For a particle in a three-dimensional box, what is the degeneracy (number of different quantum s

For a particle in a three-dimensional box, what is the...

Key Concepts

Symmetry in Energy Levels

Symmetry in the arrangement or permutation of quantum numbers plays a key role in determining energy degeneracy. For example, if multiple configurations of quantum numbers yield the same sum of squares, they will correspond to the same energy level, thus making the state degenerate. This concept underpins the analysis of degenerate states in systems with spatial symmetry.

Energy Quantization in a 3D Box

The energy of a particle in a three-dimensional box is calculated using the sum of the squares of the quantum numbers for each dimension, multiplied by a constant derived from physical parameters. This relationship makes the energy dependent on the specific set of quantum numbers and leads to a discrete energy spectrum.

Degeneracy

Degeneracy is the count of different quantum states that have the same energy level. It arises when multiple sets of quantum numbers produce the same value for the energy expression. Understanding degeneracy is important in quantum mechanics because it affects the statistical and dynamical properties of a system.

Particle in a Box

In quantum mechanics, a particle in a box is a model where a particle is confined in a perfectly rigid boundary with infinitely high potential barriers. This confinement leads to discrete (quantized) energy levels, each characterized by integer quantum numbers. It serves as a foundational example for illustrating basic quantum behavior such as wavefunctions and energy quantization.

Quantum Numbers

Quantum numbers are integers assigned to a system that describe its state. In the case of a three-dimensional box, three quantum numbers (one for each spatial dimension) determine the energy levels. These numbers indicate the number of half-wavelengths that fit into the box along each axis and are crucial for calculating the total energy.

Transcript

00:01 For a particle in a box, we are asked to find the degeneracy, the number of different quantum states that give the following energies.

00:07 So the corresponding energy state of a three quantum number system is given as the following.

00:17 So the energy is equal to h -bar squared times pi -squared, divided by two times the mass times the length of the box, l squared.

00:23 And then this is multiplied by nx squared plus n -y -squared plus n -z -squared, where n -x, n -y, and z are the three different allowed quack.

00:30 Quantum numbers.

00:32 Of course, these numbers are integers.

00:33 They can only take on a number like 0, 1, 2, 3, something like that.

00:37 And they have to be a positive integer, so 1 and above.

00:40 So for the first one, we're told that nx squared plus n y squared plus n z squared has to equal 3.

00:46 Well, the only way that this can be true is if nx, n y, and n z all equal 1.

00:58 And you can try this for yourself.

01:00 So again, nx, and n z all have to be a positive integer greater than zero so one two three or four if you plug one in for all of them one squared plus one squared plus one squared gives you three so that's where that number three out front comes from if you plug in any other number but one for just one of them so try to do one squared plus one squared plus two squared and you're going to get a number larger than three so one one and one are the only possible option so the degeneracy is equal to one there's only one possible quantum state, and that's nx, ny, and z is equal to 1.

01:32 For the next one, we have a number of 9.

01:35 So, nx plus nx squared plus n y squared plus n z squared can equal 9.

01:42 Well, the only way this can be true is if two of them equal two and one of them equal one, because two squared plus two squared is eight plus one squared is nine.

01:53 If you plugged in any other numbers except for that, for instance, if you gave one of them a value of three, then all the other values would have to be zero...

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