An electron is in a three-dimensional box with side lengths...

An electron is in a three-dimensional box with side lengths $L_X =$ 0.600 nm and $L_Y = L_Z = 2L_X$. What are the quantum numbers $n_X, n_Y,$ and $n_Z$ and the energies, in eV, for the four lowest energy levels? What is the degeneracy of each (including the degeneracy due to spin)?

Key Concepts

Particle in a Three-Dimensional Box

This concept describes a quantum mechanical system where a particle (such as an electron) is confined within a finite region of space with impenetrable boundaries. The solution of the Schrödinger equation for this setup yields wavefunctions that are sinusoidal inside the box and vanish at the walls. The energy levels are quantized due to the boundary conditions imposed by the confinement, and the problem serves as a fundamental model for quantum confinement effects.

Quantum Numbers

Quantum numbers are sets of integers that arise from the solutions to the Schrödinger equation in confined systems, and they determine the specific state of the particle. In a three-dimensional box, separate quantum numbers (nX, nY, nZ) are assigned for each spatial dimension, and they must be positive integers. These numbers dictate the number of half-wavelengths that fit along each dimension and are integral to calculating the energy levels.

Energy Quantization

Energy quantization refers to the phenomenon where a confined particle can only exist in discrete energy states. In the context of a 3D infinite potential well, the energy is a function of the squares of the quantum numbers divided by the square of the corresponding dimension lengths. The energy levels are determined by the sum of these contributions, and differences in box dimensions lead to different spacings in the quantized energy levels.

Degeneracy in Quantum Systems

Degeneracy occurs when two or more distinct quantum states share the same energy level. In quantum mechanical systems like the 3D box, degeneracy can arise from different combinations of quantum numbers that yield the same energy sum. Additionally, particles with intrinsic spin can introduce further degeneracy, as each spatial state may be occupied by multiple spin configurations (such as the two spin states of an electron).

Anisotropy in Confining Potentials

In a three-dimensional box with unequal side lengths, the system is anisotropic, meaning the physical properties vary with direction. The energy associated with confinement in each direction depends on the corresponding side length of the box, leading to non-equivalent contributions from each spatial dimension. This anisotropy affects the ordering of energy levels and the possible degeneracies of the quantum states.

Transcript

00:01 Hello friends in this problem it is given that electron in 3d box with length lx .6 nanometer and y and lz to be twice of lx we have to find the quantum number nx and y and and z and energy in electron bolt.

01:01 For lowest energy levels what is the degeneration of each let us see here as we know the energy in three -dimensional box with sight lx l y and l j is given by n x and y and j and x x x upon l x n y square upon l y square and z square upon l j s square upon l z square into pi square h square by two m say equation by part a for three -d box and y is called to l z is called to two l x equation one can be written as a n x and y and j n x square upon l x square and y square upon four l x square and z square upon four l x square into pi square h square upon two m so finally you can write energy to be for n square x and n square y n square z into pi square h square upon a time into lx square substituting the value for n square x plus and y square n square z into y square h 1 .055 10 to the power minus 34 of upon 8 into 9 .11 10 to the power minus 31 into l.

05:23 0 .6 into 10 to the power 9 whole square.

05:29 On simplifying it can be, this is to be written as 4 n square x plus n square y plus n n square z into 4 .18 into 10 to the power minus 20 jule or you may write 4 n square x plus n square y plus n square z into 0 .2615 electron bolt for the lowest value n x and y and z to be 1 .2615 electron bolt for the lowest value nx n x and y and z to be 1 so it will be 4 plus 1 plus 1 kai squared.

06:52 So it becomes 1 .569 electron bolt.

07:16 According to poly's exclusion principle, there is only one state that has that set of quantum number by including degeneracy due to spin...

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