum, denoted as \( c \), is constant and is approximately \( 3 \times 10^8 \) meters per second. The equation \( c = f \times \lambda \) directly relates the speed of light to the frequency (\( f \)) and the wavelength (\( \lambda \)) of an electromagnetic wave.

Understanding this equation is pivotal when determining properties of light, such as wavelength or frequency when the other is known. In educational context, it's essential to emphasize how constants like the speed of light provide the bedrock for calculations across various physics problems, including those involving photon energy and electromagnetic radiation.

Planck's Constant

Planck's constant, represented by \( h \), is a physical constant that plays a vital role in the field of quantum mechanics. Its value is approximately \( 6.63 \times 10^{-34} \) joule seconds (Js). Planck's constant relates the energy of a photon to its frequency through the relation \( E = h \times f \).

This equation allows us to calculate the energy of a photon if its frequency is known, which is critical for understanding how light interacts with matter. In the case of our green laser, Planck's constant helps determine the energy of the photons emitted, which relates directly to the colors we perceive.

Laser Photon Energy

Photon energy is a term that describes the energy carried by a single photon. The calculation of laser photon energy is crucial in understanding how lasers work and interact with different materials. It is determined by multiplying the frequency of the light by Planck's constant (\( E = h \times f \)).

Lasers emit photons with specific energies, which correspond to specific wavelengths of light. For example, a green laser emitting light at 532 nm has photons with an energy of approximately \( 3.74 \times 10^{-19} \) joules. This energy is the key factor in processes such as cutting, welding, or medical treatments, where lasers are utilized.

Excited State

In physics, the excited state refers to an energy level of an atom or molecule that is higher than the ground state. When an atom absorbs energy, its electrons move to higher energy levels, entering an excited state. Eventually, these electrons return to the ground state, releasing energy in the form of photons.

The energy difference between these two states is known as the energy gap, and it directly corresponds to the energy of the emitted photons when electrons transition back to a lower energy level. In a laser, this process is harnessed so that when electrons fall back from the excited state to the ground state, they emit photons with a very specific energy, creating a laser beam.