What velocity would an electron (mass = 9.11 * 10-31 kg) need for its de Broglie wavelength to be that of red light (750 nm)?

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Identify the de Broglie wavelength formula: \( \lambda = \frac{h}{mv} \), where \( \lambda \) is the wavelength, \( h \) is Planck's constant (\( 6.626 \times 10^{-34} \text{ m}^2 \text{ kg/s} \)), \( m \) is the mass of the electron, and \( v \) is the velocity.

Rearrange the formula to solve for velocity \( v \): \( v = \frac{h}{m\lambda} \).

Substitute the given values into the equation: \( h = 6.626 \times 10^{-34} \text{ m}^2 \text{ kg/s} \), \( m = 9.11 \times 10^{-31} \text{ kg} \), and \( \lambda = 750 \text{ nm} = 750 \times 10^{-9} \text{ m} \).

Calculate the velocity \( v \) using the substituted values.

Ensure the units are consistent and check the calculation for any errors.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

De Broglie Wavelength

The de Broglie wavelength is a concept in quantum mechanics that describes the wave-like behavior of particles. It is given by the formula λ = h/p, where λ is the wavelength, h is Planck's constant (6.626 x 10^-34 Js), and p is the momentum of the particle. For an electron, momentum can be expressed as p = mv, where m is the mass and v is the velocity. This relationship highlights the dual nature of matter, where particles exhibit both wave and particle characteristics.

Momentum

Momentum is a physical quantity defined as the product of an object's mass and its velocity (p = mv). In the context of the de Broglie wavelength, momentum is crucial because it directly influences the wavelength of a particle. For an electron, understanding how to manipulate its mass and velocity allows us to calculate the necessary conditions for achieving a specific wavelength, such as that of red light.

Planck's Constant

Planck's constant (h) is a fundamental constant in quantum mechanics, valued at approximately 6.626 x 10^-34 Js. It relates the energy of a photon to its frequency and plays a critical role in the de Broglie wavelength equation. This constant signifies the scale at which quantum effects become significant, allowing us to connect classical mechanics with quantum phenomena, such as the wave-particle duality of electrons.

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