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An electron trapped in a 1-D box of length 1Å having energy 338 eV. This represents
- a)ground state
- b)first excited state
- c)second excited state
- d)third excited state
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
An electron trapped in a 1-D box of length 1Åhaving energy338 eV...
En = +37.6 n2eV = 338.4eV
⇒ n2 = 9
⇒ n = 3
second excited state
Most Upvoted Answer
An electron trapped in a 1-D box of length 1Åhaving energy338 eV...
If an electron is trapped in a 1-D box of length 1, it means that the electron can only move in one dimension, along a line of length 1 unit. This scenario is often used to illustrate a simplified model of quantum confinement.
In this model, the electron is treated as a quantum particle with wave-like properties. The boundaries of the box act as potential barriers, confining the electron to the box. The wave function of the electron describes the probability distribution of finding the electron at different positions along the line.
The boundary conditions for the electron in a 1-D box are that the wave function must be zero at the boundaries of the box. This implies that the particle cannot exist outside the box.
The energy levels of the electron in a 1-D box can be calculated using the Schrödinger equation. The solutions to the equation yield quantized energy levels, which are given by the equation:
E_n = (n^2 * h^2) / (8 * m * L^2)
where E_n is the energy of the nth energy level, n is a positive integer (1, 2, 3, ...), h is Planck's constant, m is the mass of the electron, and L is the length of the box.
The wave function of the electron in a 1-D box can also be calculated using the Schrödinger equation. The wave function describes the spatial distribution of the electron's probability amplitude along the line. The square of the wave function gives the probability density of finding the electron at a particular position.
The electron in a 1-D box exhibits different energy levels and corresponding wave functions. The lowest energy level (n = 1) corresponds to the ground state, while higher energy levels (n = 2, 3, ...) correspond to excited states. The wave functions of the different energy levels have different numbers of nodes (points where the wave function crosses zero) and exhibit different spatial distributions.
Overall, the concept of an electron trapped in a 1-D box provides a simplified model to understand the quantum mechanical behavior of particles in confined spaces. It helps illustrate the quantization of energy and the wave-like properties of quantum particles.
In this model, the electron is treated as a quantum particle with wave-like properties. The boundaries of the box act as potential barriers, confining the electron to the box. The wave function of the electron describes the probability distribution of finding the electron at different positions along the line.
The boundary conditions for the electron in a 1-D box are that the wave function must be zero at the boundaries of the box. This implies that the particle cannot exist outside the box.
The energy levels of the electron in a 1-D box can be calculated using the Schrödinger equation. The solutions to the equation yield quantized energy levels, which are given by the equation:
E_n = (n^2 * h^2) / (8 * m * L^2)
where E_n is the energy of the nth energy level, n is a positive integer (1, 2, 3, ...), h is Planck's constant, m is the mass of the electron, and L is the length of the box.
The wave function of the electron in a 1-D box can also be calculated using the Schrödinger equation. The wave function describes the spatial distribution of the electron's probability amplitude along the line. The square of the wave function gives the probability density of finding the electron at a particular position.
The electron in a 1-D box exhibits different energy levels and corresponding wave functions. The lowest energy level (n = 1) corresponds to the ground state, while higher energy levels (n = 2, 3, ...) correspond to excited states. The wave functions of the different energy levels have different numbers of nodes (points where the wave function crosses zero) and exhibit different spatial distributions.
Overall, the concept of an electron trapped in a 1-D box provides a simplified model to understand the quantum mechanical behavior of particles in confined spaces. It helps illustrate the quantization of energy and the wave-like properties of quantum particles.
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An electron trapped in a 1-D box of length 1Åhaving energy338 eV.This representsa)ground stateb)first excited statec)secondexcited stated)third excited stateCorrect answer is option 'C'. Can you explain this answer?
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