Question

quantum mechanics
plancks equation

1. what is the energy and frequency of photons with a wavelength of 632 nm?

de broglie wavelength

1. an electron has the same de broglie wavelength as a 1024-nm photon. what is the speed of the electron?

Explanation:

Identify given values and constants

We identify the given values and physical constants for both questions.

For Question 3:

• Wavelength \(\lambda = 632\text{ nm} = 632 \times 10^{-9}\text{ m}\)
• Planck's constant \(h = 6.63 \times 10^{-34}\text{ J}\cdot\text{s}\)
• Speed of light \(c = 3.0 \times 10^8\text{ m/s}\)

For Question 4:

• Wavelength \(\lambda = 1024\text{ nm} = 1024 \times 10^{-9}\text{ m}\)
• Mass of an electron \(m_e = 9.11 \times 10^{-31}\text{ kg}\)
• Planck's constant \(h = 6.63 \times 10^{-34}\text{ J}\cdot\text{s}\)

Calculate frequency of the photon

We use the wave relation to find the frequency \(f\):

$$ f = \frac{c}{\lambda} $$

Substituting the values:

$$ f = \frac{3.0 \times 10^8\text{ m/s}}{632 \times 10^{-9}\text{ m}} \approx 4.75 \times 10^{14}\text{ Hz} $$

Calculate energy of the photon

We use Planck's equation to find the energy \(E\):

$$ E = \frac{hc}{\lambda} = hf $$

Substituting the values:

$$ E = (6.63 \times 10^{-34}\text{ J}\cdot\text{s}) \times (4.7468 \times 10^{14}\text{ s}^{-1}) \approx 3.15 \times 10^{-19}\text{ J} $$

Calculate speed of the electron

We use the de Broglie wavelength formula:

$$ \lambda = \frac{h}{p} = \frac{h}{m_e v} $$

Rearranging to solve for the speed \(v\):

$$ v = \frac{h}{m_e \lambda} $$

Substituting the values:

$$ v = \frac{6.63 \times 10^{-34}\text{ J}\cdot\text{s}}{(9.11 \times 10^{-31}\text{ kg}) \times (1024 \times 10^{-9}\text{ m})} $$

Calculating the denominator:

$$ m_e \lambda = 9.11 \times 10^{-31} \times 1.024 \times 10^{-6} = 9.3286 \times 10^{-37}\text{ kg}\cdot\text{m} $$

Calculating the speed:

$$ v = \frac{6.63 \times 10^{-34}}{9.3286 \times 10^{-37}} \approx 711\text{ m/s} $$

Answer:

Question 3

• Energy (\(E\)): \(3.15 \times 10^{-19}\text{ J}\)
• Frequency (\(f\)): \(4.75 \times 10^{14}\text{ Hz}\)

Question 4

• Speed of the electron (\(v\)): \(711\text{ m/s}\)