Consider the three-dimensional harmonic oscillator, for which the potential is V(r)=(1)/(2) m ω^

Consider the three-dimensional harmonic oscillator, for...

Key Concepts

Degeneracy Counting

Degeneracy in quantum mechanics refers to the occurrence of multiple independent quantum states that share the same energy level. For the three-dimensional harmonic oscillator, the degeneracy of a given energy level is determined using combinatorial methods that count the number of ways to partition the total quantum number among three independent oscillators, reflecting the system's symmetry and the underlying statistical properties.

One-Dimensional Harmonic Oscillator

The one-dimensional harmonic oscillator is a quintessential problem in quantum mechanics with well-known quantized energy levels. Its solution reveals that energy levels are equally spaced and characterized by quantum numbers; this knowledge directly facilitates the analysis of the three-dimensional case when each dimension behaves like an independent oscillator.

Energy Quantization

Energy quantization refers to the phenomenon where a quantum system can only occupy discrete energy states. In the harmonic oscillator, the energy levels are determined by an integer quantum number, leading to a spectrum where the total energy in three dimensions is a sum of the energies from the individual one-dimensional oscillators.

Three-Dimensional Harmonic Oscillator

The three-dimensional harmonic oscillator is a fundamental model in quantum mechanics that describes a particle subject to a potential proportional to the square of the distance from the origin. This model is vital for understanding systems with spherical symmetry and appears in various fields such as atomic physics, molecular vibrations, and quantum field theory.

Separation of Variables

Separation of variables is a mathematical method employed to simplify partial differential equations by expressing the solution as a product of functions, each depending on a single coordinate. In the context of the three-dimensional harmonic oscillator, using Cartesian coordinates allows the Schrödinger equation to be split into three independent one-dimensional problems, each corresponding to a separate spatial dimension.

Transcript

00:01 So in this question we are given a 3d harmonic oscillator right for harmonic oscillator i'm using h -o fine and its potential is given that v of r is equal to 1 over 2 m omega square r square right now the first part is that we need to convert this 1 3 d harmonic oscillator into three let's write it as three one d harmonic oscillators right the smallest four oscillators fine so we are required to convert one three d harmonic oscillator into three one d harmonic escalators using the separation of variables technique right so the solution is very simple but before starting with a solution you must know one thing that the energy eigen value energy eigen value of the quantum harmonic oscillator is given by the equation he is equal to n plus 1 divided by 2 the blanks constants time the omega, right? that is the angular velocity, right? now, okay, so from here we are going to start our equation.

02:02 Right.

02:02 Now, the given potential of the system is v of r is equal to 1 divided by 2m times omega square times r square, right? now since this is a cardizing coordinate system, right, so this r square is equal to x squared plus y square plus z square.

02:25 So i'm going to choose it and we would write that b is equal to 1 over 2 m omega square x square plus y square plus z square, right? let's say that this is equation 1.

02:42 Now the next thing is that as we know that we have the equation negative plank square divided by 2m del square si plus v times si is equal to e times si right so now as the time independent way function time independent wave function is this equation that is excuse me a square divided by two into partially into derivative of si double derivative of si with respect to x plus double derivative of si with respect to y plus double derivative of si with respect to y plus double derivative of si with respect to c plus v times si is equal to e times si right so over here as you can see that this wave function right it satisfies this wave function right it satisfies this wave function right it satisfies this equation fine and since um this x y and z they do not depend on each other right so it means that we can safely separate it so this would become xy z is equal to x of small x y of small y of small y z of small z right so the next step is to rewrite this equation according to this equation right so we will have an equation that looks something like this.

05:00 So a square divided by 2 y z.

05:04 Why we are not writing y z x over here is because we are taking the derivative with respect to x.

05:12 Okay.

05:14 So next as we are taking the derivative with respect to y.

05:18 So over here we would have x z.

05:22 And over here we would have d square y plus a again, as we are taking the derivative with respect to z, so we would have xy over here, and over here we would have d square z, right? plus a v times si is equal to e times si, it remains the same...

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