A particle is confined to a two-dimensional box of length L and width 2 L. The energy values are

A particle is confined to a two-dimensional box of length L...

Key Concepts

Particle in a Box

In quantum mechanics, the ‘particle in a box’ model describes a particle that is confined within an infinitely high potential well with rigid boundaries. The particle’s wavefunction must vanish at the boundaries, leading to a set of discrete, quantized energy levels. This model is a fundamental example that illustrates how spatial confinement leads to quantization of energy and other measurable properties.

Energy Quantization

Energy quantization refers to the phenomenon where only discrete energy values are allowed for a quantum system due to boundary conditions or confinement. In systems such as the particle in a box, the allowed energy levels are determined by solving the Schrödinger equation under specific boundary conditions, resulting in energies that depend on quantum numbers in a manner that reflects the dimensions and shape of the confinement.

Quantum Numbers

Quantum numbers are integers that arise from the solution of the quantum mechanical equations, such as the Schrödinger equation, for bound systems. They label the eigenstates and directly determine the energy levels of the system. In multi-dimensional problems, different quantum numbers can be associated with each dimension, and their combination describes the complete state of the system.

Degeneracy

Degeneracy in quantum mechanics occurs when two or more distinct quantum states share the same energy level. This typically arises from symmetries in the system or the geometrical properties of the confinement. The presence of degenerate energy levels is important for understanding the system’s response to perturbations and its overall symmetry properties.

Boundary Conditions

Boundary conditions are essential in determining the allowed solutions for a quantum system by imposing restrictions on the wavefunction at the edges of the confinement. For instance, in a particle in a box scenario, the requirement that the wavefunction vanishes at the walls of the box leads to the discrete set of eigenstates and quantized energy levels. These conditions ensure that the physical and mathematical solutions are consistent with the system's geometry and constraints.

Transcript

00:01 Hello, and in this question here, we have a particle in a two -dimensional infinite well potential, with the energy levels described as h -bar to be squared times pi squared, all divided by two times the mass, times the length of the box, times nx to be squared plus n y to be squared over four, where nx and n y are the quantum numbers, and they can take values of 1, 2, 3, 4, etc.

00:28 Etc.

00:30 And we want to find out the two lowest degeneracies of this system.

00:35 So to do this, let's look at the analogy of the system where we have the box that our particle in is a square.

00:46 So in this case, the energy levels would be equal to h bar squared times pi squared, times 2ml squared times nx to be squared plus n y to be squared we're going to simplify this equation here and notice that this state here we're just going to be this constant here we're going to call e0 okay so just a simplifying notation for the future of the question this here will just be e0 and our first degeneracy for this system where the box is square would occur for when nx is equal to 1 and n y is equal to 2 well this would be degenerate with the state where n y is equal to 1 and n x is equal to 2 and they would have these two states that these two states are both degenerate with each other and both have an energy e is equal to 5 times e 0 the next degenerate of this system would occur when n x is equal to 3 and n y is equal to 1 and this state would be degenerate with the state when n y is equal to 3 and n x is equal to 1 and these two states would both have an energy of e is equal to 10 e 0 so these would be the first two generacies of the two dimensional box however our our box is not square, it is instead given by this here.

02:30 So in analogy, the first two degenerate states would occur, well, they would occur when.

02:39 Well, they'd have when nx is equal to 1, and n y over 2 is equal to 2.

02:48 So if we look back up here, when we had the first state for our square box would be when nx is equal to 1.

02:56 And n y is equal to two.

02:58 And our energies would just be n y squared...

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