What is the quantum number of an electron confined in a 3.0-nm-long one-dimensional box if the electron’s de Broglie wavelength is 1.0 nm?

검증된 단계별 안내

Understand the problem: The electron is confined in a one-dimensional box, which means we are dealing with the quantum mechanical particle-in-a-box model. The quantum number (n) determines the energy state of the electron, and we are given the de Broglie wavelength (λ) and the length of the box (L).

Recall the relationship between the de Broglie wavelength and the quantum number in a one-dimensional box: The allowed wavelengths are given by the formula:

λ = 2L/n

, where L is the length of the box and n is the quantum number.

Rearrange the formula to solve for the quantum number n:

n = 2L/λ

. This equation relates the quantum number to the box length and the de Broglie wavelength.

Substitute the given values into the formula: L = 3.0 nm and λ = 1.0 nm. Ensure the units are consistent (both are in nanometers, so no conversion is needed). The equation becomes:

n = 2(3.0)/1.0

Interpret the result: The quantum number n must be a positive integer because it represents discrete energy levels. After performing the division, round the result to the nearest whole number if necessary to ensure it is an integer.

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Quantum Numbers

Quantum numbers are sets of numerical values that describe the unique quantum state of an electron in an atom. They include the principal quantum number, angular momentum quantum number, magnetic quantum number, and spin quantum number. In the context of a particle in a box, the quantum number indicates the energy level and the allowed states of the electron.

de Broglie Wavelength

The de Broglie wavelength is a concept that relates the wavelength of a particle to its momentum, expressed as λ = h/p, where h is Planck's constant and p is the momentum. For an electron in a confined space, this wavelength can be used to determine the allowed energy states and corresponding quantum numbers. It highlights the wave-particle duality of matter.

Particle in a Box Model

The particle in a box model is a fundamental quantum mechanics concept that describes a particle free to move in a small space with infinitely high potential walls. This model allows for the calculation of quantized energy levels and wave functions. The length of the box and the quantum number determine the energy levels and the corresponding wavelengths of the particle.